anatomy inverse problems

GEOPHYSICS, VOL. 65, NO. 6 (NOVEMBER-DECEMBER 2000); P. 1708–1710, 4 FIGS.

The Anatomy of Inverse Problems

John A. Scales∗ and Roel Snieder∗

A major task of geophysics is to make quantitative state-ments about the interior of the earth. For this reason, inverseproblems are an important area of geophysical research andindustrial application. Figure 1 shows how many texts presentinverse problems. The earth model is an element of a mathe-matical space that contains all allowable parameterizations ofthe earth’s properties (or at least those properties relevant to agiven experiment); this space is referred to as model space. Thephysics of the problem determines which data d correspond to agiven model m. The problem of computing the model response(synthetic “data”) given a model is called the forward problem.The corresponding data reside in a mathematical space that iscalled data space. In many applications, one records the data,and the goal is to find the corresponding model. The task iscalled the inverse problem, as shown in Figure 1.

Unfortunately, Figure 1 is wrong. There is a simple reasonfor this. In general the model that one seeks is a continuousfunction of the space variables with infinitely many degrees offreedom. For example, the 3-D velocity structure in the earthhas infinitely many degrees of freedom. On the other hand, thedata space is always of finite dimension because any real exper-iment can only result in a finite number of measurements. Asimple count of variables shows that the mapping from the datato a model cannot be unique; or equivalently, there must be el-ements of the model space that have no influence on the data.This lack of uniqueness is apparent even for problems involv-ing idealized, noise-free measurements. The problem only be-comes worse when the uncertainties of real measurements aretaken into account. Although the uniqueness question is a hotlydebated issue in the mathematical literature on inverse prob-lems, it is largely irrelevant for practical inverse problems be-cause they invariably have a nonunique solution (if by solutionwe mean an earth model). It is this nonuniqueness that makesFigure 1 deceptive, because the arrow pointing from data spaceto model space suggests that a unique model corresponds toevery data set.

A more realistic scheme of inverse problems is shown in Fig-ure 2. Given a model m, the physics of the problem determinesthe predicted data d; this is called the forward problem. Fora given data set, one determines an estimated model m. We

Manuscript received by the Editor June 8, 2000; revised manuscript received August 23, 2000.∗Colorado School of Mines, Department of Geophysics and Center for Wave Phenomena, Golden, Colorado 80401. E-mail: [email protected],[email protected]© 2000 Society of Exploration Geophysicists. All rights reserved.

refer to this as the model estimation problem. (Later, we willconsider the generalization to the problem of estimating prop-erties of models, rather than the models themselves.) Note thatthere may be many reasonable model estimates for a given dataset and that the estimation procedure may be nonlinear evenwhen the forward problem is linear. Thus, the mean of a setof numbers is a linear function of the numbers, whereas themedian is a nonlinear function. Yet both the median and themean may be reasonable estimators of the “center” of the setof numbers. Part of the art of solving inverse problems comesfrom the need to define what it means for an estimate to bereasonable.

Since the mapping from data space to model space isnonunique, the estimated model may also depend on the detailsof the algorithm that one has used for the estimation problemas well as on the regularization and model parameterizationthat has been used. In general, the estimated model m differsfrom the true model m. For example, in seismic inversion theestimated model may be a blurred version of the true model. Inaddition, the data are always contaminated with errors; theseerrors represent an additional source of discrepancy betweenthe estimated model and the true model.

One is not finished when the estimated model is constructed;it is essential to somehow quantify the error between the esti-mated model and the true model. This is called the appraisalproblem. In this problem, one determines the uncertainty in theestimated model. This uncertainty has a statistical componentrelated to the propagation of errors in the data, and a deter-ministic component that accounts for the finite resolution thatis attained in the model estimation, as well as systematic errorsin the problem.

For linear inverse problems, resolution kernels and confi-dence set analysis are powerful tools for formulating the ap-praisal problems, and the theory of linear error propagation issufficiently well developed to account for the errors in the esti-mated model due to the errors in the data. For nonlinear inverseproblems, the only tool available for the appraisal problemmay be Bayesian inversion where one estimates the statisticalproperties of the model when the data and other knowledgeare combined in a statistical sense by repeated sampling of the

1708

The Anatomy of Inverse Problems 1709

model space (e.g., Mosegaard and Tarantola, 1995; Gouveiaand Scales, 1998). These techniques are only applicable to prac-tical problems where the number of parameters is relativelysmall. This is partly due to computational cost of such methods,but is also related in a very subtle way to the behavior of prob-abilities in high dimensional spaces (see Scales and Tenorio,2001). For large-scale inverse problems such as the determi-nation of 3-D earth structure, Monte Carlo sampling methodsare not feasible. This means that there is presently no opera-tional theory to account for the appraisal problem of nonlinearinverse problems with large number of parameters (Snieder,1998). Developing such a theory is a theoretical and practi-cal challenge that is much more important than establishinguniqueness proofs of idealized mathematical problems.

In practice, one solves inverse problems with a certain goal.For example, one may use an estimated model obtained froman inversion of seismic data as a basis for deciding where todrill or how to optimally exploit a reservoir. In practice, one isnever interested in the seismic velocity at a certain spot in thesubsurface, but for a seismic interpreter it is crucial to knowwhether at a certain place a syncline or an anticline is present.This means that for practical inverse problems one is interestedin patterns that can be used in a meaningful way for makingdecisions. In practice, these decisions are usually not based ex-clusively on the estimated model m, but involve the integrationof other data as well as human expertise. In addition, the uncer-tainty in the estimated model is an important factor in makingdecisions. Thus, Figure 3 gives a more realistic view on inverseproblems, because it shows explicitly that decisions rather thanmodel estimation is the endpoint of practical inverse problems.

It is interesting to consider the relation between the appraisalproblem and the process of decision making. An important as-pect of the appraisal problem is the statistical treatment oferror propagation. This means that the appraisal problem hasprobability as an important component. Decisions are usually

FIG. 1. The conventional view of inverse problems: find themodel that predicts the measurements.

FIG. 2. An improved view of inverse problems.

based on risk rather than probability. This may be economicrisk (where to drill), environmental risk (what happens whenpollutants spread), or even academic risk (am I sure enoughto publish my results). Risk is always concerned with proba-bility plus another component such as profit or environmentalimpact. This means that in this stage those who produce themodels and their uncertainty must interface with others formaking decisions effectively.

Figure 3 offers an overview of the landscape of geophysicalinversion. Those working on wave propagation problems focuson the forward problem. Those focused on the development ofmigration algorithms work on the estimation problem. Statisti-cians and a limited number of scientists in the inverse problemcommunity are concerned with the appraisal problem. Seismicinterpretation is usually based on an estimated model, and socan be thought of as a decision-making problem. It is illuminat-ing to see how different researchers work on different parts ofthis problem. One may wonder whether these research effortscould be more effective when their activities are seen in the con-text of the anatomy of the inverse problem as shown in Figure 3.

In Figure 3, the estimated model m forms an essential partof the inversion process. But is it necessary or desirable to pro-duce an estimated model in the process of inversion of data?We, of course, are conditioned to produce models from ourdata. However, given the fact that this estimated model dif-fers from the true model, one can be led astray by features inthe estimated model that are artifacts of the inversion process.Another view on inverse problems is given in Figure 4 wherethe goal is to determine the range of models that are consistentwith the data (as well as other information). This range of mod-els can simply be a box within which all the models are believedto fit the data (there being no comparative relation among themodels in the box), or it could be a probability distribution onthe space of models P(m) (in which case one can speak of thebest or most probable model). Both the box and the probabil-ity are determined by the data, the uncertainties and whateverother data-independent information is available. As we havediscussed previously (Scales and Snieder, 1997), people takingthe former approach are called frequentists (this includes moststatisticians), whereas people taking the latter approach arecalled Bayesians (this includes most geophysicists). In eithercase, the estimation and appraisal problem is replaced by theinference problem. Inference means characterizing somehowthe set of models that explain the data (and satisfy whateverother information is available). One can then use this box, orthe probability distribution P(m), as a basis for making deci-sions. The tutorial by Scales and Tenorio (2001) shows a niceexample of a “toy” inverse problems tackled from a Bayesianand a frequentist point of view.

The reader may be put off by the idea of solving inverse prob-lems without constructing a model. Our minds are conditionedto making models from data. However, we have seen that theseestimated models can only be the endpoint of research whenone ignores the fact that models are being produced with thegoal of making decisions. When seen in this larger context (Fig-ures 3 or 4), it is worth considering whether one is not betterserved by knowing the probability distribution of the set ofmodels than by knowing a single estimated model and a mea-sure of its uncertainty.

It is interesting to consider how we would use a probabil-ity density function in a high-dimensional model space. The

1710 Scales and Snieder

FIG. 3. The inverse problem as part of a decision-makingprocess.

FIG. 4. The inverse problem as an inference problem.

simplest approach is to compute the mean and variance of eachmodel parameter, and one can visualize this information rel-atively easily. However, as we have seen, the patterns in themodel are much more interesting than the estimates of onemodel parameter. Assessing the robustness of certain patternsin the model is much more difficult, especially since this entailsthe use of the correlation between different model parameters.In a high-dimensional model space it is extremely difficult tocharacterize and interpret the correlations of the model param-eters. In order to interrogate the resulting probability densityfunction of the model in a meaningful way, research is neededinto cluster and feature analysis of (possibly multimodal) prob-ability density functions of many degrees of freedom, as wellas the development of an interface between the explorationof these high-dimensional functions and the decision-makingprocess.

It will be clear from this that in order to treat inverse prob-lems in ways that are different from current practice requiressignificant theoretical and numerical advances. However, clearstrategies for the optimal use of inverse problems in the pro-cess of decision making should also be an important item onthe agenda for researchers in inverse problems.

REFERENCES

Gouveia, W. P., and Scales, J. A., 1998, Bayesian seismic waveform in-version: Parameter estimation and uncertainty analysis: J. Geophys.Res., 103, 2759–2779.

Mosegaard, K., and Tarantola, A., 1995, Monte Carlo sampling of so-lutions to inverse problems: J. Geophys. Res., 100, 12431–12447.

Scales, J. A., and Snieder, R., 1997, To Bayes or not to Bayes: Geo-physics, 62, 1045–1046.

Scales, J. A., and Tenorio, L., 2001, Prior information and uncer-tainty in inverse problems: Geophysics, tentatively scheduled forMarch/April.

Snieder, R., 1998, The role of nonlinearity in inverse problems: InverseProblems, 14, 387–404.

GEOPHYSICS, VOL. 65, NO. 6 (NOVEMBER-DECEMBER 2000); P. 1711–1725, 9 FIGS., 3 TABLES.

Anisotropic approximations for mudrocks:A seismic laboratory study

Morten Jakobsen∗ and Tor Arne Johansen∗

ABSTRACT

An experimental technique for determining the in-situelastic properties of mudrocks with (horizontal) align-ments in the microstructure is used to study the accuracyof a set of three nested scalar anisotropic approximationsfor transversely isotropic (TI) media. Each subsequentapproximation adds one more velocity parameter andincludes the previous as a special case. These approxima-tions are convenient and robust because of their close re-lationship to standard geophysical measurements. Thereexists no good theory to predict the effects of an im-posed stress on the elasticity of mudrocks. In this study,the tensor of elastic moduli of a single test specimen ofmudrock subjected to an anisotropic stress field is de-termined from ultrasonic group velocity measurementsinvolving pointlike transducers. The mechanical char-acterization (performed at constant pore pressure) isaccompanied by detailed microscopic observations andanalysis. The method was used to obtain accurate elas-tic constants for five well-defined mudrocks with differ-ent degrees of anisotropy. These elastic constants werethen used to study the accuracy of the three anisotropicapproximations. We have found that mudrocks can besignificantly anelliptic. The agreement between the sec-ond anisotropic approximation and the exact phase ve-locity was found to be very good for qP-waves. ForqSV-waves agreement was not as good, and the thirdanisotropic approximation is required to obtain reason-able accuracy.

INTRODUCTION

The assumption that sedimentary rocks behave elasticallyas an isotropic medium is a major source of error in the seis-mic imaging of subsurface features in the earth (Sayers, 1995;Rowbotham, 1997). The nature of depositional processes tendsto produce transverse isotropy with a vertical symmetry axis

Manuscript received by the Editor October 12, 1998; revised manuscript received March 7, 2000.∗University of Bergen, Institute of Solid Earth Physics, Allegt. 41, N-5007 Bergen, Norway. E-mail: [email protected]; torarne@ ifif.uib.no.c© 2000 Society of Exploration Geophysicists. All rights reserved.

in undisturbed, horizontal, plane-layered, sedimentary rocks.Shale anisotropy, in particular, has been identified at all scalesfrom ultrasonic to seismic (Hornby et al., 1995) because of thepartial alignment of plate-like clay particles within the rock.The influence of the intrinsic anisotropy on the seismic ve-locities can often be quite strong in comparison with multi-layering effects (see Werner and Shapiro, 1998). Clearly, theinherent anisotropy of sedimentary rocks should be taken intoaccount in interpreting seismic field data. For velocity analysisand imaging, however, the full complexity of a hexagonal modelis often unnecessary (e.g., Dellinger et al., 1993; Schoenberget al., 1996; Alkhalifah et al., 1996).

Dellinger et al. (1993) have introduced a set of three nestedscalar anisotropic approximations for transversely isotropic(TI) media. Each subsequent approximation adds one morevelocity parameter and includes the previous as a special case.These approximations are convenient and robust because oftheir close relationship to standard geophysical measurements.Sayers (1993) demonstrates the applicability of the approxi-mations by using his simple model of the elastic anisotropyof shales. The purpose of this study is to check the accuracyof the approximations for anisotropic mudrocks using accu-rate elastic constants obtained from ultrasonic wave speedsmeasured on core samples under simulated in-situ stress con-ditions. This is important since mudrocks are the most abun-dant of all lithologies, constituting some 75% of sedimentaryrock sequences. We define a mudrock as a sedimentary rockcontaining admixtures of very fine sand-, silt-, and clay-sizedgrains. Fissile mudrocks are referred to as shales. [Some authorsuse shale as a synonym for mudrock (Leeder, 1982).]

The elastic stiffnesses of TI rocks are often determined byusing the pulse-transmission technique on sets of plugs coredat different known orientations with respect to the beddingnormal (Jones and Wang, 1981; Hornby, 1994; Dellinger andVernik, 1994; Johnston and Christensen, 1995). Convention-ally, ultrasonic transducers are in contact with the flat sidesof the specimens, and phase velocities are measured. In thismethod, problems often arise because of insufficient controlwith core inhomogeneities (Vernik and Liu, 1997).

1711

1712 Jakobsen and Johansen

In the method of Jakobsen and Johansen (2001) used in thisstudy, wave-speed data are obtained on a single test specimencut normal to bedding, thereby limiting the effects of inherentmaterial variability from specimen to specimen. In addition tothe ultrasonic transducers used in the conventional procedure,several pointlike transducers are used to obtain wave speedsin the bedding plane and in an off-symmetry direction of azonal section. The tensor of elastic moduli is recovered fromthe wave speeds assuming that ultrasonic group velocities aremeasured. The method is then applied to check the anisotropicapproximations of Dellinger et al. (1993) for five North Seasamples.

ANISOTROPIC APPROXIMATIONS FOR TI MEDIA

In the first anisotropic approximation of Dellinger et al.(1993), the phase velocity squared W (θ) is given by

W (θ) = WzCw + Wx Sw, (1)

where Wz and Wx are the vertical and horizontal wave speedssquared, Cw = cos2 θ , Sw = sin2

θ , and θ is the angle the wavenormal makes with the vertical. Equation (1) is consistent witha phase velocity surface being ellipsoidal.

In the second anisotropic approximation of Dellinger et al.(1993), the phase velocity squared is given by

W (θ) =(WzCw)2 + (1 + Wx NMO/Wx )(WzCw)(Wx Sw) + (Wx Sw)2

WzCw + Wx Sw

,

(2)

where Wx NMO is the square of the short-spread normal moveout(NMO) velocity we use every day in surface-to-surface dataprocessing. If Wx NMO is equal to Wx , then equation (2) reducesto equation (1).

In the third anisotropic approximation of Dellinger et al.(1993), the phase velocity squared is given by

W (θ) = (WzCw)3 + (2 + Wx NMO/Wx )(WzCw)2(Wx Sw)(WzCw + Wx Sw)2

+ (2 + Wz NMO/Wz)(WzCw)(Wx Sw)2 + (Wx Sw)3

(WzCw + Wx Sw)2,

(3)

where Wz NMO is the square of the moveout velocity measuredfor near-horizontal propagation, such as might be found in across-borehole experiment. If Wz NMO is equal to Wz , then equa-tion (3) reduces to equation (2).

In terms of the density normalized stiffnesses Wi j = ci j/ρ

of a medium of hexagonal symmetry (Musgrave, 1970), theanisotropic velocity parameters are given for qP-waves by(Dellinger et al., 1993)

Wx = W11, (4)

Wz = W33, (5)

Wx NMO = W55 + (W13 + W55)2/(W33 − W55), (6)

Wz NMO = W55 + (W13 + W55)2/(W11 − W55). (7)

For qSV-waves they are given by

Wx = W55, (8)

Wz = W55, (9)

Wx NMO = W11 − (W13 + W55)2/(W33 − W55), (10)

Wz NMO = W33 − (W13 + W55)2/(W11 − W55). (11)

DETERMINATION OF ELASTIC PROPERTIESOF MUDROCKS

Wave-speed measurements

Norsk Hydro collected 30-m-long sections of core from adepth of about 2250 m in well 31/4-A-04, which crosses a ma-jor fault zone in the Brage field, North Sea. The samples wereselected and preserved at the rig site during the coring op-eration. The 4-inch-diameter core was cut using an oil-baseddrilling mud. The oil in this drilling fluid was used to fill thevoid between the core and the core barrel. To prevent the corefrom drying out, it was ensured that no air was trapped insidethe core barrel. The core samples were then brought onshorefor storage.

It is important to leave the initial pore fluid content intact toobtain realistic acoustic measurements in the laboratory. Sed-imentary rocks dehydrate rapidly (within hours) at ambientconditions and thus detoriate. To minimize the dehydration,steps were taken to minimize the specimen’s exposure to airduring storage and preparation. An alternative approach is torestore the initial saturation by wetting the specimen with syn-thetic pore fluid. This method was not used since many miner-als have water as an integral part of their structure and dryingcracks may form irreversibly during initial dehydration.

For five different core depths, a single cylindrical specimenwas cut normal to bedding, the diameter and length being nom-inally 38 mm and 60–80 mm, respectively. End planes wereground parallel to within 0.1 mm and planar to within 0.05 mm.The specimen was analyzed in terms of density, porosity, andmineralogy, as described in Table 1. Scanning electron micro-scope (SEM) microphotography was used to analyze the mi-crostructure of the specimen (see Figure 1).

The specimen was intrumented with a permeable sleeve(strips of filter paper with sand and oil) to lower the pore fluiddraining distance and to increase the draining area. With thistechnique, an acceptable time (hours) was required for the porepressure to equilibrate after changes in applied stress, as dis-cussed in Hornby (1994).

The specimen was then instrumented with piezoelectriccrystals (see below or Jakobsen and Johansen, 2001) to helpdetermine the overall properties of the argillaceous rock, as-sumed to behave elastically as a medium of hexagonal symme-try (Kaarsberg, 1959).

Before placing the (instrumented) specimen in the pressurevessel (described in Jakobsen and Johansen, 1999), it was jack-eted with a rubber sleeve to isolate the pore-fluid-pressure sys-tem from the confining-pressure system. The measurementswere performed with separate control of confining-fluid pres-sure, axial load, and pore-fluid pressure. To simulate in-situstress conditions, we used σ11 = 13 MPa and σ33 = 20 MPa,where σ11 and σ33 denote the horizontal and vertical stress com-ponents, respectively. The pressure in the pore-fluid reservoir

Velocity Anisotropy in Mudrocks 1713

was kept constant at 1 MPa. Extensiometers (LVDTs in contactwith the jacketed specimen) were used to measure axial andradial strain, and this information was used to correct velocitymeasurements for length changes.

To measure wave speeds, we used the pulse-transmissiontechnique (e.g., Birch, 1960). The source was excited by theoutput port of a Panametrics 500PR, and the receiver was con-nected to the input port. The waveform from the receivingtransducer was digitally recorded using a Phillips PM3352Aoscilloscope. The signal was averaged 100 times to increasethe S/N ratio and transferred to a computer (using Lab View)for further processing. A schematic of the transducer/specimensystem is shown in Figure 2.

End-cap assemblies composed of 22-mm-wide transducers(500 kHz) were used to generate and receive ultrasonic pulsespropagating along the bedding normal (Figure 3a). By per-forming a reference measurement with no specimen in place,the delay in the buffer rods was found (Jakobsen and Johansen,2001).

Specially designed transducers composed of 5-mm-widePZT crystals (600 kHz) were glued on flat tracks made on thecurved side of the cylindrical specimen to generate and receiveultrasonic pulses propagating in the bedding plane (Figure 3b)and in an off-symmetry direction (Figure 3c).

Our choice for the geometry of the transducer/specimen sys-tem has implications for the type of velocity measured. In me-dia of hexagonal symmetry, the phase and group (energy) ve-locities of a wave are not, in general, equal. Only along thezonal axis and in the basal plane are the two velocities equal.For propagation along the principal symmetry axes, so-calledpure modes are generated. This allows us to use large transduc-ers along the bedding normal and smaller ones in the beddingplane.

Jakobsen and Johansen (2001) have developed a numericalmodel and have used it to demonstrate that group velocitiescan be extracted from traveltimes measured in an off-symmetrydirection in a specimen instrumented with 5-mm-wide trans-

Table 1. Description of the mudrocks examined in this study. The density was obtained from measurements of the volume andweight of a cylindrical test specimen. The porosity was determined using a standard helium expansion technique. The mineralogywas estimated using energy dispersive spectroscopy in an electron microscope.

Core depth (m) 3492 3506 3525 3536 3564

Density (kg/m3) 2428 2444 2379 2441 2408Porosity (%) 9.9 15.7 7.9 13.9 6.5Mineralogy (vol) % % % % %

Chlorite 4 20 19 27 15Illite/muscovite 9 17 8 10 15Kaolinite 4 14 23 14 6Glauconite — 2 5 8 2Biotite 1 2 4 3 3Quartz 2 20 19 11 32Albite — 6 5 2 7Ankerite 2 — — — —Barite — — — 3 —Calcite 50 — — 0 1Garnet 3 — — — —Gypsum 1 — — 0 —K-feldspar 1 7 7 4 9Siderite — — — 8 —Titanite 9 — — — —Other 14 12 10 10 10

ducers. [The work of Dellinger and Vernik (1994) is not rel-evant here, since they were dealing with a different source-receiver geometry.] The analysis of Jakobsen and Johansen isbased on simple ray theory and can be summarized as follows:The source and receiver transducers were modeled as a seriesof point transducers lying very close to one another. The resultwas a waveform that incorporated the effects of finite-widthtransducers. From analysis of synthetic waveforms and inspec-tion of associated wave surfaces, it was concluded that thefirst-break traveltimes in off-axis experiments yield group ve-locities, provided that the source–receiver separation is mea-sured from edge to edge as shown in Figure 2. This is equivalentto replacing the finite-width transducers with effective pointtransducers. A certain amount of energy (number of rays) isneeded before the first break can be detected, suggestingthere is some uncertainty in the location of the effective pointtransducers. This source of error was taken into accountin the Monte Carlo simulations discussed below.

We use reference axes coincident with rock axes so that Ox3

lies along the main symmetry axis and Ox1, Ox2 complete anorthogonal triad. We denote by vi j the speed of a pure modepropagating along axis Oxi and polarized along axis Ox j . Thegroup speed of a quasi-compressional wave propagating at anangle φ to the axis Ox3 is denoted by Vq P .

The wave speeds measured as a function of applied stress,shown in Table 2 (v11 > v33, v31 ≈ v32 ≈ v13 < v12), are consistentwith the assumption of transverse isotropy with the main sym-metry axis perpendicular to the (layering or bedding) plane ofcircular symmetry (see Schoenberg, 1994). The conclusion thatv31 is similar to v32 is not affected by the sometimes large errorsin these wave speeds since the waveforms were similar for thetwo orthogonal polarization directions (see Figure 3a). FromFigure 4, it is apparent that the wave speeds are increasing asa function of increasing stress. This can be explained by a re-duction in void space and by increased contact between grains(Jones and Wang, 1981). The anisotropy of the specimens willbe discussed in the following.


FIG. 1. Caption-see next page


Wave-speed inversion

For a medium of hexagonal symmetry (Musgrave, 1970),the nonzero elastic stiffnesses ci j are c11 = c22, c33, c44 = c55,c13 = c23, and c12 = (c11 − 2c66). The moduli c11, c33, c55, andc66 can be determined using wave-speed data measured alongthe principal axis as follows:

c11 = ρv211, (12)

c33 = ρv233, (13)

c55 = ρv213, (14)

c66 = ρv212, (15)

where ρ is the density. The modulus c13 requires at least oneoff-axis measurement for its determination. If a group wave

FIG. 1. Looking for alignments in the microstructure of five core samples (mudrocks) from various depths in a North Sea well. Thinsections were taken in a plane perpendicular to bedding. For all core depths, some alignments in the structure can be seen on the(2.18 times) natural size pictures (smaller figures). For all core depths but 3536 m, alignments in the structure can hardly be seenon the scanning electron micrographs (larger figures). Some of the pictures show that microcracks have been formed and that thestatistical distribution of heterogeneities is not locally uniform, suggesting that the ultrasonic acoustic properties of mudrocks canbe both anisotropic and weakly inhomogeneous. Core depths are (a) 3492 m, (b) 3506 m, (c) 3525 m, (d) 3536 m, and (e) 3564 m.

speed Vq P (with group angle φ) has been measured, c13 may beobtained from

c13 = N(ρ, c11, c33, c55, Vq P , φ), (16)

where N denotes a numerical procedure formulated for gen-eral anisotropic media by Every and Sachse (1990) and im-plemented for media of hexagonal symmetry by Jakobsen andJohansen (2001). The procedure corresponds to a two-stageoptimization method in which c13 is varied so as to obtain afit of measured to calculated group velocity. At each iterativestep a numerical minimization process is used to find the phaseangle and speed of waves having the required ray direction.

The above approach results in very accurate estimates of theprincipal stiffnesses c11, c33, c55, and c66 but less accurate es-timates of the critical stiffness c13. Very accurate estimates ofc13 could be obtained if a larger set of group velocities was


measured. These velocities would correspond to qP-modesmeasured in any direction or to qSV-modes associated withunfolded regions of the ray surface. Each data point wouldhave its own standard deviation, and we would minimize awell-known quantity called the chi-squared. So far, we havenot been able to work with more than one off-axis receiver.

FIG. 2. Schematic of a specimen cut normal to bedding andinstrumented with ultrasonic transducers. The vector endingat X3 lies along the main symmetry axis.

We think that the situation with several qP-receivers can berealized in future studies, although we have not solved thepractical problems that occur when we try to work with sev-eral off-axis receivers inside the pressure vessel. It should benoted that no qSV-modes were detected using our small trans-ducers. The qSV-modes are probably strongly contaminated byqP-modes.

To get an idea of how accurate c13 can be estimated (nowand in the future), we performed Monte Carlo simulations ofhypothetical experiments involving one or more qP-receivers(Jakobsen and Johansen, 2001). First we generated 1000 noisywave speed data sets; the statistical errors in the wave speedswere uncorrelated and normal distributed with a standard de-viation of 1% for the principal wave speeds and 3% for theoff-axis wave speed. Then we inverted each data set to obtainits elastic stiffnesses, and we analyzed the distribution of theresults in terms of mean values and standard deviations. In thecase of (1, 2, 3, 4) randomly located qP-receivers, the uncer-tainty in the critical stiffness c13 is (28, 25, 20, 17)%. The readermay think that Hornby (1998) is claiming larger uncertain-ties without the qSV-constraint on c13. But the reader shouldnot forget that Hornby estimated elastic parameters by usingphase slowness measurements (involving several specimen ori-entations) and obtained error bars via a simple perturbationapproach.

The recovered elastic stiffnesses for the Brage mudrocks areshown in Table 3. Clearly, most of the specimens are stronglyanisotropic, as can be noted on the wave surfaces reconstructedin Figure 6. However, for comparison and quantification ofanisotropy, it is convenient to redefine the five independentstiffnesses in terms of two vertical wave speeds and three di-mensionless anisotropy parameters. These anisotropy param-eters are given by (Thomsen, 1986):

ε = c11 − c33

2c33, (17)

γ = c66 − c44

2c44, (18)

δ = (c13 + c44)2 − (c33 − c44)2

2c33(c33 − c44). (19)

We remember that ε and γ describe the anisotropy of P- andS-waves, respectively, while δ is a critical factor that dependson the shape of the qP- and q S-wave surfaces. If (ε − δ) > 0,as in Table 3, the anellipticity is considered positive. Thismeans that the qP-wave slowness curve bulges out from the el-lipse connecting the vertical and horizontal P-wave slownessesand that the qSV-wave slowness curve is pushed inward fromthe circle that connects its horizontal and vertical slownesses(Schoenberg, 1994; Tsvankin 1996). Elliptical anisotropy canbe used in both the phase and group domains as a first stepaway from isotropy (Dellinger et al., 1993).

As an aside, let us discuss the stress dependence of thevelocity anisotropy in our mudrocks. From Table 3 it is ap-parent that the P-wave anisotropy (Thomsen’s parameter ε)(normally) decreases as a function of increasing hydrostaticstress. This can be explained by the closure of cracks and pores(Lo et al., 1986; Hudson, 1991; Hornby, 1994). The P-waveanisotropy also decreases as a function of increasing axial load


(at constant hydrostatic pressure), suggesting that bedding-parallel microcracks are getting closed. The variation in theS-wave anisotropy (Thomsen’s parameter γ ) as a function oftriaxial stress is smaller than expected, and the uncertainty inThomsen’s parameter δ (see Figure 7) is to large to draw firmconclusions about its stress dependence. At the estimated in-situ conditions, open microcracks may still contribute to theobserved anisotropy. Bedding-parallel microcracks could ex-ist in situ in most mature source shales undergoing the majorstage of hydrocarbon generation and migration (Vernik, 1994).Some of the microcracks may have been created in the pro-cess of recovering core, suggesting that our laboratory tech-nique may overestimate the in-situ elastic anisotropy of theBrage mudrocks. Nevertheless, the in-situ stiffnesses may beof great value in future modeling studies, and they are usedhere to test our favorite anisotropic approximations for TImedia.

FIG. 3. Typical ultrasonic waveforms recorded in this experi-ment. (a) Measurements along the bedding normal. (b) Mea-surements in the bedding plane. (c) Measurements in anoff-symmetry direction. The pulser/receiver settings variedwith the measurement direction. This is one reason why theoff-axis signal is often associated with a lower frequency thanthe other signals. The off-axis signal is a function of the inputelectrical transient and the off-axis response of the piezoelec-tric transducer.

TEST OF THE ANISOTROPIC APPROXIMATIONS

Figures 8 and 9 compare the three successive scalar ani-sotropic approximations of Dellinger et al. (1993) with the qP-and qSV-wave phase velocities for the Brage mudrocks, ob-tained using the densities in Table 1 and the in-situ stiffnessesin Table 3. The agreement of the first anisotropic (elliptical) ap-proximation and the exact result is relatively poor, particularlyfor qSV-waves.

The agreement between the second anisotropic approxima-tion and the exact result is good for qP-waves, particularly fornear-vertical and near-horizontal wave propagation. For qSV-waves the agreement between the second anisotropic approx-imation and the exact result is better for near-vertical prop-agation than for all other angles of propagation. The thirdanisotropic approximation works well for qSV-waves. Our find-ings essentially agree with those of Sayers (1993).


Table 2. Ultrasonic wave speeds (in units of m/s) measured on the mudrocks of Table 1 as a function of stress components σij(in GPa) with 1 MPa pore pressure. We expect that ∆v11/v11 < 1%, ∆v12/v12 < 1%, ∆v13/v13 < 1%, ∆v31/v31 < 5%, ∆v32/v32 < 5%,∆v33/v33 < 1%, and ∆VqP/VqP ≈ 3%, where ∆ indicates a standard deviation.

Depth σ11 σ33 v11 v12 v13 v31 v32 v33 Vqp(490)

3492 1.5 1.5 3444 2103 1955 1950 1950 2860 31134.0 4.0 3507 2007 1901 1900 1900 2966 31757.0 7.0 3607 2097 1971 1950 1950 3096 3287

10.0 10.0 3677 2162 1948 1950 1950 3179 329713.0 13.0 3725 2183 2058 2050 2050 3226 341313.0 16.5 3713 2183 1968 1950 1950 3270 354013.0 20.0 3740 2192 2067 2050 2050 3302 3470

3506 1.5 1.5 3383 1892 1685 1700 1700 2990 29534.0 4.0 3400 1922 1737 1750 1750 2998 28987.0 7.0 3514 1966 1767 1750 1750 3067 3019

10.0 10.0 3539 2016 1800 1800 1800 3160 308213.0 13.0 3551 2029 1828 1850 1850 3176 311413.0 16.5 3586 2056 1856 1850 1850 3253 315613.0 20.0 3593 2051 1863 1850 1850 3266 3218

3525 1.5 1.5 3156 1897 1846 1850 1850 2867 29954.0 4.0 3223 1914 1874 1850 1850 2938 30607.0 7.0 3286 1977 1939 1950 1950 3032 3151

10.0 10.0 3344 2028 1978 2000 2000 3139 318013.0 13.0 3375 2063 2007 2000 2000 3181 324613.0 16.5 3382 2078 2003 2000 2000 3243 327713.0 20.0 3416 2092 2028 2050 2050 3263 3313

3536 1.5 1.5 3610 2201 1516 1500 1500 2813 29034.0 4.0 3633 2211 1525 1550 1550 2851 29997.0 7.0 3661 2221 1532 1550 1550 2902 3054

10.0 10.0 3677 2220 1544 1550 1550 2924 306113.0 13.0 3691 2235 1545 1550 1550 2951 313913.0 16.5 3698 2222 1547 1550 1550 2973 309513.0 20.0 3717 2217 1563 1550 1550 2995 3125

3564 1.5 1.5 3341 1961 1890 1900 1900 2810 30204.0 4.0 3400 1985 1894 1900 1900 2894 30497.0 7.0 3481 2012 1924 1900 1900 2997 3163

10.0 10.0 3512 2053 1953 1950 1950 3022 317313.0 13.0 3558 2085 1982 2000 2000 3069 320813.0 16.5 3572 2094 1997 2000 2000 3097 321613.0 20.0 3592 2094 1995 2000 2000 3136 3246

CONCLUSIONS AND DISCUSSION

A procedure has been described by which an economiccharacterization of rocks with (horizontal) alignments in themicrostructure can be made, and this is important wheninterpreting seismic field data. The advantage of this proce-dure is that the elastic stiffnesses of a TI rock can be deter-mined using a single test specimen, limiting the effect of in-herent material variability from specimen to specimen. Thissame advantage becomes important in cases where specimenorientation is limited because of size and availability.

The procedure has been implemented successfully in the lab-oratory on a set of five mudrocks taken from a borehole in theNorth Sea basin. The elastic properties of the mudrocks werefound to be a function of confining pressure and axial load.The ultrasonic measurements were performed at constant porepressure. Our characterization of mechanical properties wasaccompanied by detailed microscopic observations and anal-ysis. The experimental data are useful in characterizing theanisotropic elastic properties of sedimentary rocks in terms ofdensity, porosity, mineralogy, and microstructure.

The elastic stiffnesses recovered at simulated in-situ(anisotropic) stress conditions were used to test the predic-

tions of a set of three scalar anisotropic approximations for TImedia introduced by Dellinger et al. (1993). The results of thisstudy suggest that the second and third anisotropic approxi-mations should continue to find use in replacing TI dispersionrelations in modeling, migration, and inversion programs. Wedo not recommend using an elliptical approximation for theelastic anisotropy of mudrocks.

ACKOWLEDGMENTS

We thank Norsk Hydro for their hospitality and help and forpermission to publish their data. Geir Østby provided valu-able help in the seismic laboratory. Harald Flesche providedthe mineralogy data and the SEM pictures. Dag Økland gaveus information about in-situ stress conditions. The NorwegianResearch Council and the consortium for Seismic ReservoirCharacterization at the University of Bergen are thanked foreconomic support.

REFERENCES

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FIG. 4. Ultrasonic wave speeds (compressional modes) in theBrage mudrocks measured as a function of effective verticalstress given as Pc + Pa − Pp , where Pc is the confining pressure,Pa is the axial load, and Pp is the pore pressure. Here, Pa = 0for Pc < 13 MPa and Pp = 1 MPa. Core depths are (a) 3492 m,(b) 3506 m, (c) 3525 m, (d) 3536 m, and (e) 3564 m.

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Table 3. Elastic stiffnesses and anisotropy parameters characterizing the mudrocks. The stiffnesses (in GPa) were recoveredfrom the wave speeds in Table 2, and the dimensionless anisotropy parameters were calculated using the formulas of Thomsen(1986). On the basis of Monte Carlo simulations, we expect that ∆c11/c11 < 2%, ∆c33/c33 < 2%, ∆c55/c55 < 2%, ∆c66/c66 < 2%,and ∆c13/c13 ≈ 28%, where ∆ indicates a standard deviation. The uncertainty in the anisotropy parameters is indicated in Figure 7.

Depth σ11 σ33 c11 c33 c55 c66 c13 ε γ δ

3492 1.5 1.5 28.8 19.9 9.3 10.7 3.8 0.22 0.08 0.144.0 4.0 29.9 21.4 8.8 9.8 5.4 0.20 0.06 0.087.0 7.0 31.6 23.3 9.4 10.7 5.7 0.18 0.07 0.05

10.0 10.0 32.8 24.5 9.2 11.3 5.2 0.17 0.11 −0.0413.0 13.0 33.7 25.3 10.3 11.6 6.0 0.17 0.06 0.0513.0 16.5 33.5 26.0 9.4 11.6 11.8 0.14 0.12 0.2013.0 20.0 34.0 26.5 10.4 11.7 6.9 0.14 0.06 0.05

3506 1.5 1.5 28.0 21.8 6.9 8.7 3.5 0.14 0.13 −0.184.0 4.0 28.3 22.0 7.4 9.0 0.80 0.14 0.11 −0.237.0 7.0 30.2 23.0 7.6 9.4 2.3 0.16 0.12 −0.20

10.0 10.0 30.6 24.4 7.9 9.9 2.7 0.13 0.13 −0.2013.0 13.0 30.8 24.7 8.2 10.1 2.9 0.12 0.12 −0.1813.0 16.5 31.4 25.9 8.4 10.3 2.8 0.11 0.11 −0.2013.0 20.0 31.5 26.1 8.5 10.3 4.4 0.10 0.11 −0.16

3525 1.5 1.5 23.7 19.6 8.1 8.6 4.5 0.10 0.03 0.064.0 4.0 24.7 20.5 8.3 8.7 4.8 0.10 0.02 0.057.0 7.0 25.7 21.9 8.9 9.3 5.2 0.09 0.02 0.05

10.0 10.0 26.6 23.4 9.3 9.8 4.1 0.07 0.03 −0.0313.0 13.0 27.1 24.1 9.6 10.1 5.0 0.06 0.03 0.0013.0 16.5 27.2 25.0 9.5 10.3 5.6 0.04 0.04 −0.0213.0 20.0 27.8 25.3 9.8 10.4 5.8 0.05 0.03 0.00

3536 1.5 1.5 31.8 19.3 5.6 11.8 4.7 0.32 0.55 −0.154.0 4.0 32.3 19.9 5.7 11.9 6.8 0.31 0.54 −0.087.0 7.0 32.8 20.6 5.7 12.1 7.6 0.30 0.55 −0.07

10.0 10.0 33.1 20.9 5.8 12.0 7.3 0.29 0.53 −0.0913.0 13.0 33.3 21.3 5.8 12.2 9.3 0.28 0.55 −0.0213.0 16.5 33.4 21.6 5.9 12.1 7.7 0.27 0.53 −0.0913.0 20.0 33.8 21.9 6.0 12.0 8.0 0.27 0.50 −0.08

3564 1.5 1.5 26.9 19.0 8.6 9.3 3.4 0.21 0.04 0.094.0 4.0 27.8 20.2 8.6 9.5 3.1 0.19 0.05 0.007.0 7.0 29.2 21.6 8.9 9.7 4.6 0.18 0.04 0.04

10.0 10.0 29.7 22.0 9.2 10.1 3.8 0.18 0.05 0.0113.0 13.0 30.5 22.7 9.5 10.5 3.6 0.17 0.05 0.0013.0 16.5 30.7 23.1 9.6 10.6 3.3 0.16 0.05 −0.0313.0 20.0 31.1 23.7 9.6 10.6 3.8 0.16 0.05 −0.03

Geophys. J. Internat, 107, 505–511.Jakobsen, M., and Johansen, T. A., 2001, Determination of the elas-

tic properties of shales using single test specimens, in Hood, J. A.,Ed., Advances in anisotropy—selected theory, modeling, and casestudies: Soc. Expl. Geophys., scheduled for early 2001.

Johnston, J. E., and Christensen, N. I., 1995, Seismic anisotropy ofshales: J. Geophys. Res., 100, 5591–6003.

Jones, L. E. A., and Wang, H. F., 1981, Ultrasonic velocities in Creta-ceous shales from the Williston basin: Geophysics, 46, 288–296.

Kaarsberg, E. A., 1959, Introductory studies of natural and artificialargillaceous aggregates by sound propagation and X-ray diffractionmethods: J. Geol., 67, 447–472.

Leeder, M. R., 1982, Sedimentology: Allen and Unwin Ltd.Lo, T., Coyner, K. B., and Toksoz, M. N., 1986, Experimental deter-

mination of elastic anisotropy of Berea Sandstone, Chicopee Shale,and Chelmsford Granite: Geophysics, 51, 164–171.

Musgrave, M. J. P., 1970, Crystal acoustics: Holden-Day.Rowbotham, P. S., 1997, Anisotropic migration of coincident VSP

and cross-hole seismic reflection surveys: Geophys. Prosp., 45, 683–699.

Sayers, C. M., 1993, Anelliptic approximations for shales: J. Seis. Expl.,2, 319–331.

——— 1995, Anisotropic velocity analysis: Geophys. Prosp., 43, 541–568.

Schoenberg, M., 1994, Transversely isotropic media equivalent to thinisotropic layers: Geophys. Prosp., 42, 885–915.

Schoenberg, M., Muir, F., and Sayers, C. M., 1996, Introducing ANNIE:A simple three-parameter anisotropic velocity model for shales: J.Seis. Expl., 5, 35–49.

Thomsen, L., 1986, Weak elastic anisotropy: Geophysics, 51, 1954–1966.

Tsvankin, I., 1996, P-wave signatures and notation for transverselyisotropic media: An overview: Geophysics, 61, 467–483.

Vernik, L., 1994, Hydrocarbon-generation-induced microcracking ofsource rocks: Geophysics, 59, 555–563.

Vernik, L., and Liu, X., 1997, Velocity anisotropy in shales: A petro-physical study: Geophysics, 62, 521–532.

Werner, U., and Shapiro, S. A., 1998, Intrinsic anisotropy and thinmultilayering—Two anisotropy effects combined: Geophys. J. Inter-nat, 132, 363–373.


FIG. 5. Same as in Figure 4, but for shear waves.


FIG. 6. Wave surfaces of the Brage mudrocks reconstructed us-ing the densities in Table 1 and the in-situ stiffnesses in Table 3.The curves marked with l and u were made using the lower andupper bounds on c13, respectively. The three sheets of the wavesurface (using the mean value of c13) have been overlain by ul-trasonic group-velocity measurements (Table 2). Dashed linesindicate SH -modes. Core depths are (a) 3492 m, (b) 3506 m,(c) 3525 m, (d) 3536 m, and (e) 3564 m.


FIG. 7. Indicating the uncertainty of Thomsen’s (1986) anisotropy parameters for the core samples under simulated in-situ stress.The error bars were found using a simple Monte Carlo routine.


FIG. 8. Comparison of the exact (solid lines) qP-mode phasevelocities of the Brage mudrocks with the first (dottedlines), second (dashed-dotted lines), and third (dashed lines)anisotropic approximations of Dellinger et al. (1993). Thelargest mismatch between the anisotropic approximations andthe exact result is found for the 3506-m specimen. Core depthsare (a) 3492 m, (b) 3506 m, (c) 3525 m, (d) 3536 m, and (e)3564 m.


FIG. 9. Same as in Figure 8 but for the significantly moreanisotropic qSV-mode. Core depths are (a) 3492 m, (b) 3506 m,(c) 3525 m, (d) 3536 m, and (e) 3564 m.


Potential field from a dc current source arbitrarilylocated in a nonuniform layered medium

Hedison Kiuity Sato∗

ABSTRACT

In this paper, I evaluate the potential field due to a dccurrent source located anywhere within a horizontallylayered space, all layers possessing exponentially varyingresistivities. The solution takes the form of Hankel trans-forms with their kernel expressions containing functionsdefined by recursion formulas. The resulting expressionscan be used to model any kind of resistivity array.

Specializing the general solution to full- or half-spacemodels possessing exponential dependence, I find termsI interpret as primary and secondary contributions.Analytic expressions for the secondary electrical fieldrevealed an error in the Stoyer and Wait solution for ahalf-space.

To test the derived n-layer solution based on recur-sion formulas, I modeled resistivity logs for both homo-geneous layers and intercalated homogeneous and het-erogeneous layers. Curves from the theory reproducedasymmetries and ripples observed in real resistivity logs.

INTRODUCTION

Many authors have investigated the nature of the potentialfield due to a dc current source at the surface of an earth whoseresistivities vary continuously with depth. For the particularcase of exponential dependency of resistivities, Sunde (1949),Paul and Banerjee (1970), and Stoyer and Wait (1977) havestudied the half-space model. Similar results, but for a layeredmodel, are reported by Banerjee et al. (1980a,b), Raghuwanshiand Singh (1986), and Kim and Lee (1996). The last authorsfound a recursion relation for the kernels of a Hankel transformexpression for the potential. All these investigations locatedthe current source at the surface, limiting the cases that couldbe analyzed to electrical methods using electrode arrays spreadon the earth’s surface.

Investigations using buried electrodes are reported for earthmodels having homogeneous layers. Alfano (1962) considers

Manuscript received by the Editor December 29, 1997; revised manuscript received February 25, 2000.∗CPGG/UFBA, Campus Univ. da Federacao, Instituto de Geociencias, Salvador, Bahia 40170-290, Brazil. E-mail: [email protected]© 2000 Society of Exploration Geophysicists. All rights reserved.

a three-layer model and demonstrates that, by using buriedelectrodes, the uncertainty in the interpretation of resistivitysoundings can be reduced. Daniels (1978) presents an analyti-cal solution for the n-layer model and applies his solution to themodeling of normal logs. Baumgartner (1996) uses submergedelectrodes to study underwater formations.

In this paper, I develop the analytical solution for the poten-tial field resulting from a dc current source buried in a layeredspace, where the layer electrical resistivities vary exponentiallywith depth. I limit myself to modeling normal logs for two the-oretical situations, but my solution can model any kind of re-sistivity array.

THE MODEL AND BASIC EQUATIONS

As shown in Figure 1, my system is an infinite medium slicedby n parallel horizontal planes, with the resulting consecutivelayers numbered from 0 to n. Assuming a cylindrical coordinatesystem whose z-axis is perpendicular to the interface planes andpointing downward, I assume (1) planes defined by equationsz= z0, z= z1, . . . , z= zn−1, (2) a current source with intensity Ilocated at (0, zc) in the mth layer, (3) an observer located at(r, z), and (4) electrical resistivity of the i th layer as a functionof z given by

ρi (z) = αi exp(βi z), (1)

where αi and βi are characteristic constants of the i th layer.Introducing an artificial interface defined by the plane z= zc,

the mth layer is divided into two layers, increasing the totalnumber of layers to n+ 2. With this procedure, two functionsrather than one describe the potential in the mth layer. In thisnew system, the layers are designated by 0, 1, . . . , m, m

¯, . . . ,

n− 1, n, where mrepresents the portion of the former mth layerbounded by the planes z= zm−1 and z= zc and where m

¯is the

complementary portion bounded by planes z= zc and z= zm.The thickness of each layer is defined as shown in Figure 1.

The potential Vi in each layer, except at the point (0, zc),satisfies the equation

∂2Vi

∂r 2+ 1

r

∂Vi

∂r+ ∂

2Vi

∂z2+ 1σi

∂σi

∂z

∂Vi

∂z= 0, (2)

1726

dc Field in Nonuniform Layered Medium 1727

where σi = 1/ρi is the electrical conductivity. The solution ofthis differential equation is well known. Following Kim andLee (1996), I write

Vi =∫ ∞

0

[Ai e0

+i z+ Bi e0

−i z]

J0(λr ) dλ. (3)

In equation (3), J0 is the zero-order Bessel function, Ai and Bi

are functions of λ to be determined, and

0+i =[βi +

√β2

i + 4λ2]/

2, (4)

0−i =[βi −

√β2

i + 4λ2]/

2. (5)

To compact the subsequent formulas, 10i is defined as

10i = 0+i − 0−i . (6)

BOUNDARY CONDITIONS

Convergence at infinity

To guarantee the convergence of the potential V0 when z→−∞,

B0 = 0. (7)

Similarly, the convergence of the potential Vn when z→ +∞is guaranteed by making

An = 0. (8)

FIG. 1. Layered-space model.

Conditions at the interfaces

Except at the artificial interface z= zc, the electrical potentialand the normal current density are continuous at the interfacesz= zi , where i = 0, 1, 2, . . . ,n− 1. The unknown functions Ai ,Bi of the potential Vi and Ai+1, Bi+1 of the potential Vi+1 satisfythe equations

Ai e0+i zi + Bi e0

−i zi − Ai+1 e0

+i+1zi − Bi+1 e0

−i+1zi = 0,

(9)1

ρi (zi )

[Ai0

+i e0

+i zi + Bi0

−i e0

−i zi]− 1ρi+1(zi )

×[

Ai+10+i+1 e0

+i+1zi + Bi+10

−i+1 e0

−i+1zi

]= 0. (10)

Because of the split made in layer m, special care must betaken with equations (9) and (10) at the interfaces boundingthat layer. At interface z= zm−1, i =m− 1 but i + 1 is m. Atinterface z= zm, i is m

¯while i + 1=m+ 1.

Conditions related to the current source

The physical constraints from the current source are honoredwith an approach similar to that of Sato (1996). The electricalpotentials Vm and Vm satisfy two physical conditions. The firstcondition is the continuity of the potential at the artificial planeinterface z= zc, except at r = 0. Then, the unknown functionsAm, Bm, Am, and Bm satisfy the expression

Am e0+mzc + Bm e0

−mzc − Am e0

+mzc − Bm e0

−mzc = 0. (11)

The second condition is charge conservation. Consider acylinder volume having height 2h and radius ξ , its center co-inciding with the current source of intensity I , as shown inFigure 2. The total current flowing out of the surface S of thecylinder must be equal to the current intensity for any h andξ values. In the limiting case h → 0, the current through thelateral cylindrical surface will tend to zero, assuming that thecurrent density has a finite value. Since the total current re-duces to the sum of the current flowing through the circularbase surfaces, the following equation holds:

limh→0

[∫ 2π

θ=0

∫ ξ

r=0jm

∣∣∣∣z=zc−h

· (−z r dr dθ)

+∫ 2π

θ=0

∫ ξ

r=0jm

∣∣∣∣z=zc+h

· (z r dr dθ)

]= I , (12)

FIG. 2. Cylindrical surface Saround the current source.

1728 Sato

where z is the unit vector in the z-direction and ji =−σi∇Vi ,the electrical current density in the layer i .

Using equations (1) and (3) to evaluate jm and jm in equa-tion (12), integrating over θ and r , and rearranging the terms,I obtain∫ ∞

0

2πσc

I λ3/2

{Am0

+m e0

+mzc + Bm0

−m e0

−mzc − Am0

+m e0

+mzc

− Bm0−m e0

−mzc}

(ξλ)1/2 J1(λξ) dλ= ξ−1/2, (13)

with σc= 1/ρm(zc), the electrical conductivity at the currentsource vertical position. This equation must be satisfied forany radius ξ and resembles a Hankel transform integral. UsingErdelyi (1954, p. 22), I have

Am0+m e0

+mzc + Bm0

−m e0

−mzc − Am0

+m e0

+mzc − Bm0

−m e0

+mzc

= I λ

2πσc. (14)

SOLUTION OF THE PROBLEM

The potential Vi depends on knowing the pair of functionsAi , Bi . In the present problem, n+ 2 pairs of Ai , Bi are definedand their relations are subjected to the boundary conditionsderived in the previous section. These relations comprise a setof n+ 2 linear equations.

As long as each one of these equations is related to a specificz-coordinate, the set can be organized with increasing z-values.Equation (7) is taken as the first equation and equation (8) asthe last one. Between those extremes, the second through thepenultimate equations are (a) expressions (9) and (10) writtenfor i = 0, . . . ,m− 1, (b) equations (11) and (14), and (c) equa-tions (9) and (10) written for i =m, . . . ,n− 1. Organized in thisway, the set of equations is downsized recursively to a systemof two linear equations written in terms of the Ai and Bi pair.When solved, the two linear equations determine the potentialin the i -layer.

Actually, the downsizing process consists of two sets of steps.One starts with the first equation and constitutes the downwardrecurrence, while the second, starting with the last equation,constitutes the upward recurrence. Both recurrences are de-veloped in Appendices A and B.

Each recursion process has one or two stages, depending onthe observer position relative to the current source location.For the points having z≤ zc, the downward recurrence has onestage and the upward has two. Conversely, for the points havingz≥ zc, the downward recurrence has two stages and the upwardhas just one.

Potential expressions for points z <– zc

The pair of equations (A-3) and (B-10) can be solved for Ai

and Bi , and the potential Vi , valid for the layers 0, 1, . . . ,m− 1,m, can be expressed as

Vi = I

2πσc

∫ ∞0

1− gi exp[−10i (z− zi−1)]1− gi Gi exp[−10i (zi − zi−1)]

× e−0+i (zi−z) Ri

−λ0−m

J0(λr ) dλ, (15)

remembering that zi = zc when i = m. Moreover, gi is obtainedfrom the stage 1 downward recurrence formulas (A-4) to

(A-7), and Gi and Ri is obtained from the stage 2 upwardrecurrence formulas (B-11) to (B-13), combined with equa-tions (B-7) to (B-9) and the stage 1 upward recurrence formu-las (B-2) to (B-5) evaluated for layer m

¯.

Potential expressions for points z >– zc

In a similar manner, the pair of equations (A-13) and(B-1) can be solved for Ai and Bi . The potential Vi , valid forthe layers m

¯, m+ 1, . . . ,n− 1, n, becomes

Vi = I

2πσc

∫ ∞0

1− Gi exp[−10i (zi − z)]1− gi Gi exp[−10i (zi − zi−1)]

× e0−i (z−zi−1)ri

λ

0+mJ0(λr ) dλ, (16)

remembering that zi−1= zc when i =m¯

. Furthermore, Gi is ob-tained from the stage 1 upward recurrence formulas (B-2) to(B-5), and gi and r i are obtained from the stage 2 downwardrecurrence formulas (A-14) to (A-16), combined with equa-tions (A-9) to (A-11) and the stage 1 downward recurrenceformulas (A-4) to (A-7) evaluated for layer m.

HALF-SPACE AND FULL-SPACE MODELS

If I assume a two-layer case with a nonconductive layer 0(representing the air) and a current source located at zc≥ z0,only the two potentials V1 and V1 need be considered. Followingthe directions given in the previous section, these potentials are

V1 =I

2πσc

{∫ ∞0

e−0+1 (zc−z)

0+1 − 0−1λJ0(λr ) dλ

−∫ ∞

0

(0+1/0−1)

e−0+1 (zc−z0)+0−1 (z−z0)

0+1 − 0−1λJ0(λr ) dλ

}(17)

and

V1 = I

2πσc

{∫ ∞0

e0−1 (z−zc)

0+1 − 0−1λJ0(λr ) dλ

−∫ ∞

0

(0+1/0−1)

e−0+1 (zc−z0)+0−1 (z−z0)

0+1 − 0−1λJ0(λr ) dλ

}.

(18)

Both equations (17) and (18) are a sum of two integral ex-pressions. In the equations the second integral expressions areequal and depend on z0, the coordinate of the air interface.The first integral in equations (17) and (18) does not dependon z0. The second integral tends to zero when the air inter-face recedes to −∞, meaning that this integral represents asecondary potential (V1,S). In both equations (17) and (18),the first integrals represent the primary potential (V1,P), thepotential attributable to a current source in a space having ex-ponentially varying conductivity and not the potential froma current source in a homogeneous space. Inserting the defi-nitions for 0+1 and 0−1 , I find that both primary potential ex-pressions can be integrated (Erdelyi, 1954) and represented


by a single expression:

V1,P = I e(z−zc)β1/2

4πσc× e−|β1/2|

√r 2+(z−zc)2√

r 2 + (z− zc)2. (19)

Moreover, considering the definitions for0+1 and0−1 , recogniz-ing that −λ2/0−1 =0+1 , and defining 1Z= zc− z0+ z− z0, thecomponents E1,S|r and E1,S|z of the electrical field associatedwith the secondary potential can be obtained by taking deriva-tives of the secondary potential in equations (17) or (18). Theexpressions for the E1,S|r and E1,S|z components contain onlytabulated integrals (Erdelyi, 1954). Finally, I write them as

E1,S|r = I e(z−zc)β1/2

4πσc×{

(|β1| + β1)e−|β1/2|1Z

r

−[|β1| + β11Z√

r 2 +1Z2

]e−|β1/2|

√r 2+1Z2

r

+ r e−|β1/2|√

r 2+1Z2

3√r 2 +1Z2+|β1/2|r e−|β1/2|

√r 2+1Z2

r 2 +1Z2

},

E1,S|z = I e(z−zc)β1/2

4πσc× e−|β1/2|

√r 2+1Z2

√r 2 +1Z2

×{β1

2+ 1Z[1+ |β1/2|

√r 2 +1Z2]

r 2 +1Z2

}. (20)

The term (|β1| + β1) exp(−|β1/2|1Z)/r in the E1,S|r expres-sion (20) was erroneously dropped in the analysis of Stoyerand Wait (1977). This term is null only when β1≤ 0; otherwise,it is a function of (1/r ), implying that its associated poten-tial is r -logarithm dependent. This peculiarity is analogous tothe asymptotic form for the electrical field far from a currentsource located in a conductive layer overlying an infinitely re-sistive substratum. The result can be extrapolated and used asa reminder when manipulating the n-layer earth model withits deepest layer characterized by increasing resistivity withdepth (exponential dependency or not). The logarithm termcan appear in the potential values because it was referenced toinfinity. If adequately accounted for, this fact does not consti-tute a problem in situations involving a difference of potentialsbetween points, like those found in configurations using twopotential electrodes.

NUMERICAL RESULTS

I used two numerical tests to check the theoretical results,modeling a pole–pole apparent resistivity sounding and tworesistivity logs. To evaluate the Hankel transform integrals, Icalled on the digital filtering techniques of Anderson (1975). Tomodel the resistivity logs, I specified r -values that were smallcompared with the tool size (16 or 64 inches), around 0.001 mfor the 16-inch normal tool and 0.01 m for the 64-inch tool.These r -values represent a very small deviation of the log toolalignment from the vertical. I confirmed the validity of myprocedure by comparing the resistivity log values in a homo-geneous medium obtained using the analytical expression withthose from the Hankel transform integral formula evaluated bythe digital filtering techniques. To the eye, the first test achievedresults that exactly match the pole–pole apparent resistivity re-sponses for the models used by Kim and Lee (1996).

In the second test, 16- and 64-inch normal resistivity logswere computed for two earth models. The first model had onlyhomogeneous layers; its resistivity logs are shown in Figure 3.These log curves resemble those observed in the normal resis-tivity logs shown in Daniels (1978) and in Jorden and Campbell(1986) for a thick-bed, homogeneous, three-layer model.

The second model had both inhomogeneous (varying expo-nentially with depth) and homogeneous layers. Its resistivitylogs are shown in Figure 4. In addition to the typical shape ofthe resistivity curve associated with the homogeneous layerslocated in the 50–80 m interval of the model, other featuresappeared when continuous depth variations were introduced.These included the asymmetrical shapes strongly controlledby the exponential depth dependency in the 80–150 m in-terval and ripples in the interval from 150–165 m associatedwith the fast variation of the resistivity with depth, combinedwith the large distance between the electrodes in the 64-inchtool.

CONCLUSIONS

I have found a theoretical solution for the multilayervariable-resistivity geoelectric problem. My solution repro-duces the characteristic shapes expected from idealized modelsof resistivity logs in a layered earth and are validated quantita-tively by comparison with published results. This comparisonshows that the approach used to handle the physical conditionsimposed by the current source was correct.

FIG. 3. Apparent resistivities for 16- and 64-inch normal logtools in an earth model composed of homogeneous layers.

1730 Sato

FIG. 4. Apparent resistivities for 16- and 64-inch normal logtools in an earth model composed of layers having homoge-neous or exponential depth-dependent resistivity.

Since no restrictions are imposed on the location of the ob-server and the current source, the results can be used to eval-uate the response of layered models, horizontally or verticallystratified, for any kind of tool or array used in direct or in-verse electrical modeling and tomography. My results offer anew way to overcome the difficulties (Van Dam, 1976) that

APPENDIX A

DOWNWARD RECURRENCE

My objective is to establish a linear equation with variables Ai

and Bi , incorporating the physical and geometric parametersof the layers and interfaces above layer i .

Stage 1

This stage involves only equations with variables Ai and Bi , fori = 0, 1, . . . ,m− 1,m

¯. Actually, equation (7), conditioning the

potential at −∞, is the first equation of the set. Equation (7),along with the boundary condition equations (9) and (10) atthe interface z= z0, can be simplified to an equation with thevariables A1 and B1. Repeating the process with equations (9)and (10) at the interface z= z1, and so on, the variables Ai−1

and Bi−1 satisfy an equation resembling

xi−1 Ai−1 + Bi−1 = 0. (A-1)

Equation (A-1), combined with equations (9) and (10) at theinterface z= zi−1, gives an equation with the variables Ai andBi that can be written as

1− 0+i0+i−1

si,i−11− xi−1 exp(−10i−1zi−1)

1− (0−i−1

/0+i−1

)xi−1 exp(−10i−1zi−1)

1− 0−i0+i−1

si,i−11− xi−1 exp(−10i−1zi−1)

1− (0−i−1

/0+i−1

)xi−1 exp(−10i−1zi−1)

× Ai exp(10i zi−1)+ Bi = 0, (A-2)

arise in interpreting electrical data related to gradual changesof groundwater salinity with depth.

ACKNOWLEDGMENTS

The author was supported in this research by a fellowshipfrom Conselho de Desenvolvimento Cientıfico e Tecnologico(CNPq), Brazil. I thank professors Irshad Mufti and BenClennel, who carefully read this paper and made suggestionswhich certainly improved this work. Also, I extend my grati-tude to the SEG associate editor and the reviewers for theirconstructive comments and discussions. Finally, I express mydeepest thanks to Dr. Frank Levins.

REFERENCES

Alfano, L., 1962, Geoelectrical prospecting with underground elec-trodes: Geophys. Prosp., 10, 290–303.

Anderson, W. L., 1975, Improved digital filters for evaluating Fourierand Hankel transform integrals: U.S. Geol. Survey Technical ReportPB 242 800.

Banerjee, B., Sengupta, B. J., and Pal, B. P., 1980a, Apparent resistivityof a multilayered earth with a layer having exponentiality varyingconductivity: Geophys. Prosp., 28, 435–452.

——— 1980b, Resistivity sounding on a multilayered earth containingtransition layers: Geophys. Prosp., 28, 750–758.

Baumgartner, F., 1996, A new method for geoelectrical investigationsunderwater: Geophys. Prosp., 44, 71–98.

Daniels, J. J., 1978, Interpretation of buried electrode resistivity datausing a layered earth model: Geophysics, 43, 988–1001.

Erdelyi, A., 1954, Tables of integral transforms II: McGraw-Hill.Jorden, J. R., and Campbell, F. L., 1986, Well logging II—Electric and

acoustic logging: Soc. Petr. Eng.Kim, H., and Lee, K., 1996, Response of a multilayered earth with layers

having exponentially varying resistivities: Geophysics, 61, 180–191.Paul, M. K., and Banerjee, B., 1970, Electrical potential due to a

point source upon models of continuously varying conductivity: PureAppl. Geophys., 80, 218–237.

Raghuwanshi, S. S., and Singh, B., 1986, Resistivity sounding on a hori-zontally stratified multi-layered earth: Geophys. Prosp., 34, 409–423.

Sato, H. K., 1996, Electrical current source in the interior of horizontallayers having conductivities varying potentially with depth: Ph.D.thesis, Federal Univ. Bahia (in Portuguese).

Stoyer, C. H., and Wait, J. R., 1977, Resistivity probing of an “expo-nential” earth with a homogeneous overburden: Geoexploration,15, 11–18.

Sunde, E. D., 1949, Earth conduction effects in transmission systems:Van Nostrand.

Van Dam, J. C., 1976, Possibilities and limitations of resistivity methodof geoelectrical prospecting in the solution of geohydrological prob-lems: Geoexploration, 14, 179–193.


where si,i−1= ρi−1(zi−1)/ρi (zi−1). Inspecting this equation andcomparing it to equation (A-1), I found it convenient to definexi−1= gi−1 exp(10i−1zi−2). Then equation (A-2) can be rewrit-ten in a generalized form,

gi Ai exp(10i zi−1)+ Bi = 0, (3)

with the following recursion relations:

gi =1− (0+i /0+i−1

)si,i−1 fi

1− (0−i /0+i−1

)si,i−1 fi

, (4)

fi = 1− gi−1 exp(−10i−1hi−1)1− (0−i−1

/0+i−1

)gi−1 exp(−10i−1hi−1)

, (5)

...

f1 = 1, (6)

g0 = 0. (7)

Stage 2

Equation (A-3) evaluated for i = m, combined with equa-tions (11) and (14), both related to the physical constraintsfrom the current source, gives the first equation of the secondset of equations in the downward recurrence:

gmAm exp(10mzc)+ Bm = I λ

2πσc0+m× rm exp

(−0−mzc),

(8)where

gm = 1− fm

1− (0−m/0+m) fm, (9)

rm = fm

1− (0−m/0+m) fm, (10)

fm = 1− gm exp(−10mhm)1− (0−m/0+m)gm exp(−10mhm)

. (11)

Equation (A-8) includes a nonnull independent term. There-fore, the successive combinations with the continuity equa-tions (9) and (10) at the interfaces z= zm, . . . , zn−1 give an equa-tion like

xi−1 Ai−1 + Bi−1 = yi−1. (12)

Using a procedure similar to that of stage 1, I can write anequation with the variables Ai and Bi , valid for the layers i =m

¯,

m+ 1, . . . ,n− 1, n, in the following form:

gi Ai exp(10i zi−1)+ Bi = I λ

2πσc0+m

ri exp(−0−i zi−1

),

(13)where gi and ri are defined by the following recursion relations:

gi =1− (0+i /0+i−1

)si,i−1 fi

1− (0−i /0+i−1

)si,i−1 fi

, (14)

ri = ri−1 exp(0−i−1hi−1

)[ 1− (0−i−1

/0+i−1

)fi

1− (0−i /0+i−1

)si,i−1 fi

],

(15)

fi = 1− gi−1 exp(−10i−1hi−1)1− (0−i−1

/0+i−1

)gi−1 exp(−10i−1hi−1)

, (16)

and hm should be interpreted as hm, as well as zm−1 as zc.

APPENDIX B

UPWARD RECURRENCE

In a way similar to that of Appendix A but working in the op-posite direction, my objective is to establish an equation withthe variables Ai and Bi , incorporating the physical and geo-metric parameters of the layers and interfaces below layer i .

Stage 1

Equation (8) is the first equation of the set with the variablesAi and Bi for i = n, n− 1, . . . ,m+ 1, m

¯. By combining equa-

tion (8) with equations (9) and (10), evaluated at the interfacesz= zn−1, z= zn−2, etc., I write the following equation:

Ai + Gi Bi exp(−10i zi ) = 0, (B-1)

with the recursion relations

Gi =1− (0−i /0−i+1

)Si,i+1 Fi

1− (0+i /0−i+1

)Si,i+1 Fi

, (B-2)

Fi = 1− Gi+1 exp(−10i+1hi+1)1− (0+i+1

/0−i+1

)Gi+1 exp(−10i+1hi+1)

, (B-3)

...

Fn−1 = 1, (B-4)

Gn = 0, (B-5)

where Si,i+1 = ρi+1(zi )/ρi (zi ).

Stage 2

Taking equation (B-1) evaluated for i =m¯

, along with equa-tions (11) and (14) relative to the current source, I write thefirst equation of the second set of equations in the upward re-currence as

Am + GmBm exp(−10mzc)

= − I λ

2πσc0−m× Rm exp

(−0+mzc), (B-6)

where

Gm = 1− Fm

1− (0+m/0−m)Fm, (B-7)

Rm = Fm

1− (0+m/0−m)Fm, (B-8)

Fm = 1− Gm exp(−10mhm)1− (0+m/0−m)Gm exp(−10mhm)

. (B-9)

Equation (B-6) has a nonnull independent term. Therefore,successive combinations with the continuity equations (9) and

1732 Sato

(10) at the interfaces at z= zm−1, . . . , 0 will give equations rep-resented by

Ai + Gi Bi exp(−10i zi ) = − I λ

2πσc0−m× Ri exp

(−0+i zi)

(B-10)with the following recurrence relations:

Gi =1− (0−i /0−i+1

)Si,i+1 Fi

1− (0+i /0−i+1

)Si,i+1 Fi

, (B-11)

Ri = Ri+1 exp(−0+i+1hi+1

)[ 1− (0+i+1

/0−i+1

)Fi

1− (0+i /0−i+1

)Si,i+1 Fi

],

(B-12)

Fi = 1− Gi+1 exp(−10i+1hi+1)1− (0+i+1

/0−i+1

)Gi+1 exp(−10i+1hi+1)

. (B-13)

Here, hm should be interpreted as hm and zm as zc.


A dual-grid nonlinear inversion technique with applicationsto the interpretation of dc resistivity data

Carlos Torres-Verdın∗, Vladimir L. Druskin‡, Sheng Fang∗∗,Leonid A. Knizhnerman§, and Alberto Malinverno‡

ABSTRACTWe develop a solution to the nonlinear inverse prob-

lem via a cascade sequence of auxiliary least-squaresminimizations. The auxiliary minimizations are nonlin-ear inverse problems themselves, except that they areimplemented with an approximate forward problem thatis at least an order of magnitude faster to solve thanthe algorithm used to simulate the measurements. Anygiven auxiliary minimization in the cascade is fully self-contained and yields a solution of the unknown modelparameters. This solution, in turn, is used to perform anumerical simulation of the data to quantify its agree-ment with the input measurements. If the differencebetween the measured data and the simulated data isabove the estimated noise threshold, it is input as data tothe subsequent auxiliary inverse problem in the cascade.Otherwise, the inversion is brought to a successful com-pletion. We describe the theory and operating conditionsunder which this cascade inversion approach converges

to a solution equivalent to that of a single inversion im-plemented with the original forward problem.

Depending on the choice of approximate forwardproblem and regardless of the computer algorithm em-ployed to solve the inversion, the cascade sequence ofauxiliary minimizations can be made many times moreefficient than a single inversion performed with the orig-inal forward problem. In this paper, we consider a prac-tical and flexible way to construct the approximate for-ward problem by using a subset of the finite-differencegrid used to simulate the measurements numerically, i.e.,a dual-grid construction approach.

Numerical examples of performance are describedbased on the inversion of cross- and single-well 2.5-Ddirect-current (dc) resistivity data. Our results suggestthat the proposed dual-grid minimization technique maybe able to improve greatly the performance of eventhe most effective multidimensional nonlinear inversionprocedures currently in use.

INTRODUCTIONInversion of geophysical data is a central component of the

noninvasive probing and characterization of the subsurface.Regardless of the computer resources devoted to this task,there is a permanent need for more efficient and more ro-bust procedures that can map large volumes of data accuratelyinto space and/or time distributions of the material propertyunderlying the physics of the measurements.

One popular inversion procedure consists of minimizing aquadratic cost function that emphasizes the sum of the squareddifferences between the measured and the simulated data. Be-cause in most cases the relationship between the property dis-tribution function and the simulated data is nonlinear, the min-

Manuscript received by the Editor November 4, 1996; revised manuscript received December 30,1999.∗ University of Texas at Austin, Department of Petroleum and Geosystems Engineering, Austin, Texas 78712-1061. E-mail: cverdı[email protected].‡Schlumberger-Doll Research, Old Quarry Road, Ridgefield, Connecticut 06877. E-mail: [email protected]; [email protected].∗∗Baker Atlas, 10205 Westheimer, Houston, Texas 77042. E-mail: [email protected].§Central Geophysical Expedition, Narodnogo Opolcheniya Street., House 40, Building 3, Moscow 123298, Russia. E-mail: [email protected]© 2000 Society of Exploration Geophysicists. All rights reserved.

imization is performed with a nonlinear search technique con-structed by way of a suitable number of sequential linear steps(or iterations) which eventually trace the road toward an ex-tremum of the cost function. The way this search is performedis usually a compromise between efficiency (mainly speed) ofthe computations and stability (convergence) of the method. Acommon approach is referred to as the Gauss-Newton method,in which, at a given step in the search procedure, only first-order variations of the cost function are used to direct thesearch toward the next stop point. This requires that a matrixof first-order variations of the simulated data with respect toa variation in the model parameters be computed at each iter-ation, the so-called Jacobian, or sensitivity, matrix. Generally,

1733

1734 Torres-Verdın et al.

computing the Jacobian matrix at each linear step representsthe most time-consuming element of the minimization (see,for instance, Oristaglio and Worthington, 1980). It is, then, notsurprising that a plethora of procedures has been put forth toapproximate, bypass, and/or economize the effect the Jacobianmatrix has at every linear step in the search for the extremum(see, for instance, Smith and Booker, 1991; Ellis et al., 1993;Mackie and Madden, 1993; Torres-Verdın and Habashy, 1994;and Farquharson and Oldenburg, 1995, among others).

Whereas the efficient computation of the Jacobian matrix is asubject of much-needed research, comparatively less emphasishas been placed on deriving alternative minimization schemesthat could be less taxing in the search of the extremum. In thispaper, we develop and test one such alternative scheme. Theproposed method consists of a cascade sequence of auxiliaryleast-squares minimizations. These auxiliary minimizations arenonlinear inverse problems themselves, except that they areimplemented with an approximate forward problem that ismuch faster to solve than the forward problem used to simulatethe measurements numerically. Once the approximate forwardproblem has been constructed, it remains unchanged for all theauxiliary minimizations in the cascade. Subsequently, by solv-ing each auxiliary minimization, we obtain a solution to theunknown model parameters. This solution, in turn, is used tosimulate the measurements with the original forward problem.If the difference between the simulated data and the measureddata falls below a specified threshold, then the cascade stopsand the current model solution is taken as the output of theinversion. Otherwise, this difference is input as data for thesubsequent auxiliary minimization in the cascade. We deriveoperating conditions under which this cascade process not onlyconverges but also converges to the same extremum as a min-imization scheme implemented solely with the exact forwardproblem and regardless of the computer algorithm chosen toperform the minimization. The main advantage of the cascadeinversion procedure thus lies in the fact that the exact forwardproblem needs to be computed only as many times as auxiliaryminimizations are solved before the cascade reaches conver-gence. Moreover, at no point in this process does one need toperform a minimization with the exact forward code. This es-sentially does away with the need to compute the effect of thesensitivity matrix at each stop point in the iterative solution ofthe full nonlinear inverse problem.

By way of example, in this paper we consider the construc-tion of the approximate forward problem by taking a subset ofthe finite-difference grid used to simulate the measurementsnumerically. This is why we have used the term dual-grid non-linear inversion to designate the proposed inversion procedure.The approach of choosing a subset of the original grid allowsgreat flexibility to control the efficiency and convergence prop-erties of the cascade. Our presentation concludes with severalexamples of parametric inversion drawn from the interpreta-tion of single- and crosswell direct-current (dc) resistivity datasimulated under the assumption of 2-D variations of electricalconductivity.

A TEST EXAMPLE

To focus the discussion on a practical geophysical inverseproblem, let us consider the example described in Figure 1:An array of five dc electric current sources is deployed at

10-m spacings along the borehole indicated on the left sideof the diagram. These sources are used to generate a spatialdistribution of electric potentials in the surrounding rockformations. Sampling these potentials is the first step towardestimating the distribution of electrical conductivity in thesurrounding formations. To that effect, we consider an arrayof 18 contact electrodes deployed along a second boreholelocated on the right-hand side of Figure 1. The spacingbetween contiguous electrodes is not uniform. All the voltagemeasurements acquired with the array of contact electrodesare referred to a common ground, or reference potential,located at infinity. Finally, a third borehole is assumed tobe located midway between the borehole with the currentsources and the borehole with the contact electrodes. Thisthird borehole is not equipped with electric sources or withcontact electrodes and is used solely to inject saltwater intothe surrounding rock formations within the depth intervalfrom 30 m to 80 m. An experiment with similar characteristicswas designed and implemented at the Richmond Field Stationof the University of California, Berkeley (Bevc and Morrison,1991), with the intent of monitoring in time the diffusion of theinjected saltwater into an existing aquifer. In the subsurfacemodel shown in Figure 1 and merely for the sake of simplicity,

FIG. 1. Geometric description of the subsurface test model.The figure shows three boreholes within the same cross-section.To the left, five dc electric current sources are deployed at auniform depth interval of 10 m. Contact-point electrodes aredeployed along the well shown on the right of the figure. Thethird borehole is located midway between the borehole withthe sources and the borehole with the contact electrodes and isused to inject saline water into the surrounding rock formationsalong the depth interval from 30 m to 80 m. Notice that thecontact electrodes are not assumed to be spaced evenly; thereis a total of 18 of them.

Dual-grid Nonlinear Inversion 1735

we make the additional geometric assumption that the injectedwater diffuses into the surrounding formations in a radiallysymmetric manner. This corresponds to a 2-D model of elec-trical conductivity. However, the electric potential generatedby the sources remains a 3-D scalar function, whereuponthe geometric complexity of the simulation problem can bereferred to more properly as 2.5-D.

The objective of the dc electrical experiment described inFigure 1 is to generate electric potentials in the subsurface witheach current source acting separately, and to measure thesepotentials at each of the existing contact electrodes. The set ofelectric potentials measured with all the possible combinationsof source-electrode pairs constitutes the data from which thedistribution of electrical resistivity between wells can be esti-mated in principle via an inversion process. An additional dataset is gathered along the borehole with the electric sources byplacing contact electrodes at source locations.

Synthetic (numerical) and field data examples of limited-angle dc electrical impedance tomography suggest that theinversion is severely ill-conditioned and hence at best canestimate a few salient features of the distribution of electricalresistivity between wells (see, for instance, Daily and Owen,1991; Daily et al., 1992; and Sasaki, 1992, among others).Therefore, in an attempt to reduce the number of unknownvariables in the problem but without sacrificing generality, wehave taken the approach of parameterizing the profile of waterinvasion with only a few variables. One such profile of waterinvasion is illustrated in Figure 2. The simplified way we have

FIG. 2. A sample description of the electrical resistivity modelassumed after the inception of water injection. The injection ofwater into the background resistivity model shown in Figure 1gives rise to a vertical variation in the radius of invasion becauseof existing vertical variations in permeability. We consider sixsuch variations of 1-D permeabilities along the depth intervalwhere water injection takes place. It also is assumed that theinvasion is axially symmetric and that the invaded rocks exhibita uniform electrical resistivity of 0.5 � · m.

chosen to describe the electrical resistivity distribution consistsof assuming that the rocks saturated with saltwater exhibita uniform electrical resistivity regardless of the resistivityof the uninvaded rock matrix. This is a justified assumptionbecause, according to Archie’s law, when saltwater is made tofill the available pore space in permeable rocks, the electricalresistivity of the former becomes the dominant contributionto the effective rock resistivity, especially with large values ofporosity. In actual field experiments, it is desirable to makethe injected saltwater as conductive as possible to increasethe resistivity contrast with the surrounding rock formationsand hence maximize the anomalous dc response. Assumingarbitrarily that the electrical resistivity of the invaded rocksis equal to 0.5 �.m, the remaining unknown variables are theinvasion radii at the various depth levels of the water-injectionexperiment. As indicated in Figure 2, we have chosen arbi-trarily to parameterize the vertical segment of water injectionwith six such radii of invasion. By choosing this simplifiedparameterization, we also intend to concentrate the discussionon the “mechanical” details of the inversion rather than onthe trade-offs between resolution and uncertainty that are atthe heart of any geophysical inverse problem.

NUMERICAL SIMULATION OF THE 2.5-D DC PROBLEM

The differential equation satisfied by the electric potential,U(r), is given by

∇ · [σ (r)∇U(r)] = ∇ · JS(r), (1)

where r is the observation point, σ (r) is the distribution ofelectrical conductivity, and JS(r) is the impressed dc currentsource. In this paper, a solution of equation (1) is approachedby using a finite-difference formulation in cylindrical coordi-nates via the spectral Lanczos decomposition method (SLDM)(Druskin and Knizhnerman, 1995). Details of this highly effi-cient method of solution are discussed in Appendix A. Ex-tensive self-consistency checks and benchmarking exerciseswith homogeneous and 1-D models of electrical conductivityshowed that our simulations were accurate within less than 1%.The results from one such exercise are shown in Figure 3, withplots of percent errors obtained by simulating the single- andcrosswell potentials resulting from a single current source. Forthese simulations, we considered the layered sequence of bedsshown in Figure 1 and an electric source located at a depth of45 m. We computed the potentials resulting from the layered-bed model with an accurate 1-D algorithm, and these potentialsserved as reference to calculate the percent errors caused bythe SLDM algorithm. The plots shown in Figure 3 indicate per-cent errors larger for the single-well case than for the crosswellcase, but in both cases the errors are less than 1% in absolutevalue. A single-source simulation on a 2-D grid of size 41 × 49was clocked at 10 seconds of CPU time when implemented ina Silicon Graphics Power Challenge machine.

Figure 4 is a plot of the electric potential U(r) in volts sim-ulated along the same model cross-section shown in Figure 1when the dc current source is placed at a depth of 45 m withinthe borehole shown on the left side of the same figure. Thevalues of electric potential in Figure 4 are described as a colormap and with a logarithmic scale (log10 base) in an effort to dis-play somewhat uniformly the wide range of numbers spannedby U(r). Figure 5, on the other hand, is a plot constructed in


FIG. 3. Accuracy assessment of the SLDM numerical simula-tion algorithm. The plots describe percent differences betweenthe electric potentials simulated with the SLDM algorithm andthe same potentials simulated with a 1-D algorithm. The sub-surface model of electrical resistivity is the layered sequenceof beds shown in Figure 1, and the electric source is located ata depth of 45 m. Percent potential differences are shown forboth single- and crosswell measurements.

FIG. 4. Electric potential in volts (log10-base) simulated for the1-D resistivity model shown in Figure 1 and along the samecross-section. The dc current source is located at a depth of45 m in the borehole shown on the left in Figure 1.

the same way as Figure 4 but describing the electric potentialsimulated for the model of water injection shown in Figure 2. Fi-nally, Figure 6 displays the relative percent differences betweenthe previous two simulations. Notice that the largest and mostprominent potential anomaly occurs at the boundaries of thelargest invasion radius. The remaining boundaries, at best, aresuggested slightly by the distribution of anomalous potential.

NONLINEAR INVERSION AS MINIMIZATION

Let m be the size-N vector of unknown variables that fullydescribes the underlying electrical resistivity distribution (inthis case, the radii of water invasion), and mR a reference vec-tor of the same size as m which has been determined fromsome a priori information. Often, the estimation (inversion) ofm from the measured data is undertaken by solving the equiv-alent problem of minimizing a cost function defined with somemetric. Let C(m) be one such quadratic cost function given by(Torres-Verdın and Habashy, 1994)

2C(m) = {∥∥Wd · [d(m) − dobs

]∥∥2 − χ2}+ λ‖Wm · (m − mR)‖2, (2)

where

FIG. 5. Electric potential in volts (log10-base) simulated for thewater-flood resistivity model shown in Figure 2 and along thesame cross-section. The dc current source is located at a depthof 45 m in the borehole shown on the left of Figure 2 (comparewith Figure 4).


dobs is the measured data vector, usually corrupted with somelevel of noise, Wd is the inverse of the data covariance matrix,χ2 is the prescribed value of data misfit, d(m) is the data vectornumerically simulated for specific values of m, Wm is the inverseof the model covariance matrix, and λ is a Lagrange multiplieror regularization parameter.

A common technique used to find a stationary point (orextremum), m, where the cost function attains a minimum,consists of performing a Gauss-Newton fixed-point iterationsearch (Gill et al., 1981). This method disregards second-ordervariations of the cost function in the vicinity of a local iterationpoint. The corresponding iterated formula can be written as

mk+1 = [JT(mk) · WT

d · Wd · J(mk) + λWTm · Wm

]−1

· {JT(mk) · WTd · Wd · [

d(mk) − dobs

+ J(mk) · mk] + λWT

m · Wm · mR}, (3)

subject to

li ≤ mk+1i ≤ ui .

In the above expression, the superscript k is used as an iterationcount, the superscript T denotes transpose, and J(m) is the

FIG. 6. Relative percent difference between the electric poten-tial simulated for the water-flood resistivity model shown inFigure 2 and the electric potential simulated for the 1-D back-ground shown in Figure 1. The dc current source is located ata depth of 45 m in the borehole shown on the left in Figure 1.

Jacobian matrix of C(m), given by

J(m) =

∂d1/∂m1 · · · ∂d1/∂m� · · · ∂d1/∂mN

......

. . ....

∂d j/∂m1 · · · ∂d j/∂m� · · · ∂d j/∂mN

......

. . ....

∂dM/∂m1 · · · ∂dM/∂m� · · · ∂dM/∂mN

M×N

The upper and lower bounds enforced on mk+1 are intendedto have the iterated solution yield only physically consistentvalues (in the problem at hand, for instance, a trivial choice oflower bound consists of restricting the unknown invasion radiito take only positive numbers).

When the linear system of equations embodied in equa-tion (3) is solved for subsequent values of m in the searchof a minimum of the quadratic cost function, evaluating theJacobian matrix remains the most computationally demandingoperation. The fixed-point iteration search for a minimum ofC(m) is concluded when the measured data have been fit withinthe prescribed tolerance, χ2.

There are, of course, alternative implementations of theGauss-Newton algorithm that, under particular conditions, canbe more efficient than the general Gauss-Newton procedure.For instance, Sasaki (1992) has used a formulation in whichthe source-potential reciprocity is invoked to reduce the num-ber of computations needed to assemble the Jacobian matrix.This procedure becomes extremely efficient when the mea-surements consist of a large number of independent sourcelocations. Steepest descent and/or conjugate gradient proce-dures also have been proposed by Mackie and Madden (1993),in which at most only a few elements of the Jacobian matrixare needed (and certainly no inversion of the Jacobian matrixis needed) at each step of the nonlinear inversion. All that isrequired is the product of the Jacobian matrix times a vector.Albeit efficient in their computations, such procedures maytake a very large number of linear steps before convergingto the minimum of the cost function. The specific choice ofa nonlinear minimization algorithm is determined largely bythe underlying physics of the problem, as well as by the mea-surement conditions and the assumed model parameterization(number and type of parameters). In this paper, we have nointention of pontificating a particular adaptation of the Gauss-Newton or steepest descent procedures to solve the nonlinearminimizations. Our aim is to take one step back and reformu-late the inverse problem in a manner that makes it naturallymore efficient regardless of the specific algorithm used to solvethe minimization.

A NEW MINIMIZATION PROCEDURE

In the example at hand, the forward-modeling operator d(m)consists of solving the partial differential equation (1) with afinite-difference grid. Figure 7 is a graphical description of thefinite-difference grid used to that end. It consists of 41 nodesin the radial direction (away from the injection well) and 49in the vertical direction (the figure shows only a fraction ofthe grid). This is the actual grid used to perform the numeri-cal simulations of the potential shown in Figures 3 through 5.The grid boundaries coincided with the coordinates ρ = 6 kmand z = ±6 km, respectively. Because we have assumed axial


symmetry in the electrical-conductivity distribution, the solu-tion algorithm described in Appendix A requires discretizationonly along the (ρ, z) plane (ρ ≥ 0). This is why we are showingonly the finite-difference mesh along the plane in Figure 7.

Using shorthand notation, let us identify the inverse operatorwith the formula

m = d−1(dobs), (4)

where d−1(dobs) is in general nonlinear. In analogy with theforward and inverse operators defined above, we introduceauxiliary forward and inverse operators denoted as

d = h(m) (5)

and

m = h−1(d), (6)

respectively, where d is a modified measurement vector yet tobe defined. The symbol m is used here only for labeling pur-poses to identify when a model vector has been derived fromthe operation h−1(d). Heuristically, there are two importantconditions we would like to impose on the construction of theabove auxiliary forward and inverse operators:

1) h(m) should be much faster to compute than d(m), and2) h−1 (d) should be bounded in a special way (to be defined

shortly).

FIG. 7. Detail of the 2-D finite-difference grid used to per-form the numerical simulation of single- and crosswell dc re-sistivity measurements. The complete grid consist of 41 nodesin the vertical direction and 49 in the radial direction, withouter boundaries extending to 6 km in the radial direction and±6 km in the vertical direction. The circles indicate voltagemeasurement points.

Condition (1) is highly desirable from a practical point ofview, because if a relationship exists between h−1 and d−1, thensolving for the former via repeated calls to h(m) could save,in principle, significant computer time. Condition (2), on theother hand, not only secures the possibility of inverting h(m)in a stable manner, but if the bound is designed properly, it alsocould enforce a stable contractive mapping of the differencesbetween h(m) and d(m), as is shown next.

The objective pursued here is to construct a series of approx-imations m to m by performing inversions with the operatorh−1(d). On convergence, such series of approximations shouldyield m equal to m, but the computer efficiency still shouldbe in favor of performing the repeated inversions of m. Weconstruct this series of approximations through a fixed-pointiteration recurrence given by

mk+1 = h−1[h(mk) − d(mk) + dobs] (7)

where, once again, the superscript k is used to denote iterationcount. We first remark that if the above fixed-point iteration isto converge, then, on convergence,

mk+1 = mk .

Recall also that by definition

mk = h−1[h(mk)],

whereupon it follows that

dobs − d(mk) = 0.

If d−1(dobs) has a solution equal to m, then it is obvious that

mk = m.

We therefore conclude that if the fixed-point iteration givenby equation (7) is a contractive mapping, it converges to thesolution of the original inverse problem d−1(dobs). By directinspection of equation (7), it also follows that

d = h(mk) − d(mk) + dobs . (8)

The remaining important issue to deal with here concerns theoperating conditions that will guarantee the fixed-point itera-tion given by equation (7) to converge. Evidently, the smallerthe Euclidean norm of the difference [h(m) – d(m)], the closerto global convergence the fixed-point iteration will be, but thisis hardly a quantitative statement. In Appendix B, we provethat convergence is guaranteed when the following boundingcondition is satisfied:

‖∂h−1(d) · [∂h(m) − ∂d(m)]‖ ≤ 1, (9)

where the operator ∂ designates the Jacobian matrix (Frechetderivative) of the corresponding vector function; accordingly,using the notation introduced earlier, ∂d(m) = J(m). Apartfrom being a rigorous design restriction, the above boundingcondition is not necessarily a practical recipe to construct theauxiliary forward-modeling operator h(m). The message is sim-ply that the closer the Jacobian matrix of h(m) is to the Jacobianmatrix of d(m), the closer to global convergence the recurrence(7) will be, provided that h−1(d) is bounded appropriately aswell.

We summarize the proposed nonlinear inversion procedurewith the following sequence of steps:


1) Construct the auxiliary forward-modeling operator h(m)to yield to a faster method of solution than the one usedto compute d(m), but subject to the condition

‖∂h−1(d) · [∂h(m) − ∂d(m)]‖ ≤ 1.

This construction customarily would be achieved by trialand error, but would be done only once (and even perhapsonly once for a large variety of data sets).

2) Provide an initial guess for m.3) Compute d(m); if the Euclidean norm of [d(m) − dobs] is

below the specified threshold, then the inversion stops andthe current value of m is accepted as the extremum of theleast-squares cost function. Otherwise, compute

d = h(m) − d(m) + dobs .

4) Find the extremum of the auxiliary cost function

2C(m) ={‖Wd · [h(m)− d]‖2 −χ2

}+λ‖Wm ·(m−mR)‖2.

5) Return to step (3).

Because the modified data vector d is reset every time theabove process reaches step (3) before convergence, each ofthe auxiliary cost functions of step (4) is defined in a cascadefashion whereby a given cost function depends on the resultsobtained from minimizing the previous cost function in thecascade. This is why we have adopted the term cascade mini-mization to refer to the new inversion procedure developed inthis paper.

The critical step in the cascade minimization is the construc-tion of the auxiliary operator h(m), which remains unchangedthroughout this process. Without loss of generality, in this pa-per we have chosen to define the auxiliary (approximate) for-ward operator by using a subset of the finite-difference gridused to simulate d(m). The specific subset of the original finite-difference grid should be chosen to expedite significantly thecomputation of h(m). Additionally, the choice of grid subsetshould abide by condition (9). For the problem at hand, exten-sive experimentation by trial and error with numerical simula-tions and checks to bounding condition (9) led us to the use ofthe finite-difference grid illustrated in Figure 8 for the computa-tion of h(m). The main feature of this grid is that within the areawhere the conductivity distribution is expected to change in thecourse of the minimization, the grid is as spatially fine as thegrid shown in Figure 7 that is used for the computation of d(m).Outside this area, however, the grid is assigned larger spatialsteps in the radial and vertical directions to reach the sameboundaries used in the construction of the fine grid. There is atotal of 27 nodes in the vertical direction and 26 in the radial di-rection for the “coarse” grid (compare with 41 × 49 for the finegrid). A numerical simulation with this coarse grid was clockedat approximately 2 seconds of CPU time when implementedin a Silicon Graphics Power Challenge machine (compare with10 seconds of CPU time for the fine grid). Minimization of theauxiliary least-squares cost functions was performed with theGauss-Newton procedure described by equation (3). We alsofound that for the problem at hand, the computation of theJacobian matrix within a given auxiliary minimization could besimplified by using the Broyden’s update procedure describedin Appendix C. Finally, the attending linear system of equations(3) was solved by enforcing upper and lower bound constraints

of 0 m and 60 m, respectively, using a constrained least-squaresalgorithm from Lawson and Hanson (1974).

EXAMPLES OF PARAMETRIC INVERSION

The foregoing nonlinear inversion procedure was imple-mented in the estimation of radii of water invasion for thehypothetical experiment described in Figures 1 and 2. As men-tioned earlier, there are six unknown radii of invasion, anddata consist of single- and crosswell dc resistivity experimentswith 18 contact electrodes and five dc current sources. In theestimation of the radii of water invasion, we also contaminatedthe simulated data with various amounts of random Gaussiannoise. Figure 9 shows plots of the simulated voltages plus ad-ditive noise. Noise was synthesized numerically with a zero-mean Gaussian random number generator of standard devia-tion equal to a given percentage of voltage amplitude. Threecurves are shown in that figure, for noise percentages of 0%,2%, and 5%, respectively. The data shown correspond to cross-well measurements simulated for a dc current source locatedat a depth of 45 m, and the underlying resistivity model is theone described in Figure 2.

Figure 10 shows the vertical profiles of invasion radii invertedfor the resistivity model of Figure 2. Four panels of results areshown in that figure, with the “true” invasion profile plottedfor reference with a solid line and the inverted profile witha dashed line. The upper left panel shows the initial profile

FIG. 8. Detail of the 2-D finite-difference grid used to solvethe approximate forward problem associated with the auxilaryleast-squares cost functions. The complete grid consists of 27nodes in the vertical direction and 26 in the radial direction,with outer boundaries extending to 6 km in the radial directionand ±6 km in the vertical direction (compare with Figure 7).


used to initialize the inversion (in this case uniform), and thesecond, third, and fourth panels show the vertical profile ofinvasion radii inverted from single-well, crosswell, and com-bined (single- plus crosswell) dc resistivity data, respectively.In this case, no noise was added to the simulated data beforeperforming the inversions, and the data were fit to within theaccuracy inherent to the numerical simulations. Notice that

FIG. 9. Plot of the electric potential in volts simulated along theline of contact electrodes shown in Figure 2 for a dc currentsource located at the depth of 45 m in the borehole to theleft of the same figure. The underlying resistivity model alsois described in Figure 2. Curves are shown for three differentnoise percentages added to the data. The noise was synthesizedwith a zero-mean random Gaussian number generator.

FIG. 10. Plots of the vertical profile of invasion radii invertedfrom noise-free dc resistivity data. In all panels, the originalprofile is plotted with a solid line and the inverted profile witha dashed line. Different panels show the profiles inverted fromsingle-well, crosswell, and combined (single- plus crosswell) re-sistivity data. The upper left panel displays the profile of inva-sion radii used to initialize the inversions. Each single- or cross-well experiment consisted of a set of five dc current sources and18 contact electrodes, deployed as described in Figure 2.

the vertical profile of invasion radii inverted from the cross-well data is slightly closer to the true profile than the profileinverted from the single-well data. On the other hand, the pro-file inverted from both single- and crosswell data is superior toeither profile inverted separately. We also observe that in allcases, the largest invasion profile is the one that is estimatedmore accurately. Conversely, the shortest radii of invasion arethe ones reconstructed most poorly. This behavior is consistentwith the distribution of anomalous electric potential shown inFigure 6, where the largest potential anomaly by far occurs atand about the boundary of the largest invasion radius. The factthat even with noise-free data the inverted profile does not per-fectly match the original profile indicates that the quadratic costfunction does exhibit an extremum; but in the neighborhoodof that extremum, the cost function is extremely flat. This ex-plains why multiple solution possibilities can fit the data withinthe accuracy of the forward-modeling code.

Figure 11 describes the route to convergence of the nonlin-ear inversion procedure used to obtain one of the profiles ofinvasion radii shown in Figure 10. In Figure 11a, the relativedata misfit is plotted as a function of the auxiliary least-squaresfunction defined in the cascade. We computed the relative datamisfit with the formula∥∥Wd · [d(m) − dobs]

∥∥2∥∥Wd · dobs∥∥2 ,

where the data-weighting matrix Wd is a diagonal matrix withelements equal to the inverse of the measurement times thestandard deviation of the noise (in the noise-free case, Wd is

FIG. 11. Plots of data misfit versus iteration number in thesearch of the extremum of the cost function. (a) is the nor-malized data misfit evaluated with the fine grid shown in Fig-ure 7, and the abscissas identify the corresponding auxiliaryleast-squares cost function. (b) shows the normalized data mis-fit as a function of the iteration number within one of the auxil-iary cost functions constructed with the coarse grid illustratedin Figure 8.


set to a diagonal matrix with elements equal to the inverseof the measurement). By so doing, the misfit errors, in effect,were expressed as relative errors, with the much desirable con-sequence that the inversions were not biased strongly to fit onlythe largest amplitude variations in the data. Each of the auxil-iary cost functions is identified with a number that correspondsto its location in the cascade. Figure 11b is a plot of the datamisfit with respect to the iteration number within one of suchauxiliary cost functions. For this case, the auxiliary cost functionwas minimized after 17 iterations, despite the abrupt backwardchange at iterations 3 and 4. The minimization was performedas described in the previous section, using the coarse finite-difference grid shown in Figure 8. We also encountered casesin which the auxiliary cost function was minimized in as fewas five iterations, especially when the global data misfit shownin Figure 11b was close to its expected lower bound. Becausethe number of data was much larger than the number of un-knowns, we had no need to add a regularization term to theauxiliary cost function to obtain a unique solution. In the caseof noisy measurements, we implemented a simple regulariza-tion technique which consisted in setting the matrix Wm equalto a unity diagonal matrix in equation (2). This correspondsto the well-known Wiener regularization scheme (Treitel andLines, 1982). We further set the regularization parameter, λ,to be a small percentage of the ratio between the largest andsmallest eigenvalues of matrix JT(mk) ·WT

d ·Wd ·J(mk) in equa-tion (3). The value of this percentage was chosen arbitrarily tobe proportional to the estimated noise level.

In Figure 11a, the nonlinear inversion was completed withonly four calls to the forward-modeling code implemented onthe fine grid. The complete cascade inversion was performed inapproximately 17 minutes of CPU time on a Silicon GraphicsPower Challenge machine. This represented nearly a 12-foldsavings in CPU time with respect to a single nonlinear inver-sion implemented with the original finite-difference grid. Inboth experiments, i.e., (1) using a Gauss-Newton minimizationprocedure implemented entirely with the fine finite-differencegrid (including the computation of the Jacobian matrix), and(2) using the cascade inversion with an approximate forwardproblem realized with the coarse grid, model results agreedwithin less than 1%. Sample plots of data residuals are shownin Figure 12 for the case of a dc current source located at a depthof 45 m. These residuals were constructed by subtracting thevoltages fed to the inversion as data from the correspondingvoltages simulated numerically for the inverted profiles of inva-sion radii shown in Figure 10. The panel on the right is a plot ofthe data residuals along the contact electrodes opposite to thedc source, i.e., for the case of a crosswell measurement system.On the other hand, data residuals along the contact electrodeswithin the same borehole, i.e., for a single-well measurementsystem, are plotted in the panel on the left. We remark that theresiduals are approximately two orders of magnitude smallerin amplitude than the input data.

The effect of noisy data on the inversions becomes evident inthe results described by Figure 13 and 14. In rendering both setsof inversions, zero-mean random Gaussian noise was added tothe data, as described graphically in Figure 9, with standard de-viations equal to 2% and 5%, respectively, of the measurementamplitude. These inversions show a drastic reduction in theability of noisy data to resolve the short invasion radii in com-parison with the resolution available from noise-free data. For

the most part, only the largest invasion radius remains close toits true value. Such behavior becomes most evident when onlysingle-well data are used as data to perform the inversions.

CONCLUSIONS

The algorithm for nonlinear inversion developed in this pa-per is an attempt to cast standard minimization procedures inthe form of a cascade sequence of simpler, hence more efficient,nonlinear inversions. We have chosen to construct these simple

FIG. 12. Data misfit residuals rendered by the inversions ofinvasion radii shown in Figure 10. In this subset, the dc currentsource is located at a depth of 45 m within the borehole shownon the left in Figure 2. The panel on the right is a plot of the dataresiduals along the line of contact electrodes in the oppositeborehole (crosswell experiment). Data residuals for contactelectrodes within the same borehole (single-well experiment)are shown in the panel on the left.

FIG. 13. Plots of the vertical profile of invasion radii invertedfrom 2% noisy dc resistivity data. Plotting conventions are asdescribed in Figure 10. Noise added to the measurements wassynthesized with a zero-mean random Gaussian number gen-erator of standard deviation equal to 2% of the measurementamplitude (compare with Figures 10 and 14).


FIG. 14. Plots of the vertical profile of invasion radii invertedfrom 5% noisy dc resistivity data. Plotting conventions are asdescribed in Figure 10. Noise added to the measurements wassynthesized with a zero-mean random Gaussian number gen-erator of standard deviation equal to 5% of the measurementamplitude (compare with Figures 10 and 13).

nonlinear inversions by using an approximate solution to theoriginal forward problem. Specifically, in this paper we have ex-plored the possibility of constructing an approximate forwardproblem by taking a subset of the finite-difference grid that isused to simulate the measurements numerically. Other approx-imate forward problems could be used for the same purposeas long as they abide by the necessary requirements of con-vergence. Approximations based on integral-equation formu-lations such as the ones reported by Habashy et al. (1993) andTorres-Verdın and Habashy (1994), for instance, could take therole of the auxiliary forward-modeling operator in the cascade.However, the dual-grid approach proposed here is attractive be-cause it allows great flexibility to adjust the degree of accuracyrequired to achieve a given rate of convergence. We also en-vision that the same cascade procedure can be formulated toperform the inversion over a sequence of several subsets of theoriginal finite-difference grid with different degrees of relative“coarseness.” Albeit promising, this possible extension to thedual-grid approach remains a subject for future research. Aninversion strategy such as that could well find its best applica-tion in the formulation of 3-D inverse problems, in which re-peated access to the forward-modeling code can be extremelytaxing, even in computers with parallel architecture (see, forinstance, Newman and Alumbaugh, 1996).

Through the simple examples reported in this paper, we haveshown that the dual-grid inversion approach is superior to theminimization performed solely on the grid designed to repro-duce the measurements; the difference was in the order of a10-fold reduction in CPU time. This observation stemmed fromsimple dc resistivity experiments involving very few unknownsand little or no need of regularization. However, several issuesat stake need to be further examined, particularly regardingthe link between grid design and convergence rate of the aux-iliary nonlinear inversions. The approach taken in this paperconsisted of reducing the number of nodes outside the area of

interest, but there is reason to think that even across the area ofinterest, the finite-difference grid could be made coarser. This isalso a subject for future research that will be related intimatelyto the specific nature of the partial differential equation thatgoverns the measurements.

ACKNOWLEDGMENTS

Metin Karakas of Schlumberger-Doll Research shared withus many valuable insights on the physics of fluid flow in porousmedia, particularly in the context of water-injection experi-ments. We are obliged to Greg Newman, David Alumbaugh,and Ki Ha Lee for constructive technical and editorial sugges-tions that have improved the quality of this paper.

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APPENDIX A

FINITE-DIFFERENCE SOLUTION OF THE 2.5-D dc PROBLEM FOR CYLINDRICALLY SYMMETRIC ANISOTROPICCONDUCTIVITY MODELS

Let us consider a reference frame with cylindrical coordi-nates (ρ, z, φ) chosen such that the z axis coincides with thewater-injection well described in Figure 2. We further assumethat the electrical-conductivity distribution is axially symmet-ric about the z axis and therefore can be described solely as afunction of the coordinates (ρ, z). However, we do allow bothdc current sources and contact electrode points to be locatedanywhere in space. This causes the resulting electric potentialto be a function of all three coordinates (ρ, z, φ). Simulationproblems with these characteristics, i.e., 3-D functional behav-ior and 2-D model complexity, nevertheless can be solved with2-D formulations, as we are set to show next.

A standard method of solution entails a decomposition interms of azimuthal (cylindrical) harmonics. Each one of theseazimuthal harmonics is solved via a 2-D simulation problem,and the results for all harmonics are superimposed to generatethe complete solution. However, because of the special prop-erties of the potential field arising in connection with crosswellstudies posed in a cylindrical coordinate frame, a harmonic de-composition of this nature may require computing the solutionof an excessive number of harmonics (as many as 8,000 har-monics to obtain a solution in which the current source and thecontact electrode have the same z and ρ coordinates but differ-ent φ coordinates). To circumvent this problem, we resort to analternative approach which is numerically more efficient thanharmonic decomposition and is based on the spectral Lanczosdecomposition method (SLDM). This approach has the advan-tage of implicitly rendering 2.5-D simulation problems into asingle simulation of a 2-D problem.

Let U(ρ, z, φ) designate the dc potential resulting from apoint current source δ(φ)g(ρ, z) acting in the closed spatial do-main ϒ ⊂ R3, and subject to Dirichlet boundary conditionsalong ∂ϒ . The conductivity function here is considered to havethe characteristics of an axially symmetric diagonal tensor,i.e., σ = σ(ρ, z) = diag{σρ(ρ, z), σz(ρ, z), σφ(ρ, z)}, therebylending itself to the study of an important class of conductiv-ity anisotropy. The differential equation satisfied byU(ρ, z, φ)thus is given by

∇ · [σ(ρ, z, φ) · ∇U(ρ, z, φ)] = δ(φ)g(ρ, z) (A-1)

or, in cylindrical coordinates,

1ρ

∂

∂ρ

[ρσρ

∂U

∂ρ

]+ ∂

∂z

[σz

∂U

∂z

]+ σφ

ρ2

∂2U

∂φ2= δ(φ)g(ρ, z).

(A-2)By making the change of variable

u(ρ, z, φ) = U(ρ, z, φ)√

σφ

ρ, (A-3)

equation (A-2) can be written as

A[u] + ∂2u

∂φ2= δ(φ)ϕ(ρ, z) 0 ≤ φ ≤ 2π, (A-4)

where A is a differential operator which only considers varia-tions of u with respect to ρ and z, given by

A[u(ρ, z, φ)] =√

ρ

σφ

∂

∂ρ

[ρσρ

∂

∂ρ

√ρ

σφ

u

]

+ ρ2

√1σφ

∂

∂z

[σz

∂

∂z

√1σφ

u

]

and

ϕ(ρ, z) =√

ρ3

σφ

g(ρ, z).

It can be shown easily that A[u] is symmetric and negativedefinite.

To solve for u(ρ, z, φ), we approximate the operator Ausing a five-point second-order finite-difference stencil on a2-D grid consisting of n nodes. Let matrix A denote the finite-difference discretization of the differential operator A. Equa-tion (A-4) thus becomes an ordinary linear differential equa-tion for the unknown nth dimensional vector u(φ) constructedas the approximation of u(ρ, z, φ) at the n nodes of the 2-Dgrid. By taking into account the driving term on the right-handside of equation (A-4), as well as the azimuthal symmetry ofu(ρ, z, φ), i.e.,

u(φ) = u(2π − φ),

one obtains the following matrix functional representation of u:

u(φ) = 0.5[e−φ

√−A + e(−2π+φ)√−A

]·[√−A

× (I − e−2π

√−A)]−1· ϕ 0 ≤ φ ≤ π, (A-5)

where ϕ is the nth dimensional representation of functionϕ(ρ, z) over the same 2-D grid. The computation of u(φ) viaequation (A-5) requires the evaluation of matrix

√−A and ofits subsequent exponentiation eφ

√−A for every possible valueof φ. This computation can be performed in principle if onesolves the eigenvalue problem for matrix A. After solvingthis eigenvalue problem, one could solve for vector u(φ) for asmany values of φ as needed without significant additional com-putations. We also remark that having solved the eigenvalueproblem for matrix A in principle would allow one to solve forvector u(φ) in response to several values of vector ϕ withoutan appreciable increase in computer operations. Each vectorϕ would correspond to a specific electric-source location inthe (ρ, z) plane. If the electric-source location varies only withrespect to the φ coordinate, one would not have to modify thevector ϕ as long as the electric source remains a point source(or a finite set of source points).

Although obtaining a solution of vector u(φ) via a solution ofthe eigenvalue problem of matrix A provides valuable insightto the problem, a numerical solution implemented in this waywould be impractical because of the often large size of matrixA. To solve this problem, we make use instead of the SLDMas described by Druskin and Knizhnerman (1995). With the


SLDM, vector u(φ) in equation (A-5) can be written explicitlyas

u(φ) ≈ um(φ) = 0.5‖ϕ‖Q ·[e−φ

√−H + e(−2π+φ)√−H

]

·[√−H

(I − e−2π

√−H)]−1· e1, (A-6)

where Q and H are n × m and m × m matrices, respectively,obtained after performing m steps of the Lanczos recurrenceprocess on matrix A and vector ϕ; vector e1 is the first unitvector in Rm . Under the Lanczos recurrence process (Parlett,1980), matrix H is symmetric and tridiagonal and is obtainedfrom a Gram-Schmidt orthogonalization process of A. MatrixQ, on the other hand, will lose orthogonality because of round-off errors when the Lanczos recursions are performed in finiteprecision, but a landmark theorem by Druskin and Knizhner-man (1995) shows that this phenomenon is not detrimental toconvergence.

From a computational viewpoint, the advantage of equa-tion (A-6) over equation (A-5) is that although matrix A issparse, matrix H is only tridiagonal, and therefore solvingthe eigenvalue problem of H is substantially more efficientthan solving the eigenvalue problem of A. In turn, at a givenm-step of the Lanczos process, solving the eigenvalue problemfor matrix H enables one to obtain a solution for um(φ). Yetin like manner with the eigenvalue problem of matrix A, oneof the the most important features of the SLDM solution isthat once the eigenvalue problem for matrix H is solved, re-sults for additional values of φ can be obtained with practicallyno overhead in computer efficiency. It is this very feature ofequation (A-6) that makes SLDM superior in principle to anyalternative iterative matrix solver such as conjugate gradient,for instance. However, because matrices Q and H are derivedfrom the repeated projections of vector ϕ onto matrix A, onehas to recompute both these matrices every time one changesthe source vectorϕ, i.e., every time one changes the (ρ, z) loca-tion of the electric source. Although this would not be the case

with a numerical solution obtained via the solution of the eigen-value problem of matrix A, the SLDM solution still can providea significant edge in computer efficiency when a solution is re-quired for a large collection of source vectors ϕ. Tamarchenko(1988) and Allers et al. (1994) have developed a similar solu-tion of equation (A-1) in Cartesian coordinates. In cylindricalcoordinates, however, the matrix operator A can be severelyill conditioned because of the essential singularity at ρ = 0.

Of course, one would expect that convergence of um(φ) to-ward u(φ) could be achieved for m < n. Otherwise, the SLDMsolution would run the risk of becoming as inefficient as a di-rect numerical solution based on the solution of the eigenvalueproblem of matrix A. The convergence properties of SLDM,when operated on certain matrix functionals, have been studiedat length by Druskin and Knizhnerman (1989). In their stud-ies, the number of steps m required to achieve convergenceare dictated mainly by the condition number of matrix A aswell as by the matrix functional at stake. In an effort to im-prove the convergence properties of certain matrix function-als such as the one given by

√A and which appears in equa-

tion (A-5), Druskin and Knizhnerman (1998) have advanced anovel recurrence scheme which, in similar manner to the Lanc-zos method, is based on a Krylov subspace representation ofthe matrix vector pair (A,ϕ), but which considers in additionthe extended Krylov subspace spanned by the pair (A−1 · ϕ).Druskin and Knizhnerman (1996) have coined the term ex-tended Krylov subspace method (EKSM) to designate their newrecurrence procedure. This extension of the standard Krylovsubspace spanned by (A,ϕ) in principle dramatically improvesthe convergence properties of SLDM, even considering the ex-tra expenses incurred by the preliminary computation of A−1·ϕ(via LU factorization and back substitution, for instance).

The FORTRAN code used to perform the numerical simu-lations reported in this paper is based on an implementationof the EKSM. Typical simulation errors over a 41 × 49 grid arenot higher than 1%. A single-source simulation with this sizegrid has been clocked at 10 seconds of CPU time in a SiliconGraphics Power Challenge machine.

APPENDIX B

CONVERGENCE OF THE FIXED-POINT ITERATION

In this appendix, we establish the necessary condition for therecurrence

mk+1 = h−1[h(mk) − d(mk) + dobs]

(equation 7) to be convergent. In the Cauchy sense, local con-vergence is assured whenever

‖mk+1 − mk‖≤‖mk − mk−1‖. (B-1)

From equation (7), we obtain

mk+1 − mk = h−1[h(mk) − d(mk) + dobs]

− h−1[h(mk−1) − d(mk−1) + dobs]. (B-2)

When mk+1 is close to convergence, we can approximate thislast equality as

mk+1 − mk ≈ ∂h−1(dk) · {[h(mk) − h(mk−1)]

− [d(mk) − d(mk−1)]}, (B-3)

where

dk = h(mk) − d(mk) + dobs

and the operator ∂ denotes the Jacobian matrix (Frechetderivative) of the corresponding vector function. Further sub-stitution of

h(mk) − h(mk−1) ≈ ∂h(mk) · [mk − mk−1]and

d(mk) − d(mk−1) ≈ ∂d(mk) · [mk − mk−1]yields

mk+1 −mk ≈ ∂h−1(dk) ·[∂h(mk)−∂d(mk)] ·[mk−mk−1].

(B-4)Comparison of equations (B-1) and (B-4) then shows that forthe recurrence of equation (7) to converge, the condition

‖∂h−1(dk) · [∂h(mk) − ∂d(mk)]‖ ≤ 1

is a necessary requirement.


APPENDIX C

BROYDEN’S UPDATE FORMULA

Given the relatively small number of unknowns in the in-verse problem at hand, we implemented a first-order forward-difference formula to approximate the entries of the Jacobianmatrix associated with a given auxiliary least-squares mini-mization, i.e.,

∂h j

∂m�

≈ h j (m� + m�) − h j (m�) m�

,

where m� was chosen as small as possible but without com-promising the intrinsic accuracy of the forward-modeling code(1%). These entries subsequently were updated as the valuesm� changed and the iterations progressed in their way towardthe extremum of the auxiliary cost functions. The above for-mula requires two forward-modeling runs per unknown. Inaddition, we experimented with the fact that the Jacobian ma-trix as a whole could be updated in a simple way from the valuestaken by the same matrix from previous iterations. For this, weresorted to Broyden’s rank-one update formula (Dennis andSchnabel, 1983; Scales, 1985), i.e.,

J(mk+1) ≈ J(mk) + [ h(mk) − J(mk) · mk]

· mkT

mkT · mk, (C-1)

where

h(mk) = h(mk+1) − h(mk),

mk = mk+1 − mk,

and as usual, the superscript k is used to denote iterationcount. The derivation of this last equation is based on the as-sumption that h(mk) changes in linear fashion with respectto mk along the step direction mk . Often, this remains agood approximation only if mk is close to the extremum ofthe least- squares cost function. We have found that Broyden’supdate formula can be used efficiently in conjunction with thegrid subset designed to construct a solution of the auxiliaryforward-modeling operator h(m). On occasion, we also haveused Broyden’s formula throughout a few iterations. After ob-serving no significant convergence toward the minimum, wethen proceeded to reset the Jacobian matrix by performingonce again the numerical simulations required in computing itsentries. This combination of procedures proved to be extremelyefficient.

GEOPHYSICS, VOL. 65, NO. 6 (NOVEMBER-DECEMBER 2000); P. 1746–1757, 16 FIGS., 1 TABLE.

Quasi-analytical approximations and seriesin electromagnetic modeling

Michael S. Zhdanov∗, Vladimir I. Dmitriev‡,Sheng Fang∗∗, and Gabor Hursan∗

ABSTRACT

The quasi-linear approximation for electromagneticforward modeling is based on the assumption that theanomalous electrical field within an inhomogeneous do-main is linearly proportional to the background (nor-mal) field through an electrical reflectivity tensor λ. Inthe original formulation of the quasi-linear approxima-tion, λ was determined by solving a minimization prob-lem based on an integral equation for the scattering cur-rents. This approach is much less time-consuming thanthe full integral equation method; however, it still re-quires solution of the corresponding system of linearequations. In this paper, we present a new approach tothe approximate solution of the integral equation us-ing λ through construction of quasi-analytical expres-sions for the anomalous electromagnetic field for 3-Dand 2-D models. Quasi-analytical solutions reduce dra-matically the computational effort related to forwardelectromagnetic modeling of inhomogeneous geoelec-trical structures. In the last sections of this paper, weextend the quasi-analytical method using iterations anddevelop higher order approximations resulting in quasi-analytical series which provide improved accuracy. Com-putation of these series is based on repetitive applica-tion of the given integral contraction operator, whichinsures rapid convergence to the correct result. Numer-ical studies demonstrate that quasi-analytical series canbe treated as a new powerful method of fast but rigorousforward modeling solution.

INTRODUCTION

The integral equation (IE) method is a powerful tool for elec-tromagnetic numerical modeling (Hohmann, 1975; Weidelt,

Manuscript received by the Editor July 28, 1998; revised manuscript received March 9, 2000.∗University of Utah, Department of Geology and Geophysics, Salt Lake City, Utah 84112-0111. E-mail: mzhdanov@mines,utah.edu.‡Moscow State University, Faculty of Computational Math and Cybernetics, Moscow 1169899, Russia.∗∗Formerly University of Utah, Department of Geology and Geophysics, Salt Lake City, Utah; presently Baker Atlas, 10205 Westheimer, Houston,Texas, 77042. E-mail: [email protected]© 2000 Society of Exploration Geophysicists. All rights reserved.

1975; Dmitriev and Pozdnyakova, 1992). This method is basedon expressing the electromagnetic fields in terms of an inte-gral equation with respect to the excess current within an in-homogeneity. The integral equation is written as a system oflinear algebraic equations by approximating the excess cur-rent distribution ja by the piecewise constant functions. Theresulting algebraic system is solved numerically (Xiong, 1992).The main difficulty of this technique is the size of the linearsystem of equations matrix, which demands excessive com-puter memory and calculation time to invert. This limita-tion of the integral equation technique becomes critical in in-verse problem solution which requires multiple forward mod-eling calculations for different (updated) geoelectrical modelparameters.

A novel approach to 3-D electromagnetic (EM) modelingbased on linearization of the integral equations for scatteredEM fields has been developed recently by Zhdanov and Fang(1996a, b, 1997). Within this method, called quasi-linear (QL)approximation, the excess currents are assumed to be propor-tional to the background (normal) field Eb through an electri-cal reflectivity tensor λ. In the original papers on QL approx-imations, the electrical reflectivity tensor was determined bysolving a minimization problem based on an integral equationfor the scattering currents (Zhdanov and Fang, 1996a, b). Thisproblem is much less time-consuming than the full IE method;however, it still requires solution of the corresponding systemof linear equations. In this paper, we present a new approach toestimating λ, which leads to constructing quasi-analytical (QA)expressions for the anomalous electromagnetic field for 3-Dand 2-D models. We demonstrate also the connection betweenthe QL and QA approximations and the localized nonlinear(LN) approximations introduced by Habashy et al. (1993) andTorres-Verdin and Habashy (1994) and conduct a comparativestudy of the accuracy of different approximations.

In the last sections of the paper, we extend the quasi-analytical method using iterative techniques and develop

1746

Quasi-Analytical Approximations 1747

approximations of the higher orders which provide better accu-racy than the original QA and LN approximations. Combina-tion of these iterative solutions forms the quasi-analytical serieswhich generate a rigorous solution of EM modeling problems.

TENSOR QUASI-LINEAR EQUATION

Consider a 3-D geoelectric model with the background (nor-mal) complex conductivity σ b and local inhomogeneity Dwith the arbitrary spatial variations of complex conductivityσ = σb + �σ . We assume that µ = µ0 = 4π × 10−7 H/m, whereµ0 is the free-space magnetic permeability. The model is excitedby an electromagnetic field generated by an arbitrary sourcetime harmonic as e−iωt . Complex conductivity includes the ef-fect of displacement currents: σ = σ − iωε, where σ and ε areelectrical conductivity and dielectric permittivity. The electro-magnetic fields in this model can be expressed as a sum of thebackground (normal) and anomalous fields:

E = Eb + Ea, H = Hb + Ha, (1)

where the background field is a field generated by the givensources in the model with the background distribution of con-ductivity σ b, and the anomalous field is produced by the anoma-lous conductivity distribution �σ .

It is well known that the anomalous field can be presented asan integral over the excess currents in inhomogeneous domainD (Hohmann, 1975; Weidelt, 1975):

Ea(rj) =∫∫∫

DGE(rj | r)ja(r) dv = GE(ja),

(2)

Ha(rj) =∫∫∫

DGH(rj | r)ja(r) dv = GH(ja),

where GE(rj | r) and GH (rj | r) are, respectively, the electric andmagnetic Green’s tensors defined for an unbounded conduc-tive medium with the background conductivity σ b, GE and GH

are the corresponding Green’s linear operators, and excess cur-rent ja is determined by the equation

ja = ∆σE = ∆σ (Eb + Ea). (3)

Using Green’s operators, one can calculate the electromag-netic field at any point rj, if the electric field is known withinthe inhomogeneity:

E(rj) = GE(∆σE) + Eb(rj), (4)

H(rj) = GH(∆σE) + Hb(rj). (5)

Expression (4) becomes the integral equation with respect toelectric field E(r), if rj ∈ D.

The QL approximation is based on the assumption that theanomalous field Ea inside the inhomogeneous domain is lin-early proportional to the background field Eb through sometensor λ (Zhdanov and Fang, 1996a):

Ea(r) ≈ λ(r)Eb(r). (6)

Substituting formula (6) into formula (4), we obtain the QLapproximation Ea

q1(r) for the anomalous field:

Eaq1(rj) = GE(∆σ (I + λ(r))Eb). (7)

Rewriting expression (7) gives the tensor quasi-linear (TQL)equation with respect to the electrical reflectivity tensor λ:

λ(rj)Eb(rj) = GE[∆σ λ(r)Eb] + EB(rj), (8)

where EB(rj) denotes the Born approximation:

EB(rj) = GE(∆σEb) =∫

DGE(rj | r)∆σ (r)Eb(r) dv,

(9)and GE[∆σ λ(r)Eb] is a linear operator of λ(r):

GE[∆σ λ(r)Eb] =∫

DGE(rj | r)∆σ (r)λ(r)Eb(r) dv.

(10)

The original QL approximation (Zhdanov and Fang, 1996a,b) was based on the numerical solution of a minimization prob-lem arising from the TQL equation (8):∥∥λ(rj)Eb(rj) − GE[∆σ λ(r)Eb] − EB(rj)

∥∥ = min . (11)

The advantage of this approach is that, by choosing a fineenough discretization for a function λ(rj), one can generatean accurate solution. The disadvantage, however, is that sim-ilar to the full IE method, the QL approach still requires so-lution of the corresponding system of linear equations. In thispaper, we develop a new TQL equation solution that resultsin analytical expressions for the electrical reflectivity tensorλ(rj). This technique is, obviously, much faster than the origi-nal QL approximation. However, it may be less accurate thanthe corresponding QL approximation with a fine grid for λ(rj)discretization. In other words, there is a trade-off between thesimplicity of the approximate solution and its accuracy.

QUASI-ANALYTICAL SOLUTIONS FOR 3-DELECTROMAGNETIC FIELD

In this section we analyze different approximate solutions ofthe TQL equation (8). The iterative approach to the rigoroussolution of the TQL equation is outlined in Appendix A.

Solution for a scalar reflectivity tensor

In the framework of the quasi-linear approach, we may con-sider the electrical reflectivity tensor selected to be a scalar(Zhdanov and Fang, 1996a), λ = λI, where I is the unity tensor.In this case, integral equation (8) can be rewritten as

λ(rj)Eb(rj) = GE[∆σ λEb] + EB(rj). (12)

Following Habashy et al. (1993) and Torres-Verdin andHabashy (1994), we note that Green’s tensor GE(rj | r) is sin-gular at the point where rj = r. Therefore, one can expect thatthe dominant contribution to the integral GE[∆σ λEb] in equa-tion (12) is from some vicinity of the point rj = r. Assuming thatλ(rj) is slowly varying within domain D, we write

λ(rj)Eb(rj) ≈ λ(rj)GE[∆σEb] + EB(rj)

= λ(rj)EB(rj) + EB(rj). (13)

As we seek a scalar reflectivity coefficient λ, it is useful tocalculate the dot product of both sides of equation (13) and the

1748 Zhdanov et al.

background electric field:

λ(rj)Eb(rj) · Eb(rj)

= λ(rj)EB(rj) · Eb(rj) + EB(rj) · Eb(rj). (14)

Assuming that

Eb(rj) · Eb(rj) �= 0, (15)

and dividing equation (14) by the square of the backgroundfield, we obtain

λ(rj) = g(rj)1 − g(rj)

, (16)

where

g(rj) = EB(rj) · Eb(rj)Eb(rj) · Eb(rj)

. (17)

Substituting equation (16) into equation (1), we find

E(r) = Ea(r) + Eb(r) ≈ [λ(r) + 1]Eb(r)

= 11 − g(r)

Eb(r). (18)

Therefore from equations (4) and (5), we finally determine

EaQA(rj) = E(rj) − Eb(rj)

=∫∫∫

DGE(rj | r)

∆σ (r)1 − g(r)

Eb(r) dv, (19)

and

HaQA(rj) = H(rj) − Hb(rj)

=∫∫∫

DGH(rj | r)

∆σ (r)1 − g(r)

Eb(r) dv. (20)

Formulas (19) and (20) give QA solutions for 3-D electro-magnetic fields. Note that the only difference between the newQA approximation and the Born approximation (9) is the pres-ence of the scalar function [1−g(r)]−1. Hence computationally,the QA approximation and the Born approximation are prac-tically the same. On the other hand, we show below that theQA approximation is more accurate than the Born approx-imation.

General solution for different polarizations of the anomalousand background electric fields

The QA solutions developed in the previous section werebased on the assumption that the electrical reflectivity wasa scalar. This assumption reduces the areas of practical ap-plication of the QA approximations because in this case theanomalous (scattered) field is polarized in direction parallel tothe background field within the inhomogeneity. However, ingeneral, the anomalous field can be polarized in a different di-rection than the background field. To overcome this difficulty,we introduce a tensor quasi-analytical (TQA) approximationto λ, which permits different polarizations for the backgroundand anomalous (scattered) fields.

We again assume that the product λ(r)Eb(r) is a smoothlyvarying function of the coordinates, and it can be taken outside

the integral over the anomalous domain D without substantialdiscrepancies. As a result, we obtain from the TQL equation (8)

λ(rj)Eb(rj) ≈ GE[∆σ I]λ(rj)Eb(rj) + EB(rj)

= g(rj)λ(rj)Eb(rj) + EB(rj),

or

[I − g(rj)]λ(rj)Eb(rj) = EB(rj), (21)

where

g(rj) = GE[∆σ (r)I].

Solving equation (21) yields

λ(rj)Eb(rj) = [I − g(rj)]−1EB(rj). (22)

Substituting equation (22) into equation (1), we obtain

E(r) = Ea(r) + Eb(r) ≈ [λ(r) + I]Eb(r)

= [I − g(r)]−1EB(r) + Eb(r). (23)

Therefore, from equations (4) and (5) we find

EaTQA(rj) = E(rj) − Eb(rj) =

∫∫∫D

GE(rj | r)�σ (r)

× {[I − g(r)]−1EB(r) + Eb(r)} dv, (24)

and

HaTQA(rj) = H(rj) − Hb(rj) =

∫∫∫D

GH (rj | r)�σ (r)

× {[I − g(r)]−1EB(r) + Eb(r)} dv, (25)

where

g(rj) = GE[∆σ (r)I] =∫∫∫

DGE(rj | r)∆σ (r) dv.

(26)

We call expressions (24) and (25) TQA approximationsfor an electromagnetic field. We show below that this ap-proximation provides a more accurate solution for a forwardproblem than a scalar QA approximation. However, we mustcompute the tensor multiplier [I − g(r)]−1, which is slightlymore time-consuming than calculation of the scalar coefficient[1 − g(r)]−1.

Quasi-analytical solutions for a 2-D electromagnetic field

Assume now that both the electromagnetic field and thecomplex conductivity σ in the geoelectrical model are two-dimensional (i.e., they vary only along the directions x and z ofsome Cartesian system of coordinates, and are constant in the ydirection). In this case, repeating derivations described abovefor the 3-D case, we can obtain the following QA expressionsfor a 2-D electromagnetic field:

EaQAy(rj) ≈ iωµ0

∫∫DGb(rj | r)

�σ (r)1 − g(r)

Eby(r) ds, (27)


and similarly for magnetic field components:

HaQAx(rj) ≈ −

∫∫D

∂Gb(rj | r)∂z

�σ (r)1 − g(r)

Eby(r) ds, (28)

HaQAz(rj) ≈ −

∫∫D

∂Gb(rj | r)∂x

�σ (r)1 − g(r)

Eby(r) ds, (29)

where Gb(rj | r) is a 2-D scalar Green’s function for an un-bounded conductive medium with the background conductiv-ity σ b, and

g(r) = EBy (r)

Eby(r)

. (30)

These formulas can serve as a new effective QA tool for bothdirect and inverse 2-D electromagnetic problem solutions. Nu-merical tests demonstrate that these approximations producea very accurate result for 2-D models (Dmitriev et al., 1998).

LOCALIZED NONLINEAR APPROXIMATION

The TQA approximation can be treated as a generalizationof the LN approximation introduced by Habashy et al. (1993).Let us rewrite equation (23) in the form

E(r) = [I − g(r)]−1[EB(r) − g(r)Eb(r)]

+ [I − g(r)]−1Eb(r). (31)

Taking into account once again that Green’s tensor GE(rj | r)exhibits either singularity or a peak at the point where rj = r,one can calculate the Born approximation GE[∆σ (r)Eb(r)] us-ing the formula

EB(rj) = GE[∆σ (r)Eb(r)] ≈ g(rj)Eb(rj).

This result implies

EB(r) − g(r)Eb(r) ≈ 0, (32)

and is particularly appropriate if the background field is asmoothly varying spatial function (Habashy et al., 1993).

Under assumption (32), equation (31) can be rewritten

E(r) = Ea(r) + Eb(r) ≈ [I − g(r)]−1Eb(r). (33)

Therefore, from equations (4) and (5) we find

EaLN(rj) = E(rj) − Eb(rj)

=∫∫∫

DGE(rj | r)�σ (r)[I − g(r)]−1Eb(r) dv,

(34)

and

HaLN(rj) = H(rj) − Hb(rj)

=∫∫∫

DGH (rj | r)�σ (r)[I − g(r)]−1Eb(r) dv.

(35)

Formulas (34) and (35) express the LN approximation intro-duced by Habashy et al. (1993), where

[I − g(r)]−1 = Γ(r)

is their depolarization tensor.

Thus we can see that the difference between the TQL ap-proximation and the LN approximation is determined by aterm

EaTQA(rj) − Ea

LN(rj)

=∫∫∫

DGE(rj | r)�σ (r)Γ(r)[EB(r) − g(r)Eb(r)] dv.

(36)

Note that both TQA and LN approximations use the samedepolarization tensor Γ(r), based on the idea of a localized ef-fect in the Green’s integral operator. The only difference is thatin the case of the LN approximation we use this localizationproperty twice for computing both the depolarization tensorand the expression for the Born approximation EB(rj) on theright-hand side of the TQL equation (8). In the case of TQAapproximation, we use the exact formula for EB(rj), and weconsider TQA a partially localized approximation. This differ-ence does affect the accuracy of these two approximations fordifferent geoelectrical models, illustrated below by numericalexamples.

COMPARATIVE ACCURACY STUDY

To compare the accuracy of the Born, QA, TQA, and LN ap-proximations, we conducted several numerical experiments forthe models presented in Figure 1. Model 1 consists of a conduc-tive rectangular prism embedded in a homogeneous half-spaceexcited by a horizontal rectangular loop (Figure 1, top panel).

FIG. 1. 3-D geoelectrical models used for comparative accuracystudy of different approximations.

1750 Zhdanov et al.

The frequency is 1000 Hz, and the conductivity ratio is 10. Thereceivers are located above the body along the y-axis. Model 2consists of a conductive cube with a resistivity of 1 ohm-m lo-cated at a depth of 10 m within a homogeneous half-space witha resistivity of 10 ohm-m (Figure 1, middle panel). The sidesof the cubic prism have a length of 50 m. Model 3 contains thebody with a horizontal size of 100 m × 100 m, a vertical dimen-sion of 50 m, and located at a depth of 10 m. The EM field inmodels 2 and 3 is excited by a vertical magnetic dipole locatedon the surface of the earth.

Figure 2 shows the real and imaginary parts of the horizon-tal electric and vertical magnetic components of the scatteredfield computed by solving the full integral equation and theapproximate solutions. The deviations of QA, TQA, and LNapproximations from the “true” solution are minor, but theBorn approximation fails. The next figure, Figure 3, presentsthe same approximate solutions but for an expanded verticalscale. We can see now the small differences between the variousapproximations.

To analyze more carefully the discrepancies in different ap-proximations, we consider model 2 presented in Figure 1 (mid-dle panel). In this experiment the transmitter (Tx) and receiver(Rx) geometry was fixed (transmitter-receiver separation was100 m), with profiles run over the conductive prismatic bodywith the center at a depth of 35 m below the origin of x and y co-

FIG. 2. Behavior of the anomalous electromagnetic field com-ponents computed for model 1 by solving the full integral equa-tion, Born approximation, and the scalar QA, TQA, and LNapproximations.

ordinates. For each position of the Tx/Rx system, we computedthe vertical component of the magnetic field using the rigorousfull IE method and three different approximations: (1) the QAapproximation, (2) the LN approximation, and (3) the TQAapproximation. The relative errors of approximate solutions incomparison with the rigorous solution were calculated as

ε = ‖Happr − Hfull‖2

‖Hfull‖2× 100%. (37)

The main goal of this experiment is to demonstrate that theaccuracy of approximation is not only a function of the con-ductivity contrast, frequency, and size of the anomalous body,it also depends on the relative locations of the transmitter, re-ceiver and conductive target.

Figures 4, 5, and 6 are maps of relative errors for QA, LN,and TQA approximations correspondingly at the receiver fordifferent positions of the recording system relative to the bodycenter at a depth of 35 m below the earth’s surface. Figure 7shows the profiles of errors along the line connecting transmit-ter and receiver. One can see that for all three approximationsthe errors increase when the body is located just under thetransmitter or the receiver. However, the largest discrepanciesoccur when the transmitter is near the inhomogeneity, with sig-nificantly lower (2–7%) discrepancies for all other locations.The most accurate result is delivered by TQA approximation(dotted line in Figure 7). The high level of discrepancies for

FIG. 3. Behavior of the anomalous electromagnetic field com-ponents computed for model 1 by solving the full integral equa-tion, the scalar QA, TQA, and LN approximations.


QA approximation near the transmitter can be explained bythe fact that the primary electric field is equal to zero on thevertical axes passing through the position of the transmitterdipole. Since the expression for scalar coefficient g(r) in for-mulas (17) for the QA approximation becomes singular whenEb(r) → 0, there is a significant increase of discrepancies in thenear zone below the transmitter.

Another cause of discrepancies in this zone is the fact thatQA approximation is based on a scalar reflectivity tensor. Inthe area below the transmitter, the primary electric field formsa “smoke ring” blown by the transmitting loop into the earth(Nabighian, 1979). The conductive body located just below thetransmitter distorts this field through a secondary electric field,directed at some angle with the primary field. The scalar re-flectivity tensor cannot account for this rotation of the sec-

FIG. 4. Map of the relative errors between the scalar QA ap-proximation and the full integral equation solution computedfor model 2.

FIG. 5. Map of the relative errors between the LN approxi-mation and the full integral equation solution computed formodel 2.

ondary fields which generates additional discrepancies in theapproximation. The plots in Figures 4–7 show that TQA andLN approximations handle this polarization pretty well. Anespecially accurate result is reached by a TQA solutions. Thecorresponding discrepancies do not exceed 7% and 15% in theareas under the receiver and transmitter, respectively.

An increase in discrepancies generated by LN approxima-tion in comparison with TQA approximation can be explainedby the fact that LN approximation is source independent. Whenthe receiver is closer to the transmitter, the source effect be-comes more significant, which leads to an increase in discrep-ancies. The TQA solution approximates the polarization of thesecondary field and takes into account the source position. As aresult, it produces a more accurate approximation, so the TQAapproximation is source dependent.

FIG. 6. Map of the relative errors between the tensor QA ap-proximation and the full integral equation solution computedfor model 2.

FIG. 7. The relative errors of the scalar QA approximation,TQA approximation, and LN approximations computed formodel 2.

1752 Zhdanov et al.

We analyzed also the effect of the conductivity contrast onthe accuracy of different approximations. In Figure 8, we ex-amine the relative errors for model 3 shown in Figure 1, bottompanel. The horizontal size of the conductive body in this model(100 × 100 m) is bigger than in model 2. The center of the con-ductive body is located just below the receiver. We considernow that the ratio of anomalous conductivity to backgroundconductivity varies within a range of five orders of magnitude.One can see in the plot in Figure 8 that within a conductivity ra-tio range from 10−2 to 30, the best accuracy is provided by TQAapproximations with the discrepancies less than 10%. For thehigher conductivity ratio, the accuracy of all approximationsbecomes less than 20–30% .

Figures 9 and 10 illustrate the effect of frequency on the accu-racy of approximation for model 3. The conductive rectangularprism has a resistivity of 1 ohm-m while the background resis-

FIG. 8. The relative errors of different approximations as afunction of the conductivity ratio computed for model 3.

FIG. 9. The ratio of different approximations to the full integralequation solution of the scattered Hz as a function of frequencycomputed for model 3 with a conductivity of 10.

tivity of a half-space is 10 ohm-m. The frequency range is from10−1 up to 104 Hz. Figure 9 shows the ratio of the estimatedto true (computed by IE method) amplitudes of the anoma-lous field Hz . Figure 10 presents the difference between thephases of the anomalous Hz component computed by true andapproximate solutions.

The most significant feature of all these plots is the stablebehavior of the QA approximation. The ratio of the approxi-mated and true amplitudes of the anomalous field Hz is equalto one within the entire frequency range. The phase differenceis also close to zero along the entire horizontal axis of the plotin Figure 10. At the same time both TQA and LN approxi-mations produce good amplitude estimate till the frequency of1 kHz, and good phases only for frequencies below 100 Hz. Thissimilarity in the TQA and LN data behavior can be explainedby the fact that both approximations are based on a depolar-ization tensor calculation, which is independent of the back-ground field. Therefore, these approximations cannot take intoaccount properly the background field which cause discrepan-cies in these approximations. At the same time, Figure 11 showsthat at the frequency range above 100 Hz the induction effectsbecome strong and the background field begins to vary signif-icantly from its static limit. In the case of the QA approxima-tion, we evaluate more carefully the induction effect becausethe background field is present in the expressions for scalar co-efficient g in formulas (19) and (20) for QA anomalies. That iswhy the QA approximation produces stable results for a widefrequency range.

QUASI-ANALYTICAL SERIES

The main limitation of the QA method (as well as QL and LNapproximations) is that it is still an approximate method of 3-Dforward modeling, and its practical application requires addi-tional control of the approximation discrepancies. It is possible,however, to increase the accuracy of the QA approximation byconstructing QA approximations of a higher order in a similar

FIG. 10. The difference of the phases between the approximateand the full integral equation solutions of the scattered Hz as afunction of frequency computed for model 3 with a conductivityof 10.


fashion to the QL series for QL approximations (Zhdanov andFang, 1997).

The QL series were based on a new method of constructingthe converged Born series developed by Singer and Fainberg(1995) and Pankratov et al. (1995). This method was applied inZhdanov and Fang (1997) to construct QL series that almostalways converged. In this paper, we use the same method togenerate QA series and calculate the accuracy of the QA ap-proximations. These series are built on the QA approximationas the first term of the series. As a result, the computation ofQA series becomes even easier and faster than in the case ofQL series, which required the solution of the linear algebraicequation in the first step of the iterations. Computation of theQA series does not involve any system of equation solution. Itis based on repetitive application of the given integral contrac-tion operator, which insures rapid convergence to the correctresult.

Following Zhdanov and Fang, (1997) we modify Green’s op-erator according to the formula

Gm(�σ (r)Eb(r))

=√Reσ bGE(2

√Reσ b�σ (r)Eb(r)) + �σ (r)Eb(r)

=√Reσ b

∫∫∫D

GE(r j | r) 2√Reσ b�σ (r)Eb(r) dv

+ �σ (r)Eb(r). (38)

It was proved in Zhdanov and Fang (1997) that the L2 normof this operator is always less than or equal to one:

‖Gm‖ ≤ 1. (39)

We can rewrite now the integral equation for the anomalousfield (2) as follows:

aEa = C(aEa), (40)

where C(aEa) is an integral operator of the anomalous field:

C(aEa) = Gm[βaEa] + Gm[βaEb] − βaEb, (41)

FIG. 11. Mutual coupling ratios of the real and imaginary partsof the vertical magnetic field over a homogeneous half-spaceof 10 ohm-m as a function of frequency.

and

a = 2Reσ b + �σ

2√Reσ b

, β = �σ

2Reσ b + �σ.

The solution of this integral equation can be obtained usingthe method of successive iterations, which is governed by thefollowing equation:

aEa(N) = C[aEa(N−1)], N = 1, 2, 3 . . . (42)

These iterations always converge for any lossy medium becauseC is a contraction operator (Zhdanov and Fang, 1997).

We start iterations with the QA approximation for theanomalous field:

Ea(0)qa = g

1 − gEb. (43)

In this case, the first order QA approximation is equal:

aEa(1)qa = C

(aEa(0)

qa

) = Gm[βaEa(0)

qa

]+ Gm[βaEb] − βaEb. (44)

We will call the first iteration determined by expression (44) amodified quasi-analytical approximation (MQA):

Ea(1)qa = Ea

MQA = 1a

Gm[βaEa(0)

qa

]+ 1a

Gm[βaEb] − βEb.

Taking into account the definition of the modified Green’s op-erator (38) and formula (43), we obtain

EaMQA = 2Reσ b

2Reσ b + �σ

{GE

[�σEa(0)

qa

] + GE[�σEb

]}

= 2Reσ b

2Reσ b + �σGE

[�σ

1 − gEb

]

= 2Reσ b

2Reσ b + �σEaMQA. (45)

Equation (45) shows that the modified QA approximation isequal to the original QA approximation outside inhomogene-ity D:

EaMQA(r j ) = Ea

QA(r j ), r j /∈ D, (46)

while they are different inside the geoelectrical inhomogeneity.The second-order QA approximation is equal to:

aEa(2)qa = C

(aEa(1)

qa

) = (Gmβ)2(aEa(0)qa

)+ Gm(aEBm) + aEBm,

where EBm is a modified Born approximation determined bythe formula

EBm = 1a

GmβEb − βEb = 2Reσ b

2Reσ b + �σEB .

The third-order QA approximation is given by the formula

aEa(3)qa = C

(aEa(2)

qa

) = (Gmβ)3(aEa(0)qa

) + (Gmβ)2(aEBm)

+ Gm(aEBm) + aEBm .

1754 Zhdanov et al.

Finally, the N th-order QA approximation can be treated asthe sum of N terms of the QA series:

aEa(N)qa =

N−1∑k=0

(Gmβ)k(aEBm) + (Gmβ)N(aEa(0)

qa

), (47)

where Gm is the modified Green’s operator:

(Gmβ)(aEa(0)

qa

)=

√Reσ b

∫∫∫D

GE(r j | r) 2√Reσ bbEa(0)

qa dv

+ bEa(0)qa =

√Reσ b

∫∫∫D

GE(r j | r)�σEa(0)qa dv

+ �σ

2√Reσ b

Ea(0)qa (r j ).

Note, that QA series can be built on TQA and LN approxi-mations as the first iterations. We select the approach based onQA approximation for the sake of simplicity.

ACCURACY ESTIMATION

The accuracy of the QA approximation of the N th order isestimated in the same way as in Zhdanov and Fang (1997) bythe formula

εN =∥∥aEa − aEa(N)

qa

∥∥∥∥aEa(N)qa

∥∥ ≤ ‖β‖∞1 − ‖β‖∞

rN , (48)

where Ea(0)qa = ag

1 − gEb, and rN is the relative convergence rateof the QA approximations:

rN =∥∥aEa(N)

qa − aEa(N−1)qa

∥∥∥∥aEa(N)qa

∥∥ . (49)

In particular, the accuracy of the original QA approximationEaQA can be estimated by computing, using the formula

ε =∥∥aEa − aEa

QA

∥∥∥∥aEaQA

∥∥ ≤ ‖β‖∞1 − ‖β‖∞

·∥∥aEa

QA − ag1−gEb

∥∥∥∥aEaQA

∥∥ .

(50)Thus, the accuracy estimation formula for QA solutions is ex-pressed by the QA solution itself.

Formulas (48) and (50) make it possible to obtain a quan-titative estimation of the QL approximation accuracy withoutdirect comparison with the rigorous full IE forward modelingsolution.

NUMERICAL MODELING RESULTS

We developed a computer code based on the QA series forthe electromagnetic field in a 3-D case. The algorithm wastested for 3-D geoelectrical models.

Consider a 3-D geoelectrical model (model 4), consisting ofa homogeneous half-space (with resistivity of 100 ohm-m) anda thin conductive rectangular inclusion with the resistivity of1 ohm-m (Figure 12). The electromagnetic field in this modelis excited by a horizontal rectangular loop, located 50 m to theleft of the model, with the loop 10 m on a side and a current of1 A. We have used the integral equation code for computingthe scattering current in the complex conductivity structureand the QA series code.

Figure 13 presents maps of the excess electrical currentsdistribution within the inhomogeneity obtained by a rigorous

FIG. 12. 3-D geoelectrical model of a thin conductive rectan-gular body embedded in a homogeneous half-space excited bya horizontal rectangular loop (model 4).

FIG. 13. Behavior of the scattering currents induced inside theconductive rectangular body in model 4 obtained by solvingthe full integral equation and the approximate solution afterone, 15, and 50 iterations.


integral equation solution and by QA series of different ordersfor a frequency of 1000 Hz. Note that this model is a difficultone for QA approximation because it contains a conductiv-ity contrast of 100. Nevertheless, we can see how the currentscomputed by QA series converge to the true solution. Fig-ure 14 presents a map of relative errors in the excess currentcalculations between the integral equation technique and theQA series of the order of 5 and 15. One can observe that thediscrepancies decrease during the iterations.

We applied the QA series solution to model the EM responsefor the more complicated model simulating the Kambalda-stylenickel sulfide deposit in Western Australia (Stolz et al., 1995.).The sketch of the model is shown in the top panel of Figure 15;the bottom panel of Figure 15 shows the vertical geoelectricalcross-section of the model. The model consists of a conduc-tive overburden above an inclined nickel lens. The conductivitycontrast between the nickel lens and the host rocks is 104, whichis far beyond the normal limits of QA approximations. Fig-ure 16 presents the horizontal and vertical anomalous magneticfields of forward modeling based on integral equation solutionand QA series of the order of 1, 10, 20, and 50. The very highconductivity contrast and the proximity of the anomalous bodycauses inaccuracies in the approximate solutions. However, thealways convergent series algorithm provides the correct values.

Table 1 shows a comparison of CPU time for EM modelingusing integral IE (Xiong, 1992) and the QA approximations ofthe different orders for the thin sheet and the Kambalda-stylenickel-sulfide deposit models. For 1088 cells and 50 iterations

FIG. 14. Maps of the relative residuals of scattering currentsobtained by different orders of QA series with respect to thefull integral equation solution. The number of iterations is 5(top) and 15 (bottom).

of the QA series, the algorithm spent approximately half theCPU time required for the solution of the full integral equation.For 4352 cells, the time gain is very significant. It takes about2.5 hours for the new code to run 50 iterations, which generatesthe same solution as the integral equation code (Figure 16). Ittook more than five days to reach the same result using full IEmethod.

CONCLUSION

We have generalized the QL method of forward modelingand developed a new approach to calculation of the electrical

Table 1. Comparison of the CPU time (in seconds) for EMmodeling using full integral equation solution and QA ap-proximations of the different orders for the thin sheet and theKambalda-style nickel-sulfide deposit models.

Cells Full IE 1st-order 20th-order 50th-orderCells Full IE QA QA QA

100 30 11 15 211088 1264 221 368 5892176 21 687 459 1326 29674352 490 752 911 3931 8827

FIG. 15. Kambalda-style ore deposit model with an inclineddike structure. The bottom panel shows the vertical resistivitycross-section.

1756 Zhdanov et al.

FIG. 16. Magnetic field components obtained by different num-bers of iterations of QA series and full integral equation solu-tions over the Kambalda-style ore deposit model.

reflectivity tensor based on the solution of the correspondingintegral equation.

Based on this approach, we introduced the new scalar (QA)and tensor (TQA) quasi-analytical solutions for electromag-netic fields in 3-D inhomogeneous media. We demonstratedalso that TQA approximation is a generalization of the local-ized nonlinear approximation introduced by Habashy et al.(1993). The new TQA approximation permits different polar-izations for the background and anomalous (scattered) field,which increases the accuracy of approximation for conductiv-ity contrasts below 30. The comparative accuracy study of thedifferent approximations demonstrates that TQA approxima-tion has a superb accuracy, especially in the areas close to thetransmitter and to the anomalous geoelectrical structures. Atthe same time, QA approximation generates a stable and ac-curate result (discrepancies below 3%) for a wide frequencyrange (from 10−1 up to 104 Hz).

The computational time for QA approximations is compa-rable with that required for the Born approximation, althoughthe new approximate solutions are much more accurate. Togenerate a rigorous forward-modeling result, we may applythese approximations iteratively. This approach leads us to aconstruction of the QA series.

The developed approximations of the electromagnetic fieldcan be used as effective tools for fast 3-D forward modeling.One of the attractive areas of their application is rapid comput-ing of the Frechet derivative for 3-D electromagnetic inversion.

We improved the accuracy of the QA approximation by con-structing QA approximations of a higher order in a similar wayas has been done for QL approximations in Zhdanov and Fang(1997). These series are a new fast and accurate method of3-D EM modeling that accelerate dramatically the solutionof forward EM problems in inhomogeneous 3-D geoelectricalstructures.

ACKNOWLEDGMENTS

The authors acknowledge the support of the University ofUtah Consortium for Electromagnetic Modeling and Inver-sion (CEMI), which includes Advanced Power TechnologiesInc., Baker Atlas Logging Services, BHP Minerals, Exxon Pro-duction Research Company, Inco Exploration, Japan NationalOil Corporation, Mindeco, Mobil Exploration and Produc-tion Technical Center, Naval Research Laboratory, NewmontGold Company, Rio Tinto, Shell International Exploration andProduction, Schlumberger-Doll Research, Unocal Geother-mal Corporation, and Zonge Engineering.

REFERENCES

Dmitriev, V. I., and Pozdnyakova, E. E., 1992, Method and algorithmfor computing the electromagnetic field in stratified medium witha local inhomogeneity in an arbitrary layer: Computational Mathe-matics and Modeling, 3, 181–188.

Dmitriev, V. I., Pozdnyakova, E. E., Zhdanov, M. S., and Fang, S., 1998,Quasi-analytical solutions for the EM field in inhomogeneous struc-tures based on a unified iterative quasi-linear method: 68th Ann.Mtg., Soc. Expl. Geophys., Expanded Abstracts, 444–447.

Dmitriev, V. I., and Sedelnikova, A. V., 1992, Iterative method forcomputing the anomalous electric field in a conducting medium:Computational Mathematics and Modeling, 3, 197–203.

Habashy, T. M., Groom, R. W., and Spies, B. R., 1993, Beyond theBorn and Rytov approximations: A nonlinear approach to electro-magnetic scattering: J. Geophys. Res., 98, B2, 1759–1775.

Hohmann, G. W., 1975, Three-dimensional induced polarization andEM modeling: Geophysics, 40, 309–324.

Nabighian, M. N., 1979, Quasi-static transient response of a conductinghalf-space—An approximate representation: Geophysics, 44, 1700–1705.

Pankratov, O. V., Avdeev, D. B., and Kuvshinov, A. V., 1995, Scatteringof electromagnetic field in inhomogeneous earth. Forward problemsolution: Fizika Zemli, No. 3, 17–25.

Singer, B. S., and Fainberg, E. B., 1995, Generalization of iterative dis-sipative method for modeling electromagnetic fields in nonuniformmedia with displacement currents: J. Appl. Geophys., 34, 41–46.

Stolz, N., Raiche, A., Sugeng, F., and Macnae, J., 1995, Is full 3-D in-version necessary for interpreting EM data?: Expl. Geophys., 26,167–171.

Torres-Verdin, C., and Habashy, T. M., 1994, Rapid 2.5-dimensionalforward modeling and inversion via a new nonlinear scattering ap-proximation, Radio Sci., 29, 1051–1079.

Weidelt, P., 1975, EM induction in three-dimensional structures: J. Geo-physics, 41, 85–109.

Xiong, Z., 1992, EM modeling of three-dimensional structures by themethod of system iteration using integral equations: Geophysics, 57,1556–1561.

Zhdanov, M. S., and Fang, S., 1996a, Quasi-linear approximation in 3-DEM modeling Geophysics, 61, 646–665.

———1996b, 3-D quasi-linear electromagnetic inversion: Radio Sci.,31, 741–754.

———1997, Quasi-linear series in 3-D EM modeling: Radio Sci., 32,2167–2188.


APPENDIX A

ITERATIVE METHOD

We demonstrate here that approximate expressions (16) and(22) for the electrical reflectivity tensor can be treated as thefirst iterations within an iterative solution of the TQL equation(8) solution.

Let us subtract GE[∆σ λ(rj)Eb] from both sides of equation(8):

λ(rj)Eb(rj) − GE[∆σ λ(rj)Eb]

= GE[∆σ (λ(r) − λ(rj))Eb] + EB(rj). (A-1)

One can apply an iterative process to solve equation (A-1):

λ(k+1)(rj)Eb(rj) − GE[∆σ λ(k+1)(rj)Eb]

= GE[∆σ (λ(k)(r) − λ(k)(rj))Eb] + EB(rj), (A-2)

where k is the iteration number and

λ(0)

(r) = 0. (A-3)

Note that convergence of this iteration process for a specificgeoelectrical model depends on the properties of the operator

A[λ(r) − λ(rj)] = GE[∆σ (λ(r) − λ(rj))Eb]. (A-4)

It was demonstrated by Dmitriev and Sedelnikova (1992)that the L2 norm of this operator determined on a class of theslowly varying functions is usually small, which provides theconvergence of the iteration process (A-2). This assumption isbased on the fact that for slowly varying λ(r), the difference[λ(r) − λ(rj)] is small if rj is close to r, and the kernel GE(rj | r)is small if the distance between the points rj and r is large. Inother words, one can consider that operator A[λ(r)−λ(rj)], act-ing on the class of slow varying functions, is a small operator(has a small L2 norm).

We demonstrated that by using a modified Green’s operatorwith the norm less or equal to one (Zhdanov and Fang, 1997),we can construct the modified tensor QL equation with thenorm of the operator A[λ(r) − λ(rj)] always small. So, in thissituation, the iterative process (A-2) will always converge.

In the case of a scalar electrical reflectivity tensor, integralequation (A-1) can be rewritten as

λ(rj)[Eb(rj) − EB(rj)

]= GE

[∆σ (λ(r) − λ(rj))Eb] + EB(rj). (A-5)

Calculating the dot product of both sides of equation (A-5)and the background electric field, and dividing the resultingequation by the square of the background field, we obtain

λ(rj)[1 − g(rj)] = A[λ(k)(r) − λ(k)(rj)

] + g(rj),

(A-6)

where g(rj) is determined by equation (17), and

A[λ(r) − λ(rj)] = GE[∆σ (λ(r) − λ(rj))Eb

] · Eb(rj)

Eb(rj) · Eb(rj).

(A-7)

The integral equation (A-6) can also be solved iteratively:

λ(k+1)(rj)[1 − g(rj)] = A[λ(k)(r) − λ(k)(rj)

] + g(rj).

(A-8)

As we already discussed above, these iterations will con-verge to a true solution due to the small norm of operatorA[λ(r) − λ(rj)] acting on the class of slowly varying functions.Note that these iterations will always converge if one uses themodified Green’s operator in equation (A-7).

The first iteration of equation (A-8) yields

λ(1)(rj) = g(rj)1 − g(rj)

, (A-9)

which coincides with the approximate formula (16).Thus, we can see that QA approximation can be treated as

the first iteration in the solution of the TQL equation using theiterative algorithm (A-8).

Note that in the same way one can demonstrate that TQAapproximation can be treated as the first iteration in the itera-tive solution of the TQL equation for tensor λ(r).


Directional filtering for linear feature enhancement in geophysical maps

Michael P. Sykes∗ and Umesh C. Das∗

ABSTRACT

Geophysical maps of data acquired in ground and air-borne surveys are extensively used for mineral, ground-water, and petroleum exploration. Lineaments in thesemaps are often indicative of contacts, basement fault-ing, and other tectonic features of interest. To aid theinterpretation of these maps, a versatile processing tech-nique of directional filtering, based on the 2-D “normal”Radon transform, is used to enhance or suppress spe-cific lineaments. Synthetic data and field examples us-ing electromagnetic and radiometric data are used todemonstrate the superiority of the Radon transformmethod over conventional Fourier transform filtering.The Radon transform technique is shown to be moreversatile and less susceptible to processing artefacts thanthe Fourier transform methods.

INTRODUCTION

Geophysical methods such as magnetic, gravity, radiomet-ric, electrical, and electromagnetic methods acquire measure-ments on or above the surface of the earth to assist in the inter-pretation of geological structures in the subsurface. The degreeto which geophysical maps reflect the subsurface structure islimited by data quality and sampling. It is often necessary toprocess the data in certain ways to improve the usefulness ofthe maps. Traditionally, the processing has been carried out bythe application of filters in the Fourier transform (FT) domain.Of particular interest to the geologist are the linear anoma-lies in geophysical maps which may correspond to subsurfacefaults, contacts and other tectonic features. In mineral explo-ration, large ore bodies are often accompanied by stringers,and many deposits are found along the boundaries betweenadjacent regions or in fractured zones (Palacky, 1988). Linea-ments on geophysical maps are often used to identify areasof high priority for further exploration. Pawlowski (1997) hasshown the usefulness of the Radon transform (RT) in display-ing linear features in magnetic data according to their strike

Manuscript received by the Editor August 11, 1998; revised manuscript received April 24, 2000.∗Curtin University of Technology, Department of Exploration Geophysics, Perth, Western Australia, Australia.E-mail: [email protected];[email protected]© 2000 Society of Exploration Geophysicists. All rights reserved.

direction. In his paper, a section of the transformed data ismuted out so that features having a particular orientation aredisplayed. Yunxuan (1992) applied the Radon transform tosynthetic gravity maps to demonstrate its usefulness for up-ward and downward continuation and to remove unwantedlinear features. We take these ideas further and demonstratethe usefulness of the Radon transform to enhance linear fea-tures in a geophysical map as well as attenuating unwantedacquisition artifacts.

THE 2-D RADON TRANSFORM

The Radon transform is a numerical process that is usefulfor analysing lineaments in a 2-D map. It is an integration pro-cedure that, in its most general form, maps a shape in the spacedomain into a single pixel in the spectral (transform) domain.In seismic processing, a type of Radon transform, commonlyknown as the slant stack, has been used for the generationof synthetic seismograms (Chapman, 1978), multiple suppres-sion (Taner, 1980), and ground-roll removal and separation ofrefractions (Stoffa et al., 1981). Although the theory of theRadon transform is well established and is used extensively inseismic data processing (Duranni and Bisset, 1984; Brysk andMcCowan, 1986; Beylkin, 1987), and many other fields (Deans,1983), we include a brief introduction for completeness.

The slant-stack RT operates along straight lines which aredefined by their slope p and intercept τ as shown in Figure 1(a).Off-end seismic shot records lead to this natural choice ofline parameters since, in these records, there exists an obvi-ous choice of origin at x = 0, t = 0 and a preferred directionof events that limits the range of τ and p. However, the slantstack is limited in its ability to transform steeply sloping events(Toft, 1996), and vertical lines cannot be transformed. To avoidthis limitation, it is usual practice to define two transform do-mains: one relative to the x-axis, and the other relative to thet-axis. In potential field maps, no such preferred direction ex-ists. To accommodate the full range of possible strike directionsthe “normal” RT is preferred. In this form, the origin is cho-sen as the centre of the map and the lines defined by their“normal” slope (θ) and offset (ρ) parameters [Figure 1(b)].Integration of the map amplitudes along each line (S lines

1758

Geophysical Map Enhancement 1759

in Figure 1) produces the amplitude of the spectral domainpoint located at (ρ, θ). It should be noted that the normal RTand the slant-stack RT are very closely related and performthe same function equally well. The main difference betweenthe two forms of RT is that individual points transform intostraight lines when using slant stack, but into sinusoids whenusing the normal RT. Mathematically, the spectral domain func-tion F(ρ, θ) is obtained by applying the normal RT to thespace domain function f (x, y) with the help of the followingequation:

F(ρ, θ) =∫ ∞

−∞

∫ ∞

−∞f (x, y)δ(ρ − x cos θ − y sin θ) dx dy,

(1)

where δ is the Dirac delta function. The delta function ensuresthat only points that lie on the line S contribute to the transformdomain. The RT maps linear features with significant strikelength as high-amplitude points at the appropriate location in

FIG. 1. (a) Schematic of the slant-stack Radon transform. Thespace domain function f (x, y) is integrated along S lines de-fined in terms of the parameters τ and p. The value of theintegral is assigned to the coordinate (τ, p) in the transformdomain. (b) Schematic of the normal Radon transform. Thespace domain function f (x, y) is integrated along S lines de-fined in terms of the parameters θ and ρ. The value of theintegral is assigned to the coordinate (θ, ρ) in the transformdomain.

the spectral domain. A unique correspondence exists betweenspace domain linear features and spectral domain high spots(Robinson, 1982). Thus, the analysis of the linear features in a2-D map can be carried out by examining the amplitude distri-bution of the transform domain function. The high amplitudesin the transform domain can be selectively amplified or attenu-ated to produce a corresponding enhancement or suppressionof specific linear events in the space domain.

Inversion of F(ρ, θ) to f (x, y) is achieved via the filteredback-projection method (Toft, 1996), defined as

f (x, y) =∫ π

0

∫ ∞

−∞|ν|

(∫ ∞

−∞F(ρ, θ)e−i2πρνdρ

)

× ei2πν(x cos θ+y sin θ) dν dθ, (2)

where ν is a wavenumber vector used for filtering in the Fourierdomain.

SYNTHETIC EXAMPLES

To demonstrate the usefulness of the RT as a data process-ing tool, we first use synthetic data and then follow with some“real” geophysical maps. The RT and inverse RT are numericalprocesses that are dependent on the sampling rate used in thecomputation. The implementation of the transform pair is notexact and will not recover the original amplitudes. Increasingthe sampling rate to produce a more exact result leads to in-creased CPU time and memory, which may be prohibitive insome cases. Since we are concerned with image enhancement,the recovery of exact amplitudes is not so important. However,it is important that a sufficiently dense sampling of the map isused to avoid aliasing in the transform domain and the gener-ation of transform artifacts in the new map. In the examplespresented here, we follow the sampling requirements describedby Toft (1996).

Transform integrity

Figure 2a presents a very simple test pattern comprised ofvertical and horizontal lines; Figure 2b presents its Radontransform. The linear features are clearly distinguishable in thespectral domain as high-amplitude peaks whose locations aredetermined by the corresponding line parameters (ρ, θ). Notethat the horizontal lines produces two peaks in the transformdomain. This is because θ has values ranging from −π/2 to+π/2, both of which represent a horizontal line in the spacedomain. This double sampling of horizontal features couldbe avoided by restricting the range of θ values used in theforward RT, but is considered unnecessary and undesirable.Parallel linear events in the space domain are transformedas high-amplitude peaks at different locations in the spectraldomain due to their different offset from the origin. Thisseparation in the Radon spectral domain enables the linearfeatures to be processed separately.

Figure 2c contains the map recovered by application of theforward and inverse RT transform pair; Figure 2d shows a con-tour map of the amplitude differences between the two maps. Inmost parts of the recovered map, the amplitude differences areless than 2% of the original amplitudes. The largest differencerepresents less than 10% change. The test pattern represents adifficult image to process due to the very sharp boundaries. Thecontours that represent high-percentage changes occur where

1760 Sykes and Das

there are small differences in the low amplitudes near steepedges. In most geophysical maps, the amplitude gradients areless severe and will therefore not suffer the same level of vari-ation. In terms of visual appearance, the two maps are verysimilar. This provides confidence that the RT can be used forimage processing without creating excessive distortion of theoriginal image.

It should be emphasised that excessive processing will alwaysproduce deleterious effects on the maps. Care must be takento ensure that the features of interest are enhanced as muchas possible without adversely affecting the rest of the map. Ineach of the following examples of RT processing, we presentthe maps produced using the corresponding Fourier domainprocessing for comparison.

FIG. 2. (a) Test pattern comprised of vertical and horizontal features. (b) Radon transform of (a). (c) Map produced by applicationof the forward and inverse Radon transform pair. (d) Contour surface of the difference between (a) and (c). Most of the map isrecovered within 2% of the original amplitudes. Larger variations (up to 10%) are observed near the steep edges.

Directional filtering

Directional filtering is useful for removing unwanted fea-tures in the map that have a particular orientation such asfence-line and pipeline anomalies in EM and magnetic maps.Traditionally, directional filtering is achieved by the applica-tion of a cosine tapered, pass or rejection filter to the Fourierdomain data, in the direction of choice. The filters are often re-ferred to as “pie-slice” filters. To test the Radon filtering tech-nique, we use the map in Figure 2b. Reduction of the amplitudepeaks prior to inversion will result in the attenuation of the cor-responding linear feature in the recovered map. To this end, wereduce the peaks that correspond to horizontal lines to a se-lected level prior to inversion and follow by reducing the peaks


that correspond to vertical lines. For the models presented inthis paper, we follow the recommendation of Yunxuan (1992)and replace the peak amplitudes with the mean value of theentire RT data prior to inversion. This helps to preserve the am-

FIG. 3. (a) Horizontal and vertical features attenuated using Radon domain filtering. (b) Particular verticalfeatures attenuated by Radon domain filtering. (c) Fourier domain direction filtering of horizontal and verticalfeatures produces significant “ringing.”

plitudes in the new map and reduces the severity of transformartifacts. In this implementation of the RT pair, the choice ofamplitude scaling is not particularly important. What is moreimportant is the appropriate selection of spectral data points

1762 Sykes and Das

to which the scaling is applied. If insufficient data points arescaled, the linear feature will not be completely absent fromthe recovered map. If too many data points are scaled, otherfeatures in close proximity in the spectral domain may also bealtered. Selection of the optimum scaling window is achievedby examining the amplitude distribution of the Radon domain

FIG. 4. (a) Test pattern in which linear features cross. (b) Radon transform of the test pattern and (c) the recovered maps withselected features removed by Radon domain filtering. (d) The maps produced using conventional Fourier domain directional(pie-slice) filtering. The Radon domain filtering has greater directionality and has very little effect on the unfiltered lineaments.

data. Tapering of the window edges was considered unneces-sary since the crude method of block scaling appears to workadequately for the purpose of image enhancement.

The maps shown in Figure 3a show that the RT filtering iseffective in removing horizontal and vertical features withoutadversely affecting the appearance of the map. In addition,


we show in Figure 3b that the RT filtering can be applied tospecific linear features, even if they have the same strike di-rection. In this figure, only the vertical features on one side ofthe map have been attenuated and there is very little effecton the other features in the map. For comparison, in Figure 3cwe show that conventional directional filtering in the Fourierdomain is very limited in its application to particular features.Such a filter enhances all of the features having the particularstrike direction by attenuating all others. A cosine roll-off ta-per is applied to the filter to reduce the generation of artifactscaused by the filtering. Despite this, the Fourier directional fil-tering has produced noticeably more “ringing” than the RTmethod. The FT method cannot be used to filter out specificfeatures.

To demonstrate the technique for the case where lineamentscross, another simple data set was produced (Figure 4a) thatcomprised three linear features of different strike but commonamplitude and midpoint. The Radon transform of this data set

FIG. 5. (a) A very noisy space domain map in which a sloping linear feature is barely visible and (b) its corresponding Radontransform with characteristic lineament peak. (c) The map produced after amplification of the RT peak prior to inversion. (d) Themap produced if the noise is rejected by applying a threshold cutoff to the Radon data prior to inversion. (e) The map produced bydirectional filtering in the Fourier domain.

is shown in Figure 4b. The ability of Radon domain filteringto target selected features separately and remove them fromthe map is demonstrated in Figures 4c. As a comparison, di-rectional filtering was applied to the maps shown in Figure 4ausing the more conventional Fourier domain directional filter-ing. In Figure 4d, the FT directional filtering can be seen to beless direction specific than the RT method. The amplitudes ofthe unfiltered features are affected more by the Fourier domainprocessing.

Enhancement of linear features

In Figure 5a, we show a very noisy synthetic data map con-taining a diagonal linear feature whose amplitude is equal tothe maximum noise level. The linear feature is very difficultto resolve in the space domain due to the noisy background.In real situations, such a signal-to-noise ratio is unacceptablysmall, but it is used here for demonstration purposes. We apply

1764 Sykes and Das

FIG. 6. (a) Computed AEM response of a conductive target using a towed-bird configuration showing the distorted edges (her-ringbone pattern) due to measurement asymmetry. (b) The corresponding response if a symmetric measuring system is used.(c) A spectral line (constant θ) of the Radon domain obtained from (a) showing rapid amplitude variations due to the herringbonepattern. (d) The corresponding spectral line obtained from the transform of (b) to be free of the amplitude variations. (e) Thespace domain map produced from (a) after filtering in the Radon domain to reduce the spectral line amplitude variations. (f) Theherringbone pattern removed by Fourier domain filtering.


the RT to this data to produce Figure 5b. The linear feature isnow quite recognisable as a peak in the spectral domain. To en-hance the appearance of the linear feature in the new map, thespectral data was scaled to enhance the peak prior to the inver-sion. This was achieved by selecting a threshold value equal tothe approximate background level in the spectral domain. Datapoints having a value larger than the threshold were amplified.The degree of amplification was varied until a satisfactory re-sult was achieved. To produce the map of Figure 5c, the spectralpeak was amplified by a factor of two prior to inversion. Thelinear feature is much more apparent in the new map than itwas in the original map (cf. Figure 5a).

A corollary of the previous example in which scaling wasapplied to enhance the peak in the spectral data is to reducethe portion of the spectral data which is attributable to theunwanted nonlinear features (noise) in the space domain. Inthis case, the spectral data having values less than the thresh-old level were zeroed. Figure 5d shows the map produced byscaling down the background rather than enhancing the peak.In both cases, the linear event is much more apparent after RTprocessing.

As a comparison, a directional cosine-pass filter in the direc-tion of the lineament was used to produce the image in Fig-ure 5e. In this case the directional filtering has enhanced thelineament. However, directional filtering in the Fourier do-main could not be used as effectively if the map containedtwo features having different strike directions. In such a case,one of the features would be removed from the map alto-gether. The Radon domain filtering does not suffer from thisrestriction.

Removal of data acquisition artifacts

Data maps produced from airborne geophysical measure-ments often contain artifacts attributable to flight-line direc-tion. With the fixed-wing towed-bird configurations used inmany airborne electromagnetic (AEM) surveys, the asymme-try of the measuring system often manifests itself as a herring-bone noise pattern in the mapped data. The jagged appearanceof conductor boundaries, caused by the apparent displacementof the boundary in the direction of flight, is often called a her-ringbone pattern. Since most surveys are flown in opposite di-rections along adjacent lines, the displacements alternate fromline to line and produce the herringbone effect. These effectshave a clearly defined direction and will therefore be confinedto a particular region in the RT spectral domain.

As an illustration of herringbone artifacts, a synthetic AEMmap (Figure 6a) of the vertical magnetic field component(dBz/dt) was computed for an asymmetric fixed-wing towed-bird configuration (i.e., transmitter and receiver are at differentaltitudes) over a conductive target body in a conductive hostmedium (half-space). The presence of these artifacts on a mapreduces boundary definition. In contrast, the calculated AEMmap for a symmetric acquisition system is much smoother andmore clearly represents the boundaries of the conductor (Fig-ure 6b). The herringbone pattern can been seen in the RT do-main as rapid variations along spectral lines having θ valuescorresponding to the flight-line direction. Figures 6c and d,respectively, show a plot of a spectral line obtained from eachmap. Removal of the rapid amplitude variations from Figure 6c

prior to inversion results in a new map that is free of the herring-bone pattern (Figure 6e). A variety of techniques were used toremove the rapid amplitude variations in the RT domain in-cluding zeroing, scaling, averaging along the line, interpolation,and 1-D Fourier transform filtering. All techniques were suc-cessful in producing a map without herringbone pattern. Again,the most important factor is identification of the portion of theRT domain that needs to be filtered.

Directional filtering of the flight-line direction by conven-tional Fourier cosine rejection produced the map shown inFigure 6f. Comparison of Figures 6e with f provides evidencethat the RT processing is at least as effective in removing her-

FIG. 7. (a) AEM field data map containing a significant falseanomaly. (b) The same map after Radon domain filtering toremove the unwanted feature. Note the minimal effect on therest of the map. (c) The same map to which Fourier directionalfiltering has been applied. The “pie-slice” Fourier filter hasseverely altered the nature of the map.

1766 Sykes and Das

ringbone pattern as FT directional filtering and has much lessimpact on the rest of the map.

FIELD EXAMPLES

We apply the RT enhancement procedures described aboveto three different geophysical data maps and compare withmaps produced using traditional Fourier domain processing.

FIG. 8. (a) Total count radiometric map and (b) the same map after scaling of the Radon domain peaks to enhance the lineaments.(c) and (d) The maps produced by Fourier filtering in the northeast and northwest directions, respectively.

Directional filtering to remove linear features

From inspection of a typical AEM map (Figure 7a), we seea dominant north-trending feature in the upper part of themap directed along a flight line. This is believed to be due toinstrument malfunction and not a true signal. With this in mind,the peak in the RT domain corresponding to this linear featurewas reduced. Inversion of the filtered RT domain produced the


map shown in Figure 7b. With the dominant feature removedfrom the map, the underlying features can be more clearly ob-served.

Fourier cosine rejection applied to the flight line directionproduced the map shown in Figure 7c. The Fourier filter hasremoved the unwanted feature but has severely affected therest of the map.

Lineament enhancement in radiometric data

In an attempt to enhance linear trends in a typical radiomet-ric data map (Figure 8a), we applied the Radon transform. Asdescribed earlier, selected amplitudes in the Radon domainwere amplified (doubled in this case). After inversion backto the spatial domain, the lineaments in the new map (Fig-ure 8b) are much sharper and more easily seen. As a compari-son, northwest and northeast direction filtering of the Fourierdomain was used to enhance the lineaments in the map (Fig-ures 8c and d). Both of these maps are very different from theoriginal and display lineaments in only one direction.

Removal of acquisition artefacts

Figure 9a presents a portion of a time-domain AEM mapacquired using a towed-bird Geotem system. Herringbone pat-tern, resulting from the asymmetric transmitter-receiver con-figuration, is apparent over most of the map. Figure 9b showsthe improved map after smoothing of the appropriate RTspectral lines (using a five-point averaging filter) prior to in-version back to the space domain. The herringbone noise pat-tern has been very effectively removed, allowing for improvedinterpretation of the remaining anomalies.

Application of a Fourier cosine rejection filter produced themap shown in Figure 9c. While the FT filter has removed theherringbones, it has also distorted many of the features dis-played in the map.

CONCLUSION

It has been shown that the Radon transform is a very use-ful processing tool for improving the quality of geophysicalmaps. The ability of the RT to isolate directional features inthe spectral domain enables each lineament to be processedseparately without significantly affecting the rest of the map.Scaling of selected portions of the Radon domain data can beused to enhance or attenuate specific lineaments present in themap. A number of different scaling techniques were used, andthere appears to be little advantage of one scaling techniqueover another. Selection of the optimum scaling window is bestachieved by inspection of the RT data. Tapered scaling win-dows were considered unnecessary as the RT technique is veryrobust and does not produce transform artifacts to a significantdegree. However, the inability of the RT to exactly recover theamplitudes in the reconstructed maps is an area that needs fur-ther work. This shortcoming can be tolerated in view of theoverall improvement in appearance of the maps.

In the examples presented in this paper, the RT processinghas been shown to be superior to the conventional FT domainprocessing for lineament enhancement. The Radon domain fil-tering need only be applied to a small portion of the trans-formed data. The FT domain processing is not event specific

and is therefore more likely to suffer from the production oftransform artifacts, even when using a tapered filter.

ACKNOWLEDGMENT

The authors express their gratitude to Harold Yarger and theother reviewers for their helpful suggestions for improvementof the original manuscript.

REFERENCES

Beylkin, G., 1987, Discrete Radon transform: IEEE Trans. Acoustics,Speech and Signal Processing, 35, 162–172.

Brysk, H., and McCowan, D. W., 1986, A slant-stack procedure forpoint-source data: Geophysics, 51, 1370–1386.

FIG. 9. (a) AEM map with herringbone pattern and (b) thesame map after Radon domain filtering. (c) The map producedby Fourier filtering in the flight-line direction. The FT filter hasseverely distorted the features in the map.

1768 Sykes and Das

Chapman, C. H., 1978, A new method for computing synthetic seismo-grams: Geophys. J. Roy. Astr. Soc., 54, 482–518.

Deans, S. R., 1983, Radon transform and some of its applications: JohnWiley & Sons, Inc.

Durrani, T. S., and Bisset, D., 1984, The Radon transform and its prop-erties: Geophysics, 49, 1180–1187.

Palacky, G. J., 1988, Resistivity characteristics of geologic targets, inNabighian, M. N., ed., Electromagnetic methods in applied geo-physics, Vol. 1, Theory: Soc. Expl. Geophys., 53–129.

Pawlowski, R. S., 1997, Use of slant stack for geologic or geophysicalmap lineament analysis: Geophysics, 62, 1774–1778.

Robinson, E. A., 1982, Spectral approach to geophysical inversion by

Lorentz, Fourier and Radon transforms: Proc. IEEE, 70, 1039–1053.Stoffa, P. L., Buhl, P., Diebold, J. B., and Wenzel, F., 1981, Direct

mapping of seismic data to the domain of intercept time and rayparameter—A plane-wave decomposition: Geophysics, 46, 255–267.

Taner, M. T., 1980, Long-period sea-floor multiples and their suppres-sion: Geophys. Prosp., 28, 30–48.

Toft, P. A., 1996, The Radon transform—theory and implementation:Ph.D. thesis, Technical Univ. of Denmark.

Yunxuan, Z., 1992, Application of the Radon transform to the pro-cessing of airborne geophysical data: Ph.D. thesis, Internat. Inst. forAerospace Survey and Earth Sciences.


Gas generation and overpressure: Effects on seismic attributes

Jose M. Carcione∗ and Anthony F. Gangi‡

ABSTRACTDrilling of deep gas resources is hampered by high risk

associated with unexpected overpressure zones. Knowl-edge of pore pressure using seismic data, as for instancefrom seismic-while-drilling techniques, will help produc-ers plan the drilling process in real time to control po-tentially dangerous abnormal pressures.

We assume a simple basin-evolution model with a con-stant sedimentation rate and a constant geothermal gra-dient. Oil/gas conversion starts at a given depth in areservoir volume sealed with faults whose permeabilityis sufficiently low so that the increase in pressure causedby gas generation greatly exceeds the dissipation of pres-sure by flow. Assuming a first-order kinetic reaction, witha reaction rate satisfying the Arrhenius equation, theoil/gas conversion fraction is calculated. Balancing massand volume fractions in the pore space yields the ex-

cess pore pressure and the fluid saturations. This excesspore pressure determines the effective pressure, whichin turn determines the skeleton bulk moduli. If the gen-erated gas goes into solution in the oil, this effect doesnot greatly change the depth and oil/gas conversion frac-tion for which the hydrostatic pressure approaches thelithostatic pressure.

The seismic velocities versus pore pressure and differ-ential pressure are computed by using a model for wavepropagation in a porous medium saturated with oil andgas. Moreover, the velocities and attenuation factors ver-sus frequency are obtained by including rock-frame/fluidviscoelastic effects to match ultrasonic experimental ve-locities. For the basin-evolution model used here, porepressure is seismically visible when the effective pressureis less than about 15 MPa and the oil/gas conversion isabout 2.5% percent.

INTRODUCTION

In deeply buried oil reservoirs, oil-to-gas cracking may in-crease the pore pressure to reach or exceed the lithostaticpressure (Chaney, 1950; Barker, 1990; Luo and Vasseur, 1996).Oil can be generated from kerogen-rich source rocks and canflow through a carrier bed to a sandstone reservoir rock. Ex-cess pore-fluid pressures in sandstone reservoirs are generatedwhen the rate of volume created by the transformation of oil togas is more rapid than the rate of volume loss by fluid flow. If thereservoir is sealed on all sides by an impermeable shale or lime-stone, then the condition of a closed system will be satisfied forgas generation. Because of the presence of semivertical faultplanes and compartmentalization, this condition holds for mostNorth Sea reservoirs. Moreover, gas generation significantlydecreases the relative permeability (Luo and Vasseur, 1996).

Berg and Gangi (1999) developed a simple model to cal-culate the excess pore pressure as a function of the fraction ofkerogen converted to oil and the fraction of oil converted to gas.The oil/gas conversion ratio is computed as a function of time,

Manuscript received by the Editor September 3, 1998; revised manuscript received October 25, 1999.∗Osservatorio Geofisico Sperimentale, P.O. Box 2011, 34016 Trieste, Italy. E-mail: [email protected].‡Department of Geology and Geophysics, Texas A&M University, College Station, Texas 77843-3114. E-mail: [email protected]© 2000 Society of Exploration Geophysicists. All rights reserved.

for a given sedimentation rate and geothermal gradient. ThenBerg and Gangi (1999) determined the excess pore pressureresulting from the oil/gas conversion with burial time, whichis derived by balancing mass and volume changes in the porespace. Carcione (2000) applied this model of the calculation ofthe kerogen/oil conversion to obtain the seismic properties ofoverpressured shales. Here, we use an extended version of thismodel for computing the porosity variations and fluid satura-tions as a function of the excess pore pressure. We consider twocases, when the generated gas goes into solution in the oil andwhen it remains as free gas. When gas goes into solution in deadoils, it generates live oils. The analysis is based on the equationsfor live oil given by Batzle and Wang (1992). The solubilities ofhydrocarbon gases in water are very small compared with thesolubilities of the same gases in oil.

Seismic velocities have been used to predict pore pressure,with most of the models based on empirical relations (Duttaand Levin, 1990). Attenuation has been studied mainly at ul-trasonic frequencies, and a few works have focused on verti-cal seismic profiling (VSP) data and sonic data (Hague, 1981;

1769

1770 Carcione and Gangi

Dasios et al., 1998a). Recently, attenuation obtained from sur-face seismic data has been used successfully to predict lithologyand pore pressure (Helle et al., 1993; Dasios et al., 1998b). Inthis work, the seismic properties are calculated by using a mod-ification of Biot’s theory for fully saturated porous solids. Theeffective parameters for the fluid mixture are computed by theaverage formulas given by Berryman et al. (1988).

The wave velocities of the dry rock, determined by labora-tory measurements, give estimates of its pore compressibilityand bulk moduli in the low-frequency (relaxed) regime. Be-cause dissipation mechanisms are caused mainly by grain/fluidinteractions, dry-rock velocities are frequency independent.On the other hand, measurements of ultrasonic saturated-rockvelocities provide the high-frequency-limit (unrelaxed) veloc-ities, which can be used to estimate the amount of velocity dis-persion between the seismic and ultrasonic ranges and, there-fore, the amount of attenuation. By using this information,the phase velocities and attenuation factors are computed as afunction of frequency.

GAS GENERATION AND ROCK PROPERTIES

First, We will introduce some useful definitions about thedifferent pressures considered in this work. Pore pressure, alsoknown as formation pressure, is the in situ pressure of the flu-ids in the pores. The pore pressure is equal to the hydrostaticpressure when the pore fluids only support the weight of theoverlying pore fluids (mainly brine). The lithostatic or confin-ing pressure results from the weight of overlying sediments,including the pore fluids. In the absence of any state of stress inthe rock, the pore pressure attains lithostatic pressure and thefluids support all the weight. However, fractures perpendicularto the minimum compressive stress direction appear for a givenpore pressure, typically 70–90% of the confining pressure. Inthis case, the fluid escapes from the pores, and pore pressuredecreases. A rock is said to be overpressured when its porepressure is significantly greater than hydrostatic pressure. Thedifference between pore pressure and hydrostatic pressure iscalled differential pressure. Acoustic and transport propertiesof rocks generally depend on effective pressure, a combinationof pore and confining pressures [see equation (20)]. Variousphysical processes cause anomalous pressures on an under-ground fluid. The most common causes of overpressure arecompaction disequilibrium and cracking, i.e., oil-to-gas con-version (Mann and Mackenzie, 1990; Luo and Vasseur, 1996).

Let us assume a reservoir at depth z. The lithostatic pres-sure for an average sediment density of ρ is equal to pc= ρgz,where g is the acceleration of gravity. On the other hand, thehydrostatic pore pressure is approximately pH = ρwgz, whereρw is the density of water.

For a constant sediment burial rate, S, and a constantgeothermal gradient, G, the temperature variation of a par-ticular sediment volume is

T = T0 + Gz, z= St, (1)

with a surface temperature T0 at time t = 0. Typical values of Grange from 20 to 30◦C/km, and Smay range between 0.05 and0.5 km/m.y. (m.y.= million years).

Assume that at time ti , corresponding to depth zi , the reser-voir volume has been saturated with oil which flowed from an

adjacent source rock, and that the volume is “closed.” That is,the permeability of the sealing faults is sufficiently low so thatthe rate of pressure increase caused by gas generation greatlyexceeds the dissipation of pressure by flow. Pore-pressure ex-cess is intended to be above hydrostatic.

Oil/gas generation rate

The mass of convertible oil changes with time t at a rateproportional to the mass present. Assuming a first-order kineticreaction (Luo and Vasseur, 1996; Berg and Gangi, 1999),

d Mo

dt= −ro(t)Mo(t) (2)

or

Mo(t) = Moi exp

[−∫ t

ti

ro(t) dt

], (3)

where ro(t) is the reaction rate, Mo(t) is the mass of convertibleoil at time t , and Moi is the initial oil mass. The fraction of oilconverted to gas is F(t)= [Moi − Mo(t)]/Moi :

F(t)= 1− exp

[−∫ t

ti

ro(t ′) dt′]≡ 1− exp[−8(t)]. (4)

The reaction rate follows the Arrhenius equation (Luo andVasseur, 1996)

ro(t) = Aexp[−E/RT(t)], (5)

where E is the oil/gas activation energy, R= 1.986 cal/mol ◦Kis the gas constant, A is the oil/gas reaction rate at infinitetemperature, and T(t) is the absolute temperature in ◦K givenby

T = T0 + Ht, H = GS. (6)

With this temperature dependence, the integral 8(t) be-comes

8(t) =∫ t

ti

ro(t ′) dt′ = A

H

∫ T

Ti

exp(−E/RT′) dT′,

Ti = T0 + Hti (7)

or

8(t) = A

H

[T∫ ∞

1exp(−Ex/RT)

dx

x2

− Ti

∫ ∞1

exp(−Ex/RTi )dx

x2

]. (8)

For values of E/RT greater than 10, the exponential inte-gral can be approximated by (Gautschi and Cahill, 1964, 248,Table 5.5)∫ ∞

1exp(−Ex/RT)

dx

x2∼= exp(−E/RT)

2+ E/RT, (9)

Gas Generation, Overpressure, and Seismics 1771

with an error of 1.3% or less. Then the integral 8 becomes

8(T(t)) = A

H

[T exp(−E/RT)

2+ E/RT− Ti exp(−E/RTi )

2+ E/RTi

].

(10)The activation energy and infinite-temperature rate used inthis work are E= 52 kcal/mol and A= 5.5×1026/m.y. (Luo andVasseur, 1996; Berg and Gangi, 1999).

Oil/gas conversion factor and excess pore pressure

We are interested primarily in a first-order solution whichshows the large effect of oil-to-gas conversion on seismic prop-erties. In this sense, following Berg and Gangi (1999), we as-sume:

1) The compressibilities of oil and water are independent ofpressure and temperature, and that of the rock is indepen-dent of temperature but depends on pressure. That this isthe case can be seen from the results given by Batzle andWang (1992) in their Figures 5 and 13, in which they showthat the density is almost a linear function of tempera-ture and pressure. This means that the mentioned prop-erties are approximately constant (see also their Figure 7,in which the oil compressibility remains almost constantwhen going from low temperature and low pressure tohigh temperature and high pressure).

2) The gas satisfies the van der Waals equation (Friedman,1963): (

p+ aρ2g

)(1− bρg) = ρg RT, (11)

where p is the gas pressure, ρg is the gas density, T is theabsolute temperature, and R is the gas constant. More-over, a= 0.225 Pa (m3/mole)2= 879.9 MPa (cm3/g)2,and b= 4.28× 10−5 m3/mole= 2.675 cm3/g (one mole ofmethane, CH4, corresponds to 16 g). The van der Waalsequation is a good approximation of the behavior of natu-ral gas, as shown by Berg and Gangi (1999), where the dif-ferences between the experimental data—as representedby Standing’s results (Standing, 1952)—and the van derWaals results are only about 15% over the depths of in-terest.

3) The initial pore fluids are water and oil.

The excess pore pressure at depth z is p− pH , where pH is thehydrostatic pore pressure at depth z, and p is the pore pressurewhen a fraction F of oil has been converted to gas (F = 0 andp= pi = pH at time ti ) . The mass balance is

ρgVg = Fρoi Voi , (12)

where ρoi is the initial oil density, Vg is the converted gas vol-ume, and Voi is the initial oil volume. Assumption (3) implies

Vpi = Vwi + Voi or Vpi/Vwi = 1+ v, (13)

where Vpi is the initial pore volume and v=Voi/Vwi .The compressibilities of the oil, water, and pore space are

defined, respectively, as

co = − 1Vo

dVo

dp, cw = − 1

Vw

dVwdp

, cp = − 1Vp

dVp

dpe,

(14)

where pe is the effective pressure and cp can be obtained fromthe compressibility at zero pore pressure.

In general, compressional and shear-wave velocities dependon effective pressure pe= pc − np, where n≤ 1 is the effectivestress coefficient. Note that the effective pressure equals theconfining pressure at zero pore pressure. It is found that n≈ 1for static measurements of the compressibilities (Zimmermanet al., 1986), while n is approximately linearly dependent onthe differential pressure pd = pc − p in dynamic experiments(Gangi and Carlson, 1996; Prasad and Manghnani, 1997):

n = n0 − n1 pd, (15)

where n0 and n1 are constant coefficients.We assume the following functional form for cp as a function

of effective pressure:

cp = c∞p + β exp(−pe

/p∗), (16)

where c∞p , β, and p∗ are coefficients obtained by fitting theexperimental data. On the other hand, the thermal expansioncoefficients for oil, water, and pore space are defined, respec-tively, as

αo = 1Vo

dVo

dT, αw = 1

Vw

dVwdT

, αp = 1Vp

dVp

dT. (17)

Integration from pi (pei) to p (pe) and from Ti to Ti +1T ,where 1T = T − Ti , yields

Vo(p, T) = Voi [exp(−co1p+ αo1T)],

Vw(p, T) = Vwi [exp(−cw1p+ αw1T)], (18)

and

Vp(p, T) = Vpi {exp[E(1p)+ αp1T]}, (19)

where

E(1p) = −c∞p 1pe+ βp∗[

exp(−pe

/p∗)− exp

(−pei

/p∗)]

and 1p= p− pi = p− pHi . Using equation (15), we can writethe effective pressure as

pe = pc − (n0 − n1 pc)p− n1 p2. (20)

Berg and Gangi (1999) assume a constant pore compressibilitycp. In this case, E(1p)=−cp1p.

In principle, if no gas goes into solution in the oil, the porevolume at pore pressure p and temperature T is given by

Vp(p, T) = Vpi {exp[E(1p)+ αp1T]}= Vwi [exp(−cw1p+ αw1T)]+ (1− F)Voi

× [exp(−co1p+ αo1T)]+ Vg(p, T), (21)

since (1− F) of the initial oil remains at pressure p. Substitut-ing equation (12) into equation (21) and dividing through byVwi yields


ρo = ρ0 + (0.00277p− 1.71× 10−7 p3)(ρ0 − 1.15)2 + 3.49× 10−4 p

0.972+ 3.81× 10−4(T + 17.78)1.175. (25)

(1+ 1

v

){exp[E(1p)+ αp1T]}

= 1v

exp(−cw1p+ αw1T)

+ (1− F)[exp(−co1p+ αo1T)]+ Fρoi/ρg. (22)

In the absence of water, 1/v→ 0.There are two possibilities when the generated gas goes into

solution: (1) all the gas dissolves in the oil, which occurs at thebeginning of the conversion process down to a critical depth,and (2) a fraction of gas goes into solution in the oil and therest remains as free gas. The process can be approximated, asillustrated in Figure 1: It begins at stage (1); in stage (2), avolume Vg=Vg1+Vg2 of gas is generated, with Vg1 going im-mediately into solution and Vg2 remaining as free gas (stage 3).In the following, we obtain the balance equations, equivalentto equation (22) and the fractions of solid, water, live oil, andfree gas at stage (3), for the two cases described above.

The volume ratio of liberated gas to remaining oil at atmo-spheric pressure and 15.6◦C is

RG = 0.02123Gr

[p exp

(4.072ρ0−0.00377T

)]1.205

, (23)

where Gr is the gas gravity and ρ0 is the oil density at atmo-spheric pressure and temperature (Batzle and Wang, 1992);RG, given in liters of gas/liters of oil, represents the maximumamount of gas that can be dissolved in the oil. At depth z,temperature T , and pressure p, the equivalent ratio is

R′G =ρgs

ρg

ρo

ρosRG, (24)

where ρgs, ρos and ρg, ρo are the gas and oil densities at thesurface and at depth z, respectively. The gas densities can becomputed from the van der Waals equation (11), and the oildensity is given by

FIG. 1. Oil-to-gas conversion process. Part of the generated gasgoes into solution.

Oil densities are given in g/cm3, temperature T in ◦C, and pres-sure p in MPa. Because R′G is the maximum volume of gas (Vg1)that can be dissolved in the remaining volume of dead oil, wehave

R′G =[

ρo

(1− F)ρoi

]Vg1

Voi. (26)

The volume of free gas (Vg2) is equal to the total volume ofgenerated gas (Vg) minus the maximum volume of dissolvedgas (Vg1),

Vg2 = Fρoi

ρgVoi − (1− F)

ρoi

ρoR′GVoi

={[

F

ρg− (1− F)

ρoR′G

]ρoi

}Voi ≡ δVoi , (27)

where equation (26) has been used. There is a critical depthzc for which the gas saturates the oil. Above this depth (whichcorresponds to a critical pressure pc), all the gas goes into solu-tion in the oil (note that the process is a dynamic one, in whichgas is generated continuously). This occurs when the volumeof free gas [equation (27)] is equal to zero, i.e., when a fraction

Fc = ρg R′Gρo + ρg R′G

(28)

of oil has been converted to gas, at the critical pressure pc.Below the critical depth, the pore space will contain live oilmixed with free gas. To compute zc and Fc, we assume that thesystem is subject to the pore-pressure profile obtained fromequation (22), which is the situation shown in stage (2) of Fig-ure 1, before the gas goes into solution. Then zc is the depthfor which δ= 0 [see equation (27)] or F = Fc. It can be arguedthat after the gas goes into solution, the pressure will decrease.However, if this occurs, part of the gas will go out of solutionimmediatley (as free gas) and the pressure will increase againto the previous value. It can be shown that by choosing, forinstance, the hydrostatic pressure pH as the reference pressureprofile, instead of that obtained from equation (22), the criti-cal depth will be different, but the results will not be affectedgreatly.

The density of the live oil at and below the critical depth isgiven by

ρlo = ρG + (0.00277p− 1.71× 10−7 p3)(ρG − 1.15)2

+ 3.49× 10−4 p (29)

with

ρG = (ρ0 + 0.0012Gr RG)/B0, (30)

the saturation density, and

B0 = 0.972+ 0.00038

[2.4RG

(Gr

ρ0

)1/2

+ T + 17.8

]1.175

,

(31)


the oil-volume factor (Batzle and Wang, 1992). When comput-ing the density of live oil, the temperature effect was consid-ered twice in Batzle and Wang (1982), in B0 and in their equa-tion (19) (M. Batzle, personal communication). Equation (29)gives the correct density. At the initial depth zi there is nogenerated gas, and the oil density is that of the dead oil ρoi .

When zi ≤ z≤ zc [case (1)], all the gas goes into solution, butthe oil is not saturated because it absorbs less than R′G liters ofgas; it absorbs (F/Fc)R′G. The density, ρlo, of live oil is obtainedby substituting RG by (F/Fc)RG in equations (29), (30), and(31). Because, in this range, the mass of live oil is equal to theinitial mass of dead oil, the volume of live oil is

Vlo = ρoi

ρloVoi ≡ ηVoi . (32)

Then the balance equation above the critical depth is(1+ 1

v

){exp[E(1p)+ αp1T]}

= 1v

exp(−cw1p+ αw1T)+ η. (33)

We now consider case (2), i.e., when z> zc and the pore space isfilled with live oil and free gas. The mass of live oilρloVlo is equalto the mass of remaining dead oil (1− F)ρoi Voi plus the massof gas going into solution. The latter is equal to the maximumvolume of gas Vg1 that can be dissolved in the remaining deadoil [equation (26)], multiplied by the gas density ρg. Then thevolume of live oil is

Vlo =[ρoi

ρlo(1− F)

(1+ R′G

ρg

ρo

)]Voi ≡ γVoi . (34)

Below the critical depth, the density of live oil, ρlo, is obtainedfrom equation (29). Taking into account that the volume offree gas is given by equation (27) and using (34), the balanceequation is(

1+ 1v

){exp[E(1p)+ αp1T]} = 1

vexp(−cw1p

+αw1T)+ γ + Fρoi/ρg − R′G(1− F)ρoi/ρo. (35)

Equations (33) and (35) relate the pore pressure p to theconverted oil/gas fraction F at the confining pressure pc andtemperature T . All these quantities depend on the depositiontime t . Note also that η, ρg, γ , and R′G depend on pore pressureand temperature.

Volume fractions and saturations

The fractions of live oil and water at stage (3), above thecritical depth zc (see Figure 1), are equal to the correspondingvolumes divided by the total volume Vs+Vw +Vlo. The volumeof the solid part can be obtained from the initial porosity φi

and the initial pore volume Vpi , because φi =Vpi/(Vpi + Vs);thus, Vs=Vpi (1/φi − 1), and by using (13), Vs/Voi = (1 + 1/v)(1/φi − 1). It can be shown easily that the fractions of live oil,water, and solid are given by

φlo = η [(Vs/Voi )+ (Vw/vVwi )+ η]−1, (36)

φw = 1vη

(Vw/Vwi )φlo, (37)

and

φs = 1− φlo − φw, (38)

respectively, where the volume ratios are given by equa-tion (18).

The total volume below the critical depth is given byVs+Vw +Vlo+Vg2. Using equations (18), (27), and (34), weobtain the the volume fractions of live oil, water, free gas, andsolid:

φlo = γ [(Vs/Voi )+ (Vw/vVwi )+ γ + δ]−1, (39)

φw = 1vγ

(Vw/Vwi )φlo, (40)

φ f g = δ

γφlo, (41)

and

φs = 1− φlo − φw − φ f g, (42)

respectively.The saturations are equal to the corresponding fluid volumes

divided by the pore volume or equal to the fluid fractions di-vided by the total porosity φ, which is given by

φ = Vp

Vp + Vs, (43)

where Vs is the volume of the solid part. Using equation (21)and the expression for Vs obtained above, we obtain

φ = φi exp[E(1p)+ αp1T]1− φi {1− exp[E(1p)+ αp1T]} . (44)

Then the free gas, live oil, and water saturations are

Sf g = φ f g/φ, Slo = φlo/φ, and Sw = φw/φ, (45)

respectively.Let us calculate the saturations and volume fractions when

the gas does not go into solution. As the pore pressure in-creases from pi to p, the pore volume changes from Vpi toVpi {exp[E(1p)+αp1T]} . The saturations are equal to the cor-responding volumes divided by the pore volume. Using equa-tions (13) and (21) gives, for the oil and water saturations,

So = v

1+ v (1− F)[exp(−co1p+ αo1T − E(1p)

−αp1T)] (46)

and

Sw = 11+ v [exp(−cw1p+ αw1T − E(1p)− αp1T)],

(47)

respectively. The gas saturation is simply

Sg = 1− So − Sw. (48)

On the other hand, the fluid fractions are

φo = φSo, φw = φSw, and φg = φSg. (49)


Compressibilities and dry-rock bulk moduli

The isothermal gas bulk modulus Kg and the gas compress-ibility cg= K−1

g depend on pressure. The latter can be calcu-lated from the van der Waals equation as

cg = 1ρg

∂ρg

∂p

at constant temperature. It gives

cg =[

ρg RT

(1− bρg)2− 2aρ2

g

]−1

. (50)

The pore compressibility cp is related closely to the bulk modu-lus of the dry rock Km (the compressibility cp is denoted by Cpp

in Zimmerman et al. (1986) and by K−1φp in Mavko and Mukerji

(1995), and Km corresponds to C−1bc and K−1

dry , respectively). Us-ing the present notation, cp can approximately be expressed as

cp =(

1Km− 1

Ks

)1φ− 1

Ks, (51)

where φ depends on the excess pore pressure. Because dry-rock wave velocities are practically frequency independent,the seismic bulk moduli Km and µm versus confining pressurecan be obtained from laboratory measurements in dry samples.If VP and VS are the experimental compressional and shearvelocities, the moduli are given approximately by

Km= (1− φ)ρs

(V2

P −43

V2S

), µm= (1− φ)ρsV

2S, (52)

where ρs is the grain density and φ is the porosity. We recallthat Km is the rock modulus at constant pore pressure, i.e.,the case when the bulk modulus of the pore fluid is negligiblecompared with the dry-rock bulk modulus, as, for example, airat room conditions (Mavko and Mukerji, 1995). Note that gasat high pressures can have a nonnegligible bulk modulus.

Seismic properties of the rock volume

The seismic velocities of the overpressured porous rock arecomputed by using Biot’s theory of dynamic poroelasticity(Appendix A). The phase velocities of P- and S-waves aregiven by

VP± =[

Re

(1

V∗P±

)]−1

, VS =[

Re

(1

V∗S

)]−1

, (53)

where V∗P± are the complex velocities of the fast (+ ) and slow(− ) waves, V∗S is the complex shear wave velocity, and Re takesthe real part.

EXAMPLE

We assume that the medium is saturated fully with oil andthat before oil/gas conversion occurs, the initial pressure, pi ,is hydrostatic. Because there is no water, v→∞, and equa-tion (22) becomes

exp[E(1p)+ αp1T] = (1− F)[exp(−co1p+ αo1T)]

+ Fρoi/ρg. (54)

When there is dissolved gas, the balance equations are

exp[E(1p)+ αp1T] = η, z≤ zc (55)

and

exp[E(1p)+ αp1T]= γ + Fρoi/ρg

− R′G(1− F)ρoi/ρo, z> zc. (56)

With a surface temperature of 15.6◦C, a temperature gradi-ent G= 25◦C/km, a sedimentation rate S= 0.08 km/m.y., and areservoir volume at zi = z1= 2 km, we have ti = t1= 25 m.y.and T1= 65.6◦C. After 75 m.y., the depth of burial isz2= 8 km, t2= 100 m.y., and T2= 215.6◦C. On the other hand,if ρ= 2.4 g/cm3, the confining pressure has increased from47 MPa to approximately 188 MPa, and the initial pore pres-sure is pi ' 20 MPa (assuming ρw = 1 g/cm3). If no conversiontakes place, the final pore pressure would be the hydrostaticpressure at 8 km, i.e., approximately 78 MPa.

The experimental data for oil-saturated sandstone are avail-able in Winkler (1985), in his Figures 3 and 4 and Tables 4 and 7.It is important to understand how the data were measured. Af-ter a suitable pore vacuum was achieved, confining pressurewas increased to 40 MPa. Dry-rock wave velocities then wereobtained at successive pressures as the confining pressure de-creased to 10 MPa. While still in the pressure vessel, the samplewas saturated with oil to a pore pressure of 5 MPa. Differentialpressure pc was then increased to 10, 20, and 40 MPa, corre-sponding to the values (pc, p)= (20, 10), (40, 20), and (60, 20)(in MPa). Note that Winkler calls the differential pressure theeffective stress. The experiments on dry samples correspond tozero pore pressure. Best-fit plots of the dry-rock compressibil-ity and shear modulus versus confining pressure are

K−1m [GPa−1] = 0.064+ 0.122 exp(−pc[MPa]/6.48)

(57)and

µm[GPa] = 13.7− 8.5 exp(−pc[MPa]/9.14), (58)

and cp in GPa−1 is given by equation (16), with c∞p = 0.155,β = 0.6, and p∗ = 6.48. The pore compressibility cp has beenobtained from equation (51) by assuming that the porosity isthat at hydrostatic pore pressure [this approximation is sup-ported by experimental data obtained by Domenico (1977)and Han et al. (1986)]. The best-fit plots for cp and K−1

m areillustrated in Figure 2.

To obtain the moduli for different combinations of theconfining and pore pressures, we should make the substitu-tion pc→ pe= pc− np, where we assume, following Gangi andCarlson (1996), that n depends on differential pressure as

n = n0 − n1 pd, n0 = 1, n1 = 0.014 MPa−1.

This dependence of n versus differential pressure is in goodagreement with the experimental values corresponding tothe compressional velocity obtained by Christensen andWang (1985) and by Prasad and Manghnani (1997).

Table 1 indicates the properties for Berea sandstone, wherethe values correspond to those at the initial (hydrostatic) porepressure. The oil and gas viscosities as a function of tempera-ture and pore pressure are taken from Luo and Vasseur (1996).

The procedure for computing the phase velocities and atten-uation factors is the following:

1) Compute the oil/gas conversion factor as a function of de-position time and depth by using equations (4) and (10).


2) Compute the dry-rock bulk moduli (52) from the experi-mental wave velocities VP(pc) and VS(pc). This also givesthe dry-rock bulk moduli as a function of effective pres-sure. We assume that the porosity is that at hydrostaticpressure.

3) Compute the initial gas density and the gas density atpressure p, ρg(p) from the van der Waals equation (11)and the pore compressibility from equation (51). As be-fore, we assume that the porosity is that at hydrostaticpressure.

4) Compute the excess pore pressure corresponding to theoil/gas conversion factor F by using equations (54) or (55)and (56), and the oil and gas saturations by using equa-tions (45) or (46) and (48). Overpressure occurs whenpore pressure exceeds hydrostatic pressure.

5) The low-frequency wet-rock velocities are computed byusing the porous model (Appendix A), and the dilata-tional and shear relaxation times are obtained by fittingthe experimental high-frequency wet-rock velocities atfull saturation, i.e., at the onset of the oil/gas conversion.

6) Phase velocities and attenuation factors are computed as

Table 1. Material properties for Berea sandstone.

Grainρs = 2650 kg/m3

Ks = 37 GPaµs = 39 GPa

Fluidsρ0 = 934 kg/m3

ρoi = 908 kg/m3

Ko = 2.16 GPaαo = 5 × 10−4 C−1

ηo = 800 cP∗ηg = 0.012 cP∗

Rock frameKm = 15.04 GPaµm = 13.16 GPa

φ = 0.203κ = 10−12 m2

T = 2αp = 2× 10−4 C−1

∗1 cP = 0.001 Pa·s.

FIG. 2. Best-fit plots of cp and K−1m obtained from the experi-

mental data for Berea sandstone published by Winkler (1985,Figures 3 and 4 and Tables 4 and 7).

a function of pore pressure and frequency from equa-tions (A-15) and (A-16), respectively.

Figure 3 shows the oil/gas conversion factor F [equa-tion (4)] (a) and its logarithm (b) as a function of depth andtime. The high activation energy requires either a long time ordeep burial, on the order of 4.5 to 5 km, before appreciablefractions of conversion occur, but significant fractional conver-sions occur at 3 km. The pore-pressure buildup with depth is

FIG. 3. Oil/gas conversion factor F [equation (4)] (a) andlog (F) (b) as a function of depth and time.


shown in Figure 4, for all the generated gas out of solution(continuous line), and for part of the generated gas dissolvedin the oil (broken line). The two cases yield practically thesame result. The pressure increases rapidly for very small frac-tions of oil converted to gas. This is caused by the expansionof the pore volume resulting from generation of free gas orlive oil. Beyond the lithostatic pressure, the effective pressurebecomes negative and the pore compressibility increases (seeFigure 2), making the rock highly compliant. This precludesa rapid increase of the pore pressure, which follows the litho-static pressure below 4 km depth. The pore pressure equals thelithostatic pressure when 2.5% of the oil has been convertedto gas, which occurs at a depth of approximately 4.2 km. In thecase of dissolved gas, the pore pressure equals the lithostaticpressure without generation of free gas. It is caused by the in-crease in live-oil volume, since the critical depth is zc= 5.05 kmand Fc= 20%. If the hydrostatic pressure is used as referenceprofile to solve equations (33) and (35), we obtain zc= 4.77 kmand Fc= 7% and practically the same pore-pressure buildupshown in Figure 4. Because the results with and without dis-solved gas are practically the same, we consider in the followinganalysis the case with no dissolved gas. Figure 5 shows the vari-ations of the saturations with excess pressure (a) and of theporosities with depth (b) [(equations (46), (48), and (49), re-spectively]. The arrows indicate when the pore pressure equalsthe lithostatic pressure.

The low-frequency wave velocities versus excess pore pres-sure and differential pressure are shown in Figure 6. An excesspore pressure of 60 MPa corresponds to zero differential pres-sure. The oil/gas conversion starts at a differential pressure of21.5 MPa, which corresponds to an onset time of approximately

FIG. 4. Pore-pressure buildup with depth and deposition time(continuous line), when all the generated gas is out of solution(continuous line), and when part of the generated gas is dis-solved in the oil (broken line). The hydrostatic and lithostaticpressure are represented by thin lines.

25 million years, when the sandstone is fully saturated with oil.As can be appreciated, the velocities decrease substantially af-ter an excess pore pressure of approximately 35 MPa. This re-sults partly from the replacement of oil by gas, but mainly fromthe decrease in the dry-rock bulk moduli Km and µm caused bythe decrease in effective pressure.

Winkler (1985) obtained the high-frequency compressionaland shear velocities, 4140 m/s and 2500 m/s, at a differentialpressure of 27.4 MPa. These experimental velocities, corre-sponding to 400 kHz, are fitted with the theoretical veloci-ties (A-15) to obtain the relaxation times used in equations(A-10) and (A-11). The pores are saturated with dead oil ofdensity 890 kg/m3, which corresponds to the value reportedby Winkler (1985). Figure 7 shows the normalized compres-sional (continuous lines) and shear (broken lines) velocities(a) and attenuations (b) versus frequency. The velocities arenormalized with respect to the low-frequency values 4013 m/sand 2426 m/s, respectively. The square and circle correspond

FIG. 5. Saturations versus excess pore pressure p− pH (a) andporosities versus depth (b). The arrows indicate when the porepressure equals the lithostatic pressure.


to the experimental values obtained by Winkler (1985) for thecompressional and shear waves, and the triangle is the P-waveattenuation for a differential pressure of 40 MPa [attenuationα, in dB, is related to Q factor asα' 17.372π/(2Q)]. To fit thesevalues, we use one mechanism for the shear modulus µc, cen-tered at 4 kHz and with relaxation times τε = 4.09× 10−5 s andτσ = 3.86× 10−5 s (corresponding to a mimimum quality factorof approximately 34), and a continuous spectrum of mecha-nisms for Mc from 1 Hz to 1 MHz and Q ' 25. The need for acontinuous spectrum of relaxation mechanisms to explain theP-wave attenuation values is demonstrated in Winkler (1985),but because of lack of data for the shear wave, we assume thesimplest model, a single relaxation mechanism.

The Biot relaxation mechanisms, resulting from the relativemotion between the solid and the fluids, are important beyond1 MHz. The peaks at 4 KHz result mainly from the shear vis-coelastic mechanism.

FIG. 6. Computed low-frequency wave velocities versus excesspore pressure (a) and differential pressure (b). Frequency is25 Hz. The broken lines in (b) are the dry-rock wave velocitiesderived from equations (57) and (58).

CONCLUSIONS

The variations of P-wave and S-wave velocities have beendetermined as a function of excess pressure caused by oil/gasconversion. The results for a model in which a reservoir vol-ume is buried at a constant sedimentation rate for a geother-mal gradient, which is constant in time and depth, show thatthe velocities decrease significantly when only a small amount(about 2.5%) of the oil in the closed reservoir is converted togas. If the gas goes into solution, the results are practically thesame, because the volume increase of the live oil is close to thatof the mixture of dead oil/free gas. Moreover, the differentialpressure vanishes before the oil has absorbed all the possiblegenerated gas.

FIG. 7. Normalized compressional (continuous line) and shear(broken line) velocities (a) and attenuations (b) versus fre-quency, for a differential pressure of 27.4 MPa. The veloci-ties are normalized with respect to the low-frequency values4013 m/s and 2426 m/s, respectively. They correspond to theonset of the oil/gas conversion process. The square and circleat 400 kHz correspond to the experimental values obtained byWinkler (1985) for the compressional and shear waves, and thetriangle at 400 kHz is the P-wave attenuation for a differentialpressure of 40 MPa.


That small conversion of oil to gas is sufficient to make thepore pressure equal to the confining pressure. The large changein the velocity results mainly from the fact that the dry-rockmoduli are functions of the effective pressure, with the largestchanges occurring at low effective pressures. In fact, porositychanges and fluid moduli and density have a second-order ef-fect on the velocities. The effective pressure decreases becausethe pore pressure increases as a result of conversion of high-density oil to the low-density gas or low-density live oil, in thecase of dissolved gas. Perceptible changes in velocities occurwhen the differential pressure is 15 MPa, at which about 0.6%of oil has been converted to gas. The changes become signif-icant when the differential pressure decreases to 10 MPa, atwhich about 1% has been converted.

Attenuation also will be a function of effective pressure be-cause of the changing pore-space geometry. However, exper-imental data of attenuation for partially saturated rocks as afunction of the effective pressure are not available. Realisticattenuation values were obtained only for the case with fulloil saturation, for which experimental evidence of the velocitydispersion is available.

ACKNOWLEDGMENTS

This work was supported by Norsk Hydro a.s. (Bergen) withfunds of the “Source Rock” project, and by the EuropeanUnion, under the project “Detection of overpressure zoneswith seismic and well data.” Thanks go to Hans Helle for im-portant technical comments.

REFERENCES

Barker, C., 1990, Calculated volume and pressure changes during thethermal cracking of oil to gas in reservoirs: AAPG Bull., 74, 1254–1261.

Batzle, M., and Wang, Z., 1992, Seismic properties of pore fluids: Geo-physics, 57, 1396–1408.

Berryman, J. G., Thigpen, L., and Chin, R. C. Y., 1988, Bulk elastic wavepropagation in partially saturated porous solids: J. Acoust. Soc. Am.,84, no. 1, 360–373.

Berg, R. R., and Gangi, A. F., 1999, Primary migration by oil-generationmicrofracturing in low-permeability source rocks: Application to theAustin chalk, Texas: AAPG Bull., 83, no. 5, 727–756.

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APPENDIX A

SEISMIC PROPERTIES OF A POROUS MEDIUM SATURATED WITH HYDROCARBON AND WATER

Biot’s theory of dynamic poroelasticity is used to computethe wave velocities and attenuation factors, where the porefluid is a mixture of hydrocarbon and water. The complex ve-locities of the fast (+ sign) and slow (− sign) compressionalwaves and shear wave are [see Carcione (1998)],

V∗2P± =A±

√A2 − 4M Eρcρ∗

2ρcρ∗(A-1)

and

V∗S2 = µ

ρc, (A-2)

where

A = M(ρ − 2αρ f )+ ρ∗(E + α2 M), (A-3)

ρc = ρ − ρ2f

/ρ∗, (A-4)

and

ρ∗ = Tφρ f − i

2π f

η

κ, (A-5)

with α and E elastic coefficients, f the frequency, and i =√−1.The sediment density is

ρ = (1− φ)ρs + φρ f

where ρs and ρ f are the solid and fluid densities, respectively; Tis the tortuosity; η is the fluid viscosity; and κ is the permeabilityof the medium.


The elastic coefficients are given by

E = Km + 43µ, (A-6)

M = K 2s

D − Km, (A-7)

D = Ks[1+ φ(KsK−1

f − 1)], (A-8)

α = 1− Km

Ks, (A-9)

with K f the bulk modulus of the mixture hydrocarbon/water.The stiffness E is the P-wave modulus of the dry skeleton, Mis the elastic coupling modulus between the solid and the fluid,and α is the poroelastic coefficient of effective stress.

Wave velocities generally are expected to be lower at lowfrequencies, typical of seismic measurements, than at high fre-quencies, typical of laboratory experiments. Because the mag-nitude of this effect cannot be described entirely by Biot-type theories, additional relaxation mechanisms are requiredto model the velocity dispersion. Measurements of dry-rockvelocities contain all the information about pore shapes andpore interactions and their influence on wave propagation.Low-frequency wet-rock velocities can be calculated by usingGassmann’s equation, i.e., the low-frequency limit of the dis-persion relation. High-frequency wet-rock velocities are thengiven by the unrelaxed velocities. Because dry-rock velocitiesare practically frequency independent, the data can be ob-tained from laboratory measurements

Viscoelasticity is introduced into the poroelastic equationsfor modeling a variety of dissipation mechanisms related tothe skeleton/fluid interaction. One of these mechanism is thesquirt-flow (Dvorkin et al., 1994), by which a force applied tothe area of contact between two grains produces a displace-ment of the surrounding fluid in and out of this area. Becausethe fluid is viscous, the motion is not instantaneous and energydissipation occurs. Skeleton/fluid mechanisms are modeled bygeneralizing M to a frequency-dependent modulus of the form

Mc = M

[1+ 2

π Qln(

1+ iωτ2

1+ iωτ1

)]−1

, (A-10)

where τ1 and τ2 are time constants, with τ2 <τ1, and Q definesthe value of the quality factor, which remains nearly constant

over the selected frequency band. The low-frequency limit ofMc is the Biot modulus M .

On the other hand, µ is generalized to

µc = µ

L

L∑l=1

1+ iωτεl1+ iωτσ l

, (A-11)

where τεl and τσ l are the relaxation times of the L attenuationmechanisms.

The mixture hydrocarbon/water behaves as a composite fluidwith properties depending on the constants of the constituentsand their relative concentrations. This problem has been ana-lyzed by Berryman el al. (1988), and the results are given bythe formulas

K f = (Soco + Swcw)−1, (A-12)

ρ f = Soρo + Swρw, (A-13)

where ρo is the density of the hydrocarbon,

η f = Soηo + Swηw, (A-14)

where ηo and ηw are the viscosities of the hydrocarbon and wa-ter, respectively. Equation (A-14) is a good approximation formost values of the saturations.

The phase velocity VP(S) is equal to the angular frequencyω= 2π f divided by the real wavenumber. Then

VP± =[

Re

(1

V∗P±

)]−1

, VS =[

Re

(1

V∗S

)]−1

,

(A-15)where Re takes the real part.

The magnitude of the attenuation factors (in dB) is given by

αP± = 17.372πIm(V∗P±

)Re(V∗P±

) , αS = 17.372πIm(V∗S)Re(V∗S)

,

(A-16)

where Im takes the imaginary part.Biot’s theory is strictly valid for frequencies below

fc = η f φ

2πT ρ f κ. (A-17)


The usefulness of geophone ground-couplingexperiments to seismic data

Guy G. Drijkoningen∗

ABSTRACT

Ground-coupling devices measure local conditionsaround a geophone and are therefore useful for geo-phone design. For purposes of gathering seismic data,the data from local ground-coupling experiments mustbe related to seismic data. In particular, the geophysicistwants to know whether a local measurement could helpin detecting a ground-coupling problem or in correctingthe seismic data for it. Two sets of field experiments havebeen carried out to investigate these effects. Our experi-mental data of coupling-measurement devices show thatthe behavior of well-planted spiked geophones is deter-mined by shear along the spike, while the behavior ofpoorly planted geophones is determined by its weight.However, when a link is established between the datafrom a coupling-measurement device to seismic data,it becomes clear that the usefulness of this device forpredicting problematic ground-coupling phenomena inseismic data is very limited.

INTRODUCTION

Ground coupling of geophones is still not a well-understoodproblem. Increased interest in single-sensor technology,three-component technology, four-component ocean-bottomsensors, and high-resolution seismics requires a better under-standing of geophone-coupling problems. In this paper, ex-perimental aspects of geophone coupling are investigated fur-ther, continuing from reports in the literature. A paper byKrohn (1984) is the most important in this area. The goal ofthis study was to investigate whether and how local ground-coupling measurements could help in detecting and/or cor-recting seismic data for ground-coupling problems. To answerthese questions, some basic aspects of coupling required carefulreinvestigation.

Manuscript received by the Editor February 17, 1999; revised manuscript received November 11, 1999.∗Delft University of Technology, Department of Applied Earth Sciences, Mijnbouwstraat 120, 2628 RX Delft, Netherlands. E-mail: [email protected]© 2000 Society of Exploration Geophysicists. All rights reserved.

A clear distinction is made in this paper between so-calledspike-shear coupling and weight coupling. The former termrefers to the assumed way a spiked geophone is coupled tothe ground and is important for geophone design. In weightcoupling, it is purely the weight of the detecting device thatfurnishes the coupling between the geophone and the ground.This distinction between the two ways of coupling has becomeclear after the introduction of a spike in theoretical models byTan (1987) and by Rademakers et al. (1996). In this paper, thefocus will be mainly on the distinction between spike-shear andweight coupling of geophones. This is a common problem thatthe practicing geophysicist encounters when a geophone is not“well” coupled to the ground.

The paper by Krohn (1984) about experiments on groundcoupling clearly describes the different laboratory and fieldtechniques used to determine such coupling. Here, only fieldtechniques are described briefly and used, with a focus onhow to relate these measurements to seismic data. Measure-ments have been carried out in different soils, using two typesof coupling-measurement devices. In addition, seismic-shotrecords were taken at the same locations. These results havebeen analyzed and conclusions have been drawn about the use-fulness of such measurements to seismic data.

CONCEPTS OF GROUND COUPLING

The first issue in ground coupling is to define what geophoneground coupling means. In the past, some confusion has arisenbecause theoretical models did not explain experimental re-sults about spiked geophones (see, for example, Krohn, 1984).Thus, theoreticians seemed to have a different meaning forground coupling than that used by the practicing geophysicist.The definition of geophone ground coupling used in theoreticalmodels (see, for example, Tan, 1987) is as follows: Geophoneground coupling is the difference between the velocity mea-sured by the geophone and the velocity of the ground withoutthe geophone. This definition is useful for the design of geo-phones so that optimal characteristics can be found. However,

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Usefulness of Geophone Coupling Experiments 1781

once an optimal geophone design has been obtained, the prac-ticing geophysicist must cope with a geophone that is not de-ployed appropriately. For example, the geophone is not cou-pled “well” to its surroundings. In this situation, the abovedefinition is not appropriate and should be revised as follows:Bad geophone coupling is the difference between the velocityas measured by the badly planted geophone and the velocityas measured by the well-planted geophone.

The theoretical implications of “well-coupled” and “notwell-coupled” geophones become clearer when different the-oretical models are considered. Until the publication of Tan’snote (1987), ground coupling was modeled theoretically bya cylinder resting on a half-space (Lamer, 1970; Hoover andO’Brien, 1980). Although this model is in contradiction towhat has been used for decades in the seismic industry, namelyspiked geophones, it has been used to describe geophoneground coupling. Krohn (1984) already has mentioned thatthe spike should be taken into account in the modeling, butTan (1987) was the first to carry this out. However, Tan (1987)did not quantify the ground-coupling phenomenon, which wasundertaken by Rademakers et al. (1996). In this latter model,where the spike is modeled as a cylinder, the final expressioncontains two integrals resulting from the bottom and side of acylinder, respectively. The dominant contribution is from theside. More details can be found in Rademakers (1996). Phys-ically, we learn from this model that when the geophone is“well” coupled, the term with the shear force along the spike isdominant. However, when the force along the spike is removed,the model no longer holds and the models from Lamer (1970)and Hoover and O’Brien (1980) should be used. In particu-lar, it is the weight that is dominant, hence the name weightcoupling.

A difference in the response resulting from different modelsis caused by the ground coupling showing amplitudes largerthan one around the resonance frequency in previously re-ported models (Lamer, 1970; Hoover and O’Brien, 1980), butin the model by Rademakers et al. (1996), the ground cou-pling always has an amplitude less than one. It is hypothesizedthat the model of a cylinder resting on a half-space is usefulfor weight coupling when the spike of the geophone no longerprovides coupling and is a device that uses its mass to coupleto the ground.

In practice, the problem is associated strongly with the twotypes of coupling. Spike-shear coupling generally is well be-haved for spiked geophones, but weight coupling generally isnot well behaved. Practical examples of both these “states”will be given later in this paper. Weight coupling generallyis not well behaved because the contact area between thegeophone (or another sensor) and the ground is rough. Of-ten, the only way to improve the coupling is to increase themass of the geophone so that the contact with the ground be-comes better (stronger coupling, more regular contact area).In practice, geophones are kept lightweight because of weightlimits imposed on transporting thousands of geophones. Therequirement for light weight and a good coupling conditionfor weight coupling work against each other. We investigatedwhether it is possible to detect experimentally whether geo-phones are spike-shear or weight coupled or, in practicalterms, well or not well coupled. When a “badly coupled” (thatis, weight-coupled) geophone is detected, the ideal situationwould be to change this weight-coupled geophone to one that

is spike-shear coupled. This question also is addressed in thispaper.

FIELD TECHNIQUES FOR MEASURING GROUNDCOUPLING

Different techniques for measuring ground coupling weredescribed extensively by Krohn (1984). She described labora-tory (shake-table) and field tests and compared the two meth-ods. As she found out, which is also our experience, shake-table tests show some extra scattering effects resulting frominterference of the soil with the box and other box effects.Consequently, it was decided to carry out only field tests. Inour experiments, three methods were deployed, in which oneof the methods was used only to validate another of the threemethods, as described below.

The first is a standard method available in some seismo-graphs. In our case, this is a Bison Spectra, which works as fol-lows: A signal is generated in the seismograph and sent into thegeophone; the response from the geophone then is recorded.

The second method used a setup similar to that originally de-ployed by Washburn and Wiley (1941), namely, one geophonebolted on top of another geophone. In our setup, a cylinderwas constructed between the two geophones, with the mass ofthe cylinder equal to the mass of a geophone. The upper geo-phone is used as an actuator, and the response from the lowergeophone is measured. This method was used only to validatethe response of the third method.

The third method used was a slight modification of themethod described by Hoover and O’Brien (1980). A smallweight was dropped from a fixed height, and the seismographwas triggered simultaneously with the release of the weight.Special care was taken to ensure that the geophone was notexcited outside its linear range. To be sure of the coupling re-sponse, the second method (double-geophone configuration)was deployed under the same conditions in the field as theweight-drop method. Therefore, the responses from these twomethods should be identical.

EXPERIMENTS

The aims of the experiments were twofold:

1) The first set of tests (experiment 1) was carried out tomake an inventory of the information one obtains fromthe separate measurements, such as coupling conditions(such as a geophone on its side), or to see whether ageophone was well coupled to the ground. The separatemeasurements were performed multiple times to build upa statistical basis for computing the mean and variance.From different measurements, it became clear what couldand could not be detected.

2) The second set of tests (experiment 2) was carried out toinvestigate the usefulness of ground-coupling measure-ments in relation to data obtained from seismic shots.First, the aim was to see whether the same behavior couldbe observed in the measurements of the coupling devicesand in the shot data. If this was the case, then the aimwas to see whether the shot data could be corrected forthis behavior by using the responses from the coupling-measurement devices.

1782 Drijkoningen

EXPERIMENT 1: LOCAL GROUND-COUPLINGMEASUREMENTS

As mentioned above, these tests were undertaken to seewhat could and could not be measured by local ground-coupling measurements by using the techniques described ear-lier. Let us first concentrate on the signal generated by theseismograph and sent directly into the (receiving) geophone.An example of this is given in Figure 1. On the four left traces,the response is measured from well-planted geophones. Notethat the dominant effect in the response is the resonance fre-quency of the geophone, which is a 10-Hz geophone in thiscase. The next four traces in Figure 1 show the response ofbadly planted geophones. Bad planting resulted from pickingup a well-planted geophone and letting it drop in the samehole. Care was taken to create a hole which nearly vertical(within 5%). Comparing these traces to the responses of thewell-planted geophones, no differences could be observed. De-termining the average responses and errors from 20 measure-ments, we find that the average response of the badly plantedgeophone falls within the error bars of the well-planted geo-phones. Consequently, such a system cannot be used for de-tecting badly planted geophones.

This method can be used to determine whether a geophoneis planted at all. In the last four traces in Figure 1, the responsesare given from four geophones lying on their side. Note thatthe responses are significantly different from those of a planted

FIG. 1. Geophone response to a signal from a Bison seismo-graph. Traces 1–4, well planted; traces 5–8, badly planted; traces9–12, geophones lying on their side (not planted).

geophone. Such a measurement system thus can be used for au-tomatic detection by choosing some appropriate attribute(s).

When the responses of the planted geophones are comparedwith those from geophones on their side, a small peak could beseen in the time responses of the planted geophones at approx-imately 25 ms. This is caused by an artifact of the geophone,and separate analysis showed that the different planting of thegeophone did not affect this peak.

To detect and/or correct for well- or badly planted geo-phones, the other two methods described in the section “Fieldtechniques for measuring ground coupling” must be used. Wewill describe a series of tests performed on two soils, namelypure sand, wetted, and pure clay, with some grass on top. Theexperiments were repeated 20 times to build up statistical con-fidence. These tests were undertaken only with a small mass(22.8 g), dropped on the geophone.

In Figure 2, a subset of four responses out of 20 is given forone geophone being planted well into sandy soil (beach sand),and also for four responses of badly planted geophones. Somestriking differences can be observed: The energy content (sumof squared amplitudes) of the badly planted geophone is con-sistently higher than that of the well-planted geophones; forthe 20 responses, the ratio of the average energies is 2.0. In ad-dition, the responses of the badly planted geophones containmore low-frequency energy. The average for the (statistical)first moments of the amplitude spectra was 270 Hz for thewell-planted geophone and 163 Hz for the badly planted geo-phones. This analysis also has been undertaken on 20 responsesobtained from the geophones in the clay. Four of these are givenin Figure 3. The ratio of the average energies is 1.9. The aver-age of the (statistical) first moments of the amplitude spectra

FIG. 2. Geophone response to a signal from a small weightdrop for geophones in sand. Traces 1–4, well planted; traces5–8, badly planted.


is now 243 Hz for the well-planted geophones and 151 Hz forthose that were planted badly. These measurements are consis-tent with the measurements of, for example, Krohn (1984) andFaber et al. (1994). However, from our experimental results, wecan claim something that these authors did not; that is, only two“states” can be discerned. To support this with some statisticalconfidence, the standard deviation has been determined. Therelative standard deviation (one standard deviation divided bythe average) from the well-coupled geophones in the sand is8% and for clay is 4%. For the badly planted phones, the de-viation is 16% for sand and 6% for clay. The higher values ofthe standard deviation are well known from experiments byothers. Geophones coupled by their weight need to be madeheavier because the contact area on many types of surfacesis a serious problem. Another solution is to bury the phones(Krohn, 1984). Because the amplitude spectra of the well- andbadly planted geophones have their own average and standarddeviation, we can state with statistical confidence that thereare only two distinct “states.” The literature seems to implythat the coupling could vary continuously, but this was not thecase in our measurements.

Combining these observations with the concepts obtainedfrom modeling the spike as a cylinder in the ground(Rademakers et al., 1996), we conclude that well-planted geo-phones are of the spike-shear coupled type, and badly plantedgeophones are the weight-coupled type. It should be noted herethat this observation could be used for making devices thatcan determine the coupling locally without using a coupling-measurement device. Specifically, a force should be applied thatwould be slightly larger than that necessary to compensate forthe weight of the geophone. Then, if the geophone is not lifted,

FIG.3. Geophone response to a signal from a small weight dropfor geophones in clay. Traces 1–4, well planted; traces 5–8, badlyplanted.

it is spike-shear coupled and thus well coupled. However, if thegeophone is lifted, it is weight coupled and thus badly coupled.

Another comparison was performed for the validation of anew model by Rademakers et al. (1996). Specifically, this wasthe impedance contrast between the soil and the geophone.Tan (1987) showed theoretically that the density of the spikeshould match the density of the soil to obtain the best coupling.However, in the model by Rademakers et al. (1996), the effectof spike density was not dominant. Consequently, two spikeswith different densities (stainless steel and aluminum) were de-ployed in clay and sand. For each situation, the geophone wasplanted well coupled 20 times so that a statistical base couldbe gathered. Four from each of these 20 are shown in Figure 4.The first eight are the geophones in clay, where the first four arewith the steel spike and the second four are with the aluminumspike. Although the weights are different, the responses arenot different statistically. The same conclusion can be drawnfrom the responses in the sand. Statistically, there is no differ-ence. However, when we compare the responses from the claywith those from the sand, we do see a significant difference.The (normalized) energies, with their standard deviation inparentheses, are: 4.45 (±0.47), 4.30 (±0.30), 3.26 (±0.25), and3.51 (±0.19). This effect also can be seen often in vertical seis-mic profiles (VSPs) where the geophone passes through manyrocks with different characteristics. In conclusion, the effects ofsoil variation can be determined with local devices, but the ef-fect is determined by the characteristics of the soil itself ratherthan the difference in characteristic between the soil and thespike. This confirms the model by Rademakers et al. (1996).

EXPERIMENT 2: LOCAL GROUND-COUPLINGMEASUREMENTS IN RELATION TO SEISMIC-SHOT DATA

In this set of experiments, the usefulness of local techniquesis investigated with respect to their use for seismic-shot data.As observed in the first set of experiments, there is a significant

FIG. 4. Geophone response to a signal from a small weightdrop for geophones in clay and sand, with steel and aluminumspikes. Traces 1–4, steel in clay; traces 5–8, aluminum in clay;traces 9–12, steel in sand; traces 13–16, aluminum in sand.

1784 Drijkoningen

difference between well- and badly planted geophones in re-sponses from local techniques (see Figures 2 and 3). In theseexperiments, the local techniques are combined with seismic-shot data to see whether the differences, as seen in the localmeasurements, also can be seen in the global (seismic-shot)measurements. To that end, many experiments have been car-ried out on types of soils, with four typical soils having beenchosen: peaty soil, sandy soil, gravel, and hard rock. The setdiscussed here is only a subset of a larger data set, and theexamples shown are typical.

For each soil type, a set of experiments consisted of a shotfrom a small charge located 50 m from a group of singlegeophones. The geophones were perpendicular to the shot-geophone direction so that phase differences would be mini-mal. Two of the geophones had the weight-drop device on top,with one well planted and one badly planted. Other geophoneshave been used to build up some confidence in the results ofthe observations. When the data from one shot is analyzed, thevariations seen in the seismic data should then be caused by:

1) The difference in local soil conditions (geology). Thisvariation is hard to quantify, but a qualitative conclu-sion can be obtained by looking at the variation of theresponses of the well-planted geophones from the sameseismic shot.

2) The difference in geophone-coupling conditions. Thisvariation could be determined partly from a comparisonwith the other well-planted geophones. It should be real-ized that coupling could be a 3-D effect. For bad planting,the data consisted only of the geophone with a weight-drop device on top, so that no variation could be deter-mined in this case.

FIG. 5. Response to a seismic shot and small weight drop for geophones in peaty soil. Left panel: time tracesfrom seismic shot, well planted (trace 1) and badly planted (trace 2). Upper right panel: amplitude spectra oftraces shown on the left: relative standard deviation (solid thick line), well planted (solid thin line), badly planted(dotted line). Lower right panel: amplitude spectra of responses to weight drop: well planted (solid line), badlyplanted (dotted line).

3) Differences in geophones. This variation was tested in thelaboratory and did not make any significant contributionto the total variation.

Let us look at the first set of data obtained from peaty soil.This soil was located next to a marsh area near Hannover,Germany, where measurements were taken in a dried-out por-tion of the marsh. The different responses are given in Figure 5.On the left side of the figure, two seismic traces are shown fromthe seismic-shot data, where the left one is the well-plantedgeophone and the one on the right is the badly planted geo-phone. One can see the difference in the response resultingfrom the bad planting: The first arrival is reverberatory. On theabove right of the figure, the amplitude spectra of these twotraces are given. A difference can be seen between the data atabout 100 Hz. Also shown in this panel is the relative standarddeviation in the amplitude, determined from 14 well-plantedgeophones. On the bottom right of the figure, the amplitudesof the weight-drop responses from these two particular geo-phones are given. These responses should give the differenceseen in the shot data. In these weight-drop responses, a clearmaximum (resonance) can be seen at about 75 Hz. In addition,it can be seen that the amplitude of the weight-drop responsefor the badly planted geophone has amplitude that is higherthan the amplitude for the well-planted geophone. However,there are also some differences to be seen between the spectraof the weight drop and the seismic shot. First, the differencebetween the two weight-drop responses is larger than the dif-ference between the responses to the seismic shot. Moreover,the weight-drop response for the well-planted geophone stillhas a maximum at the same frequency as the response of thebadly planted geophone. This may suggest that the resonance


is purely a soil-related resonance rather than a resonance re-lated to the coupling condition. In general, it is obvious fromFigure 5 that if we could deconvolve the seismic data of thebad planting by using the weight-drop data, the result wouldbe incorrect.

The next set of measurements was carried out on sandy soil,specifically, sand from the beach of Wassenaar, Netherlands,which was partly wet and fairly compact sand. Again, the dif-ferent data are plotted next to each other and shown in Fig-ure 6. The left trace is again from the well-planted geophone,and the right is from the badly planted geophone. In this case,there are some minor differences to be seen in the time traces.This is also the case for the amplitude spectra. Examining theresponses of these geophones to the weight drop, it can beobserved that there is no clear maximum (resonance); the re-sponses are flat. If the maximum is viewed as a resonance, thebadly planted geophone has a slight resonance at about 170 Hz.In the seismic-shot data, a difference also can be seen aroundthese frequencies, although it is not very significant. Again, itis clear that in this case, if the weight-drop data are used forcorrecting the data for bad coupling, the results will not becorrect.

The third set of measurements was carried out in a graveldeposit near Hannover, Germany. In this area, it was noteasy to plant the geophones well. The data from this area areshown in Figure 7. The time traces show only minor differ-ences. However, the amplitude spectra show some clearer dif-ferences. What is striking in this example is that the amplitudeof the well-planted geophone is higher than the amplitude ofthe badly planted geophone. This is probably caused by thegravel, because it was difficult to establish spike-shear cou-pling. The responses from the weight drop show a resonance at

FIG. 6. Response to seismic shot and small weight drop for geophones in sandy soil. Left panel: time traces fromseismic shot, well planted (trace 1) and badly planted (trace 2). Upper right panel: amplitude spectra of tracesshown on left: relative standard deviation (solid thick line), well planted (solid thin line), badly planted (dottedline). Lower right panel: amplitude spectra of responses to weight drop: well planted (solid line), badly planted(dotted line).

about 130 Hz and at frequencies where the seismic data showthe most marked differences. The weight drop indicates thatmaximum/resonance occurs at approximately the correct rangeof frequencies. In addition, the responses of the well-plantedgeophone have higher amplitudes than geophones that wereplanted badly. Again, if we use the weight-drop data to correctthe seismic data for its plant, the result will be incorrect.

The last set of measurements took place in a limestone minenear Hannover. The rock outcropped at the surface, and mea-surements were made by planting the geophones in the cracksof the limestone. The results are shown in Figure 8. In theseismic-shot data, a slight difference can be seen between theresponses of the well-coupled and badly coupled geophones.In the amplitude spectra of the seismic data, the most markeddifference can be seen at about 200 Hz. The amplitude spec-tra of the weight drop show a maximum (resonance) at about200 Hz for the well-planted geophone and about 120 Hz forthe badly planted geophone. Therefore, the weight-drop de-vice predicts a decrease in the resonance frequency for badplanting, a higher amplitude at the maximum (resonance), butit does not quantitatively predict the difference in behavior ofthe seismic data for good and bad planting. A possible explana-tion here could be that the geophones are “rocking” sidewaysbecause the geophone was planted with only the tip of thespike into the rock. Because the weight-drop device excitesthe geophone only vertically, it does not measure this sidewaysmovement.

DISCUSSION

One can query why there is no quantitative relation betweenthe responses of the coupling-measurement device and the shotdata. The reasons could be:

1786 Drijkoningen

1) Coupling-measurement devices measure coupling con-ditions and local soil characteristics. It is not possibleto discriminate between the two. The term coupling-measurement device may be confusing.

2) three-dimensional movement. The coupling-measure-ment device excites the geophone only in one direction,but the wave motion from the seismic shot is 3-D. Clearly,this is a restriction of the coupling-measurement device.A typical problem of nonvertical movement is the rock-ing (sideways) movement of the geophone. This occurswhen spiked geophones are planted in hard rock. For un-consolidated soils, Krohn (1984) partly solved the rockingproblem by recommending that the geophones be buried.However, 3-D movement will remain. It may be worth-while to design geophones with lower cross-sensitivity atthe cost of performance in the vertical direction.

3) Although theoretically there is a relation between re-flection and transmission, it is often difficult to obtaintransmission data from reflection data. Washburn andWiley (1941) gave a formula that would accomplish thisfor the weight-drop device, but it turned out not to bevery useful when applied to seismic-shot data. It may bebetter to measure the transmission of the local geophone-coupling conditions rather than to determine it from re-flection measurements.

CONCLUSIONS

Local ground-coupling-measurement devices, such as a smallweight-drop device, measure directly the most important char-acteristics of ground coupling. In particular, they show the

FIG. 7. Response to seismic shot and small weight drop for geophones in gravel. Left panel: time traces fromseismic shot, well planted (trace 1) and badly planted (trace 2). Upper right panel has amplitude spectra of tracesshown on the left: relative standard deviation (solid thick line), well planted (solid thin line), and badly planted(dotted line). Lower right panel has amplitude spectra of responses to weight drop: well planted (solid line),badly planted (dotted line).

amount of energy transferred, the maxima (resonances), andthe shifts in maxima (resonances). From the measurements dis-cussed in this paper, it has become clear what can and cannot bedetected. In addition, it was shown by using the weight-drop de-vice that well-coupled geophones are the so-called spike-shear-coupled geophones, while badly coupled spiked geophones arethe weight-coupled type. One cannot speak of a continuousvariation of coupling conditions, but mainly, two “states” canbe discerned: spike-shear or weight coupled.

However, when relating the coupling responses to seismicdata, it has become clear that the responses of the coupling-measurement devices can be used only qualitatively. When itwas used quantitatively, the following problems arose:

1) The maxima (resonances) in the amplitude spectra of thecoupling-measurement device often do not coincide withthe maximum difference seen in the seismic data.

2) A maximum in the response of the coupling-measure-ment device does not mean it is a resonance of the cou-pling condition. It also may be a feature that is purely acharacteristic of the topsoil.

3) The differences observed in the responses of the coupling-measurement device do not explain quantitatively thedifferences seen in the seismic data.

In general, one can say that coupling-measurement devicesare very suitable for small “field-laboratory” tests, as under-taken in experiment 1 of this paper. We can learn from suchexperiments, for example, that there are two distinct states forgood and bad coupling. However, quantitative use of such de-vices for seismic data, such as correcting for bad coupling con-ditions, is very limited.


FIG. 8. Response to seismic shot and small weight drop for geophones in limestone. Left panel: time traces fromseismic shot, well planted (trace 1) and badly planted (trace 2). Upper right panel has amplitude spectra of tracesshown on the left: relative standard deviation (solid thick line), well planted (solid thin line), and badly planted(dotted line). Lower right panel has amplitude spectra of responses to weight drop: well planted (solid line),badly planted (dotted line).

ACKNOWLEDGMENTS

The author would like to thank Schlumberger for its co-operation in this work, specifically Dr. Andreas Laake andDr. Wilfried Junge. In addition, my colleagues at Delft Univer-sity of Technology are acknowledged, specifically Leo de Grootand Bart Cremers. This work was supported partly by the ECThermie Programme (project no. OG150/94DE/UK/NL).

REFERENCES

Faber, K., Maxwell, P. W., and Edelmann, H. A. K., 1994, Recordingreliability in seismic exploration as influenced by geophone-groundcoupling: 56th Ann. Mtg., Eur. Assn. Expl. Geophys, ExpandedAbstracts B014.

Hoover, G. M., and O’Brien, J. T., 1980, The influence of the plantedgeophone on seismic land data: Geophysics, 45, 1239–1253.

Krohn, C. E., 1984, Geophone ground coupling: Geophysics, 49, 722–731.

Lamer, A., 1970, Couplage sol-geophone: Geophys. Prosp., 18, 300–319.

Rademakers, F., 1996, Geophone ground coupling: New elastic ap-proach: M.Sc. thesis: Catholic Univ. Leuven and Delft Univ.Tech.

Rademakers, F., Drijkoningen, G. G., and Fokkema, J. T., 1996, Geo-phone ground coupling: New elastic approach: 66th Ann. Internat.Mtg., Soc. Expl. Geophys., Expanded Abstracts, paper 4.6.

Tan, T. H., 1987, Reciprocity theorem applied to the geophone-groundcoupling problem: Geophysics, 52, 1715–1717.

Washburn, H., and Wiley, H., 1941, Effect of the placement of aseismometer on its response characteristic: Geophysics, 6, 116–131.


Estimation of fracture parameters from reflection seismic data—Part I:HTI model due to a single fracture set

Andrey Bakulin∗, Vladimir Grechka‡, and Ilya Tsvankin‡

ABSTRACT

The simplest effective model of a formation contain-ing a single fracture system is transversely isotropic witha horizontal symmetry axis (HTI). Reflection seismicsignatures in HTI media, such as NMO velocity andamplitude variation with offset (AVO) gradient, can beconveniently described by the Thomsen-type anisotropicparameters ε(V), δ(V), and γ (V). Here, we use the lin-ear slip theory of Schoenberg and co-workers and themodels developed by Hudson and Thomsen for penny-shaped cracks to relate the anisotropic parameters to thephysical properties of the fracture network and to devisefracture characterization procedures based on surfaceseismic measurements.

Concise expressions for ε(V), δ(V), and γ (V), linearizedin the crack density, show a substantial difference be-tween the values of the anisotropic parameters forisolated fluid-filled and dry (gas-filled) penny-shapedcracks. While the dry-crack model is close to ellipti-cal with ε(V)≈ δ(V), for thin fluid-filled cracks ε(V)≈ 0and the absolute value of δ(V) for typical VS/VP ratiosin the background is close to the crack density. Theparameters ε(V) and δ(V) for models with partial satu-ration or hydraulically connected cracks and pores al-ways lie between the values for dry and isolated fluid-filled cracks. We also demonstrate that all possible pairsof ε(V) and δ(V) occupy a relatively narrow triangu-lar area in the [ε(V), δ(V)]-plane, which can be used to

identify the fracture-induced HTI model from seismicdata.

The parameter δ(V), along with the fracture orienta-tion, can be obtained from the P-wave NMO ellipse fora horizontal reflector. Given δ(V), the NMO velocity of adipping event or nonhyperbolic moveout can be invertedfor ε(V). The remaining anisotropic coefficient, γ (V), canbe determined from the constraint on the parametersof vertically fractured HTI media if an estimate of theVS/VP ratio is available. Alternatively, it is possible tofind γ (V) by combining the NMO ellipse for horizontalevents with the azimuthal variation of the P-wave AVOgradient. Also, we present a concise approximation forthe AVO gradient of converted (PS) modes and showthat all three relevant anisotropic coefficients of HTImedia can be determined by the joint inversion of theAVO gradients or NMO velocities of P- and PS-waves.

For purposes of evaluating the properties of the frac-tures, it is convenient to recalculate the anisotropiccoefficients into the normal (1N) and tangential (1T )weaknesses of the fracture system. If the HTI modelresults from penny-shaped cracks, 1T gives a directestimate of the crack density and the ratio 1N/1T isa sensitive indicator of fluid saturation. However, whilethere is a substantial difference between the values of1N/1T for isolated fluid-filled cracks and dry cracks, in-terpretation of intermediate values of1N/1T for porousrocks requires accounting for the hydraulic interactionbetween cracks and pores.

INTRODUCTIONSeismic detection of subsurface fractures has important ap-

plications in characterization of naturally fractured reservoirs.It is well known that preferential orientation of fracture net-works makes the medium azimuthally anisotropic with re-

Manuscript received by the Editor April 16, 1999; revised manuscript received February 28, 2000.∗Formerly St. Petersburg State University, Department of Geophysics, St. Petersburg, Russia. Presently Schlumberger Cambridge Research, HighCross, Madingley Road, Cambridge CB 3 0EL, England. E-mail: [email protected].‡Colorado School of Mines, Center for Wave Phenomena, Department of Geophysics, Golden, Colorado 80401-1887. E-mail: [email protected]; [email protected]© 2000 Society of Exploration Geophysicists. All rights reserved.

spect to seismic wave propagation. Although the presence ofazimuthal anisotropy has a strong influence on all propaga-tion modes (i.e., P-, S-, and converted waves), most exist-ing studies concentrate on analysis of time delays or reflec-tion amplitudes of split shear waves at near-vertical incidence.

1788

Fracture Characterization for HTI Media 1789

While such measurements make it possible to map the orien-tation and intensity (or crack density) of a vertical fractureset, they are not sensitive to the fracture infill (content), un-less the fractures are corrugated (see part III of this series).Recently, it was demonstrated that the azimuthal dependenceof P-wave signatures has the potential of not only constrain-ing the fracture orientation (e.g., Corrigan et al., 1996) anddensity (Tsvankin, 1997) but also of discriminating betweendry (gas-filled) and fluid-filled fractures (Sayers and Rickett,1997; Ruger and Tsvankin, 1997). Further progress in seismicmethods of fracture detection, however, requires a better un-derstanding of the relationship between the anisotropic param-eters measured from seismic data and the physical propertiesof fracture systems.

This is the first of three papers in which we discuss anisotropicseismic signatures associated with vertical fracture sets. Thetransversely isotropic medium with a horizontal axis of sym-metry (HTI), addressed here, is the simplest anisotropic modelcaused by parallel rotationally invariant vertical fractures inisotropic host (background) rock. In the two sequel papers westudy more complicated orthorhombic and monoclinic mod-els needed to describe multiple fracture sets and/or formationswith an anisotropic background. Among the main questionswe will attempt to answer are these:

1) Do different theories of seismic wave propagation infractured media (e.g., those by Hudson, 1980, 1981;Schoenberg, 1980, 1983; Thomsen, 1995; Hudson et al.,1996) lead to the same effective anisotropic model?

2) What is the relationship between the anisotropic parame-ters that control seismic signatures and the physical prop-erties of the fractures?

3) What types of seismic data are needed to obtain informa-tion about various properties of fracture systems?

4) What characteristics of fractured formations (e.g., crackdensity, fracture azimuth, equant porosity, type of infill)can be evaluated using surface reflection data?

While our ultimate goal is to invert seismic data for thephysical characteristics of fracture networks, the data are con-trolled by the elastic parameters (e.g., by the stiffness tensor)of the effective anisotropic medium. The effective parametersdepend not only on the orientation, compliances, etc., of thefractures but also on the elastic properties of the (possiblyanisotropic) host rock. Taken together, the characteristics ofthe fractures and host rock form a set of so-called microstruc-tural parameters. Even though seismic data may provide suffi-cient information to recover the effective anisotropic model, itmay be impossible to obtain the microstructural parameters in-dividually. Indeed, with increasing complexity of the fracturedmodel (e.g., in the presence of multiple dipping fracture sets),the number of microstructural parameters can become arbi-trarily large, whereas the maximum number of the effectiveanisotropic parameters is just 21 for the most general mediaof triclinic symmetry. Still, we will show that, for a range offracture models, seismic data can be inverted for some par-ticular microstructural parameters responsible for practicallyimportant properties of the fractured medium.

To identify the fracture parameters constrained by seismicdata, it is useful to compare the results of different approachesto effective medium theory. Effective models of fractured me-dia discussed below include those based on parallel infinite

fractures with linear slip boundary conditions (Schoenberg,1980, 1983), isolated parallel penny-shaped cracks that have theform of oblate spheroids (Hudson, 1980, 1981), and partiallysaturated penny-shaped cracks or hydraulically connectedcracks and pores (Hudson, 1988; Thomsen, 1995; Hudson et al.,1996). An important result in establishing the relationships be-tween different theories is obtained by Schoenberg and Douma(1988), who prove that infinite parallel fractures with linear slip(Schoenberg, 1980, 1983) and penny-shaped cracks (Hudson,1980, 1981) yield the same structure of the effective stiffnesstensor (i.e., it is impossible to distinguish between the two mod-els just by inspecting the stiffnesses). This is a clear indicationthat seismic data are not sensitive to one of the microstruc-tural parameters: the shape of the fractures. Further general-izing this result, we demonstrate that the presence of equantporosity with hydraulic connection to the cracks and/or partialsaturation does not change the structure of the stiffness tensor.The values of the anisotropic coefficients in this case turn outto be intermediate between those for isolated fluid-filled anddry cracks.

Reflection seismic signatures in anisotropic media aremost concisely described by Thomsen-type dimensionlessanisotropic coefficients, such as the parameters ε, δ, and γ orig-inally introduced by Thomsen (1986) for transverse isotropywith a vertical symmetry axis (VTI media). These coefficientscapture the influence of anisotropy on various seismic signa-tures and therefore can be determined from seismic data. Mostexisting work on effective parameters of fractured media (e.g.,Hudson, 1981; Schoenberg, 1983), however, is formulated interms of the stiffnesses ci j or compliances si j . Better under-standing of the influence of cracks on reflection seismic signa-tures requires expressing these results through Thomsen-typeanisotropic coefficients. For the VTI medium due to a singlesystem of horizontal fractures, Schoenberg and Douma (1988)obtain the weak-anisotropy approximations for Thomsen’s(1986) anisotropic parameters and note that only two out of thethree coefficients (ε, δ, γ ) are independent. Thomsen (1995)presents more general expressions for ε, δ, and γ of a simi-lar VTI model that contains horizontal fractures hydraulicallyconnected to pore space.

For HTI media resulting from vertical, rotationally invariantfractures, reflection traveltimes and amplitudes are convenientto express through the Thomsen-type parameters ε(V), δ(V), andγ (V) described by Ruger (1997) and Tsvankin (1997). [Notethat ε(V), δ(V), and γ (V) are different from the generic Thomsenparameters defined with respect to the horizontal symmetryaxis.] Approximate expressions for ε(V) and δ(V) are derived bySayers and Rickett (1997), who use them to study the azimuthalvariation of the P-wave amplitude variation with offset (AVO)response. Here, we give a systematic analysis of the dependenceof the anisotropic coefficients of HTI media on the physicalproperties of vertical fractures.

After a brief discussion of geological data on fractures, wereview the linear slip theory following the work of Schoenberg(1983), Schoenberg and Douma (1988), and Schoenberg andSayers (1995). This theory, based on physically intuitive rela-tions between stress and discontinuity in displacement, is for-mulated in terms of the fracture compliances or weaknessesand requires no assumptions about the microstructure andmicrogeometry of fractures. Then we obtain the anisotropiccoefficients of the effective HTI solid as functions of the

1790 Bakulin et al.

fracture weaknesses and determine the bounds of ε(V) and δ(V)

for arbitrary strength of fracturing. For models with penny-shaped (spheroidal) cracks, the anisotropic coefficients andweaknesses are expressed through the microstructural parame-ters using the theories of Hudson (Hudson , 1981, 1988; Hudsonet al., 1996) and Thomsen (1995). Extending the results ofTsvankin (1997) and Contreras et al. (1999), we show that acomplete fracture characterization procedure (including eval-uation of the fluid content) can be based on (1) conventionalP-wave data recorded in wide-azimuth 3-D surveys or (2) thecombination of P-waves and converted PS-waves.

EFFECTIVE MODELS OF FRACTURED MEDIA

Geological background

A concise overview of the properties of natural fracture sys-tems can be found, for example, in Aguilera (1998). Subsurfacefractures usually occur in large populations or sets with simi-lar orientations. Fracture openings (apertures) may vary fromvery thin (0.001–0.01 mm) to relatively wide (0.1–0.5 mm), butit is believed that in-situ values do not deviate substantiallyfrom an average of 0.02 to 0.03 mm (Romm, 1985). Aguilera(1998), however, mentions that in some reservoirs where frac-tures are propped by partial mineralization, the apertures mayreach 1 inch or more.

An important parameter of a fracture set is the distance be-tween the adjacent fractures, which is much greater than thefracture opening. Conventionally, geologists describe fracturedrocks by the so-called fracture intensity,

fracture intensity = number of fracturesmeter

,

with the number of fractures counted in the direction per-pendicular to the fracture planes. Fracture intensity for verysparse sets is less than 0.75 m−1, while for tight sets it mayexceed 10 m−1 (Bagrintseva, 1982). Typical values of fractureintensity for carbonate reservoirs are estimated as 1–20 m−1

(Kirkinskaya and Smekhov, 1981).Since seismic wavelengths are on the order of tens and hun-

dreds of meters, it is clear that

seismic wavelengthÀ fracture spacingÀ fracture opening.

Therefore, in building effective seismological models we canneglect finite fracture openings and details of the spatial dis-tributions of fractures and can consider fractured blocks asequivalent or effective anisotropic solids (i.e., use the long-wavelength approximation). The parameters of such an effec-tive model will depend on the orientation and intensity of thefracture set(s) and the properties of the material filling thefractures, as well as on the elastic coefficients of the host rock.

Parallel fractures: The linear slip model

To obtain the effective parameters of fractured media,Schoenberg (1980, 1983) suggests treating the fractures, re-gardless of their shape and microstructure, as either infinitelythin and highly compliant (soft) layers or planes of weaknesswith linear-slip (nonwelded, see below) boundary conditions.While these two representations are equivalent in the long-wavelength limit, each is useful in developing effective models

of fractured rock and gaining insight into the physical meaningof the fracture parameters.

The most straightforward way to derive the parameters ofthe effective medium is to apply the approach based on the thin-layer model. The exact Backus (1962) averaging procedure forparallel thin, soft layers embedded in an isotropic matrix leadsto the following simple form of the effective compliance matrixs (the inverse of the stiffness matrix c):

s = sb + s f , (1)

where sb is the compliance matrix of the host rock and s f is theexcess compliance associated with the layers (Schoenberg andMuir, 1989; Molotkov and Bakulin, 1997). Although it wouldbe more natural to denote stiffness by s and compliance byc, the notation used here is widely cited in the literature onelasticity and wave propagation.

It may be proved by means of the reflectivity (matrix)method that for vertical layers orthogonal to the x1-axis, s f

is given by

s f = ν

s11 f 0 0 0 s15 f s16 f

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

s15 f 0 0 0 s55 f s56 f

s16 f 0 0 0 s56 f s66 f

. (2)

Here, ν is the fraction of the total volume occupied by the thinlayers and si j f are the compliances of the layers’ material. Asfollows from equations (1) and (2), only six compliance ele-ments contribute to the effective parameters. Although ν isassumed to be small, all compliances si j f are large (i.e., thematerial is soft) and the products νsi j f are finite. The generalcondition of the medium’s stability also requires that the 3× 3submatrix composed of the nonzero elements of the compli-ance matrix (2) be nonnegative definite.

Equation (2) can also be used to describe a set of parallelfractures of infinite extent with the fracture normal n parallel tothe x1-axis. Following Schoenberg (1980), we treat fractures asplanes of weakness with nonwelded boundary conditions. Forthe purpose of deriving the effective parameters, the mediumcontaining parallel planes of weakness is equivalent to the thin-layer model discussed above. Therefore, the matrix of the ex-cess fracture compliances can be written as

s f =

KN 0 0 0 KN V KN H

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

KN V 0 0 0 KV KV H

KN H 0 0 0 KV H KH

. (3)

The jumps in the stress tensor (σi j ) and displacement vector(u) across a plane of weakness satisfy the boundary condi-tions of linear slip (Schoenberg, 1980; Molotkov and Bakulin,1997):


[σ11] = [σ12] = [σ13] = 0,

[u1] = h(KNσ11 + KN Hσ12 + KN Vσ13),

[u2] = h(KN Hσ11 + KHσ12 + KV Hσ13),

[u3] = h(KN Vσ11 + KV Hσ12 + KVσ13), (4)

where h is the average distance (spacing) between the fracturesand the brackets denote the jump of the corresponding valueacross the interface (plane of weakness).

Equations (4) help explain the physical meaning of the com-pliances KN , KV , KH , KN V , KN H , and KV H . For instance, KN isthe normal fracture compliance relating the jump of the nor-mal (to the fractures) displacement u1 to the normal stress σ11.Likewise, KV and KH are the two shear compliances alongthe vertical (x3) and horizontal (x2) directions. The complianceKN V is the coupling factor between the jump of the normaldisplacement u1 and the shear stress σ13 or, equivalently, be-tween u3 and σ11. Such a coupling may be caused by a slightroughness (corrugation) of the fracture surfaces, with peaksand troughs somewhat offset from one side of the fracture tothe other (Schoenberg and Douma, 1988).

From the physical point of view, a nonzero KN V implies thatthe normal and shear slips with respect to the fracture plane(u1 and u3) are coupled. Since a nonzero element KN V in ma-trix (3) yields a nonzero compliance s15 f in equation (2), afracture set with KN V 6= 0, KN H = KV H = 0 produces a mediumof monoclinic symmetry, and is sometimes called monoclinic(Schoenberg and Douma, 1988). The same type of coupling isassociated with nonzero values of KN H and KV H . However, asshown by Berg et al. (1991), KV H (unlike KN V and KN H) canalways be eliminated by a proper rotation of the coordinateframe around the direction normal to the fractures.

c = s−1 = cb −

(λ+ 2µ)1N λ1N λ1N 0 0 0

λ1Nλ2

λ+ 2µ1N

λ2

λ+ 2µ1N 0 0 0

λ1Nλ2

λ+ 2µ1N

λ2

λ+ 2µ1N 0 0 0

0 0 0 0 0 0

0 0 0 0 µ1T 0

0 0 0 0 0 µ1T

, (9)

Using largely physical (rather than rigorous mathematical)arguments, the above results are generalized by Nichols et al.(1989) for multiple sets of fractures of infinite extent and bySchoenberg and Sayers (1995) for thin fractures of arbitraryshape and finite dimensions. In particular, the linear additionof the fracture compliances to the compliance of the host rock[equation (1)] is applied by these authors to noninteractingmultiple fracture sets.

Hereafter, we consider the simplest form of the matrix of theexcess fracture compliance by assuming no coupling betweenthe slips along the coordinate directions and a purely isotropicmicrostructure of the fracture surfaces. For fracture sets withthese properties [called rotationally invariant by Schoenbergand Sayers (1995)], the elements of the compliance matrix sat-

isfy the following relationships (Hsu and Schoenberg, 1993):

KN V = KN H = KV H = 0, KV = KH (5)

As a result, the matrix of fracture compliances (3) reduces to

s f =

KN 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 KT 0

0 0 0 0 0 KT

, (6)

where KV = KH is denoted by KT . The values KN and KT arenonnegative and have the physical meaning of the normal andtangential compliances added by the fractures to the host rock.For a purely isotropic background model characterized by theLame constants λ andµ, Hsu and Schoenberg (1993) introducedimensionless quantities

1N = (λ+ 2µ)KN

1+ (λ+ 2µ)KN, (7)

1T = µKT

1+ µKT. (8)

From this definition it is clear that both 1N and 1T , whichwe call the normal and tangential weaknesses (Bakulin andMolotkov, 1998), vary from 0 to 1.

Using equations (1) and (6) and inverting the compliancematrix yields the stiffness matrix of the effective fracturedmedium (Schoenberg and Sayers, 1995):

where cb= s−1b is the stiffness matrix of the (isotropic) host rock.

If both weaknesses (1N and1T ) are equal to zero, the mediumcontains no fractures, while the weaknesses approaching unitycorrespond to the extreme degree of fracturing. Equation (9)shows that1N = 1 (c11= 0) implies the vanishing P-wave veloc-ity in the direction normal to the fractures; likewise, for1T = 1the S-wave velocity across the fractures goes to zero.

From equation (9) it can be inferred that the stiffnesses of theeffective medium are related by c22= c33, c12= c13, c55= c66, andc23= c33− 2c44 (note that c44 is not influenced by the fractures).Hence, the effective medium is transversely isotropic with ahorizontal symmetry axis (HTI medium). Although generalHTI models are described by five independent parameters (c11,c33, c13, c44, and c55), the stiffness matrix from equation (9)

1792 Bakulin et al.

depends on just four quantities: λ and µ of the host rock andthe dimensionless fracture weaknesses1N and1T . Therefore,there exists a relationship (constraint) between the stiffnessesthat can be written as (Schoenberg and Sayers, 1995)

c11c33 − c213 = 2c44(c11 + c13). (10)

Aligned penny-shaped cracks

One of the simplest models of fractured rock contains a sin-gle set of parallel penny-shaped cracks (i.e., the cracks havethe form of oblate spheroids) embedded in an isotropic solid.If the semimajor axis of the spheroid is denoted by a and thesemiminor axis by c, we can introduce the so-called aspect ratioα≡ c/a, which is much smaller than unity for thin cracks. If thecracks are spheroidal, it is convenient to replace the fracture(crack) intensity by a similar parameter called the crack den-sity, defined as e= ξ〈a3〉, where ξ is the number of cracks perunit volume and 〈〉 denotes volume averaging (Hudson, 1980).

Considering thin, penny-shaped cracks orthogonal to the x1-axis, Hudson (1980, 1981) derives the following expression forthe stiffness matrix c of the effective medium:

c = cb − e

µ

(λ+ 2µ)2U11 λ(λ+ 2µ)U11 λ(λ+ 2µ)U11 0 0 0

λ(λ+ 2µ)U11 λ2U11 λ2U11 0 0 0

λ(λ+ 2µ)U11 λ2U11 λ2U11 0 0 0

0 0 0 0 0 0

0 0 0 0 µ2U33 0

0 0 0 0 0 µ2U33

+ O(e2), (11)

where U11 and U33 are dimensionless quantities that dependon the boundary conditions on the crack faces, infill param-eters, possible interaction of cracks, and some other factors.Quadratic and higher order terms in crack density e in equa-tion (11) are ignored. Although the underlying assumptions ofHudson’s theory differ from those of the linear slip theory, theeffective medium in both cases has the same symmetry (HTI)with the same constraint (10) on the stiffness components.

Comparison of Hudson’s and linear-slip models

Schoenberg and Douma (1988) note an even more profoundsimilarity between Hudson’s and linear-slip models that goesbeyond relationship (10). They point out that matrices (9)and (11) have the same structure and become identical if thefracture weaknesses satisfy the following relations:

1N = (λ+ 2µ)µ

U11e, (12)

1T = U33e. (13)

Equations (12) and (13) can be used to obtain explicit expres-sions for the weaknesses of dry and fluid-filled cracks. Supposethe fractures are filled with a weak solid with bulk modulusk′ and shear modulus µ′ (both moduli may be equal to zero).Substituting the expressions for U11 and U33 given by Hudson

(1981) into equations (12) and (13) yields (Schoenberg andDouma, 1988)

1N = 4e

3g(1− g)[

1+ 1πg(1− g)

(k′ + 4/3µ′

µ

)(a

c

)] ,(14)

1T = 16e

3(3− 2g)[

1+ 4π(3− 2g)

(µ′

µ

)(a

c

)] . (15)

The parameter g is defined as

g ≡ µ

λ+ 2µ= V2

S

V2P

, (16)

where VP and VS are the P- and S-wave velocities in the back-ground medium.

For dry (gas-filled) cracks both moduli of the infill materialvanish (k′ =µ′ = 0), giving

1N = 4e

3g(1− g), (17)

1T = 16e

3(3− 2g). (18)

If the cracks are filled with fluid, the shear modulus isequal to zero (µ′ = 0), but the bulk modulus k′ for water oroil may be comparable to the shear rigidity µ of the hostrock. Hence, for thin cracks with a small aspect ratio c/a,[(k′ + 4/3µ′)/µ](a/c)À 1 and 1N goes to zero. However, 1T

remains the same as for dry cracks. Therefore, for fluid-filledcracks

1N = 0, (19)

1T = 16e

3(3− 2g). (20)

Schoenberg and Sayers (1995) suggest using the ratio KN/KT

as an indicator of fluid content. They note from existing ex-periments (Pyrak-Nolte et al., 1990a, b; Hsu and Schoenberg,1993) that this ratio is close to unity for dry cracks and almostvanishes for fluid-filled ones. To confirm this observation, theyinfer from Hudson’s (1981) theory that for dry cracks

KN/KT ≈ 1− σ/2, (21)

where σ is Poisson’s ratio for the host rock.


The exact ratio KN/KT can be obtained in terms of the weak-nesses from equations (7), (8), and (16):

KN

KT= g

1N(1−1T )1T (1−1N)

. (22)

For fluid-filled cracks KN/KT = 0 because the weakness 1N

goes to zero.In the limit of small weaknesses (1N¿ 1 and1T¿ 1), equa-

tion (22) reduces to

KN

KT≈ g

1N

1T. (23)

For dry cracks, we substitute the weaknesses from equa-tions (17) and (18) into equation (23) to find

KN

KT≈ 3− 2g

4(1− g)= 1− σ/2. (24)

Thus, equation (21) is a linearized approximation valid forsmall fracture weaknesses (i.e., small crack densities). TheKN/KT ratio from equation (24) varies between 0.75 and 1for the whole range of g from 0 to 0.5, which seems to confirmthe observation of Schoenberg and Sayers (1995).

Calculations based on the exact equation (22), however, mayproduce KN/KT ratios substantially exceeding unity, even if thecrack density is small (Figure 1). The discrepancy between theexact and weak-anisotropy results is explained by the influenceof the weakness 1N in the denominator of equation (22). Ac-cording to Hudson’s (1981) theory, for dry cracks the normalweakness 1N is larger than the shear weakness 1T and theterm 1−1N may be far smaller than unity. It is still unclearwhether Hudson’s theory, which produces the large values ofKN/KT for e= 0.05− 0.07, remains accurate for these crackdensities (Ass’ad et al., 1993) because none of the existing ex-perimental measurements (Pyrak-Nolte et al., 1990a,b; Hsuand Schoenberg, 1993) report values of KN significantly higherthan those of KT .

FIG. 1. Ratio of the normal-to-shear fracture compliances inHudson’s model of dry penny-shaped cracks in isotropic hostrock. Solid lines—the exact KN/KT ratio for different crackdensities e (marked on the plot). Dashed line—linearized ap-proximation for KN/KT from equation (24).

Thomsen’s model of fractured porous media

The weaknesses 1N and 1T of subsurface fractured rocksmay differ substantially from the above estimates obtained forthe simplified model of dry or fluid-filled isolated cracks [equa-tions (17)–(20)]. On the one hand, fractures may be stiffenedby the cementation of crack (fracture) faces or by the rigidityof the fracture infill. Hudson’s theory accounts for these factorsby allowing for a nonzero ratioµ′/µ in equations (14) and (15).On the other hand, fluid-filled fractures may be weakened bypartial saturation (Hudson, 1988) or fluid flow (squirt) betweenthe fractures and pore space (Thomsen, 1995; Hudson et al.,1996). If the fluid can move under stress from the fracturesinto hydraulically connected pores, the fractures are no longerstiff enough to preserve the continuity of the normal (orthogo-nal to their faces) displacement component. Consequently, theweakness1N does not vanish (as it did for isolated cracks com-pletely filled with fluid) and can be expected to take a valueintermediate between zero and the1N for gas-filled cracks [seeequation (17)]. Thus, the weaknesses1N for fluid- and gas-filledcracks may be considered as the lower and upper bounds (re-spectively) of this parameter. This conclusion is supported bythe work of Thomsen (1995), Hudson (1988), and Hudson et al.(1996).

Thomsen (1995), extending the results of Hoenig (1979),Budiansky and O’Connell (1976), and Hudson (1981), devel-ops a formalism to account for fluid flow between cracks andspherical (isometric) pores. We consider the case of a small con-centration of pores [equations (6) and (7) of Thomsen (1995)]so that their equant porosity can be modeled as a dilute dis-tribution of spheres in an isotropic background solid. ThenThomsen’s theory also results in an effective stiffness matrixthat has the same form as in equation (9). The normal andtangential weaknesses for Thomsen’s model are given by

1N = q1Hudson,dryN (25)

and

1T = 16e

3(3− 2g). (26)

An explicit expression for the coefficient q can be found in Ap-pendix A. Note that equant porosity has no influence on thetangential weakness, and equation (26) coincides with equa-tion (18) for 1T obtained using Hudson’s (1981) theory.

Analysis of equations (25), (A-2), and (A-3) shows that inthe presence of equant porosity,1N does not vanish, even if theentire pore space is filled with fluid. The parameter q becomeslarger with increasing equant porosity, thus making 1N closerto that of dry cracks. However, q cannot exceed unity [seeequations ((A-1)–(A-3)], which means that1N for Thomsen’smodel always lies between the values for isolated fluid-filledcracks (1N = 0) and dry cracks.

To illustrate the influence of equant porosity on the weak-nesses, we computed 1N as a function of the VS/VP ratio for awide range of equant porosity values (Figure 2). All curves liebetween 1N = 0 for isolated fluid-filled cracks [equation (19)]and the dashed line that corresponds to 1N for dry cracks[equation (17)]. While 1N is almost zero for small equantporosity φp= 0.01% and φp= 0.1%, it noticeably increaseswhen the porosity φp= 1% and becomes close to that of drycracks as φp reaches 10%.

1794 Bakulin et al.

In summary, all theories examined above lead to the sameform of the elastic stiffness tensor that describes a system ofparallel, rotationally invariant fractures in a purely isotropichost rock. The effective anisotropic medium is characterizedby a special type of the HTI symmetry with four independentstiffness elements. Although each theory operates with a dif-ferent set of inherent parameters, only four parameter combi-nations can be extracted from seismic data acquired for sucha model. In principle, four measurements of the elastic coeffi-cients may be sufficient to evaluate the Lame parameters λ andµ of the host rock and two effective fracture parameters—theweaknesses 1N and 1T . An additional (fifth) measurement isredundant because of constraint (10) and may be used to checkthe validity of the assumed model.

The most general description of the effective medium is pro-vided by the linear-slip theory, which is directly formulated interms of the four measurable parameters. Below we discusshow to estimate these parameters from reflection seismic dataand use them to evaluate the physical properties of the fracturesystem.

ANISOTROPIC PARAMETERS OF HTI MEDIA RESULTINGFROM VERTICAL FRACTURES

General fracture set

First, let us consider the most general case of a vertical frac-ture set with the stiffness tensor expressed through the fractureweaknesses in equation (9). Seismic signatures in HTI mediaare most conveniently described by the vertical velocities (forinstance, the P-wave velocity VP0 and the velocity VS⊥ of theS-wave polarized orthogonal to the cracks) and the following

FIG. 2. Influence of equant porosity φp on the normal weak-ness of a crack system connected to isometric pores (dottedlines). The dashed line (on top) shows the normal weakness forisolated dry cracks; 1N = 0 corresponds to isolated fluid-filledcracks. The vertical scale is in the units of crack density e. Theparameters of the crack system are the aspect ratio α= 0.0005,the volume portion occupied by the cracks φc= 0.01%, and thecrack density e= 0.05. The cracks are filled with water that hasthe bulk modulus k′ = 2 GPa. The P-wave velocity in the hostrock is VP = 5 km/s.

set of anisotropic coefficients introduced by Ruger (1997) andTsvankin (1997) by analogy with Thomsen’s (1986) parametersfor VTI media:

ε(V) ≡ c11 − c33

2c33, (27)

δ(V) ≡ (c13 + c55)2 − (c33 − c55)2

2c33(c33 − c55), (28)

γ (V) ≡ c66 − c44

2c44. (29)

The degree of anellipticity of the model is controlled by theparameter denoted as η(V) (Tsvankin, 1997):

η(V) ≡ ε(V) − δ(V)

1+ 2δ(V). (30)

Exact expressions for ε(V), δ(V), γ (V), and η(V) in terms of thefracture weaknesses can be derived in a straightforward wayfrom equation (9) and are given in Appendix B. To gain in-sight into the relationship between the anisotropic coefficientsand the fracture weaknesses, equations (B-1)–(B-4) can be lin-earized with respect to 1N and 1T under the assumption ofweak anisotropy:

ε(V) = −2g(1− g)1N, (31)

δ(V) = −2g[(1− 2g)1N +1T ], (32)

γ (V) = −1T

2, (33)

η(V) = 2g(1T − g1N). (34)

Analogous expressions for ε(V) and δ(V) in terms of the frac-ture compliances KN and KT are given by Sayers and Rickett(1997).

Since the squared velocity ratio g≡V2S/V2

P < 0.5, we inferfrom equations (B-1)–(B-4), as well as from equations (31)–(34), that ε(V), δ(V), and γ (V) for vertical fractures are nonposi-tive:

ε(V)≤ 0, δ(V)≤ 0, and γ (V)≤ 0. (35)

An equivalent form of the expression for η(V) [derived fromequation (B-4)],

η(V) = (KT − KN)2µ2

λ+ 2µ, (36)

shows that the anellipticity parameter is controlled by thedifference between the tangential and normal compliances(Schoenberg and Sayers, 1995). Since usually KN ≤ KT , η(V)

is predominantly positive. In the model with gas-filled cracks,however, Hudson’s theory predicts that the KN/KT ratio mayexceed unity with increasing crack density (Figure 1) and η(V)

may become negative.As discussed above, the parameters of the crack-induced

HTI model satisfy an additional constraint representedthrough the stiffnesses in equation (10). Rewriting this con-straint in terms of the Thomsen-style anisotropic coefficientsyields (Tsvankin, 1997)


γ (V) = − V2P0

2V2S⊥

ε(V)[2− 1

/f (V)

]− δ(V)

1+√

1+ 2δ(V)/

f (V) + [ε(V) − f (V)δ(V)]/[

f (V)(1− f (V)

)] , (37)

where VP0 is the vertical P-wave velocity, V2S⊥ is the vertical

velocity of the S-wave polarized orthogonal to the cracks, andf (V)≡ 1−V2

S⊥/V2P0. In the linearized weak-anisotropy approx-

imation, equation (37) reduces to

γ (V) = 14g

(δ(V) − ε(V) 1− 2g

1− g

). (38)

Approximate relations analogous to equation (38) are givenby Schoenberg and Douma (1988) and Thomsen (1995) for TImedia formed by parallel horizontal cracks in an isotropic hostrock.

As discussed by Tsvankin (1997) and Contreras et al. (1999),equations (37) and (38) suggest that for crack-induced HTImodels the shear-wave splitting parameter |γ (V)| can be ob-tained from P-wave moveout data. Indeed, P-wave NMO ve-locity from horizontal and dipping reflectors can be inverted forthe vertical velocity VP0, ε(V), and δ(V) (Contreras et al., 1999).Then, if an estimate of the VP/VS ratio at vertical incidence isavailable, γ (V) can be computed from equation (37).

Possible ranges of anisotropic parameters

The Thomsen-style anisotropic coefficients in the fracture-induced HTI model are controlled by the weaknesses and theratio of λ andµ, expressed in our notation by g≡V2

S/V2P. While

the shear-wave splitting parameter γ (V) is equal to −1T/2[equation (B-3)], the exact expressions (B-1) and (B-2) for ε(V)

and δ(V) are more complicated. Therefore, it is instructive tostudy the possible ranges of ε(V) and δ(V) by varying parame-ters 1N , 1T , and g simultaneously within reasonable bounds.

We constrain g by assuming 0.35≤VS/VP ≤ 0.65 (or, equiv-alently, 0.12≤ g≤ 0.42) and use two different ranges for γ (V):|γ (V)| ≤ 0.05 (equivalently, 1T ≤ 0.1) and |γ (V)| ≤ 0.15 (equiv-alently,1T ≤ 0.3). For each pair of g and1T within the chosenrange, we vary 1N from 0 to 1, unless the maximum possiblevalue of the ratio KN/KT = 1 is reached (we ignore larger val-ues of KN/KT produced by Hudson’s theory). Figure 3 shows

FIG. 3. Possible ranges of the parameters ε(V) and δ(V) in HTI media resulting from parallel vertical fractures. (a) |γ (V)| ≤ 0.05(or 1T ≤ 0.1); (b) |γ (V)| ≤ 0.15 (1T ≤ 0.3). The VS/VP ratio is constrained by 0.12≤ g≤ 0.42, and KN/KT ≤ 1.

that the possible ranges for both ε(V) and δ(V) are relativelynarrow, especially if |γ (V)| ≤ 0.05. The areas of feasible ε(V) andδ(V) have a quasi-triangular shape and lie above the diagonalthat corresponds to the elliptical model, defined by ε(V)= δ(V)

and η(V)= 0 (KN = KT ).The values of ε(V) and δ(V) in Figure 3 were computed using

the exact equations (B-1) and (B-2), which are valid for anyvalues of1N and1T between 0 and 1 (hence, for any strengthof fracturing). For example, 1N in Figure 3b reaches an un-commonly large value of 0.92, which corresponds to extremelycompliant fractures that reduce the velocity across them by upto 75% of the background value.

Isolated penny-shaped cracks

Explicit expressions for dry and fluid-filled cracks.—For thespecial case of thin, isolated, penny-shaped cracks, we canrewrite equations (31)–(34) in terms of the crack density usingHudson ’s (1981) theory. To obtain the linearized anisotropiccoefficients for dry (or gas-filled) cracks, we substitute equa-tions (17) and (18) for the weaknesses into equations (31)–(34):

ε(V) = −83

e, (39)

δ(V) = −83

e

[1+ g(1− 2g)

(3− 2g)(1− g)

], (40)

γ (V) = − 8e

3(3− 2g), (41)

η(V) = 83

e

[g(1− 2g)

(3− 2g)(1− g)

]. (42)

Thus, in the weak-anisotropy approximation all parameters be-come simple functions of the crack density e and the squaredVS/VP ratio (g) of the host rock. If we assume that the VS/VP

ratio varies between 0.35 and 0.65, the corresponding rangesof the anisotropic coefficients are

1796 Bakulin et al.

ε(V) = −2.68e,

δ(V) = (−2.82± 0.05)e,

γ (V) = (−1.10± 0.13)e,

η(V) = (0.14± 0.05)e. (43)

According to equations (43), the HTI medium resultingfrom dry cracks is close to elliptically anisotropic (δ(V)≈ ε(V),η(V)≈ 0), with the parameters largely controlled by the crackdensity e. Although the only parameter fully independent ofg is ε(V), the variations in the other three parameters for thewhole range of g are rather insignificant. Note that the shear-wave splitting coefficient |γ (V)| is close to e (Thomsen, 1995;Tsvankin, 1997), so the time delay between the split S-wavesat vertical incidence gives a good direct estimate of the crackdensity (a well-known fact often used in fracture characteriza-tion).

For fluid-filled cracks, we substitute equations (19) and (20)into equations (31)–(34) to obtain

ε(V) = 0, (44)

δ(V) = − 32ge

3(3− 2g), (45)

γ (V) = − 8e

3(3− 2g), (46)

η(V) = −δ(V) = 32ge

3(3− 2g). (47)

Fluid saturation does not change the splitting parameter|γ (V)| [compare equations (41) and (46)] but has a strong in-fluence on the other anisotropic coefficients. If the cracks arecompletely saturated with fluid, the P-wave velocities in thedirections parallel and orthogonal to the cracks are equal toeach other, and ε(V)= 0 [equation (44)]. As follows from con-straint (38), as well as from equations (45) and (46), for van-ishing ε(V)

γ (V) = δ(V)

4g. (48)

For a typical value g= 0.25, γ (V) and δ(V) become identical.Therefore, for fluid-filled cracks and g= 0.25, the absolute val-ues of γ (V), δ(V), and η(V) are close to each other and to thecrack density e (Tsvankin, 1997; Ruger and Tsvankin, 1997).

Influence of fluid content.—The influence of fluid satura-tion on the anisotropic parameters is illustrated in Figure 4. Asmentioned above, the presence of fluid has no influence on theshear-wave splitting parameter γ (V) because the shear modulusµ′ in equation (15) for the weakness 1T goes to zero for bothdry and fluid-filled cracks.

In contrast, the values of ε(V) for fluid-filled and dry cracksare substantially different; the same conclusion holds for δ(V).The parameter ε(V) is a function of only one weakness (1N)[equation (31)], which is responsible for the jump of the nor-mal displacement across the crack face and therefore is stronglydependent on the fluid bulk modulus k′. Note that the differ-ence between ε(V) for dry and fluid-filled cracks remains thesame (8e/3) for any VS/VP ratio. For the parameter δ(V), theinfluence of the crack infill rapidly decreases with the VS/VP

ratio. For the most typical VS/VP ≈ 0.5 (g≈ 0.25), the absolutevalue of δ(V) for dry cracks is almost three times greater thanthat for fluid-filled cracks.

For models with hydraulically interconnected cracks andpores (Thomsen, 1995) or with partially saturated cracks, theanisotropic parameters always lie between the values for dryand fluid-filled isolated cracks plotted in Figure 4. This followsfrom the above analysis of equations (25) and (26) and Figure 2.

INVERSION OF SEISMIC SIGNATURES FOR THEPARAMETERS OF HTI MEDIA

Weaknesses 1N and 1T can be estimated from seismic dataif two of the three anisotropic coefficients (ε(V), δ(V), γ (V)) andthe VS/VP ratio are available. For instance, using the linearizedweak-anisotropy approximations (31) and (32) for ε(V) and δ(V),we obtain the weaknesses as

1N = − ε(V)

2g(1− g), (49)

1T = 12g

[1− 2g

1− gε(V) − δ(V)

]. (50)

Equations (49) and (50) indicate that the inversion for 1N

and 1T can be based solely on P-wave traveltime data, whichyield ε(V) and δ(V) (Contreras et al., 1999), assuming that gwas estimated independently (e.g., from well logs in the area).Other options include using the azimuthal variation of theP-wave AVO gradient (Ruger, 1997; Ruger and Tsvankin,1997) or converted (PS) waves; these various approaches arediscussed in more detail below.

Further interpretation of the weaknesses in terms of thephysical properties of the fracture set is nonunique and can-not be carried out without making assumptions about the

FIG. 4. Anisotropic parameters of the effective HTI model re-sulting from isolated penny-shaped cracks embedded in anisotropic host rock [from equations (39)–(42) and (44)–(47)].Solid lines—fluid-filled cracks. Dashed lines—dry (gas-filled)cracks. The vertical scale is in the units of the crack density e.


microstructure of the fractures and pore space. For penny-shaped cracks, possibly connected to isometric pores, the tan-gential weakness 1T depends on only the crack density (i.e.,on the strength of fracturing) and remains the same for bothdry and fluid-filled cracks [equations (18), (20), and (26)]. Thenormal weakness 1N (and the ratio 1N/1T ), in contrast, isalso a function of fluid content [compare equations (17), (19),and (25)] and thus provides information about fluid saturationin addition to the crack density (see below). Still, for porousformations1N/1T depends not just on fluid saturation but alsoon the presence of a hydraulic connection between cracks andpore space.

Reflection moveout and AVO gradient of P-waves

P-wave NMO velocity.—If the reservoir has a sufficientthickness, information about fracturing can be obtained fromthe azimuthal variation of the interval NMO velocity. In az-imuthally anisotropic media the conventional Dix differentia-tion, routinely used to recover the interval velocity, becomesinaccurate and must be replaced with a more general expres-sion given by Grechka, Tsvankin, and Cohen (1999). A detaileddiscussion of the methodology of azimuthal moveout analy-sis, including a correction for lateral velocity variation, can befound in Grechka and Tsvankin (1999).

Azimuthal variation of P-wave NMO velocity is describedby an ellipse in the horizontal plane, even for arbitraryanisotropic heterogeneous media (Grechka and Tsvankin,1998). In the special case of a plane homogeneous HTI layer,the NMO ellipse is given by (Tsvankin, 1997)

V2P,nmo(β) = V2

P01+ 2δ(V)

1+ 2δ(V) sin2 β, (51)

where VP0 is the P-wave vertical velocity andβ is the azimuth ofthe common-midpoint (CMP) line with respect to the symme-try axis (normal to the fractures). Since δ(V)≤ 0, the semimajoraxis of the NMO ellipse lies in the fracture plane.

P-wave NMO velocity measurements along three or morewell-separated azimuths can be inverted for the fracture ori-entation, the velocity VP0, and the anisotropic parameter δ(V).The value of ε(V) (or η(V)), also required for estimating frac-ture weaknesses, can be found from NMO velocities of dip-ping events (Tsvankin, 1997; Contreras et al., 1999) or fromnonhyperbolic moveout (Al-Dajani and Tsvankin, 1998).

P-wave AVO gradient.—Reflection coefficients measured atoblique incidence angles over fractured formations vary withthe azimuth of the source–receiver line. This azimuthal depen-dence can be used in AVO analysis to obtain information aboutthe orientation, density, and content of the fractures (Rugerand Tsvankin, 1997; Sayers and Rickett, 1997).

The weak-anisotropy approximation for the plane P-wavereflection coefficient at a small-contrast interface between twoHTI media with the same orientation of the symmetry axisis derived by Ruger (1997) in terms of the Thomsen-styleanisotropic parameters. The behavior of the reflection coef-ficient R at small and moderate incidence angles θ is governedby the so-called AVO gradient—the initial slope of R plottedas a function of sin2

θ . Here we consider the azimuthally vary-ing AVO gradient for a P-wave reflection from the interfacebetween isotropic and HTI media. The difference between theAVO gradients in the directions perpendicular and parallel to

the cracks (Bani) can be written as (Ruger, 1997; Ruger andTsvankin, 1997)

Bani = δ(V) − 8gγ (V)

2, (52)

where g has the meaning of the average V2S/V2

P for the two half-spaces. Substituting the approximate expressions (32) and (33)for δ(V) and γ (V) yields Bani as an explicit function of the fractureweaknesses:

Bani = g[1T − (1− 2g)1N]. (53)

An equivalent equation in terms of the (dimensional) compli-ances is given by Sayers and Rickett (1997).

For isolated penny-shaped cracks, the azimuthal variation ofthe AVO gradient can be related directly to the crack density.Equations (40), (41), (45), and (46) for δ(V) and γ (V) obtainedfrom Hudson’s (1981) theory lead to the following expressionsfor dry and fluid-filled fractures:

Banidry =

4(−8g2 + 12g− 3)3(3− 2g)(1− g)

e, (54)

Baniwet =

16g

3(3− 2g)e. (55)

Clearly, the magnitude of the azimuthal AVO response inweakly anisotropic HTI media is proportional to the crack den-sity. In addition, the value of Bani is strongly dependent on g,i.e., on the VS/VP ratio of the host rock (Figure 5). If VS/VP

takes a typical value of 0.55, Baniwet is positive and Bani

dry is close to

FIG. 5. Azimuthal variation of the AVO gradient for theP-wave reflection from the interface between isotropic andHTI media. The difference between the AVO gradients in thedirections perpendicular and parallel to the cracks (Bani) wascalculated for dry [equation (54), dashed line] and fluid-filledcracks [equation (55), solid line]. The vertical scale is in theunits of the crack density e. The dots mark the values of Bani

picked from the curves of the exact reflection coefficient forthe crack density e= 0.07. The cracks were introduced in thelower half-space using the first-order Hudson’s (1981) theory.The upper half-space and the host rock in the lower half-spacehave the same velocities, with VP = 6000 m/s and VS chang-ing in accordance with the VS/VP ratio on the horizontal axis[VS=VP(VS/VP)]. The densities of the upper and lower mediaare 2800 and 2300 kg/m3, respectively.

1798 Bakulin et al.

zero. This result agrees with the computations of Strahilevitzand Gardner (1995), Ruger and Tsvankin (1997), and Sayersand Rickett (1997), who noted that the AVO gradient for drycracks is almost independent of azimuth for models with simi-lar VS/VP ratios. The relation between Bani

wet and Banidry, however,

changes substantially for higher or lower values of VS/VP. Forinstance, Bani

dry≈−2 Baniwet when VS/VP is close to 0.4. Compari-

son with the exact reflection coefficient (Figure 5) shows thehigh accuracy of equations (54) and (55) for moderate crackdensities reaching 0.05–0.07.

Thus, the value of Bani, combined with an estimate of theVS/VP ratio, may serve as an indicator of fluid content. Quan-titative interpretation of the azimuthal AVO response, how-ever, requires additional information about the crack densityor anisotropic coefficients. Indeed, if the cracks are dry andVS/VP = 0.5 − 0.6, the P-wave AVO gradient does not varymuch with azimuth, irrespective of the degree of fracturing. Adirect estimate of the crack density (i.e., 1T ) can be obtainedfrom shear-wave splitting at vertical incidence; then Bani canbe used to find 1N [equation (53)] and evaluate fluid satura-tion. Alternatively, as suggested by Ruger and Tsvankin (1997),the azimuthal AVO response can be combined with the resultsof azimuthal velocity analysis [equation (51)] to resolve theanisotropic coefficients δ(V) and γ (V), which can be further re-calculated into the fracture weaknesses.

As discussed above, if the cracks are partially saturated orare connected to pore space, 1N is always greater than zerobut is smaller than the value for dry cracks. Equation (53) indi-cates that Bani in this case falls between the values of Bani

wet andBani

dry for the same VS/VP ratio. Therefore, the curves shown inFigure 5 define the lower and upper bounds of Bani for variousmicrogeometries of the fractured medium.

Distinguishing between dry and fluid-filled cracks.—Let usassume that two anisotropic parameters [for example, ε(V) andδ(V)] have been found and an estimate of g is available. Thenit is important to assess whether one can distinguish betweendry and fluid-filled cracks in the presence of realistic errors inε(V), δ(V), and VS/VP ratio (g).

First, consider a fracture set with the density e= 7% embed-ded in a medium with VS/VP = 0.5 (g= 0.25). For such a model,the tangential weakness1T = 0.15 for both dry and fluid-filledcracks, while the normal weakness changes from 1N = 0.50 ifthe cracks are dry (ε(V)=−0.21, δ(V)=−0.19) to 1N = 0 forfluid-filled cracks (ε(V)= 0, δ(V)=−0.07). We simulated errorsin the data by adding Gaussian noise with the standard devia-tion σ = 0.05 to the correct values of ε(V), δ(V), and VS/VP andinverted these three parameters for1N and1T using the exactequations from Appendix B. The inversion results, along withthe input anisotropic coefficients, are marked by small dots inFigure 6. Clearly, the clouds of estimated weaknesses are wellseparated by their values of1N . This implies that although er-rors in ε(V), δ(V), and VS/VP on the order of ±0.05− 0.1 intro-duce a substantial uncertainty in the value of the crack density(or 1T ), they do not prevent us from separating models withdry and fluid-filled cracks.

Figure 7 displays the results of some additional numericalexperiments. As expected, reducing errors in the data leadsto a visible tightening of the clouds of 1N and 1T (compareFigures 7a and 7b with Figures 6c and 6d).With decreasing data

error, it may become possible to distinguish between fully andpartially saturated fractures (or fractures connected to porespace).

The sensitivity of the weaknesses to errors in input data (re-flected in the size of the 1N and 1T clouds) is controlled bythe VS/VP ratio (Figures 7c–f). For a fixed magnitude of errorsin ε(V) and δ(V), the errors in 1N and 1T decrease with in-creasing VS/VP ratio. However, the value of1N for dry cracksalso decreases from 1N = 0.7 for VS/VP = 0.4 to 1N = 0.4 forVS/VP = 0.6 (Figures 7d and 7f). As a result, the separationbetween 1N for the models with dry and fluid-filled cracksis smaller for higher VS/VP ratios, and our overall ability toresolve the fluid content is almost independent of the VS/VP

ratio.Our numerical tests thus suggest that it is feasible to distin-

guish between dry and fluid-filled fractures if errors in ε(V), δ(V),and the VS/VP ratio do not exceed 0.1 and the crack densityreaches 5–7%.

Reflection moveout and AVO gradient of PS-waves

For multicomponent surveys, it is possible to devise afracture-detection algorithm based on mode-converted (PS)data. Since moveout and amplitude information averagemedium properties on different scales, combining the P-waveNMO velocity and AVO gradient in the characterization ofheterogeneous formations may prove difficult. Therefore, it isadvantageous to supplement the P-wave NMO ellipse with re-flection traveltimes of PS-waves or, alternatively to carry outjoint inversion of the AVO gradients of P- and PS-waves.

One of the split shear waves in HTI media (denoted as S‖ or Sparallel) is polarized in the isotropy (fracture) plane, while thepolarization vector of the second S-wave (S⊥) lies in the plane

FIG. 6. Noise-contaminated anisotropic coefficients ε(V) andδ(V) for (a) fluid-filled and (b) dry cracks (small dots). (c) Theinverted weaknesses1N and1T for the fluid-filled model from(a). (d) The inverted1N and1T for the dry-crack model from(b). The large dots indicate the correct parameter values.


formed by the symmetry axis and the slowness vector. At ver-tical incidence, the S⊥-wave is polarized perpendicular to thefractures and travels slower than the S‖-wave. In each verticalsymmetry plane, the P-wave is coupled to the in-plane polar-ized S-wave and generates a single converted-mode reflectionfrom a horizontal interface. Thus, in the isotropy (fracture)plane one should be able to record the reflected wave PS‖,while in the plane orthogonal to the fractures (the symmetry-axis plane) an incident P-wave is converted into the PS⊥ reflec-tion. Since either one or the other of the two converted modes isweak in the vicinity of each symmetry plane, extracting the fullazimuthal dependence of the NMO velocity or AVO gradientof the PS‖- or PS⊥-wave requires a better azimuthal coveragethan that for P-wave reflections. Here, we restrict the discus-sion to the reflection traveltimes and amplitudes of convertedwaves in the vertical symmetry planes of the HTI medium.

PS-wave traveltimes.—As follows from the analysis ofGrechka, Theophanis, and Tsvankin (1999) for the more gen-eral orthorhombic media, the azimuthal variation of the NMOvelocity of each converted wave (PS‖ and PS⊥) in a horizon-tal HTI layer is elliptical. By performing moveout (semblance)analysis in the symmetry planes, we determine the vertical trav-eltime and one of the semiaxes of the NMO ellipse for each

FIG. 7. Dependence of the inverted weaknesses 1N and 1T(small dots) for fluid-filled (left column) and dry (right col-umn) cracks on the VS/VP ratio and on the standard devi-ation σ of ε(V), δ(V), and VS/VP from their correct values.The weaknesses are the same as in Figure 6. (a) and (b):σ = 0.025,VS/VP = 0.5. (c) and (d): σ = 0.05,VS/VP = 0.4. (e)and (f): σ = 0.05,VS/VP = 0.6.

converted wave. Note that mode conversions in a horizontalHTI layer vanish at vertical incidence, and the vertical (zero-offset) traveltimes of the converted waves are found essentiallyby extrapolation using the best-fit hyperbola provided by sem-blance analysis.

Given the P-wave zero-offset reflection traveltime, the zero-offset times of the converted waves from the same interfacecan be used to obtain the ratio of the shear-wave vertical ve-locities [i.e., the parameter γ (V)] and an estimate of the ratiog≡V2

S/V2P. This implies that the P-wave NMO ellipse, which

provides δ(V), and the vertical traveltimes of the P-wave andsplit converted waves are sufficient to find γ (V), δ(V), and g and,therefore, both fracture weaknesses.

Symmetry-plane NMO velocities of the converted wavesprovide redundant information that may, however, increasethe accuracy of the parameter estimation procedure. In eachvertical symmetry plane the traveltimes of reflected waves aredescribed by the same equations as in VTI media, and the NMOvelocities of P-, S-, and PS-waves are related by the followingDix-type equation (Tsvankin and Thomsen, 1994):

[VS,nmo]2 = t (PS)0 [VPS,nmo]2 − t (P)

0 [VP,nmo]2

t (PS)0 − t (P)

0

, (56)

where t0 denotes the one-way vertical traveltime for pure (Pand S) reflections and the two-way vertical traveltime for thePS-wave. Equation (56) can be used to obtain the NMO ve-locities of the reflected shear waves S‖ (in the fracture plane)and S⊥ (in the plane perpendicular to the fractures) from Pand PS data. Since the velocity of each mode in the fractureplane is independent of the angle with the vertical, the NMOvelocity of the S‖-wave is equal to the fast shear-wave verticalvelocity VS1. The NMO velocity of the S⊥-wave in the directionperpendicular to the fractures is given by (Tsvankin, 1997)

VS⊥,nmo = VS2

√1+ 2σ (V) (57)

and

σ (V) ≡(

VP0

VS2

)2(ε(V) − δ(V)), (58)

where VP0 and VS2 are the vertical velocities of the P- andS⊥-waves, respectively. If the vertical velocities and δ(V) weredetermined from the P-wave NMO ellipse and the verticaltraveltimes, the NMO velocity of the S⊥-wave yields ε(V). Boththe fast S-wave velocity VS1 and ε(V), however, can be obtainedas well from the vertical traveltimes of the converted waves,the P-wave NMO ellipse, and the constraint on the anisotropicparameters of HTI media.

PS-wave AVO gradient.—For relatively thin target layers,interval moveout analysis becomes inaccurate; so it may beadvantageous to supplement the P-wave AVO response withprestack amplitudes of PS-wave reflections. Of course, it is alsopossible to use nonconverted shear waves, but they are seldomacquired in exploration surveys.

As before, we consider an interface between isotropic andHTI media and analyze the magnitude of the azimuthal vari-ation of the reflection coefficient. The results of Ruger (1996)show that the PSreflection coefficient at small offsets is pro-portional to sin θ , where θ is the P-wave incidence angle.

1800 Bakulin et al.

Therefore, we define the PS-wave AVO gradient as the coeffi-cient of the sin θ term and use Ruger’s (1996) weak-anisotropyapproximations to express the AVO gradient through theanisotropic parameters. The difference between the AVO gra-dients of the PS⊥-wave across the fractures and the PS‖-waveparallel to them can be written as

Bani = δ(V)2 − 4(

√g+ g)γ (V)

2

2(1+√g). (59)

Introducing the fracture weaknesses into equation (59) yields

Bani =√

g

1+√g[1T −√g(1− 2g)1N]. (60)

Despite some similarity between equation (60) and the corre-sponding P-wave expression (53), the factor Bani for the P- andPS-modes is controlled by different combinations of 1T and1N . Hence, using both P- and PS-waves in azimuthal AVOanalysis has the potential of yielding information on the weak-nesses 1T and 1N or, equivalently, the crack density and thefluid content of the fractures. Also, as mentioned, the squaredvertical-velocity ratio g may be determined from the P- andPS-wave traveltimes, unless the reservoir is too thin.

Figure 8 reproduces the numerical example from Figure 5but this time for PS-waves. The curves were generated by sub-stituting the values of 1N and 1T for dry (dashed line) andfluid-filled (solid) cracks into equation (60). Note that the accu-racy of the weak-anisotropy approximation for the PS-wave isnot much lower than that for the P-wave. As expected from theform of the analytic solutions, Bani for both P- and PS-waves ispositive in models with isolated fluid-filled cracks and changessign with increasing VS/VP for dry cracks. However, in the lat-ter case the intersection with the horizontal axis for PS-wavesoccurs at a lower value of VS/VP than for P-waves. Hence, if theVS/VP ratio is close to a typical value of 0.55 and the P-waveAVO gradient for dry cracks is almost independent of azimuth

FIG. 8. The difference between the converted-wave AVO gra-dients in the directions perpendicular (PS⊥) and parallel (PS‖)to the cracks for dry (dashed line) and fluid-filled (solid line)cracks; both curves are computed from the weak-anisotropyapproximation. The dots mark the values of Bani picked fromthe curves of the exact reflection coefficient for the crack den-sity e= 0.07. The model parameters are the same as those inFigure 5.

(Ruger and Tsvankin, 1997; Sayers and Rickett, 1997), the ad-dition of PS-waves may help identify fracturing and, moreover,discriminate between dry and fluid-filled cracks.

DISCUSSION AND CONCLUSIONS

The linear-slip theory (e.g., Schoenberg and Sayers, 1995),based on the general treatment of fractures as surfaces of weak-ness inside a solid matrix, provides a convenient framework forrelating seismic signatures to the properties of fracture systems.The inherent parameters of the linear–slip theory for rotation-ally invariant fractures are the normal (1N) and tangential(1T ) weaknesses, which can be estimated from seismic data.The theories of Hudson (1981) and Thomsen (1995), designedfor specific physical fracture models involving penny-shapedcracks, can be used to express1N and1T through parametersdependent on the microstructure of cracks and pores. Althoughdetermination of these microstructural parameters from seis-mic data requires additional information about the medium,Hudson’s and Thomsen’s models are helpful in guiding the in-terpretation of the weaknesses in terms of the crack densityand fluid content.

Vertical, parallel, rotationally invariant fractures lead toa particular type of HTI media described by four indepen-dent parameters. We obtained linearized expressions for theThomsen-style anisotropic coefficients of HTI media ε(V), δ(V),and γ (V) in terms of the weaknesses1N and1T and the VS/VP

ratio in the background medium. One of the anisotropic param-eters (ε(V), δ(V), orγ (V)) can be found from the other two and theVS/VP ratio. Interestingly, ε(V), δ(V), and γ (V) in any fracture-induced HTI model are always negative, while the anellipticityparameter η(V) is predominantly positive. All possible pairs[ε(V), δ(V)] for HTI models resulting from parallel vertical frac-tures belong to a relatively narrow area with a quasi-triangularshape on the [ε(V), δ(V)]-plane. These results can be used toidentify fracture-induced HTI media from seismic measure-ments.

If the model contains isolated penny-shaped (spheroidal)cracks, the weaknesses and the anisotropic parameters can berelated to the physical properties of the crack system usingHudson’s theory. In the linearized weak-anisotropy approx-imation, all anisotropic parameters are proportional to thecrack density, with the coefficients controlled by the VS/VP

ratio and, for ε(V) and δ(V), fluid saturation. The shear-wavesplitting parameter |γ (V)| for dry or fluid-filled cracks and awide range of the VS/VP ratios remains close to the crack den-sity. Therefore, the traveltimes or reflection amplitudes of splitS-waves at vertical incidence are strongly dependent on the de-gree of fracturing but cannot be used to discriminate betweendry and fluid-filled cracks.

In contrast, ε(V) and δ(V) are sensitive to the presence of fluidin the cracks. If the cracks are dry, the effective medium isclose to elliptical (ε(V)≈ δ(V)), with the anisotropic parameterstightly governed by the crack density and weakly dependent onthe VS/VP ratio. The absolute value of ε(V)≈ δ(V) for dry cracksis almost three times higher than |γ (V)| and the crack density.For isolated fluid-filled cracks, ε(V) practically vanishes, whileδ(V) for typical VS/VP ≈ 0.5 is close to γ (V) and (by absolutevalue) to the crack density. The influence of crack infill onδ(V), however, decreases with the VS/VP ratio. For models withinterconnected cracks and pores (Thomsen, 1995) or with


partially saturated cracks, ε(V) and δ(V) always lie between thevalues for dry and isolated fluid-filled cracks.

The simplest way to estimate δ(V) and the fracture orienta-tion is to reconstruct the P-wave NMO ellipse from a hori-zontal reflector by means of azimuthal moveout analysis. Theparameter ε(V) can also be obtained from P-wave data usingdipping events or nonhyperbolic moveout. Then ε(V) and δ(V),along with the VS/VP ratio, which must be evaluated sepa-rately (e.g., from well logs in the area), can be used to find γ (V).Alternatively, γ (V) can be determined directly from prestackP-wave amplitudes by combining the azimuthal variation ofthe AVO gradient with the NMO ellipse for horizontal events.If converted waves are available and the reservoir is sufficientlythick, it is possible to recover all relevant anisotropic parame-ters from the zero-offset traveltimes of P- and PS-waves andeither the P-wave NMO ellipse or the AVO gradients of theP- or PS-waves. For surveys acquired with shear-wave sources,it is beneficial to calibrate P-wave data with split shear waves,whose polarizations, traveltimes, and AVO response give an in-dependent estimate of the fracture orientation and parameterγ (V) and help to evaluate the VS/VP ratio.

Any two out of the three anisotropic parameters (ε(V), δ(V),and γ (V)) and the VS/VP ratio can then be recalculated into thefracture weaknesses 1N and 1T . Further interpretation of theweaknesses is not unique and must rely on the assumed phys-ical model. For penny-shaped cracks, the tangential weaknessis close to twice the crack density, while the normal weakness,1N , (or the ratio 1N/1T ) is sensitive to fluid saturation. Evenfor moderate crack densities, there is a substantial differencebetween the value of 1N for dry cracks and the vanishing 1N

for isolated fluid-filled cracks, with intermediate values of thenormal weakness corresponding to partial saturation and/orthe presence of fluid flow (squirt) between cracks and porespace. Therefore, 1N/1T is a useful indicator of fluid content,although its interpretation may be ambiguous without addi-tional information.

ACKNOWLEDGMENTS

This research was carried out during a visit of AndreyBakulin to the Center for Wave Phenomena (CWP), ColoradoSchool of Mines, in 1998. We are grateful to Leon Thomsen (BPAmoco) and members of the A (nisotropy)-team of CWP forhelpful discussions and to Andreas Ruger (Landmark) for hisreview of the manuscript. The support for this work was pro-vided by the members of the Consortium Project on SeismicInverse Methods for Complex Structures at CWP and by theU.S. Department of Energy (award #DE-FG03-98ER14908).

REFERENCES

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Al-Dajani, A., and Tsvankin, I., 1998, Nonhyperbolic reflection move-out for horizontal transverse isotropy: Geophysics, 63, 1738–1753.

Ass’ad, J. M., Tatham, R. H., McDonald, J. A., Kusky, T. M., and Jech, J.,1993, A physical model study of scattering of waves by aligned cracks:Comparison between experiment and theory: Geophys. Prosp., 41,323–339.

Backus, G. E., 1962, Long-wave elastic anisotropy produced by hori-zontal layering: J. Geophys. Res., 67, 4427–4440.

Bagrintseva, K. I., 1982, Fracturing of rocks: Nedra (in Russian).Bakulin, A. V., and Molotkov, L. A., 1998, Effective models of fractured

and porous media: St. Petersburg Univ. Press (in Russian).Berg, E., Hood, J., and Fryer, G., 1991, Reduction of the general frac-

ture compliance matrix Z to only five independent elements: Geo-phys. J. Internat., 107, 703–707.

Budiansky, B., and O’Connell, R. J., 1976, Elastic moduli of a crackedsolid: Internat. J. Solids Structures, 12, 81–97.

Contreras, P., Grechka, V., and Tsvankin, I., 1999, Moveout inversionof P-wave data for horizontal transverse isotropy: Geophysics, 64,1219–1229.

Corrigan, D., Withers, R., Darnall, J., and Skopinski, T., 1996, Fracturemapping from azimuthal velocity analysis using 3D surface seismicdata: 66th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Ab-stracts, 1834–1837.

Grechka, V., and Tsvankin, I., 1998, 3-D description of normal moveoutin anisotropic inhomogeneous media: Geophysics, 63, 1079–1092.

———1999, 3-D moveout inversion in azimuthally anisotropic mediawith lateral velocity variation: Theory and a case study: Geophysics,64, 1202–1218.

Grechka, V., Theophanis, S., and Tsvankin, I., 1999, Joint inversion ofP- and PS-waves in orthorhombic media: Theory and a physical-modeling study: Geophysics, 64, 146–161.

Grechka, V., Tsvankin, I., and Cohen, J. K., 1999, Generalized Dixequation and analytic treatment of normal-moveout velocity foranisotropic media: Geophys. Prosp., 47, 117–148.

Hoenig, A., 1979, Elastic moduli of the non-randomly cracked body:Internat. J. Solids Structures, 15, 137–154.

Hsu, C.-J., and Schoenberg, M., 1993, Elastic waves through a simulatedfractured medium: Geophysics, 58, 964–977.

Hudson, J. A., 1980, Overall properties of a cracked solid: Math. Proc.Camb. Phil. Soc., 88, 371–384.

———1981, Wave speeds and attenuation of elastic waves in materialcontaining cracks: Geophys. J. Roy. Astr. Soc., 64, 133–150.

———1988, Seismic wave propagation through material containingpartially saturated cracks: Geophys. J., 92, 33–37.

Hudson, J. A., Liu, E., and Crampin, S., 1996, The mechanical proper-ties of materials with interconnected cracks and pores: Geophys. J.Internat., 124, 105–112.

Kirkinskaya, V. N., and Smekhov, E. M., 1981, Carbonate rocks ascollectors of oil and gas: Nedra (in Russian).

Molotkov, L. A., and Bakulin, A. V., 1997, An effective model of afractured medium with fractures modeled by the surfaces of discon-tinuity of displacements: J. Math. Sci., 86, 2735–2746.

Nichols, D., Muir, F., and Schoenberg, M., 1989, Elastic properties ofrocks with multiple sets of fractures: 59th Ann. Internat. Mtg., Soc.Expl. Geophys., Expanded Abstracts, 471–474.

Pyrak-Nolte, L. J., Myer, L. R., and Cook, N. G. W., 1990a, Transmissionof seismic waves across single natural fractures: J. Geophys. Res., 95,No. B6, 8617–8638.

———1990b, Anisotropy in seismic velocities and amplitudes frommultiple parallel fractures: J. Geophys. Res., 95, No. B7, 11 345–11 358.

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———1997, P-wave reflection coefficients for transversely isotropicmodels with vertical and horizontal axis of symmetry: Geophysics,62, 713–722.

Ruger, A., and Tsvankin, I., 1997, Using AVO for fracture detection:Analytic basis and practical solutions: The Leading Edge, 10, 1429–1434.

Sayers, C., and Rickett, J. E., 1997, Azimuthal variation in AVO re-sponse for fractured gas sands: Geophys. Prosp., 45, 165–182.

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——1983, Reflection of elastic waves from periodically stratified mediawith interfacial slip: Geophys. Prosp., 31, 265–292.

Schoenberg, M., and Douma, J., 1988, Elastic wave propagation inmedia with parallel fractures and aligned cracks: Geophys. Prosp.,36, 571–590.

Schoenberg, M., and Muir, F., 1989, A calculus for finely layeredanisotropic media: Geophysics, 54, 581–589.

Schoenberg, M., and Sayers, C., 1995, Seismic anisotropy of fracturedrock: Geophysics, 60, 204–211.

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———1995, Elastic anisotropy due to aligned cracks in porous rock:Geophys. Prosp., 43, 805–830.

Tsvankin, I., 1997, Reflection moveout and parameter estimation forhorizontal transverse isotropy: Geophysics, 62, 614–629.

Tsvankin, I., and Thomsen, L., 1994, Nonhyperbolic reflection moveoutin anisotropic media: Geophysics, 59, 1290–1304.

1802 Bakulin et al.

APPENDIX A

WEAKNESSES FOR THOMSEN’S MODEL OF FRACTURED POROUS ROCK

Thomsen (1995) has developed a model that accounts forthe effect of fluid flow (squirt) between cracks and isometricpores in porous rocks. He generalizes the results of Hoenig(1979), Budiansky and O’Connell (1976), and Hudson (1981)for the model that contains two sets of hydraulically connectedinclusions (e.g, the first set is thin cracks and the second is iso-metric pores). Here we restrict the discussion to models with asmall concentration of isometric pores [equations (6) and (7)of Thomsen (1995)], where equant porosity is approximated bya dilute distribution of spherical pores in an isotropic matrix.

The expressions for the generic Thomsen parameters ε, δ,and γ for this medium (Thomsen, 1995) become identical tothose for the linear-slip model (Schoenberg and Douma, 1988),if we use the following expressions for 1N and 1T :

1N = 43

e

(1− g)g

(1− k′

λ+ 2/3µ

)Dcp,

(A-1)1T = 16

3e

(3− 2g).

Here λ and µ are the Lame parameters of the matrix, k′ is thebulk modulus of the fluid filling the cracks (µ′ = 0), e is the

crack density, g≡V2S/V2

P, and Dcp is the so-called fluid factor,designed to account for the interconnection between cracksand pores. At relatively low frequencies, where the fluid hasenough time to move from cracks to pores, Dcp is given by(Thomsen, 1995)

Dcp =[

1− k′

λ+ 2/3µ+ k′

(λ+ 2/3µ)(φc + φp)

× (Apφp + Ace)]−1

, (A-2)

where φp and φc are the fractions of volume occupied bypores and cracks, respectively [the crack porosity may be ex-pressed as φc= (4πec)/(3a)]. Coefficients Ap and Ac dependonly on the background VS/VP ratio: Ap= (3− 2g)/(2g) andAc= (4/9)× [(2− 3g)/(1− g)] (Thomsen, 1995). Comparingequation (A-1) with (17), we may represent the normal weak-ness for the Thomsen model as

1N = q1Hudson, dryN , q =

(1− k′

λ+ 2/3µ

)Dcp. (A-3)

APPENDIX B

EXACT EXPRESSIONS FOR THE ANISOTROPIC COEFFICIENTS OF TERMS OF THE WEAKNESSES

The exact expressions for the anisotropic coefficients ε(V),δ(V), and γ (V) in terms of the weaknesses 1N and 1T can bederived from equations (9) and (27)–(29) as

ε(V) = − 2g(1− g)1N

1−1N(1− 2g)2, (B-1)

δ(V) =

− 2g[(1− 2g)1N +1T ][1− (1− 2g)1N][1−1N(1− 2g)2

][1+ 1

1− g

(1T −1N(1− 2g)2)] ,

(B-2)

and

γ (V) = −1T

2. (B-3)

The exact anellipticity coefficient η(V) is proportional to thedifference between the shear and normal compliances:

η(V) = (KT − KN)

× 2µ2(λ+ µ)(1+ KN(λ+ 2µ))(λ+ µ)(λ+ 2µ)(1+ 2KNµ)2 + λ2µ(KT − KN)

.

(B-4)

Schoenberg and Sayers (1995) were the first to point out thatthe sign of anellipticity is controlled by the difference KT − KN .


Estimation of fracture parameters from reflection seismic data—Part II:Fractured models with orthorhombic symmetry


ABSTRACTExisting geophysical and geological data indicate that

orthorhombic media with a horizontal symmetry planeshould be rather common for naturally fractured reser-voirs. Here, we consider two orthorhombic models: onethat contains parallel vertical fractures embedded in atransversely isotropic background with a vertical sym-metry axis (VTI medium) and the other formed by twoorthogonal sets of rotationally invariant vertical frac-tures in a purely isotropic host rock.

Using the linear-slip theory, we obtain simple ana-lytic expressions for the anisotropic coefficients of ef-fective orthorhombic media. Under the assumptions ofweak anisotropy of the background medium (for the firstmodel) and small compliances of the fractures, all ef-fective anisotropic parameters reduce to the sum of thebackground values and the parameters associated witheach fracture set. For the model with a single fracturesystem, this result allows us to eliminate the influence ofthe VTI background by evaluating the differences be-tween the anisotropic parameters defined in the verticalsymmetry planes. Subsequently, the fracture weaknesses,which carry information about the density and contentof the fracture network, can be estimated in the sameway as for fracture-induced transverse isotropy with ahorizontal symmetry axis (HTI media) examined in our

previous paper (part I). The parameter estimation pro-cedure can be based on the azimuthally dependent re-flection traveltimes and prestack amplitudes of P-wavesalone if an estimate of the ratio of the P- and S-wavevertical velocities is available. It is beneficial, however,to combine P-wave data with the vertical traveltimes,NMO velocities, or AVO gradients of mode-converted(PS) waves.

In each vertical symmetry plane of the model withtwo orthogonal fracture sets, the anisotropic parametersare largely governed by the weaknesses of the fracturesorthogonal to this plane. For weak anisotropy, the frac-ture sets are essentially decoupled, and their parameterscan be estimated by means of two independently per-formed HTI inversions. The input data for this modelmust include the vertical velocities (or reflector depth)to resolve the anisotropic coefficients in each verticalsymmetry plane rather than just their differences.

We also discuss several criteria that can be used todistinguish between the models with one and two frac-ture sets. For example, the semimajor axis of the P-waveNMO ellipse and the polarization direction of the ver-tically traveling fast shear wave are always parallel toeach other for a single system of fractures, but they maybecome orthogonal in the medium with two fracturesets.

INTRODUCTION

This is the second part of our series of three papers on char-acterization of naturally fractured reservoirs using surface seis-mic data. The main goal of the series is to provide a connectionbetween fracture detection methods and rock physics modelsof fractured media, such as those developed by Hudson (1980,1981, 1988), Schoenberg (1980, 1983), and Thomsen (1995).In particular, we are interested in elucidating the dependence

Manuscript received by the Editor April 20, 1999; revised manuscript received March 1, 2000.∗Formerly St. Petersburg State University, Department of Geophysics, St. Petersburg, Russia. Presently Schlumberger Cambridge Research, HighCross, Madingley Road, Cambridge, CB3 0EL, England. E-mail: [email protected].‡Colorado School of Mines, Center for Wave Phenomena, Department of Geophysics, Golden, Colorado 80401-1887. E-mail: [email protected]; [email protected]© 2000 Society of Exploration Geophysicists. All rights reserved.

of reflection seismic signatures on the physical properties ofthe fractures and in developing inversion algorithms for esti-mating fracture parameters using P-wave and converted (PS)reflection data.

In the first paper of the series (Bakulin et al., 2000; hereafterreferred to as part I), we address these problems for transverseisotropy with a horizontal axis of symmetry (HTI media), whichdescribes a single set of vertical, parallel, rotationally invariantfractures in a purely isotropic background medium. HTI is the

1803

1804 Bakulin et al.

simplest azimuthally anisotropic model that provides valuableinsights into the behavior of seismic signatures over fracturedformations.

However, since the rock matrix usually exhibits some formof anisotropy, it is difficult to expect the HTI model to be ade-quate for typical fractured reservoirs. Here, we extend the ap-proach of part I to more complicated, but likely more realistic,orthorhombic models. The description of seismic signatures inorthorhombic media can be simplified significantly by applyingTsvankin’s (1997b) notation based on the analogy between thesymmetry planes of orthorhombic and TI media. For instance,P-wave velocities and traveltimes are fully controlled by theP-wave vertical velocity and five dimensionless anisotropic co-efficients, rather than by nine stiffnesses in the conventionalnotation. Also, this parameterization yields concise exact ex-pressions for the NMO velocities of pure modes reflected froma horizontal interface (Grechka and Tsvankin, 1998).

As shown by Grechka et al. (1999), eight out of the nine pa-rameters of orthorhombic media with a horizontal symmetryplane can be obtained from the NMO velocities of horizontalP- and split PS-events, provided the vertical velocities or re-flector depth are known. Reflection moveout of P-waves aloneis sufficient to find the orientation of the symmetry planes,the NMO velocities within them, and three anellipticity coef-ficients η. The parameters η, however, can be determined onlyif dipping events or nonhyperbolic moveout is available.

While orthorhombic symmetry may be caused by a varietyof physical reasons, we restrict ourselves to two orthorhombicmodels believed to be most common for fractured reservoirs:(1) a single set of vertical cracks in a transversely isotropicbackground with a vertical symmetry axis (VTI) and (2) twosystems of vertical fractures orthogonal to each other in anisotropic background medium. If the intrinsic anisotropy ofthe host rock and the anisotropy induced by the fractures areweak, the anisotropic coefficients of the background and frac-tures can be algebraically added in describing the effectivemedium. This result provides important insights into the in-fluence of fractures on the effective anisotropy and helps togeneralize the parameter estimation methodology of part I fororthorhombic media. For both orthorhombic models we re-late Tsvankin’s (1997b) anisotropic coefficients to the fracturecompliances and introduce practical fracture characterizationtechniques operating with P- and PS-waves.

MODEL 1: ONE SET OF VERTICAL FRACTURESIN A VTI BACKGROUND

Effective orthorhombic medium

A single system of vertical fractures embedded in a VTImatrix yields an effective orthorhombic medium in which oneof the symmetry planes coincides with the fracture plane. Ac-cording to the linear-slip theory (Schoenberg, 1980, 1983), theeffective compliance tensor of such a medium can be obtainedby simply adding the excess fracture compliances to the com-pliance of the background. [Part I compares the linear-sliptheory with the effective medium theories of Hudson (1980,1981, 1988) and Thomsen (1995).] The inversion of the result-ing compliance tensor yields the stiffness tensor (and the cor-responding two-index stiffness matrix) of the fracture-inducedorthorhombic model. If the fracture faces are perpendicular

to the x1-axis, the effective stiffness matrix c has the followingform (Schoenberg and Helbig, 1997; Appendix A):

c =

c11 c12 c13 0 0 0

c12 c22 c23 0 0 0

c13 c23 c33 0 0 0

0 0 0 c44 0 0

0 0 0 0 c55 0

0 0 0 0 0 c66

=(

c1 0

0 c2

), (1)

where 0 is the 3× 3 zero matrix and c1 and c2 are given by

c1 =

c11b(1−1N) c12b(1−1N) c13b(1−1N)

c12b(1−1N) c11b −1N

c212b

c11b

c13b

(1−1N

c12b

c11b

)

c13b(1−1N) c13b

(1−1N

c12b

c11b

)c33b −1N

c213b

c11b

(2)

and

c2 =

c44b 0 0

0 c44b(1−1V ) 0

0 0 c66b(1−1H )

. (3)

Here ci j b are the stiffness coefficients of the VTI background(constrained by c12b = c11b − 2c66b); 1N , 1V , and 1H are thedimensionless weaknesses of the fractures (Schoenberg andHelbig, 1997; Bakulin and Molotkov, 1998), which change fromzero (no fractures) to unity (extreme fracturing). The tan-gential weaknesses 1V and 1H provide a measure of crackdensity, whereas the normal weakness 1N contains informa-tion about the fluid content of the fractures and possible fluidflow between the fractures and pore space (Schoenberg andDouma, 1988; part I).

Matrix (1) describes a special type of orthorhombic mediawith the stiffnesses satisfying the relation (Schoenberg andHelbig, 1997)

c13(c22 + c12) = c23(c11 + c12). (4)

The existence of the additional constraint (4) stems from thefact that while general orthorhombic media are characterizedby nine independent values of ci j , the fracture-induced modelconsidered here is defined by only eight quantities (for a fixedfracture orientation): five stiffness coefficients (c11b , c13b , c33b ,c44b , and c66b) of the VTI background and three fracture weak-nesses (1N , 1V , and 1H ).

The inversion of surface data for the physical parameters ofthe fractures requires relating seismic signatures to the fractureweaknesses. The results of Grechka and Tsvankin (1999) andGrechka et al. (1999) show that, in orthorhombic media, suchcommonly used signatures as NMO velocities and AVO gradi-ents are most concisely expressed through the dimensionlessanisotropic coefficients introduced by Tsvankin (1997b). Thedefinitions of Tsvankin’s parameters ε(1,2), δ(1,2,3), and γ (1,2) in

Fractured Orthorhombic Media 1805

terms of the stiffness elements ci j are given in Appendix B.Substituting equations (1)–(3) for the stiffnesses of thefracture-induced orthorhombic model into equations (B-1)–(B-9) yields the anisotropic coefficients ε, δ, and γ as functionsof 1N , 1V , and 1H .

Weak-anisotropy approximation

The exact expressions for Tsvankin’s anisotropic parame-ters in terms of the weaknesses, however, are too compli-cated to give insight into the influence of the fractures onthe effective anisotropic model. Therefore, we assume thatThomsen’s (1986) coefficients εb, δb, and γb of the VTI back-ground, along with the weaknesses1N ,1V , and1H , are smallquantities of the same order. Linearization of the effective stiff-ness matrix in these small quantitites [equation (A-17)] showsthat for weak anisotropy the effective anisotropic coefficientsε(1,2), δ(1,2,3), and γ (1,2) should represent the sums of the VTIbackground parameters εb, δb, and γb and the correspondinganisotropic coefficients of the HTI medium resulting from thefractures in an imaginary isotropic host rock sufficiently closeto the VTI background model. This isotropic medium can becharacterized, for instance, by the P- and S-wave velocities VP

and VS equal to the vertical velocities VP0b and VS0b in the VTIbackground.

Since this result provides a theoretical basis for most of thesubsequent discussion, we prove it for one of the anisotropiccoefficients—the parameter ε(2) defined by equation (B-3).Substituting the stiffnesses from equations (1) and (2) intoequation (B-3) yields

ε(2) =c11b − c33b −1N

(c11b −

c213b

c11b

)

2c33b

(1−1N

c213b

c11bc33b

) . (5)

Using the definition of the coefficient εb in the VTI backgroundmedium [εb≡ (c11b−c33b)/(2c33b); see Thomsen (1986)], equa-tion (5) can be rewritten as

ε(2) =εb −1N

c211b− c2

13b

2c11bc33b

1−1N

c213b

c11bc33b

. (6)

Linearizing this equation with respect to εb and 1N , we dropthe anisotropic term in the denominator to obtain

ε(2) ≈ εb −1N

c211b− c2

13b

2c11bc33b

. (7)

Since (c211b− c2

13b)/(c11bc33b) is multiplied by the already small

weakness 1N , it can be replaced in the weak anisotropy ap-proximation by the corresponding isotropic quantity 4g(1− g),where g≡V2

S0b/V2

P0bis the ratio of the squared vertical S- and

P-wave velocities in the background. Thus, equation (7) canbe represented as

ε(2) ≈ εb − 2g(1− g)1N . (8)

The combination −2g(1− g)1N can be recognized as the lin-earization of the anisotropic coefficient ε(V) (Ruger, 1997;

Tsvankin, 1997a) in an HTI medium due to a single set of verti-cal fractures perpendicular to the x1-axis that are embedded ina purely isotropic rock with the squared S-to-P velocity ratiog (part I). Hence, equation (8) implies that

ε(2) ≈ εb + ε(V). (9)

We conclude that in the weak anisotropy limit the anisotropiccoefficient ε(2) reduces to the sum of the Thomsen backgroundparameter εb and the coefficient ε(V) of the HTI model due tothe fractures embedded in an isotropic medium close enoughto the VTI background medium. Similar linearizations for theother anisotropic parameters of the effective orthorhombicmodel are given below.

Symmetry plane [x2, x3] parallel to the fractures.—The lin-earized anisotropic coefficients in the symmetry plane [x2, x3],derived from equations (B-6)–(B-8), have the following form:

ε(1) = εb, (10)

δ(1) = δb, (11)

and

γ (1) = γb + 1V −1H

2. (12)

Therefore, for weak anisotropy coefficients ε(1) and δ(1) coin-cide with those of the VTI background medium. This is anexpected result; if rotationally invariant fractures are intro-duced into an isotropic matrix, the plane [x2, x3] represents theso-called isotropy plane of the effective HTI medium. In theisotropy plane the velocities of all waves are not influencedby the fractures and remain constant for all propagation direc-tions (e.g., Tsvankin, 1997a). The coefficient γ (1) does not coin-cide with γb only because, in contrast to the more conventionalmodel from part I, the fractures considered here are not rota-tionally invariant (i.e., 1V 6=1H ). The difference between 1V

and 1H , however, has no bearing on parameters ε(1) and δ(1).In addition to the basic set of anisotropic parameters intro-

duced by Tsvankin (1997b), it is useful to consider the anel-lipticity coefficients η(1,2,3) responsible for time processing ofP-wave data (Grechka and Tsvankin, 1999). For weak ani-sotropy, the η coefficient in the [x2, x3]-plane given by equa-tion (B-10) is also equal to the background value:

η(1) = ηb. (13)

Symmetry plane [x1, x3] perpendicular to fractures.—Theweak-anisotropy approximations of the anisotropic coeffi-cients in the plane [x1, x3] are given by

ε(2) = εb − 2g(1− g)1N, (14)

δ(2) = δb − 2g[(1− 2g)1N +1V ], (15)

γ (2) = γb − 1H

2, (16)

and

η(2) = ηb + 2g[1V − g1N]. (17)

Each of the expressions (14)–(17) contains two terms witha distinctly different physical meaning. The first term is thecorresponding anisotropic coefficient of the VTI background,

1806 Bakulin et al.

while the second term absorbs the influence of the fractures.As discussed above for ε(2), the fracture-related terms in equa-tions (14)–(17) are approximately equal (assuming 1H =1V )to the coefficients ε(V), δ(V),γ (V), andη(V) (respectively) derivedin part I for a fracture set embedded in an isotropic background.

The constraint on the effective stiffnesses [equation (4)]leads to an additional relationship between the anisotropic co-efficients. In the weak anisotropy limit, equation (4) can berewritten as

γ (2)−γ (1) = 14g

[δ(2)−δ(1)−(ε(2)−ε(1))1− 2g

1− g

]. (18)

It is interesting that the background parameters and the tan-gential compliance 1H have no influence on constraint (18),which depends only on the differences between the anisotropiccoefficients in the vertical symmetry planes, i.e., on ε(V), δ(V),and γ (V). As a result, equation (18) is analogous to the con-straint given by Tsvankin (1997a) and part I for HTI mediadue to rotationally invariant fractures (i.e., for 1H =1V ).

Horizontal symmetry plane [x1, x2].—The only Tsvankin’s(1997b) anisotropic coefficient defined in the horizontal planeis δ(3) [equation (B-9)]. After linearization, it becomes

δ(3) = 2g[1N −1H ]. (19)

The parameter δ(3) does not contain any background aniso-tropic coefficients because [x1, x2] is the isotropy plane of theVTI medium. Since δ(3) is defined with respect to the x1-axis,which is normal to the fractures, equation (19) coincides withthe expression for the generic Thomsen coefficient δ obtainedfor VTI media due to horizontal fractures by Schoenberg andDouma (1988).

The linearized anellipticity coefficient η(3) [equation (B-12)]has the form

η(3) = 2g[1H − g1N]. (20)

Estimation of the anisotropic parameters from reflection data

Inversion of reflection data for the effective parametersof orthorhombic media is discussed by Grechka and Tsvankin(1998, 1999), Ruger (1998), and Grechka et al. (1999). We nowgive a brief overview of their parameter-estimation methodsoperating with either wide-azimuth P-wave data or a combina-tion of azimuthally dependent signatures of P- and PS-waves.

P-wave signatures.—NMO velocity of P-waves in a hori-zontal orthorhombic layer is described by an ellipse with theaxes in the vertical symmetry planes (Grechka and Tsvankin,1998). Since for both orthorhombic models considered herethe orientation of the symmetry planes is determined bythe strike of the fractures, the axes of the P-wave NMO el-lipse yield the fracture azimuth(s). In the orthorhombic modeldue to a single fracture set, δ(1) >δ(2) [see equation (32)] andthe semimajor axis of the P-wave NMO ellipse points in thedirection of the fracture plane. This result holds for horizontaltransverse isotropy as well (part I).

If δ(1)= δ(2) and the P-wave NMO velocity from a horizontalreflector is azimuthally independent (i.e., the ellipse degener-ates into a circle), the azimuths of the symmetry planes can beobtained from the NMO ellipse of a dipping event (Grechka

and Tsvankin, 1999). P-wave reflection traveltimes from dip-ping interfaces or the azimuthal variation of nonhyperbolicmoveout (Al-Dajani et al., 1998) can also be inverted for theanellipticity coefficients η(1), η(2), and η(3) [equations (13), (17),and (20)]. Nonnegligible values of both η(1) and η(2) help todetect the presence of anisotropy in the background and dis-criminate between HTI and orthorhombic models. For HTImedia, all anisotropic coefficients (including η) in one of thesymmetry planes should go to zero.

The semiaxes V (1)P,nmo and V (2)

P,nmo of the P-wave NMO el-lipse from a horizontal reflector are given by (Grechka andTsvankin, 1998)

V (i )P,nmo = VP0

√1+ 2δ(i ), (i = 1, 2). (21)

Hence, V (i )P,nmo can be combined with the vertical velocity VP0

to determine δ(1) and δ(2). If VP0 is unknown, the P-wave NMOellipse constrains the difference χ between the two δ coeffi-cients:

χ ≡(V (2)

P,nmo

)2 − (V (1)P,nmo

)2(V (2)

P,nmo

)2 + (V (1)P,nmo

)2 =δ(2) − δ(1)

1+ δ(2) + δ(1)≈ δ(2)−δ(1).

(22)

Additional information for parameter estimation is providedby prestack amplitudes of P-waves. In the weak-anisotropyapproximation, the variation of the P-wave amplitude varia-tion with offset (AVO) gradient between the vertical symmetryplanes is governed by the expression δ(2)− δ(1)− 8g(γ (2)− γ (1))(Ruger, 1998). Therefore, if the squared velocity ratio g isknown and δ(2)− δ(1) has been found from the P-wave NMOellipse [equation (22)], the AVO gradient yields an estimate ofγ (2)− γ (1).

P- and PS-wave signatures.—Although P-wave data alonemay be used to estimate a subset of the effective parameters oforthorhombic media, it is highly beneficial to combine P-wavetraveltimes or amplitudes with the signatures of shear or con-verted waves. Since S-waves are not generated in most explo-ration surveys, we emphasize the joint inversion of P-wavesand P- to S-mode conversions. The vertical traveltimes of P-and PS-waves give a direct estimate of the vertical-velocityratio needed to compute the fracture weaknesses. The shear-wave splitting parameter at vertical incidence, conventionallyevaluated from the time delays between the fast and slow shearor converted waves, is close to γ (2) − γ (1). Also, the differencebetween the AVO gradients of the PS-wave in the vertical sym-metry planes can be combined with the azimuthal variation ofthe P-wave AVO gradient or the P-wave NMO ellipse to ob-tain another estimate of γ (2) − γ (1). A more detailed discus-sion of the azimuthal AVO inversion of PS-waves is given inpart I.

Grechka et al. (1999) outline the following methodology ofthe joint moveout inversion of P- and PS-waves. Similar topure modes, the azimuthally varying NMO velocity of eithersplit PS-wave in a horizontal orthorhombic layer is describedby an ellipse aligned with the vertical symmetry planes. Thesemiaxes of the NMO ellipses of the P- and two split PS-waves(i.e., the symmetry-plane NMO velocities) can be used to re-construct the NMO ellipses of the pure shear waves S1 and S2.Assuming that wave S1 represents an SV (in-plane polarized)mode in the [x2, x3]-plane [the superscript (1)], the semiaxes of


the shear-wave ellipses are expressed as

V (2)S1, nmo = VS1

√1+ 2γ (2) = V (1)

S2, nmo, (23)

V (1)S1, nmo = VS1

√1+ 2σ (1), (24)

V (2)S2, nmo = VS2

√1+ 2σ (2), (25)

and

V (1)S2, nmo = VS2

√1+ 2γ (1) = V (2)

S1, nmo. (26)

Here σ (1) and σ (2) are the anisotropic coefficients

σ (1) ≡(

VP0

VS1

)2(ε(1) − δ(1)) (27)

and

σ (2) ≡(

VP0

VS2

)2(ε(2) − δ(2)), (28)

and VS1 and VS2 are the vertical velocities of the split shearwaves S1 and S2:

VS1 = VS2

√1+ 2γ (1)

1+ 2γ (2)(29)

and

VS2 = VS0. (30)

If one of the vertical velocities or the reflector depth is knownand the anisotropic parameters δ(1,2) have been obtained fromP-wave moveout, the shear-wave NMO ellipses make it pos-sible to find four additional coefficients—ε(1,2) and γ (1,2). Theonly anisotropic parameter not constrained by conventional-spread normal moveout in a horizontal orthorhombic layer isδ(3).

Estimation of fracture parameters

Inversion for the weaknesses.—Since we are mostly inter-ested in evaluating the properties of the fracture set, it is con-venient to remove the influence of the background medium atthe outset of the inversion procedure. Inspection of equations(10)–(17) shows that the contribution of the VTI backgroundparameters can be eliminated by computing the difference be-tween the anisotropic coefficients in the planes parallel andorthogonal to the fractures:

ε(2) − ε(1) = −2g(1− g)1N, (31)

δ(2) − δ(1) = −2g[(1− 2g)1N +1V ] ≈ χ, (32)

γ (2) − γ (1) = −1V

2, (33)

η(2) − η(1) = 2g[1V − g1N]. (34)

Equations (31)–(34) are identical to the expressions for ε(V),δ(V), γ (V), and η(V) (respectively) in HTI media due to a singleset of vertical rotationally invariant fractures with the weak-nesses 1N and 1V (part I). Therefore, the weaknesses can bedetermined from equations (31)–(34) using the HTI expres-sions described in detail in part I. We emphasize that obtaining1N and 1V requires knowledge of any two of the differencesε(2) − ε(1), δ(2) − δ(1), γ (2) − γ (1), and η(2) − η(1).

As discussed, χ ≈ δ(2)− δ(1) can be estimated from the elon-gation of the P-wave NMO ellipse for horizontal events [equa-tion (22)]. Also,η(2) andη(1) can be obtained using the NMO ve-locities of dipping P-events or nonhyperbolic moveout. Thus,P-wave moveout data provide sufficient information to deter-mine the weaknesses1N and1V from equations (32) and (34):

1N = −(δ(2) − δ(1)

)+ (η(2) − η(1))

2g(1− g), (35)

1V = 12(1− g)

[1− 2g

g

(η(2) − η(1))− (δ(2) − δ(1))].

(36)In principle, the inversion based on equations (35) and (36)requires knowledge of the squared vertical-velocity ratio g,which cannot be found without shear or converted-wave data.However, numerical tests (and the results of part I) show thateven a relatively rough estimate of g is sufficient for recoveringthe weaknesses with acceptable accuracy.

P-wave moveout data may provide not just the differencebetween η(2) and η(1) but also the individual values of these co-efficients. Therefore, in addition to substituting η(2)− η(1) intothe equations for the weaknesses, we can use the approximaterelation (13), η(1)= ηb, to determine the η coefficient in thebackground VTI model.

If the P-wave moveout information is limited to the NMOellipse from a horizontal reflector, it is possible to obtain thedifference γ (2)− γ (1) by including the P-wave AVO gradientsin the directions parallel and perpendicular to the fractures.The algorithm based on the NMO ellipses of horizontal eventsand AVO gradients is identical to the one described in part Ifor HTI media. The presence of anisotropy in the backgroundhas no influence either on estimating δ(2)− δ(1) and γ (2)− γ (1)

or on inverting these differences [equations (32) and (33)] forthe weaknesses 1N and 1V .

In multicomponent surveys, the vertical traveltimes of PS-waves (in combination with P-wave data) provide estimates ofg and the splitting coefficient γ (2)− γ (1). Thus, another possi-ble set of input parameters includes the P-wave NMO ellipsefrom a horizontal reflector (yielding δ(2)− δ(1)) and the verticaltraveltimes of converted modes.

Note that the weakness1H does not enter any of the differ-ences (31)–(34), despite the fact that it appears in equations(12) and (16) for γ (1) and γ (2). However, even the individualvalues of γ (1) and γ (2) constrain the combination γb−1H/2 butnot 1H separately. The only source of information about 1H

is η(3) or δ(3) [equations (19) and (20)], which can be estimatedusing the NMO velocities of P-wave reflections from dippinginterfaces (Grechka and Tsvankin, 1999):

1H = η(3)

2g+ g1N . (37)

If dipping events are not available, the data cannot be in-verted for the coefficient η(3) and, therefore, for the weakness1H . In this case, a reasonable simplifying assumption is thatthe fractures are rotationally invariant and1V =1H . Then thenumber of independent medium parameters reduces to seven,and δ(3) and η(3) can be expressed as

δ(3) = δ(2) − δ(1) − 2(ε(2) − ε(1)), (38)

η(3) = η(2) − η(1). (39)

1808 Bakulin et al.

Interpretation of the weaknesses in terms of the physicalproperties of the fractures is discussed in part I. For rotation-ally invariant penny-shaped cracks, the tangential weakness1V =1H gives an estimate of the crack density, while the nor-mal weakness 1N represents a sensitive, albeit nonunique, in-dicator of fluid content.

Numerical examples.—Although the weak-anisotropy ap-proximations provide useful insight into the parameter esti-mation problem, it is preferable to use the exact equations foractual inversion. In the examples below, we compute the frac-ture weaknesses 1N , 1V , and 1H , along with the anisotropiccoefficient ηb of the background, by inverting the values of χ[equation (22)], η(1), η(2), and η(3) presumably extracted fromP-wave seismic data. The goal of this numerical study is to ex-amine the sensitivity of the inverted parameters to errors in theinput data and in the parameters of the background medium.

Our nonlinear inversion algorithm is based on equations (1)–(3) for the stiffness elements and on the exact expressions forthe anisotropic coefficients given in Appendix B. The weak-anisotropy approximations (13) and (35)–(37) are used onlyto obtain the initial guesses for the weaknesses and the co-efficient ηb. The parameters of the background medium areg=V2

S0b/V2P0b= 0.25, δb= 0.2, andγb = 0.1 (εb can be expressed

through δb and ηb). Although the only background parame-ter in equations (35)–(37) is the squared velocity ratio g, theanisotropic coefficients δb and γb of the VTI medium are con-tained in the exact equations for the stiffnesses and effectiveanisotropic parameters and must be specified for the inversion.

FIG. 1. Numerical inversion of χ , η(1), η(2), and η(3) for the fracture weaknesses 1N (dotted line), 1V (dash-dotted), 1H (dashed),and the coefficient ηb of the background medium (solid). Errors of ±0.1 were introduced in (a) χ , (b) η(1), (c) η(2), and (d) η(3),with the other input parameters held at the correct values. In the absence of errors, the inversion yields the correct values of theweaknesses and ηb (large dots).

As shown below, however, the inversion results are insensitiveto variations of g, δb, and γb within the range important in prac-tice. The fracture set is defined by the weaknesses1N = 0.5 and1V =1H = 0.2, which approximately correspond to values forpenny-shaped gas-filled cracks (see part I).

The inversion results are displayed in Figure 1. For the cor-rect input parameters (and the correct background parametersg, δb, and γb), the inversion algorithm produces accurate valuesof the weaknesses and ηb. Errors in the measured anisotropicparameters of up to±0.1 result in similar (often smaller) errorsin the inverted values, indicating that the parameter estimationprocedure is reasonably stable. As expected from the analyticresults, different inverted parameters are most sensitive to er-rors in different measured quantities. For example, errors in χproduce comparable distortions in all three weaknesses, reach-ing and sometimes exceeding±0.1 (Figure 1a). The anelliptic-ity coefficient ηb is quite sensitive to errors in η(1) (Figure 1b)and is almost independent of the other input parameters, inaccordance with the weak-anisotropy approximation (13). Er-rors in η(1) and η(2) of about ±0.1 cause smaller errors (±0.05)in the tangential weaknesses 1V and 1H than in the normalweakness 1N (±0.1; Figures 1b,c). The weakness 1H is sen-sitive primarily to errors in η(3) (Figure 1d), as suggested byequation (37). In the weak-anisotropy limit,1N and1V are in-dependent of η(3) [equations (35) and (36)], which is confirmedby Figure 1d.

In the second test, we examine the influence of errors inthe background parameters g, δb, and γb (which were assumedknown a priori in the previous example) on the inversion


results. Substantial errors in g of ±0.1, or up to 40%, leadto relatively small distortions of only about ±0.03 in all in-verted parameters except 1N (for 1N the errors reach ±0.1;Figure 2a). The anisotropic coefficients δb and γb have an evensmaller influence on the estimated quantities, especially on thetangential compliances (Figure 2b,c). This result is especiallyencouraging because δb and γb are difficult to estimate fromsurface seismic data without information about the vertical ve-locities or reflector depth.

Estimation of background parameters.—As discussedabove, the individual values of the anisotropic parameters ε(1,2),δ(1,2), and γ (1,2) can be found by combining P-wave moveoutdata with the reflection traveltimes of PS-waves (provided thereflector depth is known). Then it is possible to estimate not justthe fracture weaknesses but also the squared vertical-velocityratio g and the background anisotropic parameters εb, δb, andγb [equations (10)–(12); to find γb, we assume 1V =1H ].

For the numerical example in Figure 3, the input data in-cluded the vertical and NMO velocities of P- and split S-waves,where the shear-wave signatures were supposedly determinedfrom P and PS data. The NMO velocity of each mode wascomputed in three azimuthal directions with a step of 45◦ andapproximated with an ellipse to find the NMO velocities in thesymmetry planes. Then the vertical velocities and symmetry-plane NMO velocities were combined to determine ε(1,2), δ(1,2),

FIG. 2. Influence of the background parameters (a) g=V2S0b/V2

P0b, (b) δb, and (c) γb on the inversion results from Figure 1 for the

correct values of χ and η(1,2,3). Dotted line—1N ; dash-dotted—1V ; dashed—1H ; solid—ηb.

and γ (1,2) from equations (21) and (23)–(26). To check the in-fluence of measurement errors, we added Gaussian noise witha standard deviation of 2% to the vertical and NMO velocitiesand performed the inversion for 200 sets of the distorted inputparameters. The algorithm is based on the exact expressionsfor the anisotropic coefficients derived from equations (1)–(3),with the starting model computed using the weak-anisotropyapproximation. Since the inversion errors in Figure 3 are lim-ited by±0.05, the estimation of both weaknesses and all back-ground parameters is sufficiently stable.

FIG. 3. Inversion of the vertical and NMO velocities of the P-and split S-waves for the parameters of the fractures and VTIbackground. The dots mark the correct values of the parame-ters; it is assumed that1V =1H . The bars correspond to± onestandard deviation in the inverted quantities caused byGaussian noise in the input data with a standard deviationof 2%.

1810 Bakulin et al.

MODEL 2: TWO ORTHOGONAL FRACTURESETS IN ISOTROPIC ROCK

Another practically important fractured model with or-thorhombic symmetry is composed of two orthogonal verticalfracture sets embedded in an isotropic or VTI background. Tosimplify the parameter estimation procedure, we assume thatthe fractures are rotationally invariant and the background ma-trix is purely isotropic. Choosing the x1-axis to be perpendicularto the first set of fractures, we obtain the compliance matricesfor both fracture systems in equations (A-3) and (A-14) where,because of the rotational invariance, the compliances satisfyKV1= KH1≡ KT1 and KV2= KH2≡ KT2. Building the effectivematrix of the excess compliance [equation (A-12)] and addingthe compliance of the isotropic background yields the effectivestiffness matrix [equation (A-1)]

c =

c11 c12 c13 0 0 0

c12 c22 c23 0 0 0

c13 c23 c33 0 0 0

0 0 0 c44 0 0

0 0 0 0 c55 0

0 0 0 0 0 c66

= ˜c1 0

0 ˜c2

,

(40)

where 0 is the 3× 3 zero matrix and the matrices ˜c1 and ˜c2 aregiven by

˜c1 =

1d

(λ+ 2µ)l1m3 λl1m1 λl1m2

λl1m1 (λ+ 2µ)l3m1 λl2m1

λl1m2 λl2m1 (λ+ 2µ)(l3m3− l4)

,(41)

˜c2 =µ(1−1T2) 0 0

0 µ(1−1T1) 0

0 0 µ(1−1T1)(1−1T2)

(1−1T11T2)

.(42)

Here, λ and µ are the Lame parameters of the backgroundmedium and

l1 = 1−1N1, l2 = 1− r1N1, l3 = 1− r 21N1,

l4 = 4r 2g21N11N2,

m1 = 1−1N2, m2 = 1− r1N2, m3 = 1− r 21N2,

(43)g = µ/(λ+ 2µ) = V2

S

/V2

P, r = 1− 2g,

d = 1− r 21N11N2.

The values1Ni and1T i (i = 1, 2) are the normal and tangentialfracture weaknesses (see part I) related to the fracture compli-ances by the equations

1Ni = KNi (λ+ 2µ)1+ KNi (λ+ 2µ)

(44)

and

1T i = KT iµ

1+ KT iµ, (45)

which represent a special case (valid for an isotropic back-ground) of equations (A-4).

Since both fracture planes and the horizontal plane consti-tute three orthogonal planes of mirror symmetry, the effectivemedium must be orthorhombic with the nine stiffness elements[equation (40)]. Our model, however, represents a special caseof general orthorhombic media with only six independent pa-rameters: λ, µ,1N1,1T1,1N2, and1T2. As follows from equa-tions (41) and (42), the three additional relationships (con-straints) between the stiffnesses are

c12(c33 + c23) = c13(c22 + c23), (46)

2 (c11 + c13) =(c11c33 − c2

13

)(c44 + c55

c44c55− 1

c66

), (47)

and

2(c22 + c23) = (c22c33 − c223

)(c44 + c55

c44c55− 1

c66

). (48)

Models with two identical orthogonal fracture sets, character-ized by fewer independent parameters, are discussed in Ap-pendix C.

Anisotropic coefficients in the weak-anisotropy limit

Equations (40)–(43), combined with the definitions fromAppendix B, can be used to express Tsvankin’s (1997b)anisotropic coefficients in terms of the fracture weaknesses1T i

and 1Ni . Here we restrict ourselves to linearized expressionsobtained in the limit of small fracture compliances 1T1,2¿ 1and1N1,2¿ 1. The result of this linearization can be predictedfrom equation (A-13): since in the linear approximation frac-ture systems are not influenced by each other, the effective or-thorhombic medium is composed of two HTI media producedby each fracture set individually.

Symmetry plane [x2, x3].—The anisotropic parameters withthe superscript (1) are defined in the symmetry plane [x2, x3],which is parallel to the first set of fractures and orthogo-nal to the second one. In the absence of the second set, the[x2, x3]-plane would coincide with the isotropy plane of theHTI medium associated with the first fracture system. There-fore, we can expect these parameters to be largely influencedby the second set of fractures orthogonal to the x2-axis. In-deed, the linearized ε, δ, γ , and η coefficients in the [x2, x3]plane depend only on the weaknesses 1N2 and 1T2:

ε(1) = −2g(1− g)1N2, (49)

δ(1) = −2g[(1− 2g)1N2 +1T2], (50)

γ (1) = −1T2

2, (51)

η(1) = 2g[1T2 − g1N2], (52)

where g≡V2S/V2

P is the ratio of the squared S- and P-wave ve-locities in the background. Equations (49)–(52) coincide withthe expressions for the anisotropic coefficients ε(V), δ(V), γ (V),and η(V) of the HTI model associated with the second fractureset.


Symmetry plane [x1, x3].—Likewise, the linearized aniso-tropic coefficients in the symmetry plane [x1, x3] are governedby the weaknesses of the first fracture set. As follows from thesymmetry of the model, the expressions for ε(2), δ(2), γ (2), andη(2) are fully analogous to equations (49)–(52):

ε(2) = −2g(1− g)1N1, (53)

δ(2) = −2g[(1− 2g)1N1 +1T1], (54)

γ (2) = −1T1

2, (55)

η(2) = 2g[1T1 − g1N1]. (56)

Symmetry plane [x1, x2].—The remaining anisotropic coef-ficient δ(3) defined in the horizontal symmetry plane [x1, x2] isgiven by

δ(3) = 2g[1N1−1T1]− 2g[(1− 2g)1N2+1T2]. (57)

The linearized parameter δ(3) is equal to the sum of the genericThomsen coefficient δ caused by horizontal fractures in VTImedia (the term 2g[1N1 −1T1]; see Schoenberg and Douma,1988) and the HTI coefficient δ(V) is due to vertical fractures(part I). To explain this result, recall that δ(3) is defined withrespect to the x1-axis, which is orthogonal to the first fracture set(so this set becomes horizontal if the x1-axis is made vertical inVTI media) and lies in the planes of the second set of fractures(making this set vertical).

The anellipticity coefficient η(3) [equation (B-12] in the hor-izontal plane is

η(3) = η(1) + η(2). (58)

Relations between anisotropic coefficients.—The nine stiff-nesses of the effective orthorhombic model contain only six in-dependent quantities, which leads to the three constraints (46)–(48). Rewriting equations (46) and (47) through Tsvankin’s(1997b) parameters yields

γ (i ) = 14g

[δ(i ) − ε(i ) 1− 2g

1− g

], (i = 1, 2), (59)

where quadratic and higher-order terms in the anisotropic co-efficients were dropped. The same result can be obtained fromthe linearized expressions for the anisotropic coefficients givenabove. Equation (59) is identical to the relationship betweenε(V), δ(V), and γ (V) of HTI media (Tsvankin, 1997a; part I).

The remaining constraint (48) takes the form

δ(3) = δ(1) + δ(2) − 2ε(2). (60)Equations (59) and (60) show that in Tsvankin’s notation ourorthorhombic model can be fully described by the two verticalvelocities and four anisotropic coefficients δ(1,2) and ε(1,2).

In the special case of penny-shaped gas-filled fractures, thenormal and tangential compliances are equal to each other(KN1= KT1 and KN2= KT2) and, as noticed by Schoenberg andSayers (1995), the anellipticity parameters in the symmetryplanes go to zero:

η(1) = η(2) = η(3) = 0. (61)


The structure of equations (49)–(56) for the anisotropic co-efficients suggests that estimation of the weaknesses of the

orthogonal fracture sets can be decomposed into two HTI-type inversions in the vertical symmetry planes discussed inpart I. Here, we verify the accuracy of the weak-anisotropy ap-proximations and outline two possible strategies for obtainingthe weaknesses from surface reflection data. One is based onazimuthally dependent P-wave reflection moveout alone; theother uses both P- and converted-wave data. In contrast to themodel with a single fracture set, we must know one of the ver-tical velocities (or reflector depth) because the inversion algo-rithm requires the individual values of Tsvankin’s anisotropiccoefficients rather than their differences.

P-wave inversion.—As for the model with a single fractureset, the fracture orientation can be determined directly fromthe P-wave NMO ellipse (or the NMO ellipses of convertedwaves), unless the ellipse degenerates into a circle. This andsome other special cases are discussed in Appendix D. Supposethe parameters δ(1) and δ(2) have been found using the semiaxesof the P-wave NMO ellipse from a horizontal reflector and thevertical velocity [equations (21)]. Combining the δ coefficientswith the anellipticity parameters η(1) and η(2) (also determinedfrom P-wave moveout) allows us to solve equations (50), (52),(54), and (56) for the weaknesses

1N1 = −δ(2) + η(2)

2g(1− g), (62)

1T1 = 12(1− g)

[1− 2g

gη(2) − δ(2)

], (63)

1N2 = −δ(1) + η(1)

2g(1− g), (64)

and

1T2 = 12(1− g)

[1− 2g

gη(1) − δ(1)

]. (65)

The equations for each pair of weaknesses [(62)– (63) and (64)–(65)] are identical to those obtained in part I for HTI mediadue to a single set of rotationally invariant fractures.

To test the accuracy of the weak anisotropy approximations(62)–(65), we computed the exact anisotropic coefficients δ(1,2)

and η(1,2) for three fractured models in Table 1 [using equations(40)–(42) and the definitions from Appendix B] and invertedthem for the weaknesses in the limit of weak anisotropy. Table1 shows that approximations (62)–(65) give reasonably goodestimates of the fracture weaknesses. The only significant er-ror, in the tangential weakness of the second fracture set, isunrelated to the first set because it remains the same for thecorresponding HTI model (third row in Table 1).

Inversion of P- and PS-wave data.—If dipping events arenot available and CMP spreads are not sufficiently long forusing nonhyperbolic moveout, it may be possible to findthe anisotropic coefficients ε(1,2), δ(1,2), and γ (1,2) by combin-ing P- and PS-wave data [equations (23)–(26)]. Since theseanisotropic parameters depend on only four fracture weak-nesses, P- and PS-wave data (including the vertical veloci-ties or reflector depth) provide useful redundancy in the in-version procedure. In the weak-anisotropy approximation, the

1812 Bakulin et al.

weaknesses can be computed, for example, as

1T1 = −2γ (2) = 1−(

V (2)S1,nmo

VS1

)2

, (66)

1T2 = −2γ (1) = 1−(

V (1)S2,nmo

VS2

)2

, (67)

1N1 = − 11− 2g

[1T1 + δ

(2)

2g

], (68)

1N2 = − 11− 2g

[1T2 + δ

(1)

2g

]. (69)

Figure 4 shows the results of numerical inversion basedon NMO equations (23)–(26) and the exact expressions forTsvankin’s anisotropic coefficients in terms of the fractureweaknesses. Similar to the numerical example in Figure 3,the vertical velocities and azimuthally varying NMO veloci-ties of P- and S-waves (input data) were distorted by Gaussiannoise with a standard deviation of 2%. The initial model wasfound from the weak-anisotropy approximations (66)–(69).The standard deviations for the inverted weaknesses are rela-tively small, with errors in the tangential compliances1T1 and1T2 being somewhat lower than those in 1N1 and 1N2.


Applying the linear-slip theory developed bySchoenberg (1980, 1983), we studied two types of orthorhom-bic media believed to be representative of naturally fracturedreservoirs. The first model contains a single set of vertical frac-tures embedded in a VTI background (e.g., the background

Table 1. Comparison of the actual fracture parameters withthose estimated by inverting the exact anisotropic coefficients[equations (40)–(42) and Appendix B] using the linearizedequations (62)–(65); the squared velocity ratio g = 0.25. Thefirst model is composed of two orthogonal fracture sets in anisotropic background; the second and third models contain onlyone fracture set (i.e., they have the HTI symmetry).

Model Fracture parameters 1N1 1T1 1N2 1T2

1 Actual 0.30 0.15 0.60 0.30Estimated 0.28 0.14 0.66 0.21

2 Actual 0.30 0.15 0 0Estimated 0.30 0.14 0 0

3 Actual 0 0 0.60 0.30Estimated 0 0 0.67 0.21

FIG. 4. Inversion of the P- and S-wave vertical and NMO ve-locities for the model with two orthogonal fracture sets. Theinput data were contaminated by Gaussian noise with a stan-dard deviation of 2%, and the inversion was repeated 200 timesfor different realizations of the input parameters. The bars cor-respond to± one standard deviation in the inverted quantities.

anisotropy may result from fine layering), while the second isproduced by two orthogonal systems of rotationally invariantvertical fractures in an isotropic host rock. For both effectivemodels we obtained Tsvankin’s (1997b) anisotropic parame-ters, which capture the combinations of the stiffness coefficientsresponsible for commonly measured seismic signatures.

To gain a better understanding of the influence of fractureson the effective medium, we simplified the anisotropic param-eters under the assumption of weak background and fracture-induced anisotropy. For the model with a single fracture set,the anisotropic parameters of the HTI medium due to the frac-tures are added to the background coefficients to produce thelinearized effective parameters of the orthorhombic medium.Therefore, by computing the difference between the effectivecoefficients defined in the vertical symmetry planes, we elim-inate the influence of the background and reduce the inverseproblem to that for horizontal transverse isotropy (part I). Theinformation necessary for this inversion procedure can be ob-tained from azimuthally dependent P-wave reflection travel-times alone if dipping events or nonhyperbolic moveout areavailable (also, it is necessary to have an estimate of the ra-tio of the P- and S-wave vertical velocities). Alternatively, theinversion can be performed by combining the P-wave NMO el-lipse from a horizontal reflector with other data, such as the az-imuthally varying P-wave AVO gradient or the vertical travel-times and, possibly, NMO velocities of the split converted (PS)modes. For weak anisotropy, the algorithm based on the NMOellipses and AVO gradients of P- and PS-waves reflected fromhorizontal interfaces is identical to that outlined in part I forHTI media (i.e., it is independent of the presence of anisotropyin the background).

In the case of two orthogonal fracture sets, the linearizedexpressions for the effective anisotropic coefficients in eachvertical symmetry plane contain only the contribution of thefractures orthogonal to this plane. As a result, the inversionfor the fracture weaknesses splits into two separate inversionprocedures in the symmetry planes, which can be carried outusing the HTI algorithm of part I. In contrast to the modelwith a single fracture set, however, determination of the weak-nesses of both fracture sets requires knowledge of the verticalvelocities in addition to the azimuthally varying surface seismicsignatures.

Although both effective media considered here have or-thorhombic symmetry, the results of our analysis indicate sev-eral possible ways to identify the correct underlying physicalmodel. In both cases, the stiffness tensor is described by fewerindependent parameters than for general orthorhombic media,and the anisotropic coefficients for the models with one andtwo fracture sets satisfy different constraints. Another usefulcriterion is the sign of the anisotropic parameters. The coeffi-cients ε(1,2), δ(1,2), and γ (1,2) for the model with two orthogonalsystems of fractures in an isotropic background are negative[equations (49)–(51) and (53)–(55)]. In contrast, ε and γ (andoften δ as well) defined in the fracture plane of the model witha single fracture set in a VTI background are usually positive.Also, we can distinguish between the two models by compar-ing the polarization direction of the vertically traveling fastshear wave with the orientation of the semimajor axis of the P-wave NMO ellipse. For one set of fractures (in either isotropicor VTI background), they always coincide with each other;for two systems of fractures with different fluid content, theymay become orthogonal. This happens, for example, if the two


inequalities1T1 >1T2 and (1− 2g)1N1+1T1 < (1− 2g)1N2+1T2 (or, equivalently, δ(2) >δ(1)) are satisfied simultaneously.

ACKNOWLEDGMENTS

This research was carried out during a visit of AndreyBakulin to the Center for Wave Phenomena (CWP), Col-orado School of Mines, 1998. We are grateful to members ofthe A(nisotropy)-team of CWP for helpful discussions and toAndreas Ruger (Landmark) for his review of the manuscript.The support for this work was provided by the members ofthe Consortium Project on Seismic Inverse Methods for Com-plex Structures at CWP and by the U.S. Department of Energy(award #DE-FG03-98ER14908).

REFERENCES

Al-Dajani, A., Tsvankin, I., and Toksoz, N., 1998, Nonhyperbolicreflection moveout in azimuthally anisotropic media: 68th Ann.Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1479–1482.

Alkhalifah, T., and Tsvankin, I., 1995, Velocity analysis in transverselyisotropic media: Geophysics, 60, 1550–1566.

Bakulin, A. V., and Molotkov, L. A., 1998, Effective models of frac-tured and porous media: St. Petersburg Univ. Press (in Russian).

Bakulin, A., Tsvankin, I., and Grechka, V., 2000, Estimation of fractureparameters from reflection seismic data–Part I: HTI model due to asingle fracture set: Geophysics, 65, 1788–1802, this issue.


——— 1999, 3-D moveout velocity analysis and parameter estimationfor orthorhombic media: Geophysics, 64, 820–837.

Grechka, V., Theophanis, S., and Tsvankin, I., 1999, Joint inversion ofP- and PS-waves in orthorhombic media: Theory and a physical-modeling study: Geophysics, 64, 146–161.

Hudson, J. A., 1980, Overall properties of a cracked solid: Math. Proc.

Camb. Phil. Soc., 88, 371–384.——— 1981, Wave speeds and attenuation of elastic waves in material

containing cracks: Geophys. J. Roy. Astr. Soc., 64, 133–150.——— 1988, Seismic wave propagation through material containing

partially saturated cracks: Geophys. J., 92, 33–37.Molotkov, L. A., and Bakulin, A. V., 1997, An effective model of a

fractured medium with fractures modeled by the surfaces of discon-tinuity of displacements: J. Math. Sci., 86, 2735–2746.


Ruger, A., 1997, P-wave reflection coefficients for transverselyisotropic models with vertical and horizontal axis of symmetry: Geo-physics, 62, 713–722.

——— 1998, Variation of P-wave reflectivity with offset and azimuthin anisotropic media: Geophysics, 63, 935–947.


——— 1983, Reflection of elastic waves from periodically stratifiedmedia with interfacial slip: Geophys. Prosp., 31, 265–292.

Schoenberg, M., and Douma, J., 1988, Elastic wave propagation in me-dia with parallel fractures and aligned cracks: Geophys. Prosp., 36,571–590.

Schoenberg, M., and Helbig, K., 1997, Orthorhombic media: Modelingelastic wave behavior in a vertically fractured earth: Geophysics, 62,1954–1974.




——— 1995, Elastic anisotropy due to aligned cracks in porous rock:Geophys. Prosp., 43, 805–830.

Tsvankin, I., 1997a, Reflection moveout and parameter estimation forhorizontal transverse isotropy: Geophysics, 62, 614–629.

——— 1997b, Anisotropic parameters and P-wave velocity for or-thorhombic media: Geophysics, 62, 1292–1309.

Winterstein, D. F., 1990, Velocity anisotropy terminology for geophysi-cists: Geophysics, 55, 1070–1088.

APPENDIX A

COMPLIANCE FORMALISM FOR FRACTURED MEDIA

Here we review both exact and approximate methods ofobtaining effective parameters of fractured media using theresults of Schoenberg (1980, 1983), Schoenberg and Douma(1988), Schoenberg and Muir (1989), Nichols et al. (1989), andMolotkov and Bakulin (1997). A discussion of different ap-proaches to effective medium theory for fractured models canbe found in part I.

To simplify the derivation of the effective elastic parametersof fractured media, it is convenient to use the compliances s in-stead of the stiffnesses c. The effective compliance matrix of afractured medium can be written as the sum of the backgroundcompliance sb and the so-called matrix of excess fracture com-pliance s f :

c−1 ≡ s = sb + s f . (A-1)

If the background is VTI, then the stiffness matrix is given by

cb ≡ s−1b =

c11b c12b c13b 0 0 0

c12b c11b c13b 0 0 0

c13b c13b c33b 0 0 0

0 0 0 c44b 0 0

0 0 0 0 c44b 0

0 0 0 0 0 c66b

,

(A-2)

where c12b= c11b− 2c66b. The matrix s f of the excess compli-ance of a fracture set with the normal in the x1-direction canbe written as

s f =

KN 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 KV 0

0 0 0 0 0 KH

, (A-3)

where KN is the normal fracture compliance and KV and KH

are the two shear compliances in the vertical and horizontal di-rections. The matrix s f is no longer diagonal if the fractures arecorrugated (part I; for more details, see part III of this series).Since KV 6= KH , fractures described by matrix (A-3) are some-times called orthorhombic (Schoenberg and Douma, 1988).

It is convenient to replace the excess fracture compliancesKN , KV , and KH by the following dimensionless quantities in-troduced by Schoenberg and Helbig (1997):

1N = KNc11b

1+ KNc11b, 1V = KVc44b

1+ KVc44b,

(A-4)1H = KH c66b

1+ KH c66b.

1814 Bakulin et al.

Then equation (A-3) becomes

s f =

1N

c11b(1−1N)0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 01V

c44b(1−1V )0

0 0 0 0 01H

c66b(1−1H )

.

(A-5)

We call 1N , 1V , and 1H the normal, vertical, and hori-zontal weaknesses introduced by the fractures (Bakulin andMolotkov, 1998). The weaknesses vary from 0 to 1, with thezero value corresponding to unfractured media and unity de-scribing heavily fractured media in which the P- (for 1N = 1)or S-wave (for 1V = 1 or 1H = 1) velocity vanishes for propa-gation across the fractures (part I).

Substituting equations (A-2) and (A-5) into equation (A-1)yields the effective stiffness matrix for a single fracture set em-bedded in a VTI background (Schoenberg and Helbig, 1997):

c =(

c1 0

0 c2

), (A-6)

where

c1 =

c11b(1−1N) c12b(1−1N) c13b(1−1N)

c12b(1−1N) c11b −1N

c212b

c11b

c13b

(1−1N

c12b

c11b

)

c13b(1−1N) c13b

(1−1N

c12b

c11b

)c33b −1N

c213b

c11b

,

(A-7)

c2 =

c44b 0 0

0 c44b(1−1V ) 0

0 0 c66b(1−1H )

, (A-8)

and 0 is the 3× 3 zero matrix.Alternatively, the same result can be obtained using the se-

ries

c ≡ [sb + s f ]−1 = [(I+ s f sb−1)sb

]−1 = cb[I+ s f cb]−1

= cb

∞∑k=0

(−s f cb)k, (A-9)

where I is the 6× 6 identity matrix. Series (A-9) converges toequation (A-6) if all eigenvalues of the matrix s f cb have abso-lute values less than 1. This is always the case if the weaknesses

1N ,1V , and1H , which define nonzero eigenvalues of the ma-trix s f [see equation (A-5)], are sufficiently small.

Equation (A-9) can serve as the basis for developing use-ful approximations for the effective stiffness matrix c. If thecrack density (or fracture intensity) is small and the weaknesses{1N,1V ,1H } ¿ 1, we may truncate series (A-9) by keepingonly linear terms with respect to 1N , 1V , and 1H :

c ≈ cb − cbs f cb. (A-10)

It is interesting to note that equation (A-10) becomes exactif we replace the matrix s f [equation (A-5)] by its linearizedversion:

slinf =

1N/c11b 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 1V/c44b 00 0 0 0 0 1H/c66b

.

(A-11)

Approximation (A-10) can be used to obtain two impor-tant results. First, it can be extended in a straightforward wayto multiple fracture sets (Nichols et al., 1989). If the effectivecompliance represents the sum of the excess compliances of Nfracture sets,

s f =N∑

i=1

s f i , (A-12)

then for the effective stiffness c we have from equation (A-10)

c ≈ cb −N∑

i=1

cbs f i cb. (A-13)

The compliance matrix s f i cannot be described by equa-tion (A-3) if the normal ni to the i th fracture set does not coin-cide with the x1-axis. The matrices s f i for arbitrary orientationof ni may be obtained from equation (A-3) using the so-calledBond rotation (Winterstein, 1990). For example, the compli-ance matrix for a fracture set with the normal ni = [0, 1, 0] hasthe form

s f i =

0 0 0 0 0 0

0 KN 0 0 0 0

0 0 0 0 0 0

0 0 0 KV 0 0

0 0 0 0 0 0

0 0 0 0 0 KH

. (A-14)

Second, equation (A-10) is well suited for deriving the weak-anisotropy approximation for the stiffnesses of effective mediaformed by fractures embedded in an anisotropic background.We assume that the background medium is weakly anisotropic,so that

cb = cisob + εcani

b , (A-15)

where cisob is the stiffness matrix of a purely isotropic solid that

approximates cb in some sense, canib is the anisotropic portion


of cb, and ε¿ 1. We also assume that the elements of the frac-ture compliance matrix are of the same order as ε and can bedenoted as εs f . If we represent the effective stiffness matrix cin a form similar to equation (A-15), equation (A-10) can berewritten as

ciso + εcani ≈ cisob + εcani

b −(ciso

b + εcanib

)εs f

(ciso

b + εcanib

).

(A-16)

Collecting the terms linear in ε yields

cani = canib − ciso

b s f cisob . (A-17)

This equation, valid in the weak anisotropy limit, can be looselyinterpreted in the following way: the anisotropy cani of an ef-fective medium containing fractures in an anisotropic back-ground described by cb is equal to the sum of the backgroundanisotropy cani

b and the anisotropy caused by the fractures em-bedded into any isotropic medium with ciso

b sufficiently closeto cb [see equation (A-13)]. The matrix ciso

b must satisfy theinequality |(||cb||/||ciso

b ||)− 1|<ε.

APPENDIX B

ANISOTROPIC PARAMETERS FOR ORTHORHOMBIC MEDIA

The basic set of the anisotropic parameters for orthorhom-bic media has been introduced by Tsvankin (1997b). His no-tation contains the vertical velocities of the P-wave and oneof the S-waves and seven dimensionless Thomsen-type (1986)anisotropic coefficients. The definitions of those parameters interms of the stiffnesses ci j and density ρ are given below.

VP0—the P-wave vertical velocity:

VP0 ≡√

c33

ρ(ρ is density). (B-1)

VS0—the velocity of the vertically traveling S-wave polarizedin the x1-direction:

VS0 ≡√

c55

ρ. (B-2)

ε(2)—the VTI parameter ε in the [x1, x3] symmetry planenormal to the x2-axis [this explains the superscript (2)]:

ε(2) ≡ c11 − c33

2c33. (B-3)

δ(2)—the VTI parameter δ in the [x1, x3] plane:

δ(2) ≡ (c13 + c55)2 − (c33 − c55)2

2c33(c33 − c55). (B-4)

γ (2)—the VTI parameter γ in the [x1, x3] plane:

γ (2) ≡ c66 − c44

2c44. (B-5)

ε(1)—the VTI parameter ε in the [x2, x3] symmetry plane:

ε(1) ≡ c22 − c33

2c33. (B-6)

δ(1)—the VTI parameter δ in the [x2, x3] plane:

δ(1) ≡ (c23 + c44)2 − (c33 − c44)2

2c33(c33 − c44). (B-7)

γ (1)—the VTI parameter γ in the [x2, x3] plane:

γ (1) ≡ c66 − c55

2c55. (B-8)

δ(3)—the VTI parameter δ in the [x1, x2] plane (x1 plays therole of the symmetry axis):

δ(3) ≡ (c12 + c66)2 − (c11 − c66)2

2c11(c11 − c66). (B-9)

These nine parameters fully describe wave propagation ingeneral orthorhombic media. In particular applications, how-ever, it is convenient to operate with specific combinations ofTsvankin’s parameters. For example, P-wave NMO velocityfrom dipping reflectors depends on three coefficients η, whichdetermine the anellipticity of the P-wave slowness in the sym-metry planes (Grechka and Tsvankin, 1999). The definitions ofη(1,2,3) are analogous to that of the Alkhalifah–Tsvankin (1995)coefficient η in VTI media:η(1)—the VTI parameter η in the [x2, x3] plane:

η(1) ≡ ε(1) − δ(1)

1+ 2δ(1). (B-10)

η(2)—the VTI parameter η in the [x1, x3] plane:

η(2) ≡ ε(2) − δ(2)

1+ 2δ(2). (B-11)

η(3)—the VTI parameter η in the [x1, x2] plane:

η(3) ≡ ε(1) − ε(2) − δ(3)(1+ 2ε(2)

)(1+ 2ε(2)

)(1+ 2δ(3)

) . (B-12)

Vertical transverse isotropy may be considered as a specialcase of orthorhombic media, where

ε(1) = ε(2) = ε, (B-13)

δ(1) = δ(2) = δ, (B-14)

γ (1) = γ (2) = γ, (B-15)

δ(3) = 0, (B-16)

and, as a consequence,

η(1) = η(2) = η, (B-17)

η(3) = 0. (B-18)

1816 Bakulin et al.

APPENDIX C

TWO IDENTICAL FRACTURE SETS

If the model includes two identical orthogonal fracture setsin a purely isotropic background, only four effective stiffnessesout of nine remain independent because the stiffness matrixdepends on just four quantities: λ, µ, 1N1=1N2≡1N , and1T1=1T2≡1T . From equations (40)–(43) it follows that inthis case c11= c22, c13= c23, and c44= c55. Constraints (47) and(48) become identical and, together with equation (46), maybe reduced to

c12(c33 + c13) = c13(c11 + c13) (C-1)

and

2(c11 + c13)c44c66 = (2c66 − c44)(c11c33 − c2

13

). (C-2)

We can call this medium quasi-VTI because its stiffness ma-trix is close to the one for vertical transverse isotropy. Un-like real VTI media, however, the stiffnesses of the quasi-VTImodel satisfy constraints (C-1) and (C-2), which replace theVTI relationship c11= c12 + 2c66. Indeed, the combination ofthe stiffnesses that must vanish in VTI media can be written inthe exact form as

c11 − 2c66 − c12 = 4µ2(KT − KN)(1+ 2µKN)(1+ 2µKT )

. (C-3)

Hence, because of the difference between the normal and shearcompliances, the model with two identical fracture sets does nothave the VTI symmetry.

The anisotropic coefficients of quasi-VTI media are de-scribed by equations (B-3)–(B-5), (B-13)–(B-15), and

δ(3) = −

4µ2(λ+ µ)(KT − KN)

2µ+ λ

1+ 2(λ+ µ)KN

×[µ+ 2µ(λ+ µ)(KT − KN)+ λ(1+ 4µKN)

1+ 2µKN

+ 2µ2KT1+ 2(λ+ µ)KN

1+ 2µKN

]−1

. (C-4)

Interestingly, quasi-VTI media have two additional verticalsymmetry planes at 45◦ with respect to planes [x1, x3] and[x2, x3]. Clearly, expressions for anisotropic coefficients ε(1,2),δ(1,2), and γ (1,2), which could be defined within these symmetryplanes, are different from those given by equations (49)–(58).

Two identical scalar fracture sets

If we further assume that both fracture sets are filled withgas [i.e., KN = KT ; Schoenberg and Sayers (1995); part I], theeffective medium becomes VTI. The effective stiffnesses givenbelow depend only on the background parameters λ andµ andthe weakness 1T = (KTµ)/(1+ KTµ) [equation (45)]:

c11 = (λ+ 2µ)(1−1T )[1+1T (3− 4g)]

D, (C-5)

c13 = λ1−12T

D, (C-6)

c33 = (λ+ 2µ)1−1T (8g− 5)

D, (C-7)

c44 = µ(1−1T ), (C-8)

c66 = µ1−1T

1+1T, (C-9)

where

D ≡ (1+1T )[1+1T (2− 3g)]. (C-10)

The five stiffness elements of this three-parameter VTImedium are related by two constraints that can be ob-tained by substituting the VTI relation c11= c12+ 2c66 intoequations (C-1) and (C-2). In terms of Thomsen’s (1986)anisotropic coefficients, these constraints take the form

ε = δ = 4γ (1+ 2γ )(1− g). (C-11)

The equality ε= δ indicates that the effective medium is ellipti-cally anisotropic. In the weak-anisotropy limit, equation (C-11)yields

ε = δ ≈ 4γ (1− g). (C-12)

For a typical value g= 0.25, ε= δ≈ 3γ .

APPENDIX D

FRACTURE CHARACTERIZATION FOR TWO ORTHOGONAL FRACTURE SETS: SPECIAL CASES

Azimuthally independent P-wave NMO velocity

In contrast to the case of a single fracture set in an isotropicor VTI background, the orientation of two orthogonal fracturesystems cannot always be found from the P-wave NMO ellipsefrom a horizontal reflector. If δ(1)= δ(2) or, equivalently,

(1− 2g)1N1 +1T1 = (1− 2g)1N2 +1T2 (D-1)

[equations (50) and (54)], the P-wave NMO ellipse degener-ates into a circle. This equation does not necessarily imply thatthe two systems of fractures are identical because it can besatisfied for fracture sets with different crack densities (e.g.,1T1 >1T2) and different fluid saturations (1N1 <1N2). Theazimuths of fractures in this case can be found from the shear-

wave polarization directions or by using P-wave reflectionsfrom dipping interfaces and/or the azimuthal variation of P-wave nonhyperbolic moveout.

Equal tangential weaknesses

If the tangential weaknesses are equal to each other(1T1=1T2), the anisotropic coefficients γ (1) and γ (2) also be-come identical [see equations (51) and (55)] and there is noshear-wave splitting at vertical incidence. Hence, the fractureorientation cannot be determined from shear-wave splitting.However, if the fracture sets have different normal weaknesses(1N1 6=1N2), the fracture azimuths can be found using P-waveNMO ellipses from a horizontal or a dipping reflector.


Identical fracture sets (quasi-VTI medium)

In the case of two identical orthogonal fracture sets, neitherthe P-wave NMO ellipse from a horizontal reflector (it degen-erates into a circle because δ(1)= δ(2)) nor the shear-wave po-larization directions (γ (1)= γ (2), see Appendix C) can be usedto detect the fracture orientation. Still, since δ(3) (or η(3)) differsfrom zero, the orientation can be found from normal moveoutof dipping P events or from P-wave nonhyperbolic moveout.The rest of the inversion procedure is similar to that for thegeneral case of two different fracture systems.

Despite the absence of shear-wave splitting in the verticaldirection, S-waves can still be used for estimating fracture pa-rameters. The wavefront of the slow shear wave in quasi-VTImedia always has cusps at 45◦ with respect to the symmetryplanes [x1, x3] and [x2, x3] (Figure D-1) if c11− 2c66− c12 > 0[or, equivalently, KT > KN ; see equation (C-3) and part I]. Thiswavefront structure by itself can be used to identify the sym-metry directions (and, therefore, the fracture strikes) if a suf-ficient number of azimuthal measurements is available. Oncethose directions are found, we can use equations (23)–(29) for

FIG. D-1. Group velocity surfaces of the (a) fast and (b) slow shear waves for the model with two identical orthogonal fracturesets. The background velocities are VP = 2 km/s and VS= 1 km/s. The fracture weaknesses 1N = 0 and 1T = 0.15 approximatelycorrespond to fluid-filled penny-shaped cracks with a crack density of 7%.

the NMO velocities of the S1- and S2-waves in the vertical sym-metry planes and estimate the fracture parameters based onequations (66) and (68). It is interesting that in quasi-VTI me-dia the S-wave NMO velocities from a horizontal reflector aredefined, strictly speaking, only within the vertical symmetryplanes because the shear-wave singularity makes the S-waveoffset-traveltime curve nondifferentiable in any other azimuth.

Identical gas-filled fracture sets (VTI medium)

Finally, if two identical fracture sets are gas filled, the ef-fective medium becomes VTI (Appendix C). Since the VTImodel is azimuthally isotropic, it is impossible to find the frac-ture orientation from seismic data. The single parameter 1T

(or KT ) needed to describe the fractures can be found from anyanisotropic coefficient given by equations (49)–(56) if the back-ground VS/VP ratio is known. Estimating several anisotropiccoefficients provides a redundancy that can be used to verifythe validity of this model [see equation (C-11)]. Indeed, thisVTI medium is elliptical (ε= δ) and has a specific relationship(C-12) between ε and γ .


Estimation of fracture parameters from reflection seismic data—Part III:Fractured models with monoclinic symmetry


ABSTRACT

Geophysical and geological data acquired over nat-urally fractured reservoirs often reveal the presence ofmultiple vertical fracture sets. Here, we discuss modelingand inversion of the effective anisotropic parameters oftwo types of fractured media with monoclinic symmetry.The first model is formed by two different nonorthog-onal sets of rotationally invariant vertical fractures inan isotropic host rock; the other contains a single set offractures with microcorrugated faces.

In monoclinic media with two fracture sets, the shear-wave polarizations at vertical incidence and the orienta-tion of the NMO ellipses of pure modes in a horizontallayer are controlled by the fracture azimuths as well asby their compliances. While the S-wave polarization di-rections depend only on the tangential compliances, theaxes of the P-wave NMO ellipse are also influenced bythe normal compliances and therefore have a differentorientation. This yields an apparent discrepancy betweenthe principal anisotropy directions obtained using P and

S data that does not exist in orthorhombic media. Byfirst using the weak-anisotropy approximation for theeffective anisotropic parameters and then inverting theexact equations, we devise a complete fracture charac-terization procedure based on the vertical velocities ofthe P- and two split S-waves (or converted PS-waves)and their NMO ellipses from a horizontal reflector. Ouralgorithm yields the azimuths and compliances of bothfracture systems as well as the P- and S-wave velocitiesin the isotropic background medium.

In the model with a single set of microcorrugated frac-tures, monoclinic symmetry stems from the coupling be-tween the normal and tangential (to the fracture faces)slips, or jumps in displacement. We demonstrate that forthis model the shear-wave splitting coefficient at verti-cal incidence varies with the fluid content of the frac-tures. Although conventional fracture models that ig-nore microcorrugation predict no such dependence, ourconclusions are supported by experimental observationsshowing that shear-wave splitting for dry cracks may besubstantially greater than that for fluid-filled ones.

INTRODUCTION

This work completes our series of three papers on seismiccharacterization of naturally fractured reservoirs. The first twopapers are devoted to the model with a single system of rota-tionally invariant fractures in an isotropic background (Bakulinet al. 2000a; hereafter referred to as part I) and to fracturedmodels with orthorhombic symmetry (Bakulin et al. 2000b;part II). Parts I and II contain a detailed review of recent pub-lications on fracture characterization that is not repeated here.All three papers strive to build a bridge between rock physicstheories of fractured media and seismic methods of fracturedetection. They also attempt to develop efficient fracture char-acterization methodologies based on surface reflection data.

Manuscript received by the Editor June 16, 1999; revised manuscript received March 3, 2000.∗Formerly St. Petersburg State University, Department of Geophysics, St. Petersburg, Russia. Presently Schlumberger Cambridge Research, HighCross, Madingley Road, Cambridge CB 3 0EL, England. E-mail: [email protected].‡Colorado School of Mines, Center for Wave Phenomena, Department of Geophysics, Golden, Colorado 80401-1887. E-mail: [email protected]; [email protected]© 2000 Society of Exploration Geophysicists. All rights reserved.

Theoretical tools (the so-called effective medium theories)for modeling the seismic response of fractured media haveexisted since the early 1980s (e.g., Schoenberg, 1980, 1983;Hudson, 1980, 1981, 1988; Thomsen, 1995). In particular,Schoenberg (1980, 1983) and Schoenberg and Muir (1989) sug-gest treating fractures as highly compliant surfaces inside asolid host rock. According to their linear-slip theory, the ef-fective compliance of a rock mass with one or several frac-ture sets can be found as the sum of the compliances of thehost (background) rock and those of all the fractures. Thenthe background and fracture parameters can be related tothe effective Thomsen-type anisotropic coefficients, which gov-ern the influence of anisotropy on various seismic signatures.

1818

Fractured Monoclinic Media 1819

This formalism can be used to invert seismic signatures forthe fracture compliances and to make inferences about thephysical properties of the fracture network. (Schoenberg andDouma (1988) and part I show that fracture compliances ef-fectively absorb such information about the microstructureof fractures as their shape, possible interaction, partial sat-uration, the presence of equant porosity, etc. This informa-tion cannot be obtained unambiguously from seismic dataalone.)

In part I we implement these ideas for transversely isotropicmedia with a horizontal symmetry axis (HTI) formed by a sin-gle set of vertical rotationally invariant fractures. A system ofvertical fractures in a VTI (transversely isotropic with a ver-tical symmetry axis) background or two orthogonal fracturesets in an isotropic host rock lead to a medium of orthorhom-bic symmetry examined in part II. Further complications intro-duced into the model may lower the symmetry of the effectivemedium to monoclinic. First experimental evidence of mono-clinic symmetry in the subsurface is provided by Wintersteinand Meadows (1991), who analyze walkaway vertical seismicprofiling (VSP) data over a fractured reservoir.

sb=

λ+ µµ(3λ+ 2µ)

− λ

2µ(3λ+ 2µ)− λ

2µ(3λ+ 2µ)0 0 0

− λ

2µ(3λ+ 2µ)λ+ µ

µ(3λ+ 2µ)− λ

2µ(3λ+ 2µ)0 0 0

− λ

2µ(3λ+ 2µ)− λ

2µ(3λ+ 2µ)λ+ µ

µ(3λ+ 2µ)0 0 0

0 0 01µ

0 0

0 0 0 01µ

0

0 0 0 0 01µ

. (2)

Here, we discuss two types of fracture-induced monoclinicmedia. The first model contains two different nonorthogonalsets of rotationally invariant fractures in an isotropic back-ground. We prove that all fracture parameters of the effectivemonoclinic medium with a horizontal symmetry plane can beestimated using the vertical and NMO velocities of the P- andtwo split S(or PS)-waves reflected from horizontal interfaces.It should be emphasized that shear-wave data alone do notcontain enough information to constrain the model parame-ters (Liu et al., 1993). Also, we confirm the result of Grechkaet al. (2000) that the polarization directions of the verticallypropagating shear waves generally are not aligned with theaxes of the P-wave NMO ellipse (which cannot happen in thehigher symmetry HTI or orthorhombic media). A similar con-clusion is drawn by Sayers (1998) who examines the azimuthalvariation of the P-wave phase-velocity function in monoclinicmedia. This gives a plausible theoretical explanation for the dis-crepancies in the fracture orientation estimated from P- andS-wave data by Perez et al. (1999).

The second monoclinic model examined here has a verticalsymmetry plane and consists of a single set of microcorrugated(rotationally noninvariant) fractures in an isotropic matrix. Wederive the shear-wave splitting coefficient as a function of the

compliances and show that it is substantially different for fluid-filled and dry fractures.

TWO SETS OF VERTICAL FRACTURES

Effective elastic parameters

Within the framework of the linear-slip theory, the effec-tive compliance matrix s of a rock with multiple fracture setscan be determined as the sum of the compliance sb of the back-ground medium and the fracture compliances s f (Nichols et al.,1989; Schoenberg and Muir, 1989; Molotkov and Bakulin, 1997;Bakulin and Molotkov, 1998). Considering two different arbi-trarily oriented fracture sets with the compliances s f 1 and s f 2 ina purely isotropic background, we can represent the effectivecompliance as

s = sb + s f 1 + s f 2 ≡ c−1, (1)

where c is the stiffness matrix of the effective medium. Thecompliance sb of the isotropic background, written in terms ofthe Lame parameters λ and µ, has the form

Assuming that both fracture sets are rotationally invariant,their matrices s f i (i = 1, 2) can be described by the normaland tangential (with respect to the crack faces) compliancesKNi and KT i . If the normal ni to the i th fracture set points inthe direction of the x1-axis, the compliance matrix is given by(Schoenberg and Douma, 1988; Schoenberg and Sayers, 1995;part I)

sx1f i =

KNi 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 KT i 0

0 0 0 0 0 KT i

. (3)

Rotation of the fracture normal by the angle φi around thex1-axis changes the matrix s f i according to the so-called Bondtransformation,

s f i = N(φi ) sx1f i NT(φi ), (4)

where the 6× 6 matrix N (NT is the transpose) is explicitly writ-ten in Winterstein (1990). The exact expressions for s f i , taken

1820 Bakulin et al.

from Schoenberg et al. (1999), are given in Appendix A. Itseems to be natural to choose the coordinate frame in such away that one of the fracture sets is orthogonal to the axis x1

(or x2), as is done by Liu et al. (1993). As we discuss below,however, a simpler effective stiffness matrix can be obtainedby aligning the horizontal coordinate axes with the polariza-tion directions of the vertically traveling S-waves. In this case,neither fracture system is orthogonal to one of the horizontalcoordinate directions.

Equations (1)–(4) make it possible to compute the com-pliance (s) and stiffness (c) matrices of the effective mediumformed by two vertical fracture sets in an isotropic background.Analysis of the compliance matrix [see equations (A-1)–(A-9)for a single fracture set] shows that this medium is monoclinicwith a horizontal symmetry plane. Below, we discuss the in-version of the effective anisotropic coefficients of monoclinicmedia for the fracture parameters.

Anisotropic coefficients of monoclinic media

Dimensionless anisotropic parameters, first introduced byThomsen (1986) for VTI media, proved to be extremely usefulin seismic velocity analysis and inversion. The main advantageof Thomsen’s notation is in capturing the combinations of thestiffness coefficients responsible for a wide range of seismicsignatures (Tsvankin, 1996). Thomsen-style notation for HTImedia was introduced by Ruger (1997) and Tsvankin (1997a).(In part I we discuss the inversion of the anisotropic coefficientsε(V), δ(V), and γ (V) of fracture-induced HTI media for the frac-ture compliances.) Tsvankin (1997b) shows that orthorhombicmedia can be conveniently described by seven anisotropic co-efficients and two vertical velocities (of the P- and one of thesplit S-waves), with a total of only six parameters fully respon-sible for the P-wave kinematics. Tsvankin’s notation simplifiesthe expressions for azimuthally varying NMO velocities in or-thorhombic media (Grechka and Tsvankin, 1998) and helps toidentify the subset of the medium parameters that can be de-termined using P-wave surface reflection data (Grechka andTsvankin, 1999).

An extension of Thomsen parameters to monoclinic mediais suggested by Grechka et al. (2000). They note that the ex-pression for NMO ellipses of P- and S-waves from horizontalreflectors take a particularly simple form if the horizontal coor-dinate axes x1 and x2 coincide with the polarization directionsof the vertically propagating split shear waves S1 and S2. In thisnatural coordinate frame the stiffness coefficient c45 vanishes(Helbig, 1994; Mensch and Rasolofosaon, 1997),

c45 = 0, (5)

and the number of independent stiffnesses reduces from 13to 12.

Grechka et al. (2000) replace the stiffness elements with thevertical velocities of P-waves (VP0) and one of the S-waves(VS0) and 10 anisotropic coefficients denoted as ε(1,2), δ(1,2,3),γ (1,2), and ζ (1,2,3) (Appendix B). These parameters can be di-vided into two different groups. The first contains the verticalvelocities and the ε, δ, and γ coefficients, which are definedexactly in the same way as Tsvankin’s (1997b) parameters fororthorhombic media. These nine quantities mainly control thesemiaxes of the NMO ellipses of waves P, S1, and S2 reflectedfrom horizontal interfaces. The second group includes the three

ζ coefficients responsible for the rotation of the NMO ellipseswith respect to the coordinate axes (Grechka et al., 2000).The parameter ζ (3) determines the orientation of the P-waveNMO ellipse, whereas ζ (1) and ζ (2) govern the rotations of theS1- and S2-ellipses, respectively [see equations (B-10)–(B-12)].This definition of the ζ coefficients distinguishes the notationof Grechka et al. (2000) from the Thomsen-style parameteriza-tion of arbitrary anisotropic media introduced by Mensch andRasolofosaon (1997).


Grechka et al. (2000) show that 11 (out of 12) parameters ofmonoclinic media [VP0, VS0, ε(1,2), δ(1,2), γ (1,2), and ζ (1,2,3)] canbe estimated in a stable way using the vertical velocities andNMO ellipses from horizontal reflectors of the P- and two splitS-waves. If the survey is acquired with P-wave sources, pureshear reflections can be replaced by the converted waves PS1

and PS2; this option is especially practical for offshore data.Our goal is to demonstrate that the 11 effective quantities

listed above can be inverted for the following physical parame-ters of the model: the P- and S-wave velocities VP and VS in theisotropic background, the azimuths φ1 and φ2 of the normals tothe fracture faces, and dimensionless fracture weaknesses de-noted by1Ni and1T i (i = 1, 2). The weaknesses are related tothe fracture compliances KNi and KT i as (Hsu and Schoenberg,1993; part I)

1Ni = (λ+ 2µ) KNi

1+ (λ+ 2µ) KNiand 1T i = µ KT i

1+ µ KT i,

(i = 1, 2), (6)

where1Ni and1T i are always positive and vary from zero (nofracturing) to unity (extreme degree of fracturing).

For isolated penny-shaped cracks, vanishing values of theratio 1Ni/1T i correspond to fluid-filled fractures, whereas1Ni/1T i ≈ (λ+ 2µ)/µ (or KNi ≈ KT i ) indicate that the cracksare dry. Also, regardless of the type of crack infill, the tangen-tial weakness1T i is close to twice the crack density ei (part I).If we ignore the influence of one fracture system on the other(a reasonable assumption for small crack density), these rela-tions hold for the weaknesses of each system.

Following Grechka et al. (2000), we choose the axis x1 tocoincide with the polarization direction of the fast shear waveS1 at vertical incidence (Figure 1). In this coordinate frame,c45 vanishes [equation (5)], providing an additional constraintfor the fracture parameters. This constraint can be obtainedby analyzing the nonzero elements of the effective compliancematrix s [equation (1)] and the matrices sb [equation (2)], s f 1,and s f 2 [equations (A-1)–(A-9)]. Since s44, s45, and s55 are theonly nonzero elements of the fourth and fifth columns and thefourth and fifth rows of s, the inverse (stiffness) matrix c has ablock (

c44 c45

c45 c55

)=(

s44 s45

s45 s55

)−1

.

Hence, c45 = 0 requires that

s45 = 0, (7)


which can be satisfied only if [see equations (1), (2), and (A-7)]

KT1 sin 2φ1 + KT2 sin 2φ2 = 0. (8)

Replacing the compliances with the weaknesses [equation (6)]yields

1T1

1−1T1sin 2φ1 + 1T2

1−1T2sin 2φ2 = 0. (9)

Thus, we have a total of 12 equations [11 monoclinic param-eters VP0, VS0, ε(1,2), δ(1,2), γ (1,2), and ζ (1,2,3) plus constraint (9)]to be solved for eight unknowns [VP, VS, 1Ni , 1T i , and φi

(i = 1, 2)]. The nonlinear relations between the anisotropic co-efficients and fracture parameters make it necessary to applynumerical methods and study the uniqueness of the solution.As shown below, our inversion algorithm converges toward val-ues close to the actual model parameters, given a reasonablemagnitude of errors in the input data.

Equations (8) and (9) allow us to make two important con-clusions about the fracture orientation even prior to the in-version. First, equation (9) can be satisfied only if the fractureazimuths φ1 and φ2 (assumed to lie between −π/2 and π/2)have opposite signs because both 1T1 and 1T2 are nonnega-tive (Figure 1). Therefore, the angle between the normals (andthe two crack systems themselves) can be found as φ1−φ2; theabsolute values of φ1 and φ2 are equal only if 1T1=1T2.

Second, equation (8) indicates that for a fixed angle betweenthe fractures, the azimuths φ1 and φ2 are controlled only by theratio of the tangential compliances KT1/KT2. The shear-wavepolarization directions bisect the angles between the fracturesif KT1= KT2. To find the fracture orientation relative to theshear-wave polarization directions when KT1 6= KT2, we needto supplement equation (8) with the condition

c55 > c44. (10)

Inequality (10) ensures that the fast shear wave S1 is polarizedalong the x1-axis (Figure 1). Using equations (1)–(4), we canrewrite condition (10) in the form

KT1 cos 2φ1 + KT2 cos 2φ2 < 0. (11)

FIG. 1. Two sets of parallel vertical fractures form an effectivemonoclinic medium. The fracture normals make the angles φ1and φ2 with the axis x1 that coincides with the polarizationdirection of the vertically traveling S1-wave. To define φ1 andφ2 in a unique fashion, we assume that −π/2≤φ1 <π/2 and−π/2<φ2≤π/2.

Combined with equation (8), inequality (11) unambiguouslydefines the directions of both fracture sets for given values ofKT1, KT2, and the angle φ1−φ2.

Let us assume that the first fracture set has a higher tangen-tial compliance (KT1 > KT2). Analysis of equations (8) and (11)shows that in this case the normal to the first fracture set lieswithin the intervalπ/4<φ1 < 3π/4. Hence, the polarization di-rection of the fast (S1) shear wave (i.e., the x1-axis) is alwayscloser to the strike of the more compliant fractures. Accord-ing to the quantitative estimates in Figure 2, for KT1/KT2 > 3the S1-polarization direction does not deviate by more than10◦ from the strike of the first fracture system. In the limit ofKT1À KT2, the medium becomes effectively HTI, and the po-larization of the S1-wave is parallel to the first fracture set; thisresult is well known for horizontal transverse isotropy (e.g.,Winterstein, 1990).

Weak-anisotropy approximation.—If the fracture density issmall and the weaknesses 1Ni¿ 1 and 1T i¿ 1, the effectivemedium is weakly anisotropic. In this case, it is possible tosimplify the expressions for the anisotropic coefficients by lin-earizing them in the weaknesses 1Ni and 1T i (Appendix C).Equations (C-1)–(C-11) show that all ε, δ, and γ coefficientsare even functions of the fracture azimuths φ1 and φ2, and in-formation about the signs of the azimuths can be obtained onlyfrom the ζ coefficients. Since ζ (1,2,3) determine the rotation ofthe P- and S-wave NMO ellipses with respect to the shear-wavepolarization directions (Grechka et al., 2000), this result indi-cates the importance of carefully measuring the orientation ofthe elliptical axes.

The linearized expressions (C-1)–(C-11) can be used to esti-mate all fracture parameters following, for instance, the al-gorithm outlined in Appendix D. To check the accuracy ofthe weak-anisotropy approximation, we computed the exactanisotropic coefficients for the fracture parameters given inTable 1 and carried out the inversion using the equations fromAppendix D. For typical moderate values of the tangential

FIG. 2. Azimuth of first fracture system (|φ1− 90◦|) as a func-tion of the ratio of the tangential compliances KT1/KT2. Eachcurve corresponds to a different angle between the fracture sys-tems: φ1−φ2= 20◦ (circles), 40◦ (triangles), 60◦ (dashed line),and 80◦ (solid line).

1822 Bakulin et al.

compliances for both fractures systems (if the fractures arepenny-shaped, the crack densities are e1≈1T1/2= 0.06 ande2≈1T2/2= 0.10), our approximation leads to noticeable er-rors in two weaknesses (1T1 and 1N2). However, it can stillbe used to obtain a good initial guess for the nonlinearinversion.

Special case: Equal tangential weaknesses.—It is instructiveto examine the special case of equal tangential weaknesses ofthe two fracture systems:

1T1 = 1T2 ≡ 1T . (12)

Then, as follows from equation (9),

φ1 = −φ2 ≡ φ, (13)

and the S1-wave polarization direction bisects the angle be-tween the fracture sets (see Figure 2 for KT1= KT2, which corre-sponds to1T1=1T2). Substituting relations (12) and (13) intoequations (C-1)–(C-11), we obtain equations (E-3)–(E-13),which can be inverted for the fracture parameters in a rela-tively straightforward fashion. The results of Appendix E indi-cate that the fracture azimuth φ and the weaknesses 1T , 1N1,and 1N2 can be found just from the vertical velocities of theP- and S-waves and the P-wave NMO ellipse; the shear-waveNMO ellipses provide redundant information.

If the tangential weaknesses of the two fracture sets areequal, the orientations of the pure-mode NMO ellipses arerelated in a simple way to the fracture azimuths. For instance,the azimuth θP of the semimajor axis of the P-wave NMO el-lipse can be written in the weak-anisotropy approximation as(Grechka et al., 2000)

tan 2θP = 2ζ (3)

δ(2) − δ(1). (14)

Substituting equations (E-7), (E-8), and (E-13) into equa-tion (14), we find

tan 2θP = AP tan 2φ, (15)

where

AP = (1N2 −1N1)(1− 2g)(1N1 +1N2)(1− 2g)+ 21T

. (16)

Note that |AP| ≤ 1, which implies that the axes of the P-waveNMO ellipse deviate from both fracture azimuths (except forthe special case of orthogonal fractures discussed below). Also,if1N2 6=1N1, the angle θP 6= 0 and the P-wave ellipse is rotatedwith respect to the polarization directions of the verticallypropagating shear waves, as noted by Grechka et al. (2000) (seealso Sayers, 1998). This may explain the discrepancies in thefracture orientation estimated from the P-wave NMO ellipse

Table 1. Comparison of the actual fracture parameters withthose estimated from the exact anisotropic coefficients usingthe linearized formulas from Appendix D.

Parameter VS/VP 1N1 1T1 φ1 1N2 1T2 φ2

Actual 0.5 0.25 0.12 30◦ 0.00 0.20 −13◦Estimated 0.5 0.20 0.04 40◦ 0.08 0.22 −5◦

and S-wave polarizations described by Perez et al. (1999). Thesame conclusion is true for the NMO ellipses of the S1- andS2-waves whose rotation angles are determined by the param-eters ζ (1) and ζ (2) (Grechka et al., 2000). As follows from equa-tions (E-11) and (E-12) for the fracture-induced monoclinicmodel, ζ (1) 6= 0 and ζ (2) 6= 0.

If the two fracture systems are identical [i.e., 1N1=1N2 inaddition to 1T1=1T2], the axes of the P-wave NMO ellipseare aligned with the shear-wave polarization directions (AP =θP = 0). In this case, the medium becomes orthorhombic, andS-wave polarization vectors (which bisect the fractures) lie inthe vertical symmetry planes of the model. For 1N1=1N2, allthree ζ coefficients vanish, so the shear-wave NMO ellipses arecooriented with the P-wave ellipse.

Another special case of orthorhombic symmetry is that ofdifferent but orthogonal fracture sets (2φ= 90◦). If only thenormal weaknesses1N1 and1N2 are different (but1T1=1T2),shear waves do not split at vertical incidence and have no de-fined polarization directions (part II).

For monoclinic media, the axes of the P-wave NMO el-lipse and the polarization vectors of the vertically travelingS-waves are parallel only if the ratio of the tangential and nor-mal compliances is the same for both fracture systems (i.e.,ζ (3)= 0 if KN1/KT1= KN2/KT2 or 1N1/1T1=1N2/1T2). Thisresult, valid only in the weak anisotropy approximation, canbe obtained from equations (8), (9), and (C-11). For example,ζ (3)= 0 if the normal and tangential compliances are equal toeach other, which agrees with the result of Sayers (1998).

Arbitrary fracture sets.—The main significance of the lin-earized approximations for the fracture parameters is in pro-viding insight into the behavior of the anisotropic coefficientsand a good initial model for the inversion procedure. Herewe discuss the inversion of the anisotropic parameters for thefracture compliances and orientations based on the exact equa-tions (1)–(4) for the stiffnesses and the definitions from Ap-pendix B. We assume that the quantities VP0, VS0, ε(1,2), δ(1,2),γ (1,2), and ζ (1,2,3) have been estimated from the vertical andNMO velocities of the P- and split S-waves (or converted PS-waves). To examine the stability of the inversion algorithm, weintroduce errors in all quantities [similar to those shown in Fig-ures 2 and 4 in Grechka et al. (2000)] caused by Gaussian noisewith a variance of 2% added to the NMO velocities. The vari-ances of the errors in the effective parameters are as follows:2% in VP0 and VS0, 0.01 in ζ (1) and ζ (2), and 0.03 in all otheranisotropic coefficients. The fracture parameters that give thebest fit to the error-contaminated values of the vertical veloc-ities and anisotropic coefficients are found by minimizationusing the simplex method.

Figure 3 displays typical inversion results for the VP/VS ra-tio in the isotropic background and weaknesses 1Ni and 1T i .The standard deviation in the estimated VS/VP ratio (3.1%)is somewhat higher than that in the input vertical velocitiesVP0 and VS0 but is still quite acceptable. The errors in theweaknesses 1T i are about twice as large as those in the in-put anisotropic coefficients. This error amplification can beunderstood from the weak-anisotropy approximations (D-6)and (D-8), which indicate that 1T i are derived from the γ co-efficients multiplied by a factor of 2; similar results are obtainedfor models with one system of fractures in parts I and II.


The accuracy of the inverted normal weaknesses1Ni is evenlower than that of 1T i . This can also be explained using theweak-anisotropy approximations (C-1)–(C-11), which containterms in the form1T i ±a1Ni with |a|< 1. Therefore, the sameerrors in the input anisotropic coefficients can be compensatedby greater variations in 1Ni compared to those in 1T i . Eventhough the errors in 1Ni are rather significant, Figure 3 showsthat we can still distinguish between dry fractures [the firstsystem with KN1≈ KT1 or1T1(1−1N1)≈ g1N1(1−1T1)] andfluid-filled ones with 1N2≈ 0. The unphysical values 1N2 < 0appear in Figure 3 because random errors added to the datamay produce anisotropic coefficients that do not correspondto any fractured media.

MICROCORRUGATED FRACTURES ANDFLUID-DEPENDENT SHEAR-WAVE SPLITTING

Shear-wave splitting at vertical incidence traditionally hasbeen regarded as the most reliable measure of fracture intensityfor a set of parallel vertical fractures embedded in an isotropicor VTI host rock. The fractional difference between the S-wavevertical velocities VS1 and VS2 is expressed through the so-calledshear-wave splitting parameter γ (S), defined as

γ (S) ≡ V2S1 − V2

S2

2V2S2

≈ VS1 − VS2

VS2. (17)

It has been common knowledge among researchers working onfracture characterization that γ (S) depends only on the crackdensity and does not contain information about the fluid con-tent of the fractures. This conclusion seems to have been con-firmed both theoretically (Hudson, 1981; Thomsen, 1995) andexperimentally (e.g., Martin and Davis, 1987; Winterstein andMeadows, 1991). On the other hand, Guest et al. (1998) presenta case study where the splitting parameter for gas-filled cracksproves to be significantly higher than that for brine-filled ones,which clearly contradicts the existing understanding. Here, wegive a possible theoretical explanation of these observationsby obtaining the effective parameters of a fracture set withmicrocorrugated faces. The idea of the theory is to allow thecoupling between the tangential slip along fracture faces (re-sponsible for shear-wave splitting) and the normal slip knownto depend on the fluid content.

FIG. 3. Results of the inversion for the parameters of two frac-ture systems and the background velocities. The dots mark theactual values of the fracture parameters; the bars correspond to± one standard deviation in the inverted quantities. Not shownare the standard deviations in the estimated background veloci-ties VP and VS (2.0% and 2.5%, respectively) and in the fractureazimuths (9◦; the actual values are φ1= 30◦ and φ2=−12.8◦).

Linear-slip model for fractures with microcorrugated faces

As in the previous section, our analysis is based on the linear-slip theory of Schoenberg (1980), Schoenberg and Muir (1989),and Schoenberg and Sayers (1995). For simplicity, we examinea single system of parallel fractures with the normal n= [1, 0, 0]in an isotropic background medium. The matrix of the excessfracture compliance in this case has the form (Schoenberg andDouma, 1988; part I)

s f =

KN 0 0 0 KN V KN H

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

KN V 0 0 0 KV KV H

KN H 0 0 0 KV H KH

, (18)

where KN is the normal fracture compliance that relates thejump of the displacement normal to the fracture (i.e., the nor-mal slip) to the normal stress in the direction [1, 0, 0]. The val-ues KV and KH are the tangential compliances relating the slipsand stresses in the vertical [0, 0, 1] (KV ) and horizontal [0, 1, 0](KH ) directions. The compliance KN V couples the normal slipto the tangential vertical stress or, equivalently, the tangentialslip in the direction [0, 0, 1] to the normal stress. Likewise, thecompliances KN H and KV H couple the horizontal stress in the[0, 1, 0] direction to the normal and vertical slips.

The conventional conclusion about the shear-wave splittingcoefficient γ (S) being independent of fracture infill is basedon the assumption that the normal and tangential slips aredecoupled (KN V = KN H = KV H = 0) and the matrix (18) is di-agonal. An alternative model of cracks with microcorrugatedfaces (see Figure 4) is suggested by Schoenberg and Douma(1988). Since the stresses in either the normal (x1) or vertical

FIG. 4. Cross-section of a vertical fracture with microcorru-gated faces. This microstructure leads to the coupling betweenthe normal and tangential slips (after Schoenberg and Douma,1988).

1824 Bakulin et al.

(x3) direction applied to such a fracture produce slips in bothx1- and x3-directions, the compliance component KN V must benonzero. In contrast, the component KV H can always be setto zero by the appropriate rotation of the coordinate system(Berg et al., 1991). Below, we show that fractures characterizedby the compliances

KN H = KV H = 0 and KN V 6= 0 (19)

cause infill-dependent shear-wave splitting.

Effective model of fractured media

If a fracture system with the compliances described by equa-tions (18) and (19) is embedded in an isotropic rock, the effec-tive stiffness matrix has the form [see equation (1)]

c−1 = sb + s f , (20)

where the background compliance matrix sb is given by equa-tion (2). Evaluating the components of the matrix c yields

c =

c11 c12 c12 0 c15 0

c12 c33 c23 0 c35 0

c12 c23 c33 0 c35 0

0 0 0 c44 0 0

c15 c35 c35 0 c55 0

0 0 0 0 0 c66

. (21)

Equation (21) describes a monoclinic medium with the verticalsymmetry plane [x1, x3]. The stiffness elements are given by

c11 = (λ+ 2µ)1+ EV

D, c12 = λ1+ EV

D,

c15 = −√µ(λ+ 2µ)

EN V

D,

c33 = (λ+ 2µ)(1+ EV )(1+ νEN)− νE2

N V

D,

(22)

c23 = λ (1+ 2gEN)(1+ EV )− 2gE2N V

D,

c35 = −λ√

gEN V

D,

c44 = µ, c55 = µ1+ EN

D, and c66 = µ

1+ EH,

where

D = (1+ EN)(1+ EV )− E2N V, ν = 4µ(λ+ µ)

(λ+ 2µ)2,

g = µ

λ+ 2µ,

and λ and µ are the Lame constants of the host rock. Thedimensionless compliances EN , EV , EH , and EN V are definedas follows (Hsu and Schoenberg, 1993):

EN = (λ+ 2µ)KN, (23)

EV = µKV , (24)

EH = µKH , (25)

EN V =√µ(λ+ 2µ)KN V. (26)

The stability condition requires that matrix (18) be nonneg-ative definite. In our case, this condition implies that all dimen-sionless compliances EN , EV , and EH are nonnegative and

EN EV − E2N V ≥ 0. (27)

Shear-wave splitting coefficient

The velocities of the split shear waves traveling in the verticaldirection in the monoclinic medium described by equation (21)are given by

V2S1 =

µ

ρ(28)

and

V2S2 =

2ρ

c33c55 − c235

c33 + c55 +√

(c33 − c55)2 + 4c235

, (29)

where ρ is the density. While the velocity VS1 of the fast shearwave is simply equal to the background S-wave velocity, theslower velocity VS2 depends on the fracture compliances EN ,ET , and EN V .

Assuming weak anisotropy (EN¿ 1, ET¿ 1, and EN V¿ 1)and keeping the linear and quadratic terms in the dimension-less compliances, we obtain the shear-wave splitting coefficient[equation (17)] as

γ (S) ≈ EV

2− E2

N Vg(3− 4g)2(1− g)

. (30)

Equation (30) shows that the coupling between the normal andtangential slips (i.e., EN V 6= 0) always reduces the value of γ (S).

To analyze the influence of fluid content on shear-wave split-ting for fractures with microcorrugated faces, we generalize thecriterion given by Schoenberg and Sayers (1995) for isolatedpenny-shaped cracks. They point out that the ratio KN/KV mayserve as an indicator of fluid saturation because it vanishes forfluid-filled cracks and is equal to unity if the cracks are dry.The result of Schoenberg and Sayers (1995), however, is for-mulated for KN V = 0 and must be modified for microcorrugatedfractures.

Let us consider the 2× 2 submatrix

s f =(

KN KN V

KN V KV

)(31)

of the compliance matrix (18). Note that the Schoenberg-Sayers criterion for KN V = 0 is equivalent to the statement thatthe fractures are fluid filled if one eigenvalue of s f is equal tozero, whereas the fractures are dry if the matrix s f has twoequal eigenvalues.

We assume that the same relationship between fluid satu-ration and the eigenvalues of s f holds for the more generalcase with KN V 6= 0. To justify this assumption, we recall thatthe excess compliance matrix s f relates the stress applied tothe fracture faces to the slip or the jump of the displacementacross the fractures (Schoenberg, 1980). The eigenvalues of s f

or s f are the coefficients that relate the magnitudes of the slipand the stress vectors in the principal directions. Intuitively, weexpect dry fractures to be equally compliant in all directions(there is no material inside to stiffen them), so the eigenvaluesof the fracture compliance matrix are supposed to be equal. In


contrast, fluid-filled fractures are noticeably stiffer in a particu-lar direction where the applied stress tends to squeeze the fluid.Thus, the eigenvalue corresponding to this direction should besignificantly smaller than the other eigenvalues.

The submatrix (31) has equal eigenvalues if and only ifKN = KV and KN V = 0, which means [see equation (26)] that

EN V = 0. (32)

For one of the eigenvalues of s f to go to zero, it is required thatK 2

N V = KN KV or [equations (23), (24), and (26)]

E2N V = EN EV . (33)

Since E2N V ≤ EN EV [inequality (27)], equations (32) and (33)

impose strict bounds on possible absolute values of EN V .Substituting equations (32) and (33) into equation (30) yields

the shear-wave splitting coefficients for dry and fluid-filledcracks:

γ(S)dry ≈

EV

2(34)

and

γ(S)wet ≈

EV

2

[1− ENg(3− 4g)

1− g

]. (35)

Hence, for microcorrugated fractures the splitting coefficientis always higher for dry than for fluid-filled fractures, which isconsistent with the observations of Guest et al. (1998). Figure 5illustrates the influence of the dimensionless compliance EN V

on the shear-wave splitting coefficient. In this particular exam-ple, γ (S) decreases by about 30% as the fluid saturation changesfrom zero (EN V = 0) to 100%. Also note that the approxima-tion (30) gives a reasonably accurate qualitative description ofγ (S).

FIG. 5. Shear-wave splitting coefficient γ (S) computed from theexact equations (17), (28), and (29) (solid) and the approxima-tion (30) (dashed) as a function of the dimensionless compli-ance EN V . The model parameters are g= 0.16, EN = 1.3, andEV = 0.25.


The objective of this series of papers was to analyze the de-pendence of seismic signatures on the physical parameters offracture networks and to develop fracture characterization al-gorithms operating with surface seismic data. In part I we con-sidered the simplest model of a single vertical fracture systemin a purely isotropic host rock (HTI medium) and showed thatthe fracture compliances, or the dimensionless fracture weak-nesses, are the only quantities that can be unambiguously esti-mated from seismic data. The microstructure of the fracturedformation (e.g., the shape of fractures, their possible interac-tion, the presence of equant porosity, etc.), however, cannotbe evaluated without additional information. For example, ifthe fractures are known to be penny-shaped and isolated frompore space, the weaknesses give a direct estimate of the crackdensity and fluid content of the fracture system.

By deriving the relationships between the weaknesses andThomsen-type anisotropic coefficients, we were able to studythe behavior of surface seismic signatures in fracture-inducedHTI (part I) and orthorhombic (part II) media. The analyticresults for HTI media provided the basis for inversion algo-rithms designed to estimate the orientation and weaknesses ofa vertical fracture set using P-wave reflections alone or a com-bination of P and converted (PS) data. In part II we extendedour parameter estimation methodology to orthorhombic me-dia formed either by a single fracture set in an anisotropic (VTI)host rock or by two orthogonal fracture sets in an isotropicbackground.

This paper has investigated the inverse problem for an ef-fective monoclinic medium caused by two nonorthogonal setsof rotationally invariant fractures. The weaknesses and az-imuths of both fracture systems, along with the velocities inthe isotropic background, can be obtained using the verticalvelocities and NMO ellipses (from horizontal interfaces) of theP-wave and two split S-waves. In principle, pure S reflectionscan be replaced in the inversion procedure by the converted(PS) waves. Using the weak anisotropy approximation, we ob-tain simple, linearized expressions for the fracture parameters,which can serve as a good initial guess for the nonlinear inver-sion algorithm. Numerical analysis shows that the tangentialcompliances are generally estimated with a higher accuracythan the normal ones, but the difference between the normalcompliances of dry and fluid-filled cracks can still be detectedin the presence of noise in the data.

Parameter estimation is particularly convenient to carry outin the natural coordinate frame associated with the polarizationdirections of the vertically propagating S-waves. These direc-tions are controlled only by the weaknesses 1T i tangential tothe fracture faces and are independent from the normal weak-nesses 1Ni . In the special case of equal tangential weaknesses1T1=1T2, shear-wave polarization directions bisect the anglesbetween the fracture systems, and all fracture weaknesses canbe obtained just from the shear-wave splitting coefficient γ (S)

and the P-wave NMO ellipse. For different tangential compli-ances, the polarization vector of the fast shear wave deviatestoward the strike of the more compliant fracture system.

It is important to note that the axes of the P-wave NMOellipse generally do not coincide with either the polarizationdirections of the vertically traveling S-waves or the fracturestrike (the same is true for the S-wave NMO ellipses). This

1826 Bakulin et al.

result may explain the observations of Perez et al. (1999) whofind that the predominant fracture orientation obtained us-ing P-wave azimuthal moveout analysis does not agree withthat inferred from shear-wave polarizations.

We also examine another monoclinic model that contains asingle system of vertical fractures with microcorrugated facesin an isotropic host rock. An important feature of this modelis the coupling between the slips (jumps in displacement) inthe directions normal and tangential to the fractures. Thiscoupling leads to the dependence of the splitting coefficientγ (S) of the vertically traveling S-waves (determined by thetangential slip or tangential weakness) on the fluid content ofthe fractures (which influences the normal slip). For typicalfracture parameters, γ (S) may noticeably decrease with fluidsaturation, which is consistent with the case study of Guestet al. (1998).

ACKNOWLEDGMENTS

This work was performed during Andrey Bakulin’s visit tothe Center for Wave Phenomena (CWP), Colorado School ofMines, in 1998. We thank Sean Guest and Cees van der Kolkof Shell for their provocative discussion of the shear-wavesplitting coefficient. We are also grateful to members of theA(nisotropy)-Team of CWP for helpful discussions and toAndreas Ruger (Landmark) for his review of the manuscript.The support for this work was provided by the membersof the Consortium Project on Seismic Inverse Methods forComplex Structures at CWP and by the U.S. Department ofEnergy (award #DE-FG03-98ER14908).

REFERENCES

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——— 2000b, Estimation of fracture parameters due to reflection seis-mic data—Part II: Fractured models with orthorhombic symmetry:Geophysics, 65, 1803–1817, this issue.

Bakulin, A. V., and Molotkov, L. A., 1998, Effective models of frac-tured and porous media: St. Petersburg Univ. Press (in Russian).

Berg, E., Hood, J. A., and Fryer, G. J., 1991, Reduction of the generalfracture compliance matrix Z to only five independent elements:Geophys. J. Internat., 107, 703–707.


——— 1999, 3-D moveout velocity analysis and parameter estimationfor orthorhombic media: Geophysics, 64, 820–837.

Grechka, V., Contreras, P., and Tsvankin, I., 2000, Inversion of normalmoveout for monoclinic media: Geophys. Prosp., 48, 577–602.

Guest, S., van der Kolk, C., and Potters, H., 1998, The effect of fracturefilling fluids on shear-wave propagation: 68th Ann. Internat. Mtg.,Soc. Expl. Geophys., Expanded Abstracts, 948–951.

Helbig, K., 1994, Foundations of anisotropy for exploration seismics,in Helbig, K., and Treitel, S., Eds., Handbook of geophysical explo-ration 22: Pergamon Press.

Hsu, C.-J., and Schoenberg, M., 1993, Elastic waves through a simu-lated fractured medium: Geophysics, 58, 964–977.

Hudson, J. A., 1980, Overall properties of a cracked solid: Math. Proc.Camb. Phil. Soc., 88, 371–384.

——— 1981, Wave speeds and attenuation of elastic waves in materialcontaining cracks: Geophys. J. Roy. Astr. Soc., 64, 133–150.

——— 1988, Seismic wave propagation through material containingpartially saturated cracks: Geophys. J., 92, 33–37.

Liu, E., Crampin, S., Queen, J. H., and Rizer, W. D., 1993, Behaviorof shear waves in rocks with two sets of parallel cracks: Geophys. J.Internat., 113, 509–517.

Martin, M. A., and Davis, T. L., 1987, Shear-wave birefringence: A newtool for evaluating fractured reservoirs: The Leading Edge, 6, 22–28.

Mensch, T., and Rasolofosaon, P., 1997, Elastic-wave velocities inanisotropic media of arbitrary symmetry—generalization of Thom-sen’s parameters ε, δ, and γ : Geophys. J. Internat., 128, 43–64.

Molotkov, L. A., and Bakulin, A.V., 1997, An effective model of afractured medium with fractures modeled by the surfaces of discon-tinuity of displacements: J. Math. Sci., 86, 2735–2746.


Perez, M. A., Grechka, V., and Michelena, R. J., 1999, Fracture de-tection in a carbonate reservoir using a variety of seismic methods:Geophysics, 64, 1266–1276.

Ruger, A., 1997, P-wave reflection coefficients for transverselyisotropic models with vertical and horizontal axis of symmetry: Geo-physics, 62, 713–722.

Sayers, C., 1998, Misalignment of the orientation of fractures and theprincipal axes for P and S waves in rocks containing multiple non-orthogonal fracture sets: Geophys. J. Internat., 133, 459–466.


——— 1983, Reflection of elastic waves from periodically stratifiedmedia with interfacial slip: Geophys. Prosp., 31, 265–292.

Schoenberg, M., and Douma, J., 1988, Elastic wave propagation in me-dia with parallel fractures and aligned cracks: Geophys. Prosp., 36,571–590.



Schoenberg, M., Dean, S., and Sayers, C., 1999, Azimuth-dependenttuning of seismic waves reflected from fractured reservoirs: Geo-physics, 64, 1160–1171.


——— 1995, Elastic anisotropy due to aligned cracks in porous rock:Geophys. Prosp., 43, 805–830.

Tsvankin, I., 1996, P-wave signatures and notation for transverselyisotropic media: An overview: Geophysics, 61, 467–483.

——— 1997a, Reflection moveout and parameter estimation for hori-zontal transverse isotropy: Geophysics, 62, 614–629.

——— 1997b, Anisotropic parameters and P-wave velocity for or-thorhombic media: Geophysics, 62, 1292–1309.

Winterstein, D. F., 1990, Velocity anisotropy terminology for geophysi-cists: Geophysics, 55, 1070–1088.

Winterstein, D. F., and Meadows, M. A., 1991, Shear-wave polariza-tions and subsurface stress directions at Lost Hills field: Geophysics,56, 1331–1348.

APPENDIX A

COMPLIANCE OF AN ARBITRARY ORIENTED VERTICAL FRACTURE SET

The compliance matrix of a fracture system orthogonal tothe x1-axis is given in equation (3). Applying Bond transfor-mation (4) [see Winterstein (1990)] to matrix (3), Schoenberget al. (1999) obtain the compliance s f (for brevity, the number iof the fracture set is omitted) of a fracture system with the nor-mal n= [cosφ, sinφ, 0] (Figure A-1). The nonzero elements ofthe matrix s f have the following form:

s11 f =3KN + KT

8+ KN

2cos 2φ + KN − KT

8cos 4φ,

(A-1)

s12 f =KN − KT

8(1− cos 4φ), (A-2)

s16 f =KN

2sin 2φ + KN − KT

4sin 4φ, (A-3)

s22 f =3KN + KT

8− KN

2cos 2φ + KN − KT

8cos 4φ,

(A-4)


FIG. A-1. Set of vertical fractures with the normal making theangle φ with the x1-axis. The azimuth φ is positive in the coun-terclockwise direction.

APPENDIX B

ANISOTROPIC PARAMETERS OF MONOCLINIC MEDIA

Using the coordinate frame in which the x1- and x2-axes coin-cide with the polarization directions of the vertically travelingshear waves, Grechka et al. (2000) define anisotropic param-eters for monoclinic media by generalizing Thomsen’s (1986)and Tsvankin’s (1997b) notations for VTI and orthorhombicsymmetry systems. Expressions for these parameters in termsof the stiffness coefficients and density ρ are given below.

VP0—P-wave vertical velocity:

VP0 ≡√

c33

ρ. (B-1)

VS0—velocity of the vertically traveling S1-wave, which ispolarized in the x1-direction:

VS0 ≡√

c55

ρ. (B-2)

Dimensionless anisotropic parameters:

ε(1) ≡ c22 − c33

2c33, (B-3)

δ(1) ≡ (c23 + c44)2 − (c33 − c44)2

2c33(c33 − c44), (B-4)

γ (1) ≡ c66 − c55

2c55, (B-5)

ε(2) ≡ c11 − c33

2c33, (B-6)

δ(2) ≡ (c13 + c55)2 − (c33 − c55)2

2c33(c33 − c55), (B-7)

γ (2) ≡ c66 − c44

2c44, (B-8)

δ(3) ≡ (c12 + c66)2 − (c11 − c66)2

2c11(c11 − c66). (B-9)

ζ (1)—the parameter responsible for the rotation of S1-waveNMO ellipse:

ζ (1) ≡ c16 − c36

2c33. (B-10)

ζ (2)—the parameter responsible for the rotation of S2-waveNMO ellipse:

ζ (2) ≡ c26 − c36

2c33. (B-11)

ζ (3)—the parameter responsible for the rotation of P-waveNMO ellipse:

ζ (3) ≡ c36

c33. (B-12)

Parameters (B-1)–(B-9) are identical to those defined byTsvankin (1997b) for the higher-symmetry orthorhombic me-dia. The additional anisotropic coefficients ζ (1,2,3) are responsi-ble for the rotation of the NMO ellipses of the waves P (ζ (3)),S1(ζ (1)), and S2(ζ (2)) with respect to the coordinate axes. Thecoefficient ζ (3) is analogous to the parameter χz introduced byMensch and Rasolofosaon (1997).

APPENDIX C

WEAK-ANISOTROPY APPROXIMATION FOR MEDIA WITH TWO VERTICAL FRACTURE SYSTEMS

Here, we use equations (1)–(4) and the expressions for thecompliance matrix from Appendix A to obtain linearized ap-proximations for the anisotropic parameters defined in Ap-pendix B. We assume that the weaknesses of both frac-ture systems are small (1Ni¿ 1 and 1T i¿ 1) so that the

anisotropy of the effective monoclinic medium is weak. Keep-ing only terms linear in 1Ni and 1T i leads to the expres-sions listed below. The parameter δ(3) is not given here sinceit has no influence on the NMO velocities of horizontalevents.

s26 f =KN

2sin 2φ − KN − KT

4sin 4φ, (A-4)

s44 f = KT1− cos 2φ

2, (A-5)

s45 f = KTsin 2φ

2, (A-6)

s55 f = KT1+ cos 2φ

2, (A-7)

s66 f =KN + KT

2− KN − KT

2cos 4φ. (A-8)

1828 Bakulin et al.

VP0 = VP

[1− (1− 2g)2

2(1N1 +1N2)

]. (C-1)

VS0 = VS

[1− 1T1

4(1+ cos 2φ1)− 1T2

4(1+ cos 2φ2)

].

(C-2)

ε(1) = −2g{[

(1− g)1N1+ (1T1− g1N1) cos2 φ1]sin2 φ1

+ [(1− g)1N2 + (1T2 − g1N2) cos2 φ2]sin2φ2

}.

(C-3)

ε(2) = −2g{[

(1− g)1N1+ (1T1 − g1N1)sin2φ1]cos2φ1

+ [(1− g)1N2+ (1T2− g1N2)sin2φ2]cos2φ2

}.

(C-4)

δ(1) = −2g{[(1− 2g)1N1 +1T1]sin2φ1

+ [(1− 2g)1N2 +1T2]sin2φ2}. (C-5)

δ(2) = −2g{[

(1− 2g)1N1 +1T1]cos2φ1

+ [(1− 2g)1N2 +1T2]cos2φ2}. (C-6)

γ (1) =[

2(1T1 − g1N1)cos2φ1 − 1T1

2

]sin2φ1

+[

2(1T2 − g1N2)cos2φ2 − 1T2

2

]sin2φ2. (C-7)

γ (2) =[

2(1T1 − g1N1)sin2φ1 − 1T1

2

]cos2φ1

+[

2(1T2 − g1N2)sin2φ2 − 1T2

2

]cos2φ2. (C-8)

ζ (1) = g

4[2g(1N1 sin 2φ1 +1N2 sin 2φ2)

− (1T1 − g1N1)sin 4φ1 − (1T2 − g1N2)sin 4φ2].

(C-9)

ζ (2) = g

4[2g(1N1 sin 2φ1 +1N2 sin 2φ2)

+ (1T1 − g1N1)sin 4φ1 + (1T2 − g1N2)sin 4φ2].

(C-10)

ζ (3) = g(1− 2g)(1N1 sin 2φ1 +1N2 sin 2φ2). (C-11)

The values VP and VS are the P- and S-wave velocities in theisotropic background, and

g ≡ V2S

V2P

. (C-12)

Since the parameters of the effective monoclinic model de-pend on only eight parameters of the fractures and backgroundmedium, not all of the anisotropic coefficients are independent.Combining equations (C-9)–(C-11) yields

ζ (3)

ζ (1)+ ζ (2)= 1

g− 2. (C-13)

It can be proved that this result remains valid for any strengthof the anisotropy. Another relation between the effective co-efficients, which follows from equations (C-3)–(C-8), has theform

δ(1) − δ(2) = 4g(γ (1)− γ (2))+ 1− 2g

1− g

(ε(1)− ε(2)).

(C-14)Equation (C-14) coincides with the constraint derived in part IIfor the effective orthorhombic medium caused by one systemof fractures in a VTI background.

APPENDIX D

ESTIMATION OF FRACTURE PARAMETERS IN THE WEAK-ANISOTROPY LIMIT

The linearized approximations from Appendix C can be usedto invert the anisotropic parameters of monoclinic media forthe orientation and compliances of both fracture sets. A con-venient way of performing this inversion procedure is outlinedhere.

Three ζ -coefficients combined in the form [equation (C-13)]

ζ (3)

ζ (1) + ζ (2)= 1

g− 2 (D-1)

give an estimate of g or the VS/VP ratio [see equation (C-12)] inthe background. To avoid errors stemming from dividing twosmall numbers, g can be approximately found from the ratio ofvertical velocities: g≈V2

S0/V2P0 [see equations (C-1) and (C-2)].

Two combinations,

δ(1) + δ(2) = −2g[(1− 2g)(1N1 +1N2)+ (1T1 +1T2)]

(D-2)

and

ε(1) + ε(2) + g(γ (1) + γ (2)) = −g

2[4(1− g)(1N1 +1N2)

+ (1T1 +1T2)], (D-3)

can be used to find the sums of the normal and tangentialcompliances:

1N1 +1N2 = A− B

3− 2g≡ S1N (D-4)

and

1T1 +1T2 = 4 (g− 1)A+ (1− 2g)B

3− 2g≡ S1T , (D-5)

where

A = δ(1) + δ(2)

2gand B = 2

g

[ε(1)+ε(2)+g

(γ (1)+γ (2))].


Equations (C-7) and (C-8) can be rewritten in the form

1T1 cos 2φ1+1T2 cos 2φ2= 2(γ (1)− γ (2))≡Dγ . (D-6)

Combined with the linearized equation (9),

1T1 sin 2φ1 +1T2 sin 2φ2 = 0, (D-7)

equation (D-6) can be solved for 1T1 and 1T2:

1T1 = Dγ sin 2φ2

sin 2(φ2 − φ1), 1T2 = −Dγ sin 2φ1

sin 2(φ2 − φ1). (D-8)

Substituting equation (D-8) into equation (D-5) yields a rela-tionship between the fracture azimuths:

Dγ cos(φ2 + φ1) = S1T cos(φ2 − φ1). (D-9)

Note that the denominator in equations (D-8) does not vanishfor1T1=1T2 because the azimuths φ1 and φ2 in this case haveopposite signs (φ1=−φ2).

Similarly, equations (C-3), (C-4), (C-9), and (C-10) can becombined to obtain

1N1 cos 2φ1 +1N2 cos 2φ2 = ε(1) − ε(2)

2g(1− g)≡ Dε (D-10)

and

1N1 sin 2φ1 +1N2 sin 2φ2 = ζ (2) + ζ (1)

g2≡ Sζ . (D-11)

Equations (D-10) and (D-11) give the following expressionsfor the normal weaknesses:

1N1 = Dε sin 2φ2 − Sζ cos 2φ2

sin 2(φ2 − φ1),

(D-12)1N2 = −Dε sin 2φ1 + Sζ cos 2φ1

sin 2(φ2 − φ1).

Substituting equation (D-12) into equation (D-4), we find asecond relation between the azimuths:

Dε cos(φ2 + φ1)+ Sζ sin(φ2 + φ1) = S1N cos(φ2 − φ1).

(D-13)

Equations (D-9) and (D-13) can be solved for φ2 + φ1 andφ2 − φ1:

φ2 + φ1 = tan−1[DγS1N −DεS1T

S1TSζ

]. (D-14)

Using equation (D-9), we find

φ2 − φ1 = cos−1[ DγS1T

cos(φ2 + φ1)]. (D-15)

Equations (D-14) and (D-15) make it possible to determinethe fracture azimuths φ1 and φ1.

After recovering the fracture orientation, we can computethe four weaknesses 1T i and 1Ni from equations (D-8) and(D-12). Finally, equations (C-1) and (C-2) yield the P- andS-wave velocities in the background.

APPENDIX E

SPECIAL CASE OF EQUAL TANGENTIAL WEAKNESSES

Suppose the tangential weaknesses of the crack systems areequal, so that

1T1 = 1T2 ≡ 1T . (E-1)

Then it follows from equation (9) that

φ1 = −φ2 ≡ φ. (E-2)

Substituting equations (E-1) and (E-2) into the weak-anisotropy approximations (C-1)–(C-11) yields

VP0=VP

[1− (1− 2g)2

2(1N1 +1N2)

], (E-3)

VS0=VS

[1− 1T

2(1+ cos 2φ)

], (E-4)

ε(1)=−2g{(1− g)(1N1 +1N2)

+ [21T − g(1N1 +1N2)]cos2φ}sin2φ, (E-5)

ε(2)=−2g{(1− g)(1N1 +1N2)

+ [21T − g(1N1 +1N2)]sin2φ}cos2φ, (E-6)

δ(1)=−2g{(1− 2g)(1N1 +1N2)+ 21T }sin2φ, (E-7)

δ(2)=−2g{(1− 2g)(1N1 +1N2)+ 21T }cos2φ, (E-8)

γ (1)= {2[21T − g(1N1 +1N2)]cos2φ −1T}sin2φ,

(E-9)

γ (2)= {2[21T − g(1N1 +1N2)]sin2φ −1T}cos2φ,

(E-10)

ζ (1)= 2g2(1N1 −1N2)sinφ cos3φ, (E-11)

ζ (2)= 2g2(1N1 −1N2)sin3φ cosφ, (E-12)

and

ζ (3) = g(1− 2g)(1N1 −1N2)sin 2φ. (E-13)

Equations (E-3)–(E-13) are much easier to invert for thefracture parameters than the more general expressions of Ap-pendix C. Having estimated g from either equation (D-1) orthe VS0/VP0 ratio, we obtain the fracture azimuth as

δ(1)

δ(2)= ζ (2)

ζ (1)= tan2φ. (E-14)

Note that the angle between the two systems of cracks is equalto 2φ. The tangential weakness 1T can be found from theshear-wave splitting coefficient γ (S) which, in the limit of weak

1830 Bakulin et al.

anisotropy, reduces to the difference between γ (2) and γ (1):

γ (S) = γ (2) − γ (1) = −1T cos 2φ. (E-15)

Then any of equations (E-11)–(E-13) for the ζ coefficients givethe difference between the normal weaknesses1N1−1N2. Thesum 1N1 +1N2 can be determined, for instance, from δ(1) and

δ(2) [equations (E-7) and (E-8)]:

1N1 +1N2 = 12g− 1

(δ(1) + δ(2)

2g+ 21T

). (E-16)

Combined with the difference 1N1−1N2 obtained earlier,equation (E-16) yields the individual values of the normalweaknesses 1N1 and 1N2.

GEOPHYSICS, VOL. 65, NO. 6 (NOVEMBER-DECEMBER 2000); P. 1831–1836.

Deconvolving the ghost effect of the watersurface in marine seismics

Santi Kumar Ghosh∗

ABSTRACT

The ghost filters arising from the effect of the watersurface on both source and receiver sides have a commontime domain representation that consists of a unit im-pulse followed by its ghost, which is a delayed, negativeunit impulse. The origin of the difficulties of deghostinglies in the zeroes in the spectrum of the ghost filter, whichrender incorrect any deghosting through least-squaresinverse filtering in the time domain. Another shortcom-ing of the time domain approach is that the digital de-scription of the ghost filter is inexact when a samplinginstant does not coincide with the instant of the onsetof the ghost impulse. A frequency domain approach, onthe other hand, is straightforward and accurate becauseit can avoid the zeroes of the filter either by explicitlychoosing a recording band that excludes the zeroes orby recording at two depths. These two depths should beselected according to the criterion that their highest com-mon measure is small enough to prevent zeroes at a com-mon frequency of the two recordings. As the source-sideand the receiver-side ghost filters have the same form, thecriterion derived for the selection of the depths of the re-ceivers would also hold for the selection of the depths oftwo sources whose aggregate signature is desired to haveno zeroes in the spectrum, within the operative band. Animportant ramification of the analysis consists of the dis-proof of a prevalent conjecture that the zeroes in thespectrum of a wavelet make its autocorrelation matrixsingular; actually, the zeroes cause an inexact and unac-ceptable least-squares inverse, although the matrix itselfis well conditioned.

INTRODUCTION

Technical reasons require both the source and the receiverto be placed at a certain depth underwater in marine reflectionseismics. The water surface, however, is responsible for distort-

Manuscript received by the Editor April 24, 1996; revised manuscript received March 27, 2000.∗National Geophysical Research Institute, Uppal Road, Hyderabad 500 007, India. E-mail: [email protected]© 2000 Society of Exploration Geophysicists. All rights reserved.

ing both the source waveform as well as the data recorded inthe receiver in a manner to be described.

As the wave is recorded after it has traveled a long paththrough the seawater and the layers within the earth and asthe source-receiver offset is small in comparison, one can ap-proximately treat this as a normal incidence propagation inwhich the spherical divergence effects are nearly the same forthe source (or receiver) and its nearby image. The similaritybetween the two spherical divergence effects is a consequenceof a long path of travel from the source to the receiver.

The contact between the water surface and air acts like aperfect reflector whose reflection coefficient is very nearly−1.As a result, the effective source includes not only the actualsource but also its ghost or negative image, which is situatedabove the water surface at a height equal to the depth of thesource. A similar effect arises on the receiver side as well. Ahydrophone detects not only an upcoming reflection but alsoits negative image reflected from the water-air contact. Botheffects arising in the source and receiver sides can be treatedin terms of two filters of identical forms.

The ghost effect of the water surface is a well-known obstaclein exploration and has been mentioned by Ziolkowski (1971),Giles and Johnston (1973), Brandsaeter et al. (1979), andWaters (1987), among many others. Lindsey (1960) discusses adeghosting solution when the reflection coefficient, unlike thepresent case, has a magnitude smaller than unity. In describingthe ill effects of ghosting, Schneider et al. (1964) note how anoverlap between the desired data and its ghost can compoundthe subsurface interpretation and how they can interfere de-structively, masking subsurface information.

Both ghost filters, arising on source and receiver sides, havezeroes in their frequency response at many frequencies includ-ing the d.c. According to Ziolkowski (1971), a number of airguns at different depths can eliminate zeroes in the amplitudespectrum of the source, making possible an eventual decon-volution of the source signature. The present work presentsa comprehensive remedy to eliminate the zeroes in the spec-trum of the data caused by the receiver side ghost, an issuenot addressed by Ziolkowski (1971). The remedy consists ofrecording at two depths chosen according to the criterion that

1831

1832 Ghosh

the respective ghost filters do not have common zeroes intheir spectra. A similiar criterion applies to the selection ofthe depths of the sources, a crucial aspect not in Ziolkowski(1971).

Numerical examples will demonstrate that although the filteris completely known, the conventional time domain method ofconstructing a least-squares inverse is poor in deconvolving theeffect of the ghost. The analysis has important ramifications inthat it yields interesting insights into the relation between thezeroes in the spectrum of a wavelet and its least-squares inverse.A conjecture by Ziolkowski (1971) that zeroes in the spectrumof a wavelet make its autocorrelation matrix singular turns outto be false. Nevertheless, the inexactness of the least-squaresinverse of a wavelet is shown to have its origin in the zeroes inthe spectrum of the wavelet. This suggests that a frontal assaulton the problem may be more appropriate in the frequencydomain. Indeed, analysis corroborates this conjecture.

DEFINITION OF THE PROBLEM

Ziolkowski (1971) explains the nature of the distortion insource and receiver waveforms caused by the water surface.The discussion here pertains to the distortion on the receiverside; that on the source side is obtainable in an identical mannerwith an appropriate substitution of the source depth for thereceiver depth.

A pressure-sensitive hydrophone is kept at certain depth un-der the sea surface. The fundamental reason for this is that thefree surface of water is always pressure free and excess pres-sure cannot be measured there. The practical reason is to avoidnoise generated on the surface due to sea waves. According toPieuchot (1984), the hydrophone must be operated at a depthbetween 10 and 20 m, whereas Waters (1987) prescribes a depthof 12–13 m. In contrast, operational depths for marine seismic-energy sources are commonly between 6 and 9 m (Johnston andCain, 1982). While keeping a hydrophone at depth is essential,it causes distortion of the signal in the following way.

As stated earlier, the upcoming reflected wavefield observedat the receiver can be approximated by normally incidentplane-wavetrains. I designate this wavefield by s(t). The ghostarising from the water-air contact is also recorded at the re-ceiver as a negative image of s(t) with a time delay equal to thetraveltime from the sensor at depth to the water surface andback.

In other words, instead of s(t), the receiver senses f (t), givenby

f (t) = s(t)− s(t − to), (1)

where the delay time to= 2D/Vw, D= depth of sensor, andVw = velocity of sound in water=1500 m/s, approximately. Theeffect can also be described by a filter, namely,

f (t) = s(t) ∗ h(t), (2)

where

h(t) = δ(t)− δ(t − to), (3)

where δ(t) is the familiar Dirac delta function and ∗ denotes aconvolution.

It is instructive to investigate the characteristics of the filterin the frequency domain. The Fourier transform of h(t) is given

by

H(ω) = 1− exp(iωto), (4)

where ω is the angular frequency and is according to the con-vention that Fourier transform G(ω) of a function g(t) is givenby

G(ω) =∫ ∞−∞

g(t) exp(−iωt) dt. (5)

Equation (4) can be rewritten as

H(ω) = 2 sin(ωto/2) exp[i (π/2− ωto/2)], (6)

in which the exponential term stands for the phase spectrumand the rest for the amplitude spectrum.

In the minimum delay deconvolution model, all the fil-ters involved in the process of propagation, beginning withthe source pulse, must have a minimum phase characteristics(Robinson, 1967). The ghost filters, both at the source and atthe receiver, are evidently not minimum phase filters. This is be-cause a minimum phase filter must have a stable one-sided fre-quency inverse (Robinson and Treitel, 1980), and the ghost fil-ters do not possess a frequency inverse owing to having notchesin the spectrum at many frequencies (including the d.c. fre-quency), as evident from equation (6). Therefore, the effect ofthese filters cannot be removed and the lost frequencies can notbe restored by conventional deconvolution, even in principle.This is in contrast with other earth filters such as the reverber-ation filter, which is known to have a minimum phase with nonotches in the spectrum and which has a standard deconvolu-tion operator. One also notes that no linear filter can restorethe frequencies lost due to the ghost effect. This would neces-sitate design of a deconvolution scheme specific to the presentproblem.

A comprehensive solution must also reconcile two conflict-ing requirements. While a hydrophone depth between 10 and20 m would cause the first non d.c. zero of the ghost spec-trum between 37.5 and 75 Hz, the current emphasis on highresolution requires a band width of 250 Hz (Pieuchot, 1984).Thus on one hand, the requirement of high frequencies forimproving resolution and the consequent need to push the ze-roes of the ghost filter beyond the Nyquist range enforce asmall depth for the hydrophone. On the other, the minimiza-tion of sea wave–generated noise requires a reasonably largedepth for the hydrophone. These conflicting demands wouldbe reconciled satisfactorily if one could, by some means, retaina large depth for the hydrophone without compromising therequirement of a large bandwidth.

TIME DOMAIN DECONVOLUTION

After ascertaining the description of the ghost filter from aknowledge about the depth of the hydrophone, one can makea first attempt at deghosting in the time domain by an inversefilter which is optimal in the least-squares sense. In other words,given the ghost filter h(t), one designs an inverse filter x(t)such that its convolution with h(t) is as close to δ(t) as possible.Recalling equation (2), it is seen that then the desired datas(t) is, in principle, recoverable by the convolution of f (t), therecorded data, with x(t).

It is worthwhile to begin with a realistic example involvinga hydrophone at a depth of 12 m under the sea surface and a

Surface Deghosting in Marine Seismics 1833

sampling interval of 4 ms. This yields a two-way time to of 16 msfor the travel from the sensor to the water surface and back.Under these circumstances, the ghost filter h(t) has a discretetime representation

ht = (1, 0, 0, 0,−1). (7)

One can construct an inverse of ht by the standard method offinding the least-squares inverse (Robinson and Treitel, 1980).Here the input is ht , and the output is dt = (1, 0, 0, 0 . . .), thediscrete representation of δ(t). What is sought is xt , the optimalinverse of ht . The autocorrelation of ht , symmetric about zerotime lag is given by

φhh(τ ) = (2, 0, 0, 0,−1) (8)

for positive time lag. The corresponding autocorrelation matrixΦhh turns out to be

Φhh =

2 0 0 0 −1

0 2 0 0 0

0 0 2 0 0

0 0 0 2 0

−1 0 0 0 2

. (9)

The inverse of this matrix is given by

Φ−1hh =

2/3 0 0 0 1/3

0 1/2 0 0 0

0 0 1/2 0 0

0 0 0 1/2 0

1/3 0 0 0 2/3

. (10)

If the successive sample values of xt constitute a column vector,

X =

x0

x1

x2

x3

x4

,

then the solution of the normal equations yield

X = Φ−1hhφdh, (11)

whereφφdh is the crosscorrelation vector between dt and ht . Thiscrosscorrelation turns out to be dt itself for nonegative timelags. Since dt = (1, 0, 0 . . .), xt = (2/3, 0, 0, 0, 1/3), i.e., merelythe successive entries in the first column of the inverse of theautocorrelation matrix in equation (10).

The performance of the filter xt is readily evaluated by com-puting the actual output at obtainable through the convolutionof ht and xt :

at = (2/3, 0, 0, 0,−1/3, 0, 0, 0,−1/3). (12)

The poor spiking entails a normalized mean-square error of1/3, a reduced amplitude at zero time, and two negative ghostsof appreciable magnitude. Investigations reveal that neither in-creasing the length of the inverse filter nor delaying the desiredoutput improves the result.

The autocorrelation matrix has an inverse and is far frombeing singular. Indeed, it is well conditioned. The conditionnumber, defined as the ratio of the moduli of the largest andthe smallest eigenvalues turns out to be 3. This goes against theconjecture of Ziolkowski (1971) that the zeroes in the spectrumof a wavelet lead to a singular autocorrelation matrix.

The poor quality of the inverse filter indeed is related to thezeroes in the spectrum of the wavelet. The aforesaid ghost fil-ter (1, 0, 0, 0,−1) with a sampling interval of 4 ms had zeroesat 62.5 Hz (half of the Nyquist frequency) and at 125 Hz (theNyquist frequency itself). This fact is responsible for the poorspiking. To demonstrate this claim, consider a hypothetical ex-ample with a hydrophone depth of 3 m, a rather unrealisticchoice in noisy sea conditions. It is easy to verify that with thesame sampling interval of 4 ms the ghost filter now assumes theform

ht = (1,−1), (13)

which has one zero at the d.c. frequency and whose other zeroesare well beyond the Nyquist frequency. It will now be shownthat this filter has a least-squares inverse that is exact in anasymptotic sense. Following a standard exercise of spiking, onecan design an inverse of any length. By progressively increasingthe length of the filter, one discovers a pattern in the structureof the inverse filter. The pattern reveals that an inverse filterof length n has the representation

xt = [n/(n+1), (n−1)/(n+ 1), . . . 2/(n+1), 1/(n+1)].

(14)The actual output at , obtainable by a convolution of ht and xt ,turns out to be

at = [n/(n+ 1),−1/(n+ 1),−1/(n+ 1), . . . −1/(n+ 1)].

(15)The normalized error energy becomes

E = 1/(n+ 1). (16)

Evidently, as n tends to infinity, at approaches a spike of unitamplitude whose tail assumes zero values everywhere. Onecan argue that it is the absence of zeroes in the Nyquist rangein the frequency response of the filter which lends it an exactinverse and makes it akin to a minimum phase filter. For areasonable performance, the filter length must be long, as seenfrom equation (16).

In addition to the fact that the depth of the hydrophone isunrealistically shallow, there is an implementational problemwith this inverse filter. The undesired tail in the actual outputin equation (15) is nothing but a negative and scaled rectanglefunction, rich in d.c. and near-d.c. frequencies. The area un-der the actual output wavelet is zero, indicating the lack of d.c.component in it. The negative tail has the effect of reducing theamplitudes of the very low frequencies. This is because whenconvolved with the data, the tail has a low-pass effect becauseof its smoothing property, and a negative sign in effect sub-tracts the low-frequency amplitudes from the principal spikeat zero time, which contains all frequencies in the band. Theresult is a suppression of very low frequencies whose phasedifference with the main spike is small enough to permit arith-metic subtraction of Fourier amplitudes in place of vectorialsubtraction. Higher frequencies will undergo phase distortionin addition to the amplitude distortion. In short, the effect of

1834 Ghosh

the tail compromises the very purpose of deconvolution, whichaims at an equal emphasis and a zero phase for all frequencies.

There is another difficulty with deghosting in the time do-main. As seen from equation (3), the representation of theghost filter h(t) involves delta functions in continuous time. Adiscrete representation implies band limiting. A delta functionband limited between the frequencies −νm and +νm assumesthe form Ib(t) given by

Ib(t) = (1/π) sin(2πνmt)/t. (17)

At t = 0, Ib(t) is maximum and equals 2νm. If one wishes to sam-ple this function, the sampling interval 1t can be legitimatelychosen as 1/2νm. The zeroes of Ib(t) occur at

t = ±n1t, n = 1, 2, . . . . (18)

Thus, except for t = 0, all other sample values of Ib(t) are iden-tically zero. That is the reason that the delta function has anexact discrete representation (1, 0, 0 . . .) when one ignores ascale factor of 2νm. As long as the sampling instant coincideswith the instant of onset of the impulse, the delta function willcontinue to have such finite representation. When that is notso, the exact representation will contain an infinity of nonzerosample values extending in both positive and negative timedirection. For actual computation, such a sequence must betruncated at the cost of error. If one chooses 10 m as the depthof the hydrophone and a sampling interval of 4 ms, then thevalue of to in the representation of h(t) in equation (3) wouldbe 13.3 ms, which will not coincide with a sampling instant.This will require a truncated representation of h(t), involvingcomplication and error.

All of the three preceding examples argue against carryingout the deghosting in the time domain. The next section showsthat the frequency domain offers better prospects.

FREQUENCY DOMAIN DECONVOLUTION

Since the difficulties of deghosting originate from the zeroesin the response of the ghost filter, a frontal assault on the prob-lem is more apt in the frequency domain. A frequency domainsolution is readily possible if the recording is done in a fre-quency band where the lowcut point is a little above zero (4 or5 Hz) and the highcut limit is below 1/to. This band containsno zeroes in the filter response, and the effect of the filter canbe removed. From the frequency domain equivalent of equa-tion (2), one can write

S(ω) = F(ω)/H(ω), (19)

where H(ω) is given by equation (6). Operation (19) is sta-ble because H(ω) is nonzero within the band. Next, s(t), thedesired data, can be recovered by taking an inverse Fouriertransform of S(ω). This, however, has limited applications, asin the case below.

In marine seismics intended for the study of gross geologicfeatures, one operates in a lower frequency range, typically witha lowcut at 4 Hz and a highcut at 64 Hz. In such a case, a depthof 10 m would produce a second zero in the filter response at75 Hz. This is tolerable for the objective. Also, a depth of 10 mis often adequate for avoiding surface noise.

If, however, the restriction of a smaller bandwidth is nottolerable, a comprehensive solution to the problem is possibleby making the following modification in the data acquisition

procedure. The only additional requirement is to record not justat one depth, but at two successive depths. Let these depthsbe D1 and D2 with D2 > D1. Now the upcoming signal s(t)will reach D2 and, after some delay, D1. If one fixes D2 as thereference, then the upcoming signal at D2 will be s(t) and thatat D1 will be s[t − (D2− D1)/Vw]. In addition, each sensor willrecord the respective delayed images of the signal. Therefore,the recorded trace at D2 would be

f2(t) = s(t) ∗ [δ(t)− δ(t − t2)], (20)

and that at D1 would be

f1(t) = s[t − (D2 − D1)/Vw] ∗ [δ(t)− δ(t − t1)], (21)

where t1= 2D1/Vw and t2= 2D2/Vw .After transformation into the frequency domain, the last two

equations would assume the following forms:

F1(ω) = exp[−iω(D2 − D1)/Vw]S(ω)[1− exp(−iωt1)]

(22)and

F2(ω) = S(ω)[1− exp(−iωt2)]. (23)

A rearrangement of equation (22) followed by its addition toequation (23) yields

F1(ω) exp[iω(D2 − D1)/Vw]+ F2(ω)

= S(ω)[2− exp(−iωt1)− exp(−iωt2)]. (24)

From equation (24), one obtains

S(ω) = {F1(ω) exp[iω(D2 − D1)/Vw]+ F2(ω)}/[2− exp(−iωt1)− exp(−iωt2)]. (25)

The process of computing S(ω) is now stable except atω= 0.This is unavoidable but is inconsequential in practice, as thezero frequency is never recorded. Except for this pathologiccase, the denominator in the last equation can always be madenonvanishing in principle by a proper choice of D1 and D2. Asshown in the Appendix, the proper choice turns out to be thecriterion that D1 and D2 should have no common measure, (i.e.,D1/D2 should be irrational). In practice, however, D1 and D2

will have common measures and the second zero will occur ata finite frequency given by

ν=Vw/(2× the highest common measure of D1 and D2),

(26)as established in the Appendix.

For example, if D2= 13 m and D1= 10 m, their highest com-mon measure is 1 m. This yields ν= 750 Hz, which is well be-yond any seismic frequency recorded in practice.

Ziolkowski (1971) suggests putting sources at multipledepths as a means of eliminating the zeroes of the ghost fil-ter on the source side, though he does not provide a criterionfor choosing the depths. Analysis done here for deghosting onthe receiver side obtains this criterion that applies to the sourceside as well because both ghost filters have identical forms. It isevident that two sources at depths D1 and D2 are sufficient toavoid zeroes in the frequency response of the aggregate ghostfilter, provided that D1 and D2 are chosen so that their high-est common measure is as low as possible in order to push the

Surface Deghosting in Marine Seismics 1835

zeroes of the aggregate ghost filter beyond a frequency bandwhose width is given by equation (26). An aggregate signatureof the sources and their ghosts recorded at large enough depthcan then be used for an effective signature deconvolution.


The ghost filters arising from the effect of the water surfaceon both source and receiver sides are quite similar to eachother and dissimilar to the filters of the convolutional model.Whereas the earth filters are known to have a minimum phasecharacter, the ghost filters do not have minimum phase charac-teristics when the filter response contains zeroes in the Nyquistrange.

The ghost filters both on the source and the receiver sideshave zeroes in their frequency response. It is shown that thestandard time-domain method of inverse filters cannot effec-tively get rid of the ghost. For realistic hydrophone depths andfor a reasonably large bandwidth, the ghost filter must con-tain zeroes in the signal spectrum in addition to the manda-tory zero at the d.c. frequency. These additional zeroes areresponsible for an imperfect deghosting, which consists of asplit of the ghost into two smaller ghosts while the main spikeis only two-thirds of its desired amplitude. Both increasing thelength of the inverse filter and delaying the temporal positionof the desired spike turn out to be fruitless exercises in thiscase.

Though unrealistic, one can conceive of a shallow hy-drophone depth which would restrict the ghost filter’s length totwo samples, thus avoiding zeroes in the filter spectrum withinthe Nyquist range except for the essential zero at the d.c. fre-quency. In this case, the inverse filter’s performance improves,as its length increases, approaching that of the exact asymp-totically. The inverse filter’s problem, however, lies in actualimplementation, when the filter length must be finite. Then,the actual output turns out to be a spike followed by a negativetail of uniform magnitude and whose length equals the lengthof the filter. When convolved with the actual data, the tail hasa detrimental effect: the lower frequencies suffer attenuationand even their phases are altered, compromising the very aimof deconvolution.

The last two instances also make a point regarding the in-verse of the ghost filter. When the ghost spectrum containszeroes, the inverse filter performs poorly, giving rise to ratherunsatisfactory deghosting. On the other hand, when the spec-trum has no zeroes in the Nyquist range, the inverse filter be-comes exact as its length tends to infinity. The conjecture inZiolkowski (1971) that zeroes in the spectrum of a waveletmake its autocorrelation matrix singular is belied in the treat-ment of the first instance. The autocorrelation matrix turnsout to be well conditioned. The link, however, lies in the poorspiking ability of the autocorrelation matrix when the waveletspectrum has zeroes.

Another problem associated with the least-squares deghost-ing in the time domain is the error and the complication whichmay arise in the digital description of the ghost filter in whichthe onset of the ghost impulse does not coincide with a samplinginstant. In this case, the filter’s exact digital representation isnot finite, and any truncation for computational purposes leadsto error and complication.

In view of the above, a frequency domain deghosting is moreappropriate. When the recording band has a relatively small

width and does not include the frequencies at which the ghostfilter has zeroes, a frequency domain deghosting is readily pos-sible. When, on the other hand, one aims at a wide recordingband for greater resolution, one must record the seismic fieldat two depths. The depths are chosen such that the respectiveghost filters do not have common zeroes in their spectrum. Acriterion which yields this condition is that the depths D1 andD2 should have a highest common measure that is as low aspossible with the width of the recording band being given byequation (26).

Equation (25) is based on the assumption of a reflectedplane wavefront traveling vertically upward. When the wave-front normal (the ray) is not vertical but away from it by anangle θ , the effective delay time of the ghost can be shownto be to cos θ , where to [one recalls from equation (1)] is thedelay time for normal incidence. Whereas for normal inci-dence the delay of the ghost corresponds to the two-way ver-tical distance from the hydrophone to the water surface, foroblique propagation the delay is produced along an obliqueray which is initially parallel to the ray carrying from belowthe true signal to the hydrophone and which after reflectionat the free surface comes down to the hydrophone to pro-duce the ghost. It is easy to verify that the decrease in de-lay time is by about 3% for an emergence angle of 15◦ andaround 6% for an angle of 20◦ with the vertical. One, of course,bears in mind that an emergence angle of 20◦ in water wouldcorrespond by Snells’s law for a horizontally stratified earthto near-critical angles of reflection in the deeper sedimentswhose compressional velocity would often be more than two-and-a-half times that of water. This would impose tolerablelimits on the error caused on equation (25) by nonverticalpropagation.

To implement this method of deghosting one must have twoparallel recording arrangements. This requires operating twostreamers at two preset depth levels constantly monitored bydepth transducers incorporated in the streamers. Also, an ac-curate knowledge of the depths of the hydrophones is essentialfor deghosting. The two recording systems must also be iden-tical in their electromechanical response. All this will surelyincrease the cost of operation, but it offers the benefit of asuperior deghosting result.

In suggesting multiple sources at different depths,Ziolkowski (1971) was aiming at avoiding common zeroes inthe source spectrum. He, however, provided no criterion forselecting the depths. Analysis here points to the requirementof only two sources at two depths chosen according to the crite-rion, as in the case of receiver depths, that the highest commonmeasure of the two source depths should be as small as possiblein order to yield a source spectrum as broad as required.

ACKNOWLEDGMENTS

It is a pleasure to thank Dr. H. K. Gupta, Director, NationalGeophysical Research Institute, for his kind permission to pub-lish this paper and for his constant encouragement.

This work was originally submitted to The Leading Edge. Dr.Sven Treitel, the then SEG Editor, was considerate enough tothink it appropriate for GEOPHYSICSand to put it in the reviewprocess.

The suggestions by P. G. Kelamis, Associate Editor, andC. G. Macrides, the reviewer, have rendered the manuscriptmore readable.

1836 Ghosh

Thanks are due to Mr. V. Subrahmanyam for typing the orig-inal manuscript. I am grateful to Mr. G. Ramakrishna Rao forkeying the revised manuscript into a word processing system.

REFERENCES

Brandsaeter, H., Farestveit, A., and Ursin, B., 1979, A new high-resolution or deep penetration airgun array: Geophysics, 44, 865–879.

Giles, B. F., and Johnston, R. C., 1973, System approach to air-gun arraydesign : Geophys. Prosp., 21, 77–101.

Johnston, R. C., and Cain, B., 1982, Marine seismic energy sources:Acoustic performance comparison: Presented at the 14th OffshoreTech. Conf.

Lindsey, J. P., 1960, Elimination of seismic ghost reflections by meansof a linear filter: Geophysics, 25, 130–140.

Pieuchot, M., 1984, Seismic instrumentation: Geophysical Press.Robinson, E. A., 1967, Multichannel time series analysis with digital

computer programmes: Holden-Day.Robinson, E. A., and Treitel, S., 1980, Geophysical signal analysis:

Prentice-Hall, Inc.Schneider, A. W., Larner, K. L., Burg, J. P., and Backus, M. M.,

1964, A new data-processing technique for the elimination ofghost arrivals on reflection seismograms: Geophysics, 29, 783–805.

Waters, K. H., 1987, Reflection seismology: John Wiley & Sons, Inc.Ziolkowski, A., 1971, Design of a marine seismic reflection profile sys-

tem using airguns as a sound source: Geophys. J. Roy. Astr. Soc., 23,499–530.

APPENDIX

DERIVATION OF THE CRITERION FOR CHOOSING THE HYDROPHONE DEPTHS

The computation of S(ω) is unstable whenever the denomi-nator in equation (25) is zero, i.e.

2− exp(−iωt1)− exp(−iωt2) = 0. (A-1)

For nonzero frequencies, this would imply that

ωt1 = 2kπ (A-2)

and

ωt2 = 2nπ, (A-3)

where both k and n are integers. In terms of frequency ν, equa-tions (A-2) and (A-3) can be combined to

ν = k/t1 = n/t2. (A-4)

Recalling that t1= 2D1/Vw and t2= 2D2/Vw , equation (A-4)can be readily rewritten as

D1/D2 = k/n. (A-5)

The last equation provides the condition for the denominatorin equation (25) to become zero.

If D1 and D2 have no common measure (i.e., if D1/D2 is ir-rational), then condition (A-5) would never be satisfied. This,however, may not be realistic for an actual marine seismic ex-periment, and one allows D1 and D2 to have common measuresin a tolerable manner. Thus, one permits D1=ak and D2=an,such that

D1/D2 = (ak)/(an) = k/n, (A-6)

where k and n are two integers with no common factor betweenthem. Consequently, a is the highest common measure betweenD1 and D2. One can then find the lowest frequency ν at whichthe denominator would become zero. Using equations (A-4)and (A-6), one can write

ν = Vw/(2D1/k) = Vw/2a. (A-7)

Equations (A-7) and (26) make identical statements.


An approach to upscaling for seismic waves in statisticallyisotropic heterogeneous elastic media

N. Gold∗, S. A. Shapiro‡, S. Bojinski∗, and T. M. Muller‡

ABSTRACT

For complex heterogeneous models, smoothing is of-ten required for imaging or forward modeling, e.g., raytracing. Common smoothing methods average the slow-ness or squared slowness. However, these methods areunable to account for the difference between scatter-ing caused by fluctuations of the different elastic moduliand density. Here, we derive a new smoothing methodthat properly accounts for all of the parameters of anisotropic elastic medium. We treat the geophysical prob-lem of optimum smoothing of heterogeneous elastic me-dia as a problem of moving-window upscaling of elasticmedia. In seismology, upscaling a volume of a hetero-geneous medium means replacing it with a volume of ahomogeneous medium. In the low-frequency limit, thisreplacement should leave the propagating seismic wave-field approximately unchanged. A rigorous approach to

upscaling is given by homogenization theory. For ran-domly heterogeneous models, it is possible to reduce theproblem of homogenization to calculating the coherentwavefield (mean field) in the low-frequency limit. Afterderiving analytical expressions for the coherent wave-field in weakly heterogeneous and statistically isotropicrandom media, we obtain a smoothing algorithm. We ap-ply this algorithm to random media and to deterministicmodels. The smoothing algorithm is frequency depen-dent, i.e., for different dominant frequencies, differentsmooth versions of the same medium should be consid-ered. Several numerical examples using finite differencesdemonstrate the advantages of our approach over com-mon smoothing schemes. In addition, using a numericaleikonal equation solver we show that, in the case of com-plex heterogeneous media, appropriate initial smooth-ing is important for high-frequency modeling.

INTRODUCTION

The modeling of wave propagation in heterogeneous mediais increasingly important in seismic exploration. For example,from vertical and horizontal log-based geostatistical modelingof reservoirs, extremely complex models of media can be ob-tained. Upscaling of heterogeneous media is necessary to sim-plify models of the earth. In seismology, upscaling a volumeof a heterogeneous medium usually means replacing it witha volume of a homogeneous medium. This substitution shouldnot change the propagating seismic wavefield. A typical upscal-ing problem for seismic waves is the smoothing of heteroge-neous models. In seismic processing it is often useful to smoothmodels for later application of imaging or ray-tracing methods(Grubb and Walden, 1995). Since ray-tracing methods can becorrectly applied only to media with spatial fluctuations largerthan the wavelength (Cerveny et al., 1977), smoothing is nec-

Published on Geophysics Online May 30, 2000. Manuscript received by the Editor June 23, 1997; revised manuscript received April 7, 2000.∗Karlsruhe University, Geophysical Institute, Hertzstrasse 16, 76187 Karlsruhe, Germany. E-mail: [email protected]; [email protected].‡Freie Universitat Berlin, Fachrichtung Geophysik, Malteserstrasse 74-100, Haus D, 12249 Berlin, Germany. E-mail: [email protected]; [email protected]© 2000 Society of Exploration Geophysicists. All rights reserved.

essary for media containing small-scale inhomogeneities. Usu-ally, smoothing methods average the slowness or the squaredslowness. However, since fluctuations of the Lame parametersand the density cause different scattering effects, averagingmethods using only the velocities cannot smooth the mediumin a way that properly accounts for the effects of stiffness anddensity fluctuations.

A rigorous procedure for smoothing must correctly accountfor scattering effects of heterogeneities of the small spatial scale(a scale much smaller than the dominant wavelength). Thismeans the difference between the wavefield propagating in theactual medium and the wavefield propagating in a smoothedversion of this medium must be minimized.

A rigorous approach to smoothing in heterogeneous media ishomogenization theory. The mathematical background of thistheory was summarized and further developed by Bensoussanet al. (1978) and Sanchez-Palencia (1980). The homogenization

1837

1838 Gold et al.

approach is briefly explained as follows. In heterogeneous me-dia the elastic parameters and the density (let us denote themas functions fi ) depend on the location r, fi = fi (r). To char-acterize the nonsmooth behavior of fi (r) we must introduce alength characterizing our observation system, i.e., the dominantwavelength λ. Additionally, we introduce a small parameter ν.With respect to our observation system, the functions fi willbe nonsmooth if they significantly fluctuate on the scale νλ.Apart from the fluctuations on the small scale, the functions fi

can fluctuate independently on a larger scale. Therefore, theyare functions of two arguments, fi = gi (r/ν, r), where the firstargument describes the fluctuating character of fi on the smallscale.

The functions gi define the linear differential operatorLg(r/ν, r) of the dynamic equation (i.e., Lame’s equation):

Lg

(rν, r)

u(

rν, r)= 0, (1)

where u(r/ν, r) is the displacement. In the homogenization ap-proach a solution of this equation is sought in the form of thefollowing series:

u(

rν, r)= u0(r)+ νu1

(rν, r)+ ν2u2

(rν, r)+ · · · . (2)

Under some conditions (which are applicable to usualgeological heterogeneous structures) homogenization theoryshows that the first term, u0(r), in series (2) satisfies the follow-ing new Lame equation:

L0(ν, r)u0(r) = 0, (3)

where L0(ν, r) is the corresponding linear differential Lameoperator. This equation describes the main part of the wave-field [on the order of 1, while other terms of series (2) are on theorder of ν and higher], which propagates in an elastic mediumwithout small-scale fluctuations. This medium is a smoothedcounterpart of the original medium. Its density and elastic pa-rameters [which now define the new Lame operator L0(ν, r)] donot depend on the small-scale variable r/ν. One can show thatif ν→ 0, then u→ u0 and Lg→L0. Therefore, homogenizationtheory provides a rigorous smoothing algorithm.

Moreover, in nonsmooth heterogeneous media equation (3)is subject to further approximations such as geometrical op-tics or ray tracing. However, the operator L0 depends onthe small parameter ν which is the ratio between the char-acteristic size of the small spatial-scale fluctuations and thewavelength. Therefore, L0 is wavelength dependent. In otherwords, for different frequencies one must work with differ-ently smoothed versions of the same nonsmooth medium. Thishas sound physical consequences. For instance, the ray-tracingapproximation must be wavelength dependent because raysets are different in different smooth versions of the samemedium.

In the general case of an arbitrary heterogeneity, the ho-mogenization approach does not provide any closed analyticalrecipe for smoothing. Therefore, to arrive at a general and sim-ple smoothing rule, we must introduce some simplifications.Let us consider equation (3) in a spatial domain which is stilllarger than the characteristic length of the small-scale fluctua-

tions but which is so small that elastic moduli and density areindependent of the large spatial-scale variable r. In this caseequation (3) has the form

L0(ν)u0(r) = 0. (4)

This equation implies that the smooth version of the mediumprovided by the homogenization approach is a homogeneousmedium. Because the density and the elastic parameters ofthis homogeneous medium are found in the long-wavelengthlimit, it is often called the effective medium. Therefore, in thislimit the problem of smoothing is reduced to the problem ofupscaling elastic properties in heterogeneous media. Thus, ifwe carefully scale up the elastic properties of the medium in avery small spatial window and then move this window in spaceand replace the elastic parameters in the center of the windowby the corresponding upscaled parameters, we arrive at a well-substantiated smoothing procedure.

Generally, even in the case of equation (4) the homoge-nization approach does not provide a simple analytic solution.However, for randomly heterogeneous media equation (1) canbe solved for the averaged (coherent) wavefield 〈u〉, where av-eraging stands for statistical or spatial averaging on the smallspatial scale νλ. Moreover, Keller (1977) shows that 〈u〉=u0 ifν→ 0. Thus, instead of solving the homogenization problem,we can look for the coherent wavefield in the low-frequencyrange. Assuming random weak heterogeneities, an analyticalrecipe for upscaling arbitrary isotropic structures can be ob-tained. This is the strategy of our paper.

For weakly heterogeneous acoustic random media, analyti-cal solutions for the coherent part of the wavefield already exist(Rytov et al., 1987; Shapiro et al., 1996). In general, the effec-tive medium, i.e., a fictitious homogeneous medium where thewavefield equals the coherent field, is dispersive and absorb-ing. However, in the long-wavelength limit the result for theeffective medium is neither absorbing nor dispersive. In thispaper we derive the coherent wavefield in weakly fluctuatingelastic random media by means of the elastic Bourret approx-imation (see Appendix A). Using the low-frequency limit ofthis theory, we find a frequency-dependent smoothing methodwhere the smoothing area is small compared to the wavelength.To demonstrate the usefulness of our smoothing method, weshow results of finite-difference modeling in original and dif-ferently smoothed media. Finally, using a finite-difference so-lution of the eikonal equation, we show that for complex het-erogeneous media appropriate initial smoothing is importantfor high-frequency modeling.

EFFECTIVE SLOWNESSES

The elastic Bourret approximation (Appendix A) providesthe effective Green’s function for wavefields in an elasticisotropic random medium, assuming small fluctuations of themedium parameters and low frequencies. As a result, we obtainthe effective wavenumbers of the P- and S-waves, dependingon frequency, the spatial correlation functions of the mediumparameters, and their crosscorrelation functions.

To calculate the wavenumbers, the parameters of the randommedium are assumed to be the sum of the parameters of ahomogeneous background medium, where density and Lameparameters are spatially averaged mean values (λo, µo, ρo), andof the fluctuations of the medium parameters (ελ, εµ, ερ):

Upscaling for Seismic Waves 1839

λ(r) = λo(1+ ελ(r)); 〈ελ〉 = 0,

µ(r) = µo(1+ εµ(r)); 〈εµ〉 = 0, (5)

ρ(r) = ρo(1+ ερ(r)); 〈ερ〉 = 0.

As a precondition, the statistical average of the fluctuations εvanishes.

We find the following results for the slownesses of the co-herent P-wave and S-wave in a 3-D random medium in thelow-frequency and weak uctuation limits:

Spe = Sp

o

[1+ 1

2λ2

o

(λo + 2µo)2σ 2λλ +

23

λoµo

(λo + 2µo)2σ 2λµ

+ 25

µ2o

(λo + 2µo)2σ 2µµ +

415

µo

(λo + 2µo)σ 2µµ

], (6)

Sse = Ss

o

[1+ 1

5σ 2µµ +

215

µo

λo + 2µoσ 2µµ

]. (7)

For 2-D media we find

Spe = Sp

o

[1+ 1

2λ2

o

(λo + 2µo)2σ 2λλ +

λoµo


+ 34

µ2o

(λo + 2µo)2σ 2µµ +

14

µo

(λo + 2µo)σ 2µµ

], (8)

Sse = Ss

o

[1+ 1

4σ 2µµ +

14

µo

λo + 2µoσ 2µµ

]. (9)

In these equations σ 2xz denotes the normalized covariance of

fluctuations of arbitrary parameters x and z, σ 2xz=〈εxεz〉, and

Spo and Ss

o are the slownesses of P-waves and S-waves in thehomogeneous background medium, respectively:

Spo =

[ρo

λo + 2µo

]1/2

, (10)

Sso =

[ρo

µo

]1/2

. (11)

The result depends only on the background medium parame-ters, the variances, and the covariances of the Lame parame-ters. Density fluctuations have no effect on the properties ofthe effective medium in the case of low frequencies becausevariances of ρ(r) do not occur in equations (6)–(9). Again,we emphasize those results have been obtained in the low-frequency and small-contrast (i.e., weak-fluctuation) approxi-mations. Therefore, all terms with third and higher powers ofthe medium-parameter fluctuations ε have been neglected. Wedenote such terms as O(ε3).

The medium parameterization given by equation (5) is notthe only one possible. In the case of another parameteriza-tion, equations (5)–(11) must be changed. For instance, let usdescribe the random medium using the compliances insteadof the Lame parameters. Now the parameters of the randommedium are the sum of the parameters of a homogeneousbackground medium, where the density and the compliancesm= (λ+ 2µ)−1 and χ =µ−1 are spatially averaged mean val-ues (mo, χo, ρo) and of the fluctuations of the medium para-

meters (εm, εχ , ερ):

m(r) = mo(1+ εm(r)); 〈εm〉 = 0,

χ(r) = χo(1+ εχ (r)); 〈εχ 〉 = 0, (12)

ρ(r) = ρo(1+ ερ(r)); 〈ερ〉 = 0.

To obtain the new forms of equations (6)–(9), we must expressthe Lame parameters in terms of the new parameterization:

λ = χo − 2mo

moχo

[1+ 2mo

(εχ − ε2

χ

)− χo(εm − ε2

m

)χo − 2mo

], (13)

µ = 1χo

[1− εχ + ε2

χ

]. (14)

In these two equations we neglected all terms of order O(ε3)and higher. The statistical averaging of these equations pro-vides us with the properties of the old background medium:

λo = χo − 2mo

moχo

[1+ χoσ

2mm− 2moσ

2χχ

χo − 2mo

], (15)

µo = 1χo

[1+ σ 2

χχ

]. (16)

Now equations (13)–(16) can be used directly to express thevariances of the Lame parameters in terms of the variances ofthe compliances:

σ 2λλ =

χ2oσ

2mm+ 4m2

oσ2χχ − 4moχoσ

2mχ

(χo − 2mo)2, (17)

σ 2µµ = σ 2

χχ , (18)

σ 2λµ =

χoσ2mχ − 2m2

oσ2χχ

χo − 2mo. (19)

In the next step, to express the effective slownesses of P- andS-waves in terms of the new parameterization, we must substi-tute results (15)–(19) into equations (6)–(11). This procedurewill provide us with a new form for equations (6)–(9). The phys-ical meaning and the values of slowness will not change [withaccuracy up to the terms on the order of O(ε3) and higher].

Any other parameterization of the medium can be chosen. Inthe following we use our parameterization and equations (5)–(11), respectively. This parameterization is convenient in thesense of its direct applicability to the Bourret approximation.

UPSCALING AND SMOOTHING

In this section we establish the smoothing method for anarbitrary point O in an isotropic elastic random medium. Firstwe define a circular area around this point. Using the gridpointsin this area, we determine the mean values, the variances, andthe covariances in equations (6)–(9):

xo = 16a

∫6a

x(r) d2r, (20)

zo = 16a

∫6a

z(r) d2r, (21)

σ 2xz =

1xozo6a

∫6a

(x(r)− xo)(z(r)− zo) d2r, (22)

1840 Gold et al.

where6a denotes the circular area (spherical volume) of radiusa around point O,

6a =a

,

and x and z are arbitrary properties of the medium. Numeri-cally, a summation is performed in these equations for all gridpoints in the area6a. Using equations (6)–(9), we calculate theeffective slownesses for point O. Using the averaged densityand the rule ρe= ρo, we can replace the real medium parame-ters by the upscaled medium parameters:

µe = ρo(Ss

e

)2 , (23)

λe = ρo(Sp

e)2 − 2µe. (24)

The relation of the radius of the averaging area to the wave-length is a subject of heuristic choice. However, the theo-retical approach is based on the low-frequency assumption.Therefore, the radius can be considered small compared to thewavelength. In the following we discuss this issue using sev-eral numerical examples. Finally, by moving the circular area(spherical volume in three dimensions) 6a and repeating the

FIG. 1. A realization of the random medium with an expo-nential correlation function and a correlation length of 10 m.The area of the medium is 500× 500 m2; the grid spacing is1 m. Bright spots denote areas with large parameter valuescompared with the mean value; in dark areas, parameters aresmaller than the mean value.

FIG. 2. Parameters of medium 1 at a depth of 200 m. The standard deviation of the stiffness tensor elements is 20%; M = λ+ 2µand µ are perfectly correlated.

above procedure for all points of a model, the smoothing willbe performed.

NUMERICAL RESULTS

To demonstrate our smoothing method and to estimate thesmoothing volume around a point, we performed several nu-merical tests using an explicit finite-difference program. As acriterion for the accuracy of our smoothing algorithm com-pared to other methods, we considered the traveltimes inthe original and smoothed media. An appropriate smoothingmethod should preserve both the amplitude as well as the trav-eltimes of the propagating modes. However, we restricted ourattention to the traveltimes of P-waves because they are ofmajor importance for ray-tracing or imaging problems.

For all numerical simulations we used the same backgroundmedium with a P-wave velocity of 4000 m/s, an S-wave ve-locity of 2300 m/s, and density of 2.5 g/cm3 and added fluctu-ations of the elastic parameters. As a source we used a planewave (Ricker wavelet) propagating from the surface into depththrough the medium. Significant frequencies were in the rangeof 35 to 150 Hz with a maximum in the amplitude spectrum at75 Hz. This corresponded to a wavelength of about 50 m forthe P-wave. The theoretical result for our smoothing algorithmwas valid in the case of a smoothing radius that was small com-pared to the wavelength, so an averaging radius of 30 m wasslightly beyond the validity range. However, even in this casewe found good results using our upscaling method. Figure 1shows a realization of the random medium with an exponen-tial correlation function and a correlation length of 10 m.

Medium 1: Fluctuations of λ and µ

The first medium we used had fluctuations of the Lame pa-rameters λ and µ but no density fluctuations. The fluctuationsof λ andµwere perfectly correlated. The perfect correlation offluctuations provided a most serious test of equations (6)–(9)because of the complete contributions of terms with crossvari-ances. Figure 2 shows the medium parameters at a depth of200 m. Because of the spatial fluctuations of the medium pa-rameters, the initial plane wave becomes distorted when prop-agating through the medium. Figure 3 shows snapshots of thewavefield at different times. Dark and light areas indicate neg-ative and positive displacement values in the vertical direction.Applying the 2-D smoothing algorithm to the medium shownin Figure 2, we obtained the smoothed medium parametersshown in Figure 4. Note the change in axis scale in comparison


with Figure 2. From the propagation of the elastic wave in thesmoothed medium, as shown in Figure 5, it is obvious thatthe scattered part of the wavefield is much smaller—an effectcaused by the reduced scattering at small-scale fluctuations. Wecan also see the numerical effect of the absorbing boundaries,which is unimportant for further considerations.

Now we had to find out whether the wavefield in thesmoothed medium gives better results for traveltimes than amedium smoothed by an alternative method. For this reason weapplied several other smoothing methods and performed thesame finite-difference modeling as shown in Figures 3 and 5. Tocompare the seismic records, we stacked them along the hor-izontal axis, resulting in a single seismogram. Figure 6 showsthe stacked seismic sections obtained for the original and thesmoothed media.

We see very good agreement between the traveltimes ofthe unsmoothed medium and the medium smoothed by usingequations (8) and (9). In the case of medium 1, the smooth-

FIG. 3. Wavefield at times of 25 ms (left), 75 ms (center), and 125 ms (right), showing increasing wavefield distortion.

FIG. 4. Parameters of medium 1 after smoothing using the algorithm according to equations (8) and (9). Note the significantreduction of fluctuations.

FIG.5. Wavefield at times of 25 ms (left), 75 ms (center), and 125 ms (right) in the smoothed medium. The initial wavefield parametersare identical to those in Figure 3.

ing method using equations (8) and (9) resulted in a mediumwhere the wave propagates slightly faster than in the mediumobtained by averaging the slowness. This can be verified in Fig-ure 6.

Medium 2: Density fluctuations

A different case is that of density fluctuations (Lame param-eters are constant), as shown in Figure 7. The upscaling methodis then equivalent to the squared slowness averaging, as seenin Figure 8.

Medium 3: Correlated density and stiffness fluctuations

The previous examples showed the advantages of our ap-proach. To demonstrate its limitations, we developed the fol-lowing model. The medium fluctuations were such that the ve-locities were constant in the whole medium (4000 and 2300 m/sfor P- and S-wave velocity). Since density and stiffness values

1842 Gold et al.

fluctuate, the effective medium had a slightly different velocitythan the velocities at every point.

Using the same smoothing radius as before, we found nogood agreement of the wavefield in the original and smoothedmedia (see Figure 9). The reason for this effect lies in the higherorder terms that were neglected in the zero-frequency limit ofthe Bourret approximation [see equations (A-19–A-20) and(A-28–A-29)]. Since those terms depended on the crosscor-relation functions between density and stiffness parameters,they were irrelevant for the previous models. For the modelingresults depicted in Figure 11, we used a smoothing radius of8 m, being about 1/2π of the wavelength. In geological me-dia we normally observe some correlation between stiffnessand density fluctuations; fortunately, in most cases the densityfluctuations are small compared with the stiffness fluctuations.

FIG. 6. (left) Stacked sections of the wavefield transmitted in the original (dashed line) and smoothed by the upscaling method(solid line) media. Good agreement can be observed for the traveltimes. The difference in amplitude is caused by scattering fromsmall spatial-scale inhomogeneities. (right) Results of different smoothing methods: (1) the stacked section for averaging of theinverse squared velocity compared with the stacked section of the unsmoothed medium (dashed line), (2) the stacked section forinverse velocity averaging, (3) upscaling method, (4) velocity averaging, and (5) squared velocity averaging, which corresponds tostiffness averaging.

FIG. 7. A medium with density fluctuations. The standard deviation of the density is 40%. This corresponds to velocity fluctuationsof about 20%.

FIG. 8. The same as Figure 6, but for medium 2.

Moreover, the correlation between these parameters is neverperfect. Thus, we expected the upscaling method to yield betterresults than other methods for a smoothing radius smaller than1/2π of the wavelength. For decreasing smoothing radius, theresults improved significantly, as shown in Figures 10 and 11.The favorable effect of a small averaging radius compensatedfor neglecting higher order terms with density–stiffness corre-lations.

Medium 4: Uncorrelated density and stiffness fluctuations

To support the hypothesis that the differences in model 3 be-tween wave propagation in smoothed and unsmoothed mediaare because of perfect correlation between density and stiff-ness parameters, we used the same fluctuations of mediumparameters as in model 3 but with uncorrelated density and


stiffness fluctuations. Figure 12 shows the medium parameters.For the wavefield we obtained very good agreement betweenthe original and smoothed media using the upscaling method(see Figure 13). This suggested that the upscaling method couldapply for smoothing areas smaller than half of the wavelength,provided that density and stiffness fluctuations were weaklycorrelated. If fluctuations were highly correlated, the upscal-ing method had to be applied using smaller smoothing areas,as seen for model 3.

Marmousi model

We then applied the upscaling method to the 2-D Marmousimodel. In the following sections the smoothing effect on the

FIG. 9. The results for the correlated density fluctuation model with 40% standard deviation of the medium parameters. (left)Comparison between the stacked sections of the original medium and the medium smoothed by the upscaling method (dashed andsolid line, respectively). Smoothing was performed in an area with a radius of 30 m. (right) Stacked sections for the original medium(dashed), (1) a homogeneous medium where the P-wave velocity is 4000 m/s (smoothed squared slowness), (2) upscaling method,and (3) the section for a medium where the velocity was smoothed (all solid lines). In the latter case because of constant velocity,the velocity averaging yields a homogeneous medium with Vp= 4000 m/s.

FIG. 10. The results for the correlated density fluctuation model with a smoothing radius of 15 m. The same delineation of curvesas in Figure 9 is chosen. The difference between the sections of the smoothed and original media becomes significantly smaller.

FIG. 11. The results for the correlated density fluctuation model with a smoothing radius of 8 m. The same delineation of curves asin Figure 9 is chosen. Compared with the results of velocity smoothing, the upscaling method now yields the best result.

wavefield is observed on individual traces rather than on themean field (i.e., stacked traces). For the modeling procedure,we used the central part of the model shown in Figure 14.The sketch of the stiffness parameters and the density on theright side shows that these parameters are highly correlated,but stiffness fluctuations are much larger than density vari-ations. Therefore, with a smoothing length of wavelength/3,we expected the upscaling method to yield good results. Theseismograms for the three subsurface locations indicated inFigure 14 are displayed in Figures 15–17.

The outcome of the experiment with respect to the bestsmoothing method strongly depends on the recording position.This is because the assumption of a random medium with anisotropic correlation function is not valid for the Marmousi

1844 Gold et al.

model. If we consider a wave propagating from the source tothe receiver at position 1, areas with constant medium param-eters are approximately oriented along the propagationdirection, so we expect a behavior of the wavefield tendingtoward isostrain rather than toward an isotropic randommedium. On the other hand we expect an isostress behaviorfor receiver position 3 since the medium changes significantlyin the propagation direction. These two effects are seen in

FIG. 12. A medium with uncorrelated density and stiffness fluctuations.

FIG. 13. Stacked seismic sections for a medium with uncorrelated density and stiffness fluctuations. (left) Comparison between thestacked sections of the original medium and the medium smoothed by the upscaling method (dashed and solid lines, respectively).(right) As a result of the smoothing method, (1) the stacked section for averaging the inverse squared velocity compared withthe stacked section of the unsmoothed medium (dashed line), (2) the stacked section for inverse velocity averaging, (3) upscalingmethod, (4) velocity averaging, and (5) squared velocity averaging.

FIG. 14. (left) Central part of the Marmousi model, showing the stiffness tensor element M . The crosses indicate the source position(S) and receiver positions 1, 2, and 3. (right) A vertical section through the model at 600 m. Note the different axes labels.

(1) Figure 15, where the velocity in the medium smoothed byour algorithm is smaller than the velocity in the unsmoothedmedium, and (2) Figure 17, where we see a tendency to theopposite effect. This comparison of smoothing methods showsthat for some positions one can find different methods whichproduce the optimum results. However, our approach alwaysproduces acceptable results and never shows the maximumerror compared to other methods.


SMOOTHING FOR HIGH-FREQUENCY MODELING

In this section we show the performance of the upscalingmethod after applying a subsequent high-frequency wavefrontconstruction method. As an example, we initially chose twolaterally homogeneous blocks with overall dimensions of 500by 500 m. In the upper block, the P-wave velocity increasedfrom 3600 m/s at the surface up to 4000 m/s at a depth of 400 mwith a constant velocity gradient. The S-wave velocity equallyincreased from 2052 to 2280 m/s. At the same time, λ and µ,which were perfectly correlated, increased linearly with depth.

FIG. 15. Comparison between seismic records for original (dashed line) and smoothed medium (solid line) at a depth of 600 m. Fromleft to right we see the seismic records in the medium smoothed with the upscaling method, inverse squared velocity averaging, andinverse velocity averaging. The seismograms are recorded at position 1.

FIG. 16. Equivalent to Figure 15, but at location 2.

FIG. 17. Equivalent to Figure 15, but at location 3.

In the lower block of 100 m thickness, velocities remained con-stant at 4000 and 2280 m/s, respectively. Random fluctuationsof λ and µ with a standard deviation of 20% were superim-posed on the entire model, resulting in a standard deviation ofthe P-wave velocity of about 10% (see Figure 18). The densityρ remained constant everywhere in the medium at 2.5 g/cm3.

We centered a point source at the surface of the model wherethe model was smoothed by different methods with a smooth-ing radius of 20 m. To observe wave propagation of the firstarrivals, we used both finite-difference modeling to solve theelastodynamic wave equation and a stabilized finite-difference

1846 Gold et al.

eikonal solver (Pasasa et al., 1998; Zhao et al., 1998). Fig-ure 19 displays a windowed snapshot of the first arrivals, cal-culated by the finite-difference eikonal algorithm for a varietyof smoothed media as well as for the original medium.

The first arrivals calculated by the finite-difference eikonalsolver in the original medium are far earlier than in the

FIG. 18. Vertical section (well log) of the medium with a veloc-ity gradient superposed by statistical variations. The gradientin the upper block and the constant velocity in the lower partare delineated by straight lines.

FIG. 19. Windowed snapshot of first-arrival isochrones in me-dia with different smoothing schemes, calculated by a finite-difference eikonal solver, after 120 ms. The propagation di-rection is from top to bottom, originating from a point sourcelocated at a depth of 3 m and centered horizontal position(250 m). The wavefronts are delineated according to differentsmoothing algorithms: averaged inverse squared velocity (lightblue), inverse velocity (white), upscaling method (solid black),averaged velocity (green), and averaged squared velocity (darkblue). Note the result for the unsmoothed medium (dashedblack) which differs significantly from the other isochrones.The background shows the unsmoothed P-wave velocity.

smoothed media. This can be explained by the properties of themodeling algorithm: similar to ray tracing, the eikonal solverseeks the quickest path for the wave to propagate. In the orig-inal medium, maximum values that exist from statistical vari-ation of the elastic parameters can be exploited fully by thealgorithm. These maximum values can be seen in Figure 18and vanish during any smoothing process with a reasonablylarge smoothing radius. However, the question arises: whichfirst-arrival curve is most adequate? To answer this question wecompare these results with the outcome of the finite-differenceexperiment that solves the elastodynamic wave equation inthe original and smoothed media. The source is located at thesame position as before, and the frequency content of thewavelet is equal to that in previous examples. Figure 20shows snapshots of the wavefronts after 120 ms in differ-ent smoothed media. We again observe results similar tothose for medium 1. The upscaling method yielded a satis-factory representation of the original case compared to othersmoothing methods in terms of traveltimes as well as am-plitudes. The overall results were also confirmed by othertraces of the same section. Further, the wavefields in the orig-inal and smoothed media (at least in the smoothed versionobtained by our upscaling method) were very close to eachother. Therefore, their first arrivals must also be close to eachother. However, if we compare the first occurrence of energyfrom the finite-difference synthetics with the first arrivals fromthe finite-difference eikonal calculations, we see reasonablygood agreement, except for the finite-difference eikonal re-sults in the original medium. Thus, the eikonal solver gavea good estimate of the first-arrival time when it was applied

FIG. 20. Comparison between the results obtained from ordi-nary finite-difference modeling and from Figure 19 for a singletrace cut out at 250 m horizontal position. Dashed lines indi-cate amplitudes in the smoothed media; the solid line repre-sents the nonsmooth case. The smoothing methods 2, 3, and 4best approximate the original traces. (1) The inverse squaredvelocity is averaged; (2) inverse velocity; (3) result from theupscaling method; (4) averaged velocity; (5) averaged squaredvelocity. The positions of first arrivals (right) are calculated bya finite-difference eikonal solver applied to the correspondingmedia.


to the correctly smoothed model rather than to the original one.The corresponding first-arrival isochrone is the most adequateone.

This comparison shows that, in principle, a high-frequencyapproximation is incapable of correctly constructing the first-arrival isochrone for complex unsmoothed heterogeneousmedia. An appropriate (frequency-dependent) smoothingprocedure makes it possible. Therefore, appropriate initialsmoothing is of high importance for the performance of high-frequency modeling.

CONCLUSIONS

The analytical description of wave propagation in randommedia, obtained from the Bourret approximation, provides aquick and physically justified smoothing algorithm for elasticisotropic media. Compared to methods that average the slow-ness or the squared slowness, the upscaling method proposedhere is sensitive to the nature of medium fluctuations. Sincethe smoothing radius depends on the wavelength, the upscalingmethod is also frequency dependent. Using synthetic data cal-culated by finite-difference modeling, our approach gave verygood results if the smoothing radius was small enough and themedium fluctuations were isotropic. Over the whole range ofcalculations, we found that this approach proved superior toother smoothing methods, even if the assumption of isotropicfluctuations and low frequencies were not valid. Moreover,because of its well-understood physical background, the up-scaling method should be preferred to other intuitive meth-ods, even in the case of results of equal quality. We alsoshowed that smoothing significantly improves results of high-frequency modeling. Finally, the upscaling formulas derivedhere [equations (6)–(9)] can be applied in more general situa-tions than just for smoothing. For example, they could be usefulfor blocking 2-D or 3-D statistically isotropic heterogeneousmodels.

ACKNOWLEDGMENTS

Critical and constructive comments of Jerry Schuster, MikeSchoenberg, and two anonymous reviewers greatly improvedthe manuscript. We thank Linus Pasasa for providing us withthe finite-difference eikonal solver code and Stefan Buskefor proofreading. Being a part of the collaborative researchproject SFB 381, we gratefully acknowledge the support of theDeutsche Forschungs-gemeinschaft (DFG). In part, this workhas been also supported by the sponsors of the WIT-consortiumproject.

REFERENCES

Bensoussan, A., Lions, J. L., and Papanicolaou, G., 1978, Asymptoticanalysis for periodic structures: North Holland Publ. Co.

Cerveny, V., Molotkov, I. A., and Psencik, I., 1977, Ray method inseismology: Univ. Karlova.

Christensen, R. M., 1979, Mechanics of composite materials: JohnWiley & Sons, Inc.

Grubb, H. J., and Walden, A. T., 1995, Smoothing seismically derivedvelocities: Geophys. Prosp., 43, 1061–1082.

Gubernatis, J. E., Domany, E., and Krumhansl, J. A., 1977, Formal as-pects of the theory of the scattering of ultrasound by flaws in elasticmaterials: J. Appl. Phys., 48, 2804–2811.

Katsube, N., 1995, Estimation of effective elastic moduli for compos-ites: Internat. J. Solids Structures, 32, 79–88.

Keller, J. B., 1977, Effective behaviour of heterogeneous media, inLandman, B., Ed., Statistical mechanics and statistical methods intheory and application: Plenum Press, 631–644.

Pasasa, L., Wenzel, F., and Zhao, P., 1998, Prestack high-resolutionimaging with non-Fermat traveltimes: 68th Ann. Internat. Mtg., Soc.Expl. Geophys., Expanded Abstracts, 1965–1968.

Rytov, S. M., Kravtsov, Y. A., and Tatarskii, V. J., 1987, Principles ofstatistical radiophysics: Springer-Verlag New York, Inc.

Sanchez-Palencia, E., 1980, Non-homogeneous media and vibrationtheory: Springer-Verlag New York, Inc.

Shapiro, S. A., Schwarz, R., and Gold, N., 1996, The effect of randomisotropic inhomogeneities on the phase velocity of seismic waves:Geophys. J. Internat., 127, 783–794.

Zhao, P., Uren, N. F., Wenzel, F., and Hatherly, P. J., 1998,Kirchhoff diffraction mapping in media with large velocity contrasts:Geophysics, 63, 2072–2081.

APPENDIX A

EFFECTIVE WAVENUMBERS

The following calculation is analogous to the treatment byRytov et al. (1987), which is done for the case of acoustic ran-dom media. Instead of the Helmholtz equation, we use theelastic wave equation

ρ(r)ω2ui (r)+ (Ci jkl (r)uk,l (r)), j = 0. (A-1)

Throughout this paper we use standard tensor index conven-tion. Indices that occur twice imply summation; the subscript,i denotes differentiation with respect to the i th coordinate.In the following calculation we assume that the medium isisotropic and that the fluctuations of the medium parametersare small compared with their mean values.

Equation (A-1) can be written in the following operatorform:

Lik(r)uk(r) = 0, (A-2)

with

Lik(r) ≡ ρ(r)ω2δik + ∂ j Ci jkl (r)∂l ,

= ρoω2δik + ∂ j C

oi jkl ∂l

(=Loik

)+ δρ(r)ω2δik + ∂ j δCi jkl (r)∂l (=δLik(r)).

Here, we have separated the operator into a mean part, con-taining the mean values of the elastic parameters Ci jkl and ρ,and a fluctuating part.

A Green’s function approach to the wavefield gives the fol-lowing equation:

ui (r) = uoi (r)+

∫d3(r′)Go

i j (r− r′)δL jk(r′)uk(r′), (A-3)

according to Gubernatis et al. (1977). Here, Go stands for theGreen’s function of a homogeneous medium, where δCi jkl = 0and δρ= 0.

1848 Gold et al.

Substituting Go for uo in equation (A-3) and expanding thisequation, we find the following Born expansion of the Green’sfunction in the medium with fluctuating parameters:

Gim(r, ro) = Goim(r, ro)

+∫

d3r′Goi j (r, r′)δL jk(r′)Go

km(r′, ro)

+∫ ∫

d3r′d3r′′Goi j (r, r′)δL jk(r′)Go

kl

× (r′, r′′)δLln(r′′)Gonm(r′′, ro)+

∫ ∫ ∫· · · .

(A-4)

To find the effective (i.e., averaged) Green’s function, we aver-age equation (A-4). If we take into account that the mean valueof δLik equals 0 and if we assume that the random medium is sta-tistically homogeneous, then we find for the effective Green’sfunction

Gim(r− ro) = Goim(r− ro)+

∫ ∫d3r′d3r′′Go

i j (r− r′)

×〈δL jk(r′)Gokl(r′ − r′′)δLln(r′′)〉

×Gonm(r′′ − ro)+

∫ ∫ ∫· · · . (A-5)

Rytov et al. (1987) apply the Feynman technique to regroup thevarious scattering terms. The correspondence between the rep-resentation in diagrams and integral expressions is exactly oneto one, so the result of reordering the Feynman diagrams canbe directly translated back to analytic formulas. In other words,this treatment is totally independent of the exact formulas forthe Green’s function and the medium fluctuation terms. So it ispossible to use the diagram result from Rytov et al. for elasticrandom media. Translated back to integral equations, we havethe following expression for the effective Green’s function:

Gim(r− ro) = Goim(r− ro)+

∫ ∫d3r′d3r′′Go

i j

× (r− r′)Qjk(r′ − r′′)Gkm(r′′ − ro). (A-6)

This equation is called the Dyson equation. The quantity Qjk

is called the kernel of mass operator:

Qik(r′ − r′′) =⟨δLi j (r′)Go

jl (r′ − r′′)δL jk(r′′)

+∫

d3r′′′δLi j (r′)Gojl (r′ − r′′)δLlm

× (r′′)Gomn(r′′ − r′′′)δLmk(r′′′)

+∫ ∫· · ·⟩. (A-7)

The problem is now to find a solution of equation (A-6). Sincethis equation contains two convolutions, it is easier to solve itsFourier transform. We obtain

gim(κ) = goim(κ)+ (8π3)2go

i j (κ)qjk(κ)gkm(κ). (A-8)

Here, go, g, and q are the Fourier transforms of Go, G, and Q,respectively. Therefore, we have

gim(κ) = W−1ik go

km(κ), (A-9)

where W−1ik is the ikth element of the matrix W−1, and

Wik =(δik − (8π3)2go

il (κ)qlk(κ)). (A-10)

The Green’s function of a homogeneous isotropic medium isknown:

Goik(r− r′) = 1

4πρω2

{(β2δik + ∂i ∂k

)eiβr

r− ∂i ∂k

r iαr

r

}.

(A-11)

Here, α and β are the wavenumbers of the compression andshear waves in the background medium. The Fourier transformof this function is

goik =

−1ρoω2

18π3

{β2δik − κi κk

β2 − κ2+ κi κk

α2 − κ2

}. (A-12)

Therefore, it seems to be obvious to use the ansatz

gik = −1ρeω2

18π3

{β2

eδik − κi κk

β2e − κ2

+ κi κk

α2e − κ2

}(A-13)

for the Fourier transform of the effective Green’s function.Here, αe and βe denote the wavenumbers of the effectiveGreen’s function and ρe is the effective density (recall thatρe= ρo). Inserting equations (A-12) and (A-13) into equation(A-9) yields a lengthy equation containing the unknown quan-tities αe and βe. This equation should be valid for all valuesof κ if the Fourier transform of the effective Green’s functionhas exactly the form as in equation (A-13). Since this is notnecessarily the case, we solve this equation for those valuesof κ where the Fourier-transformed effective and background-medium Green’s functions have the largest amplitudes or sin-gularities, respectively. In this way, using κ =α and κ =β inequation (A-9), we find solutions for the effective wavenum-bers of P- and S-waves. With κ chosen in the z-direction, whichis reasonable for a homogeneous isotropic medium, they read

αe = α(

1+ 8π3

ρoω2q33(κ)

)−1/2

,

≈ α(

1− 4π3

ρoω2q33(κ)

); (A-14)

βe = β(

1+ 4π3

ρoω2(q11(κ)+ q22(κ))

)−1/2

,

≈ β(

1− 2π3

ρoω2(q11(κ)+ q22(κ))

). (A-15)

This approximation is valid even in the low-frequency limitbecause the dependence of equations (A-14) and (A-15) onω−2

vanishes as a consequence of equation (A-17) and frequencydependencies of terms in equations (A-19) and (A-20).


The Bourret approximation

Equation (A-7) shows an expansion of the quantity Qi j interms of the medium fluctuations δLi j and the Green’s func-tion of the homogeneous background medium. Assuming smallfluctuations of the medium parameters, the higher order (inrespect to δLi j ) terms of equation (A-7) can be neglected.Therefore, we obtain

Qik(r′ − r′′) = ⟨δLi j (r′)Gojl (r′ − r′′)δL jk(r′′)

⟩. (A-16a)

Operator δLi j (r′′) is shown in equation (A-2). Assuming anisotropic elastic medium, we obtain

Ci jkl = λδi j δkl + µδikδ j l + µδi l δ jk . (A-16b)

Thus, combining equation (A-16b) with equation (A-2), weobtain

Qik(r′ − r′′) = ⟨(δρ(r′)ω2δi j + ∂i δλ(r′)∂ j + ∂ j δµ(r′)∂i

+ ∂mδi j δµ(r′)∂m)Go

jl (r′ − r′′)

× (δρ(r′′)ω2δlk + ∂l δλ(r′′)∂k

+ ∂kδµ(r′′)∂l + ∂nδlkδµ(r′′)∂n)⟩, (A-17)

where δρ= ρoερ, δλ= λoελ, and δµ=µoεµ. We can see that Qik

consists of a sum of 16 terms. The fluctuations of the mediumparameters at positions r′ and r′′ appear in the form of spatialcorrelation functions and crosscorrelation functions. We usethe following definition of a spatial correlation function:

Bxz(1r) ≡ 12〈(δx(r+1r)δz(r)+ δz(r+1r)δx(r))〉,

(A-18)where x and z are arbitrary medium parameters.

Using equations (A-14) and (A-15) combined with the spa-tial Fourier transform of equation (A-17), we obtain the fol-lowing result:

αe=α + α

2ρoω2

∫d3r e−iαz× {ω4 Bρρ(r )Go

33(r)

−α2 Bλλ(r )Gojl , j l (r)− 4α2 Bµµ(r )Go

33,33(r)

+ 4iω2αBρµ(r )Go33,3(r)+ 2iρω2αBλρ(r )Go

m3,m(r)

− 4α2 BλµGo3m,3m(r)

}, (A-19)

βe=β + β

4ρoω2

∫d3r e−iβz

× {ω4 Bρρ(r )(Go

mm(r)− Go33(r)

)− 4β2 Bµµ(r )

(Go

mm,33(r)− Go33,33(r)

)+ 4iω2βBρµ(r )

(Go

mm,3(r)− Go33,3(r)

)}. (A-20)

We used partial integrations to shift the spatial derivations frommedium parameters to Green’s functions. We also assumed astatistical isotropic medium. Therefore, the correlation func-tions depend only on r = |r|.

Low-frequency limit

To find the low-frequency limit of the effective wavenum-bers, we calculated the corresponding limit of expressions likeGo

33,33 using the spatial Fourier transform of the Green’s func-tion:

goik =

−1ρoω2

18π3

{β2δik − κi κk

β2 − κ2+ κi κk

α2 − κ2

}. (A-21)

To compute the Fourier transform of spatial derivatives of theGreen’s function, we used the well-known property

A(r)⇒ a(κ),

∂i A(r)⇒ −i κi a(κ).

As an example let us consider the term related to density fluc-tuations. The following equation will be helpful for this:

go33 =

−1ρoω2

18π3

{β2 − κ2

3

β2 − κ2+ κ2

3

α2 − κ2

},

= −1ρoω2

18π3

{β2 − θ2κ2

β2 − κ2+ θ2κ2

α2 − κ2

},

= −1ρoω2

18π3

{(1− θ2)

β2

β2 − κ2+ θ2 α2

α2 − κ2

}.

(A-22)

Here, θ denotes the cosine of the angle between the directionof the z-axes and the direction of the vector κ, cos(θ). In thelast step, we used

β2 − θ2κ2

β2 − κ2= θ2 + β

2 − θ2β2

β2 − κ2. (A-23)

To calculate Go33 and to get the lowest order terms in the low-

frequency limit, we can make an estimation of the order of Go33

in terms of wavenumbers α and β by neglecting the angularbehavior of go

33. Then we obtain

Go33 = O

(α2

4πρoω2

eiαr

r

)+ O

(β2

4πρoω2

eiβr

r

).

An explicit computation of Go33 using the inverse Fourier

transform leads to the same result. From the equation aboveand equation (A-19), we see that the contribution of densityfluctuations to 1α=αe−α behaves like α3 for low frequen-cies. Consequently, in the low-frequency limit this term has noinfluence on the effective wavenumber, which is on the orderof α.

For the other derivatives of the Green’s function in equa-tion (A-19), a treatment analogous to the calculation in equa-tion (A-22) can be performed. We find, among other terms,

go33,33 =

18π3ρω2

{β2(θ2 − θ4)− θ2 β4

β2 − κ2

+α2θ4 − θ4 α4

α2 − κ2

}, (A-24)

gojl , j l =

18π3ρω2

{α2 − α4

α2 − κ2

}, (A-25)

1850 Gold et al.

go3 j, j 3 =

18π3ρω2

{θ2α2 − θ2 α4

α2 − κ2

}. (A-26)

In these equations the underlined terms provide the lowestorder contributions to the effective wavenumber. In respectto κ, they show only angular dependency. We expect that bythe inverse Fourier transform they behave like δ-functions (seethe explanation below). For these terms 1α is proportional toω (i.e., to α). Therefore, their contributions are significant. Itis not necessary to calculate the Fourier transform of θn ex-plicitly. Indeed, we only have to find the low-frequency limitsof the integrals in equations (A-19) and (A-20). The follow-ing calculation shows how to do this. First, we assume thatn is an arbitrary even number. We then consider an arbi-trary integral term in equation (A-19). Taking into accountthe form of the lowest order (in respect to ω) contributionsin equations (A-24)–(A-26) and that in the same approxima-tion the exponential factor under the integral (A-19) is equalto 1, one can see that such an integral term can be written asfollows:∫

d3rX(r )∫

d3κeiκrθn =∫

d3κθn∫

d3rX(r ) eiκr,

= 8π3∫

d3κX(−κ)

∫dθ θn∫dθ

,

= 8π3

n+ 1

∫d3κX(−κ) eiκ0,

= 8π3

n+ 1X(0), (A-27)

where X(r ) is a function of the absolute value of the radius vec-tor r. The third line in this derivation was obtained by chang-ing the coordinates of the spatial inverse Fourier transform tospherical ones and by taking into account that the spectrumX(κ) is independent of θ . We see that the Fourier transformof θn (n is even) behaves like a δ-function when applied to afunction without angular dependence. Performing such calcu-lations for all terms in equations (A-19) and (A-20), we find

the following final equations for the effective wavenumbers:

αe = α[

1+ 12

λ2o

(λo + 2µo)2σ 2λλ +

23

λoµo


+ 25

µ2o

(λo + 2µo)2σ 2µµ +

415α2

β2σ 2µµ

], (A-28)

βe = β[

1+ 15σ 2µµ +

215α2

β2σ 2µµ

]. (A-29)

In the low-frequency limit, since the change in wavenum-ber has no imaginary part, the effect of the medium fluctu-ations results only in a frequency-independent velocity shift.The results above coincide with equations (3.17) and (2.23)from Christensen (1979) and equations (40) and (43) fromKatsube (1995), calculated for the case of a low concentrationof randomly distributed spherical inclusions in a backgroundmedium.

An analogous calculation can be made for the 2-D case. Onecan also calculate the 2-D case from the 3-D case under theassumption that the wavefield and the medium parameters areconstant along one axis. Under this assumption we find

αe = α[

1+ 12

λ2o

(λo + 2µo)2σ 2λλ +

λoµo


+ 34

µ2o

(λo + 2µo)2σ 2µµ +

14α2

β2σ 2µµ

], (A-30)

βe = β[

1+ 14σ 2µµ +

14α2

β2σ 2µµ

]. (A-31)

Therefore, by applying both the weak fluctuations and the low-frequency approximations, we arrive at formulas for effectivewavenumbers of P- and S-waves in statistically isotropic mediawith heterogeneities of an arbitrary geometry. In our deriva-tions, by interchanging the order of these two small-parameterexpansions, we have implicitly assumed they both are homo-geneously converging.

GEOPHYSICS, VOL. 65, NO. 6 (NOVEMBER-DECEMBER 2000); P. 1851

Special Section on mining geophysics—Introduction

Michael Asten∗, Guest Editor

The Exploration 97 Conference held in Toronto in 1997 was adecennial event which provided a forum for review of advancesin geophysics applied to mineral exploration and resource eval-uation. This Special Section of GEOPHYSICSpublishes a set of16 papers, originally presented at that conference, whichdemonstrate recent advances in mineral exploration and minegeophysics.

One of the most dramatic changes in our profession, illus-trated by five of these papers, is the successful cross-fertilizationbetween petroleum and mining geophysics, whereby advancesin seismic methods, in particular 3-D data acquisition andprocessing, have been brought into routine application inthe structurally-complex environments associated with min-eral provinces and ore-deposit geometry. A further three pa-pers describe developments in underground or crosshole seis-mic, radar, and radio-wave imaging methods. Just as theseexamples have extended traditional “petroleum” technolo-gies into mineral-related problem solving, it is likely that thehigh-resolution techniques and the integration of electromag-netic methods with seismic methods will prove of benefit topetroleum geophysicists.

Papers in this Special Section are reprinted from Geophysics and Geochemistry at the Millennium, Proceedings of Exploration 97, the FourthDicennial International Conference on Mineral Exploration. Used with permission.∗Monash University, Department of Earth Sciences, Clayton, Victoria 3168, Australia. E-mail: [email protected]© 2000 Society of Exploration Geophysicists. All rights reserved.

Four papers review the state of the art in acquiring and imag-ing airborne electromagnetic and spectrometric gamma-raydata. The quality of the data and images now achievable showwhy these techniques are becoming essential tools for geologicmapping and reconnaissance, as well as for their traditionalrole of direct ore-body target location.

One paper on 3-D inversion of induced-polarization dataintroduces a new advance (from 2-D into 3-D geometry) in oneof the most significant breakthroughs of the last generation insurface mineral geophysics: the stable inversion of resistivity-IP data.

Finally, three papers discuss the application of geophysicsand borehole logging to mine planning and ore-body evalua-tion. Acceptance of geophysical technology in these areas hasbeen slow, and it is notable that these case histories are able todemonstrate how geophysics can bring significant savings onthe order of millions of dollars to the field of mineral extraction.

I thank former GEOPHYSICSEditor Sven Treitel for initiatingthis Special Section; Laurie Reed, Peter Annan, Ian MacLeod,and the Technical Committee of Exploration 97 for their co-operation; and the 60 authors involved for their hard work andpatience in bringing the set of papers together.

1851


Seismic reflection imaging of mineral systems: Three case histories

Barry J. Drummond∗, Bruce R. Goleby∗, A. J. Owen∗, A. N. Yeates∗,C. Swager‡, Y. Zhang∗∗, and J. K. Jackson§

ABSTRACT

Mineral deposits can be described in terms of theirmineral systems, i.e., fluid source, migration pathway,and trap. Source regions are difficult to recognize in seis-mic images. Many orebodies lie on or adjacent to majorfault systems, suggesting that the faults acted as fluid mi-gration pathways through the crust. Large faults oftenhave broad internal zones of deformation fabric, whichis anisotropic. This, coupled with the metasomatic ef-fects of fluids moving along faults while they are active,can make the faults seismically reflective. For exam-ple, major gold deposits in the Archaean Eastern Gold-fields province of Western Australia lie in the hanging-wall block of regional-scale faults that differ from othernearby faults by being highly reflective and penetratingto greater depths in the lower crust. Coupled thermal,mechanical, and fluid-flow modeling supports the the-ory that these faults were fluid migration pathways fromthe lower to the upper crust. Strong reflections are alsorecorded from two deeply penetrating faults in the Pro-terozoic Mt. Isa province in northeastern Australia. Bothare closely related spatially to copper and copper–golddeposits. One, the Adelheid fault, is also adjacent to the

large Mt. Isa silver–lead–zinc deposit. In contrast, otherdeeply penetrating faults that are not intrinsically reflec-tive but are mapped in the seismic section on the basisof truncating reflections have no known mineralization.Regional seismic profiles can therefore be applied in theprecompetitive area selection stage of exploration. Ap-plying seismic techniques at the orebody scale can bedifficult. Orebodies often have complex shapes and re-flecting surfaces that are small compared to the diame-ter of the Fresnel zone for practical seismic frequencies.However, if the structures and alteration haloes aroundthe orebodies are targeted rather than the orebodiesthemselves, seismic techniques may be more successful.Strong bedding-parallel reflections were observed fromthe region of alteration around the Mt. Isa silver–lead–zinc orebodies using high-resolution profiling. In addi-tion, a profile in Tasmania imaged an internally nonre-flective bulge within the Que Hellyer volcanics, suggest-ing a good location to explore for a volcanic hosted mas-sive sulfide deposit. These case studies provide a pointerto how seismic techniques could be applied during min-eral exploration, especially at depths greater than thosebeing explored with other techniques.

INTRODUCTION

Deep seismic reflection programs around the world aremostly directed toward understanding the tectonic evolutionof the regions studied and therefore have often led only indi-rectly to an improved understanding of their mineral potential.In contrast, a seismic transect of the Mt. Isa inlier of northeast-ern Australia, sponsored by the Australian Geodynamics Co-operative Research Centre, was deliberately designed to place

Manuscript received by the Editor March 22, 1999; revised manuscript received June 4, 2000.∗Australian Geological Survey Org., P.O. Box 378, Canberra, A.C.T. 2601, Australia. E-mail: [email protected]; [email protected].‡Formerly Geological Survey of Western Australia, 100 Plain Street, Perth, Western Australia 6000, Australia; presently North Ltd., P.O. Box 231,Cloverdale, Western Australia 6105, Australia.∗∗CSIRO, Div. of Exploration and Mining, Nedlands, Western Australia 6009, Australia.§Formerly Mount Isa Mines Exploration, GPO Box 1042, Brisbane, Queensland 4001, Australia; presently Sons of Gwalia, 16 Parliament Place,West Perth, Western Australia 6005, Australia.c© 2000 Society of Exploration Geophysicists. All rights reserved.

major orebodies in the inlier into their regional geodynamicframework (Drummond et al., 1997).

The results from the Mt. Isa transect, together with thefindings of Drummond and Goleby (1993) from the EasternGoldfields province of Western Australia, suggest that the seis-mic profiling technique could be imaging fluid migration path-ways within the crust. Higher resolution studies (e.g., Milkereitet al., 1996; Yeates et al., 1997; and Goleby et al., 1997) suggestthat seismic techniques can also be successful at the orebody

1852

Seismic Imaging of Mineral Systems 1853

scale, especially if they are used to image the structures andalteration zones around the orebody rather than the orebodyitself.

In this paper, we place the seismic results from three unre-lated and widely separated terranes of different ages, i.e., theEastern Goldfields province (Drummond and Goleby, 1993),the Mt. Isa Inlier (Drummond et al., 1997) and Tasmania(Yeates et al., 1997), into a mineral system framework. Al-though the mineral deposits studied differ in many ways, theapproach of simplifying mineral systems provides pointers onhow seismic techniques can be effective in the precompetitivearea selection stage and in direct exploration.

MINERAL SYSTEMS

Mineral systems are analogous to petroleum systems, whichdescribe the genetic relationship between a petroleum sourcerock and an accumulation (Magoon and Dow, 1991). Using thisdefinition, a mineral system can be described in its simplestform as a fluid source, a migration pathway, and a trappingmechanism that scavenges the minerals from the fluids. Thefluid source could be basin brines in the case of strata-bounddeposits, or it could be more deep-seated lower crustal or evenupper mantle hydrated rocks in the case of other deposits.

Migration pathways are needed to focus the fluids from theirsource into the trap. Many mineral deposits lie on, or adjacentto, major fault zones, suggesting a causal relationship. Fault sys-tems provide fracture porosity as well as a focusing mechanism.In the case of basin brines, the general distribution of perme-able and impermeable rocks of the basin strata also allows fluidflow and influences its form.

Trapping mechanisms take a variety of forms and requirethe superposition of physical barriers to fluid flow, e.g., localstructure, stratigraphy, and permeability, whether intrinsic orfracture induced, with the appropriate chemical, thermal, andprobably palaeogeographic settings for the minerals to be de-posited. Hence, studies that describe mineral deposits ratherthan mineral systems and that focus mainly on the trappingmechanisms tend to describe the unique and often complexcombinations of elements in the trapping mechanism for eachdeposit but do not see the underlying unifying elements of themineral system.

Mineral systems are usually triggered by a thermal pulse,which in turn can often be related to intraplate tectonics re-sulting from interplate activity (Loutit et al., 1994). Whereaspetroleum systems are usually characterized according to theage and type of source rock (the fluid source) (Bradshaw, 1993),the ages of mineral deposits are often less certain. Mineral sys-tems may be characterized according to the age of the hostrocks or the age of the thermal event that triggered them. Just asa sedimentary basin can have several superimposed petroleumsystems reflecting the maturing through time of a number ofstacked source rocks and their associated fluid pathways andtraps, so also can a mineral province be host to several mineralsystems.

Identifying fluid source regions in seismic images may be dif-ficult. The dehydration of a large area of crust to create miner-alizing fluids will not necessarily leave an observable physicalimprint on the rocks that distinguishes that region from anyother region, especially in metamorphic rocks. This is becausethe physical effects of dehydration may be similar to those ofmetamorphism (higher densities and seismic velocities).

Large volumes of rock can be effectively dehydrated overtime by relatively low fluid flux rates. But if the fluids areconcentrated into fracture-induced permeability zones alongfaults, higher flux rates will occur along the faults. This canlead to wide alteration haloes along faults and metasomatismwithin the fault zone. Where the fault zone is the focus of highstrain, mylonite zones develop. They characteristically have awell-developed anisotropic fabric (e.g., Jones and Nur, 1984;Siegesmund and Kern, 1990). Mylonite zones can be good re-flectors (Jones and Nur, 1984; Goodwin and Thompson, 1988).The seismic reflectivity results from the constructive interfer-ence of reflections from the bands of altered and strainedanisotropic rock within the mylonite zones. The case histo-ries presented in this paper describe the seismic effect of themineral system in terms of migration pathway and trappingmechanism.

In the first case history, the crustal structure of the East-ern Goldfields is interpreted in terms of linked fluid pathways.Drummond and Goleby (1993) interpret some elements ofcrustal reflectivity in the Archaean Eastern Goldfields provinceof Western Australia in terms of fluid pathways through thecrust based on the geometry of the fault zones and their spatialrelationship to known mineralization. They make no attemptto link this regional study with local studies at the orebodyscale.

The second case history, at Mt. Isa, tries to link from theregional scale into the orebody scale. Salisbury et al. (1996)demonstrat that many of the sulfide minerals that typicallymake up the bulk of mineralization constituting orebodies haveseismic velocities similar to felsic and mafic igneous rocks, andalso some sedimentary rocks, but much higher densities. Pyritehas both higher density and higher seismic velocity. Therefore,in many cases the orebody should have a significant impedancecontrast with country rock of most compositions. However,orebodies can have very complex shapes and often lie in com-plexly folded or deformed host rocks. Orebody reflections maybe lost among the reflections and interference signals from thesurrounding host material. Orebodies are often very small insize compared to the wavelength of seismic energy returnedfrom the earth. Therefore, to maximize the chance of successin using seismic methods at the trap or orebody part of themineral system, we recommend targeting the controlling struc-tures around the trap and perhaps the broader alteration haloesaround the orebodies.

In the Mt. Isa study, the strategy was not to try to image theorebody itself but rather to target a known larger alterationzone characteristic of the environment where mineralizationmight occur—the fluid migration pathways adjacent to the ore-body and the structure of the trap. This is analogous to the ap-proach used in petroleum exploration, where the structure ofthe reservoir would be the seismic target rather than the poolof oil it might contain.

The third case history, in Tasmania, studies a totally differentstyle of ore environment and demonstrates that by targetingthe mineral system rather than the orebody, seismic methodscan be applied successfully in a range of environments.

SEISMIC METHODOLOGY CONSIDERATIONS

Typically, the geology and structure of the three mineralizedregions described in these case histories is complex, with arange of lithologies subjected to at least three deformational

1854 Drummond et al.

and/or metamorphic events prior to the mineralizing event.However, in all cases the available geological control is good,with information from mining in the region, deep drillholes,and detailed surface geological mapping. Two-dimensional andsome low-fold 3-D seismic reflection as well as 2-D and 3-Dcrustal-scale refraction techniques were used. However, onlyseismic reflection results are presented here.

The case studies used regional transects that were focused onstructure within the middle to upper crust and higher resolutionseismic surveys of mine-scale structures.

Explosive charges in deep drillholes provided the energysources. The seismic data were collected with 96 or 120 chan-nels, and quality control was primarily through field monitorsand in-field data processing to at least brute stack stage, espe-cially for the high-resolution data. Typically, the station spac-ing was 40 m for the regional surveys and 10–20 m for thehigher resolution surveys. Shot-hole spacings were variable,but a nominal stacking fold of between 12 and 24 was achieved.Symmetrical split-spread geometries were used, which resultedin a maximum shot–receiver offset of 2400 m for the regionalsurveys.

In this type of project, the main data processing problemsresult from difficult static corrections and large velocity vari-ations. Detailed refraction static analysis is required to adjustfor the effects of a highly variable regolith in most parts ofAustralia, especially with the higher frequencies needed in highresolution studies. Near-surface velocity variations are high,ranging from around 1000 m/s within parts of the regolith up to7000 m/s in metamorphosed ultramafic bedrock in the EasternGoldfields province.

The rocks are mostly highly deformed, so reflector continu-ity, although variable, is usually far shorter that that encoun-tered within sedimentary basins. The amplitudes of reflectionsare often excellent, but they may not be primary reflections.We adopted a strategy of identifying regions of similar reflec-tor coherency and dip. We correlated those regions with thesurface geology or with seismic velocities from crustal-scalerefraction or tomography studies to assign rock type. Then weuse, the geometry and spatial relationships of regions of similarreflection character to infer tectonic processes. Interpretationsare usually confirmed with both qualitative and quantitativeinterpretation of gravity and magnetic data, supplemented byavailable geological evidence.

CASE HISTORY 1: THE EASTERN GOLDFIELDSOF WESTERN AUSTRALIA

The Yilgarn craton in Western Australia (Figure 1) consists ofseveral geological provinces. Gneissic granitoid with granitoidplutons and greenstone supracrustal rocks are common in allprovinces. Each province can be divided into a number of ter-ranes, each defined by the distinct stratigraphy of its volcanicand sedimentary supracrustal rocks. The Eastern Goldfieldsprovince is host to much of the region’s known gold deposits,most of which occur in the west of the province. The Ida faultseparates it from the Southern Cross province to the west.

A regional-scale, 213-km-long seismic reflection traversewas positioned east–west across the regional strike (Figure 1)(Goleby et al., 1993). The interpretation of the shallow part ofthe seismic data is given by Swager et al. (1997). The green-stone supracrustal rocks lie above a subhorizontal detachment

between 1.5 and 2.5 s (4.5 and 7.5 km) and therefore havea tectonic boundary with the underlying, presumably felsicgneissic basement (Figure 2). Many of the faults in the green-stones, e.g., the Zuleika shear (Figure 2), are not reflective andare interpreted by their truncations of greenstone stratigra-phy. These faults can be mapped laterally over considerabledistances within the greenstones, but they are not deeply pen-etrating and sole on the detachment surface.

However, several faults penetrate the detachment. Thesefaults are often reflective. Within the seismic section, the Idafault and the Bardoc shear are the prominent examples (Fig-ure 2). The Ida fault dips approximately 30◦ to the east andextends to 25–30 km depth. The Bardoc shear dips west, pen-etrates the detachment surface, and truncates against the Idafault at about 15 km depth. Bottomhole cuttings from the shotholes along the traverse were chemically analyzed. Those fromnear the Ida fault and Bardoc shear have comparable alteration

FIG. 1. Major structural subdivisions of the southern EasternGoldfields province, Yilgarn block, Western Australia. Positionof 1991 seismic transect is also shown (modified from Swagerand Griffin, 1990).


patterns, indicating that similar or the same fluids moved alongboth of the faults in the past. This supports the seismic obser-vation that the faults are probably linked at depth (Golebyet al., 1993).

Many of the faults in the region, including those that do notpenetrate the detachment, can be associated spatially with golddeposits. However, the Bardoc shear and its southern extension(Boorara shear near Kalgoorlie, Lefroy fault near Kambalda)

FIG. 2. Portion of the 1991 deep seismic transect recorded within the Archaean Yilgarn block. Arrows indicate fluid-flow directionspredicted by Drummond and Goleby (1993). D= diffraction, S= sill. Detailed images of the interpretation, particularly the top 2 s,are found in Goleby et al. (1993) and Swager et al. (1997).

are associated spatially with major gold districts, including theGolden Mile at Kalgoorlie and the Kamdalda–St. Ives deposits.Many of the gold deposits lie to the west of the shear, i.e., inthe hanging-wall block.

Based on near-surface fluid flow patterns suggested byGoleby et al. (1993), Drummond and Goleby (1993) proposedthat mineralizing fluids migrating from the lower crust to higherlevels in the greenstones followed a path—first into and along


east-dipping shear zones within the lower crust, then intothe Bardoc shear. From there they were able to percolateinto the hanging-wall block (arrows, Figure 2). Some leakedalong the detachment and into other faults within the green-stone sequence that splay from the detachment (e.g., Zuleikashear; Figure 2), but most concentrated in the greenstonesimmediately above and to the west of the Bardoc shear.

Numerical modeling of fluid flow within the EasternGoldfields province supports this linked fluid-pathway model(Figure 3) (Upton et al., 1997). The numerical modeling usedthe crustal structure defined by the seismic profiling; otherphysical properties of the crust were assumed. The order ofdeformational and thermal events was derived from the geo-logical record. The modeling predicted an initial, single crustal-scale convection cell in which fluids were driven up the Ida faultfrom depth and down the Ida fault from the surface. Thesemixed and flowed up the Bardoc shear, resulting in a highdegree of chemical alteration and hence mineral depositionwithin the upper crust, particularly within the greenstones. Astemperatures dropped, the single convection cell broke downinto smaller cells within the upper crust. These concentratedthe mineral species in the upper crust. Coupled deformation-fluid flow modeling predicted the occurrance of east-dippingfaults within the crust. These were seen in the seismic data. Themodel also predicted focused fluid flow into the east-dippingshear zones.

FIG. 3. Fluid-flow modeling of the Yilgarn block. The fault geometry is derived from the seismic data. Crustal layering is impliedfrom reflectivity patterns along the profile. The topmost layer was added to account for crust removed by erosion since the oredeposits were formed. Arrows represent fluid-flow vectors. East-dipping zones of longer vectors in the greenstones and base-ment are predicted by the modeling and coincide with weak reflections in the seismic data. LC= lower crust, UC= upper crust,G= greenstones, C= crust removed by erosion.

CASE HISTORY 2: MT. ISA

The Proterozoic Mt. Isa inlier of northern Australia is recog-nized for its world-class silver–lead–zinc and copper–gold oredeposits. The inlier consists of an Eastern fold belt, a Westernfold belt, and the central Kalkadoon block (Figure 4). A ma-jor east–west deep seismic traverse 255 km long was recordedjust to the south of Mt. Isa (Figure 4). The seismic reflectionsection shows a marked difference in the structure of the top5–10 km between the Eastern fold belt and the Western foldbelt (Drummond et al., 1997; MacCready et al., 1999).

Seismic data from the Marimo region (Figure 5) combinedwith detailed structural mapping show that the sediments ofthe Eastern fold belt were emplaced by thin-skinned thrustingfrom the east along several stacked and probably contempora-neous subhorizontal thrusts. Further shortening then occurredalong steeper east-dipping reverse faults. They are mostly rec-ognized in the seismic data by the offset they created on the re-flective lowermost thrust detachment (MacCready et al., 1999)and extend to depths of 15 to 18 km where they intersect a re-gion of high seismic velocities, probably mafic in composition(Drummond et al., 1997).

The Marimo fault (M , Figure 5) is intrinsically reflective inthe upper 6 km (2 s two-way time). Prior to the seismic survey,no fault had been mapped in this region. Structural mappingundertaken to support the interpretation of the seismic data


FIG. 4. Tectonic provinces of the Mt. Isa Inlier and the location of the seismic transect.

FIG. 5. Portion of migrated seismic data from the Eastern fold belt, Mt. Isa inlier, showing the earlier shallow detachment cut bylater steeper faults imaged within the Marimo region. Contours (in kilometers per second) show position of high-velocity body inmidcrust. R= reflections from high-velocity body; LVL= low-velocity layer; M=Marimo fault (from Drummond et al., 1997).


found an east-dipping, 200-m-wide zone of hydrothermal alter-ation. The reflectivity of the fault is believed to result from thebroad zone of alteration. The spatial relationship of this faultto mineral deposits in the region is seen as further evidencethat this fault acted as a fluid pathway. The Mount McNamaracopper–gold mine lies near the transect, and the Hampden,Mount Dore, Selwyn, and Osborne copper–gold mines lie ona linear trend to the south (Figure 4).

Farther west, the seismic transect imaged a sequence offolded and faulted reflectors representing sequences within theLeichhardt River fault trough of the Western fold belt. Thoseof the Eastern Creek volcanics (ECV) are marked in Figure 6(MacCready et al., 1997). The Mt. Isa, Adelheid, and Sybellafaults and a number of other minor faults form part of an anas-tomosing fault system that extends for many tens of kilometersalong the western side of the Leichhardt River fault trough.

To the north of the transect, four major lead–zinc(–silver)and copper deposits lie close to the Mt. Isa fault. The faultdips west at 70◦ and extends into the upper to middle crust(Figure 6). However, it is not reflective. Its surface outcrop is<1 kilometer to the east of the Adelheid fault. The data inFigure 6 are not migrated and show that the Adelheid faultnot only has strong P-wave reflections, which are interpretedin the figure, but also S-wave reflections which are unmarkedand lie between the Adelheid and Mt. Isa faults. The strongestreflections in this pseudotrue-amplitude section are seen on thefault below the word vortex at the top of the figure. Heinrichet al. (1989) propose that the lead–zinc deposits in the regionformed from brines circulating within the Leichhardt Riverfault trough. The seismic data indicate that the Adelheid faultprobably acted to focus these brines into the anastomosingfault set near the present-day surface.

FIG. 6. Portion of seismic line from the Western fold belt, Mt. Isa inlier, showing the interpretation of MacCready et al. (1999). TheAdelheid fault is highly reflective, but other faults are not. Reflections between the Adelheid and Mt. Isa fault are interpreted asS-wave reflections from the Adelheid fault. ECV=Eastern Creek volcanics (from Drummond et al., 1997).

Both 2-D and 3-D high-resolution seismic data were re-corded between the Mt. Isa and Hilton mines (1–2 km northof Mt. Isa, Figure 4). They were designed to test predictedcross-sections just north of the copper and lead–zinc orebodies(Figure 7) (Neudert and Russell, 1981). Locally, lead–zinc min-eralization tends to be in steeply west-dipping lenses parallelto bedding within the Urquhart Shale. Copper mineralizationlies deeper. Both the lead–zinc and copper mineralization lieabove the Paroo fault.

The Paroo fault and alteration haloes above the copperdeposits were the targets for the high-resolution survey; theUrquhart Shale was expected to have strong impedance con-trast with the underlying basement of Eastern Creek volcanics.A 2-D data section is shown in Figure 8. Reflections wererecorded from the subhorizontal part of the Paroo fault inthe east of the section; farther west it is mapped using trun-cations of the reflections from the Urquhart Shale. The dataalso show unexpectedly strong, west-dipping reflections fromwithin the Urquhart Shale; they are parallel to bedding andcorrespond to predicted zones of alteration and sulfide mineralenrichment.

CASE HISTORY 3: TASMANIA

The Dundas trough and its constituent Mount Read vol-canics in Tasmania (Figure 9) are host to a number of world-class mineral deposits (Large, 1992). A seismic reflection sur-vey across the region in 1995 included both regional deepand shallow high-resolution seismic profiles. Drummond et al.(1996) summarize the results of the regional profiles. They showthe Paleozoic section of the Dundas trough and Mount Readvolcanics to be a highly folded and faulted succession with a


total thickness of at least 4 km. Individual stratigraphic unitswithin the Dundas Group and Mount Read volcanics are gen-erally not differentiated in the regional profiles, probably be-cause of their highly deformed nature and the low impedancecontrasts between the units.

The high-resolution data were expected to overcome theseproblems. They were interpreted using drillhole control (J. Silic,A. McNeill, and S. Richardson, Aberfoyle Pty. Ltd., personalcommunication, 1996) (Figure 10 and Yeates et al., 1997). Re-flectors at about 900 and 1150 m below shotpoint 1055 are in-terpreted as the top and base of the Que River Shale. This unitoverlies the Que-Hellyer volcanics. The base of the volcanics isinterpreted as the reflector at about 1500 m. The Que-Hellyervolcanics therefore have a noticeable bulge, similar in geome-try to the mound-type zinc–lead–copper (silver, gold) Hellyerdeposit (Large, 1992) within the Que-Hellyer volcanics sev-eral kilometers to the south. From the known lithologies in thearea, the strong reflections above this zone, at 1150 m, inferthe presence of carbonates, dolerite, or massive sulfides. Re-flections within the bulge are weak, inferring a zone of strongalteration that produced homogeneity within the volcanics inthe bulge.

DISCUSSION

The relation of fault reflectivity to anisotropy within faultzones and alteration caused by fluids is both observationallybased and supported by modeling. In the Eastern Goldfields

FIG. 7. Schematic section across the Mt. Isa Valley lead–zincand copper mineral field, showing the geological structurearound and within the orebody (after Fallon et al., 1997).

province, some faults are reflective and others are not. Thosethat are reflective penetrate to greater depths. The surface out-crop of the Bardoc shear is hundreds of meters wide and showshigh strain (Swager and Griffin, 1990). It has several gold de-posits in its hanging wall in the region of the transect. The geo-chemical alteration signatures from shot-hole samples supportthe seismic interpretation that it links in the crust with the Idafault. The surface outcrop of the Marimo fault in the Easternfold belt of the Mt. Isa Inlier has extensive hydrothermal alter-ation and lies along strike from operating copper–gold mines.In northwestern Tasmania, the loss of reflectors in the bulge ofthe Que-Hellyer volcanics would infer alteration.

The physical model of shear zones of Jones and Nur (1984)suggests that anisotropy is the primary cause of the reflectivityfrom two crustal-scale shear zones. The model is based on mea-surements of the physical properties of samples collected fromsurface outcrops of mylonite zones. It consists of anisotropicrock with low impedance normal to the shear zone, interlay-ered with isotropic rock with impedance similar to the pro-tolith on either side of the shear zone. The model requiredconstructive interference to create amplitudes comparable tothose observed. To do this, the layers had to be 110–150 mthick. Goodwin and Thompson (1988) also discuss reflectivityattributable to mylonite zones. Their physical models are basedon logs from a conveniently located well, and their layers aremuch thinner (about 30 m). They also note tuning of the layerthickness is important to achieve the observed amplitudes, butthat lateral variations in layer velocity and/or thickness overdistances of about 100 m are also important.

FIG. 8. Migrated high-resolution seismic data from the Mt. IsaValley, with interpreted positions of the Mt. Isa and Paroofaults.


Neither of these studies compares the relative effects ofanisotropy resulting from strain and alteration caused by meta-somatism. Siegesmund and Kern (1990) find that reflectivity asa result of anisotropy cannot account for the amplitudes ofall observed lower crustal reflections. Klemperer and BIRPS(1986) study the Outer Isles thrust in Britain, and find thatmetasomatism is more important than anisotropy in causingimpedance contrasts. Anisotropy, although present, is disorga-nized at scales of 100 m. Whereas Jones and Nur (1984) findthat the bulk physical properties of anisotropic rock are nearthose of the protolith, Jones (1986) finds that both velocity anddensity increase from protolith through mylonite to ultramy-lonite, accompanying a reduction in silicon and an increase iniron, calcium, and magnesium.

The details of the physical properties on which these modelsare based differ from study to study and from fault to fault, indi-cating that local factors are important. However, the principlesestablished by these studies are that on a regional, deep crustalscale, fault-zone reflectivity can be related to strain-inducedanisotropy and changes in bulk physical properties resultingfrom alteration. These principles also apply to structures andalteration at the mine scale.

FIG. 9. Geology of Tasmania and the locations of two deepseismic transects. The high-resolution seismic line in Figure 10is coincident with and just to the west of the bend in line 2 (afterDrummond et al., 1996).

However, the alteration products may be different becausethey are formed at the end of the fluid migration pathway. Forexample, the alteration haloes around the copper orebody atMt. Isa contain significant amounts of pyrite, whereas the lead–zinc orebodies at Mt. Isa lie within an altered zone containingless pyrite. Salisbury et al. (1996) find that massive sulfide de-posits can have impedances much higher than most commonhost rocks, and Milkereit et al. (1986) confirm that massive sul-fide orebodies, if thick enough, can be good reflectors. Pyrite,in particular, has a higher density than the host rocks of theMt. Isa orebodies. If present in sufficient quantities in the al-teration haloes, it will result in an impedance contrast betweenthe host rocks and the alteration zone. This is probably whatcaused the high-amplitude reflections in Figure 8 and possiblyalso over the bulge in the data from northwestern Tasmania(Figure 10).

Not all strong reflections in the seismic sections are from faultzones. Fault zone identification often depends on whether thezones link with surface outcrop and, in the case of reflectorswhich do not reach the surface, whether the structure mappedat depth in the seismic section makes structural sense if thereflectors are or are not interpreted as faults.

We believe that reflectivity of faults is one parameter thatcould be used to indicate where fluids have fluxed from thedeeper crust. In our studies, the seismic methodology was usedto image the main structures that either control the orebody

FIG. 10. Portion of the 1995 high-resolution seismic sectionrecorded within the northern Mount Read VHMS district.The data show a bulge in the Que Hellyer volcanics (QHV)and weaker reflections in the bulge, suggesting alteration.High-amplitude reflectors above the bulge could representdense rocks—perhaps carbonate, dolerite, or massive sulfides.QRS=Que River Shale; BA=Bouguer anomaly gravity val-ues in micrometers per second squared (after Yeates et al.,1997).


itself or bound the extent of mineralization. In our experience,in fold-belt terrains the strongest and most laterally continuousreflection zones often can be interpreted as faults and shearzones. In the few occasions where we applied this principle atthe mine scale, our results were encouraging.

Many tens of thousands of kilometers of high-quality, deepseismic profiles now exist for a wide range of tectonic envi-ronments around the world. Most are in the public domain. Inalmost all cases, the data have not been examined extensivelyfor pointers to areas of enhanced prospectivity.

ACKNOWLEDGMENTS

This paper was originally presented in an expanded format the Exploration’97 Conference in Toronto in 1997 (Golebyet al., 1997). We thank the organizers of that conferencefor recommending its republication in GEOPHYSICS. GeorgeThompson and Simon Klemperer provided valuable point-ers to published and unpublished material on reflectivity ofthe lower crust and fault zones. We thank two reviewersfor their suggestions. We also thank Joe Mifsud for draftingthe figures. The Mt. Isa seismic data are published with thepermission of the Director, Australian Geodynamics Cooper-ative Research Centre. A. Owen, B. Goleby, A. Yeates, andB. Drummond have permission to publish from the ExecutiveDirector, Australian, Geological Survey Org.

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major ore-controlling structures in the Eastern Goldfields, WesternAustralia: Expl. Geophys., 24, 473–478.

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Drummond, B. J., Korsch, R. J., Barton, T. J., and Yeates, A. V., 1996,Crustal architure in northwest Tasmania revealed by deep seismicreflection profiling: Austr. Geol. Surv. Org. Res. Newslett., 25, 17–19.

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crustal structure from seismic reflection profiling, Eastern Gold-fields, Western Australia: Austral. Geol. Surv. Org. record 1993/15.

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Application of 3-D seismics to mine planning at Vaal Reefsgold mine, number 10 shaft, Republic of South Africa

C. C. Pretorius∗, W. F. Trewick∗, A. Fourie∗, and C. Irons‡

ABSTRACT

During 1994, a 3-D seismic reflection survey was un-dertaken at Vaal Reefs No. 10 shaft with the objectiveof mapping the detailed structure of the Ventersdorpcontact reef gold orebody. This would provide vital in-put into future mine planning and development. Thesurvey benefitted from 10 years of 2-D seismic expe-rience and one previous 3-D mine survey, conducted inthe Witwatersrand Basin.

The seismic survey at No. 10 shaft accurately and spec-tacularly delineated the 3-D structure of the Ventersdorpcontact reef at depths ranging from 1000 to 3500 m,imaging faults with throws in the 20- to 1200-m range.

The resultant structure plans were satisfactorily vali-dated by subsequent surface drilling and undergroundmapping mining operations during the period 1994 to1996. These plans have been merged with drillhole, un-derground, and sampling data into an integrated minemodeling, gold reserve estimation, and mine schedulingpackage.

The geology department now manages the planningfunction at No. 10 shaft, and 3-D seismics has playeda significant role in placing this important responsibil-ity firmly within the geologists’ domain. Building on thesuccess of the No. 10 shaft survey, two other 3-D seis-mic surveys were concluded over mines during 1996 and1997.

INTRODUCTION

The Gold Division of the Anglo American Corp. of SouthAfrica (AAC) has successfully used seismic reflection tech-niques in its Witwatersrand basin analysis programs since 1983.The emphasis during the first ten years was on reconnaissance2-D seismic surveys for subsurface structural mapping of theWitwatersrand Triad rocks, particularly the auriferous CentralRand Group within the main Witwatersrand Basin (Figure 1).Building on the success of these 2-D surveys, AAC’s first 3-Dseismic survey for mine planning and development took placeat Western Deep Levels gold mine in 1993, with the secondfollowing one year later at Vaal Reefs No. 10 shaft (Figure1).

In many respects the No. 10 shaft survey represents themature application of seismics to detailed structural mappingin a deep AAC gold mine. Seismic structural maps of theVentersdorp contact reef gold orebody at No. 10 shaft weresatisfactorily validated by subsequent drilling and mining op-erations between 1994 and 1996. These positive results encour-aged two more AAC mines to conduct 3-D seismic surveys in1996–97, with the latest survey, at Western Ultra Deep Levels,eight times the size of the No. 10 shaft survey and probably one

Manuscript received by the Editor February 10, 1999; revised manuscript received June 2, 2000.∗Anglo American Corp. of South Africa, Ltd., Marshalltown 2107, South Africa. E-mail: [email protected]; [email protected].‡Irons Geophysical Consulting, Henley on Thames, Oxfordshire, United Kingdom.c© 2000 Society of Exploration Geophysicists. All rights reserved.

of the largest mine geophysical surveys undertaken to date inthe mineral industry.

This paper summarizes the 3-D seismic data acquisition, pro-cessing, and interpretation methodologies developed at No. 10shaft and illustrates how this geophysical technique can make apowerful contribution to optimizing orebody extraction in thedynamic Witwatersrand mining environment.

GEOLOGICAL SETTING AND ITS RELATIONSHIPTO SEISMIC STRATIGRAPHY

Descriptions of Witwatersrand geology and associated seis-mic stratigraphy appear in Pretorius et al. (1987, 1994), de Wetand Hall (1994), and Weder (1994). Most of the gold in the MainWitwatersrand basin occurs in thin auriferous conglomerates,anomalously termed reefs, within the predominantly arena-ceous Central Rand Group. The principal reef mined at No. 10shaft is the Ventersdorp contact reef. The Ventersdorp contactreef is found at the base of the Klipriviersberg Group, whichis the basal group of the Ventersdorp Supergroup (Figure 2).The Ventersdorp contact reef is conformable with the overly-ing lavas but unconformable on the underlying quartzites of

1862

3-D Seismics for Mine Planning 1863

the Witwatersrand Supergroup. Figure 3 is a generalized geo-logical section through the No. 10 shaft lease area, showing theunconformable relationship of the various lithostratigraphicunits. This presents more stratigraphic detail within the Cen-tral Rand Group than could be displayed on the generalizedstratigraphic column in Figure 2.

The Ventersdorp contact reef is a highly channelized reefwith thicknesses varying from 20 to 400 cm. It can be dividedinto two major reef types—plateau reef and channel reef—withthe channel reef being further subdivided into three subfacies.There is a very strong correlation between facies type and goldgrade, as described by Trewick (1994).

Figure 4 shows part of a previous northwest–southeast 2-Dseismic section passing close to the No. 10 shaft 3-D survey area.It clearly illustrates the seismic stratigraphy of the Ventersdorpand Witwatersrand Supergroups at this locality. Note espe-cially the strong angular unconformity (event 2) where thePlatberg Group sediments are draped over underlying, tiltedfault blocks of Klipriviersberg Group lavas. The drop in bothP-wave velocity (from 6300 to 5800 m/s) and density (from 2.9

FIG. 1. Regional location, surface, and subsurface geology of the Witwatersrand basin [after Pretorius (1986) with modification].

to 2.67 g/cm3−) as seismic waves pass from the Klipriviersberg

lavas into the underlying Central Rand Group quartzites pro-duces a strong reflection coefficient, which fortuitously coin-cides with the Ventersdorp contact reef. Reflections at thislava contact (event 3) can therefore be used to map theVentersdorp contact reef orebody. The Central Rand Groupbetween events 3 and 4 maintains a seismically transparentcharacter in this portion of the Witwatersrand basin. The nu-merous positive and negative reflections in the JeppestownSubgroup of the West Rand Group below event 4 arecaused by alternating shales and quartzites. This stripy re-flectivity characterizes most of the West Rand Group ex-cept for the Bonanza Formation quartzites of the Govern-ment Subgroup, between events 5 and 6. These understandablyhave a similar seismic signature to the Central Rand Groupquartzites.

Note how a large, pre-Platberg normal fault such as f1– f ′1

(2000 m throw) on Figure 4 is imaged not only where it dis-places the Ventersdorp contact reef but also where it traversesthe reflective West Rand Group. However, it is not easy to

1864 Pretorius et al.

pick the exact position of this fault throughout the sectionsince certain West Rand Group reflectors still appear to passthrough it undisturbed. This phenomenon is because out-of-plane events are imaged on this section; it illustrates a majordrawback of 2-D seismics, which prevents its use as a detailedstructural mapping tool. Another excellent example of suchsideswipe is the steeply dipping fault plane reflector, labeledfpr, just above the interpreted position of f1– f ′

1. This reflectorappears to originate from a strike extension of f1– f ′

1, severalhundred meters north of the 2-D section line.

To map structure at the minimum resolution (20 m) expectedby the mining clients and potentially available from the seis-mic data at the Ventersdorp contact reef level, it is crucial torecord 3-D seismic data, followed by full 3-D depth migrationat the processing phase. This will restore reflectors such as fprto their true subsurface position in 3-D space. Further discus-sions on the need for 3-D seismic imaging in a structurallycomplex hard-rock mineral exploration environment appearin Milkereit and Eaton (1996).

SEISMIC DATA ACQUISITION

Three-dimensional seismic survey design criteria must ad-dress issues such as the subsurface area to be imaged, the re-quired spatial resolution, bin dimensions, fold of cover and therequired source and receiver configurations to achieve this,

FIG. 2. Lithostratigraphic columns in the Witwatersrand basin.

migration apertures, and static control. Ashton et al. (1994)have published an interesting article on 3-D seismic surveydesign, including oil industry examples. Notes on appropriatemodifications to seismic survey design criteria when addressingthe hard-rock Witwatersrand environment appear in Pretoriuset al. (1987).

We knew from previous 2-D seismic surveys undertakenat No. 10 shaft that the peak frequencies achievable at theVentersdorp contact reef would be between 60 and 75 Hz. As-suming an average velocity of 6000 m/s and quarter-wavelengthvertical resolution criteria limits, we decided that 20 m verticalresolution was possible. We hoped this resolution could be fur-ther improved with the benefit of 3-D migration and graphicworkstation interpretation techniques such as 3-D visualiza-tion of horizon picks and related seismic attribute data.

The 3-D survey design was oriented toward maximizing thestructural resolution on the Ventersdorp contact reef within the5-km− subsurface core area shown on Figure 5. This area cov-ers proposed underground development plans until the year2012 on a complex Ventersdorp contact reef at depths rangingbetween 1000 and 3500 m. The seismic acquisition parametersrequired to achieve the imaging are summarized in Table 1, andthe required surface coverage is displayed in Figure 5. The dataacquisition was undertaken by Geoseis (Pty.) Ltd. between De-cember 1993 and February 1994.

While a full discussion of the acquisition methodology is be-yond the scope of this paper, it is pertinent to stress certainimportant aspects of the design.

Target bin size.—The target bin size of 20 m was chosento ensure that spatial aliasing would not occur at the typicalstratigraphic dips (<30◦) and maximum frequencies (<75 Hz)in the survey area. The spatial sampling parameters will, intheory, allow 3-D migration of dips up to 56◦ at frequencies upto 90 Hz.

Table 1. Vaal Reefs No. 10 shaft 3-D seismic survey dataacquisition parameters.

Field crew GeoseisInstrument type SN368/CS2502No. of channels 240Record length 3 sSample rate 2 msFold 2000%

Nominal Vibroseis source parameters

Vibrator type Pelton Mk IIPattern 4 Vibs in-line 62◦ W of NVP interval 40 mArray length 30 mSweep length 16 sSweep frequency 10–90 HzGain 6 dB/octave boostTaper 0.3 s

Nominal receiver array parameters

Geophone type GCRGeophone frequency 10 HzStation interval 40 mPattern in-lineSpread 4 lines of 60 receivers


Migration aperture.—To migrate dips of up to 30◦ in the in-line (northwesterly) direction at depths of 3500 m, a migrationaperture of 2000 m was selected for the core area shown inFigure 5. In contrast to this, the main concern in the cross-linedirection was to ensure that diffraction hyperboloids originat-ing at faulted contacts were sufficiently sampled to allow themto be satisfactorily collapsed during migration and to provideadequate spatial resolution of the diffraction points. Exper-imental migration of partially sampled, computer-simulateddiffractions was used to estimate a satisfactory strike migra-tion aperture of 700 m.

The surface survey area shown in Figure 5 provides full foldcoverage of the core area, and it includes the strike and dipmigration apertures and the stack-on zone for both in-line andcross-line fold build-up. The total surface coverage required tosatisfactorily image the 5 km core area is 17 km.

Fold.—Fold decimation exercises were undertaken on theoriginal 48-fold 2-D seismic sections, and the results were usedto determine that nominal 20-fold coverage would probably beadequate for the 3-D survey.

Recording time.—A recording time of 3 s was consideredadequate to allow for full 3-D migration of seismic data downto the Jeppestown subgroup of the West Rand Group. Thiswas necessary to ensure that faults affecting the Ventersdorpcontact reef could be mapped at depths where they displacethe upper West Rand Group reflectors (Figure 4).

Recording geometry.—Figure 6 displays the acquisition ge-ometry used at No. 10 shaft. The staggered brick wall tech-

FIG. 3. (a) Generalized geological section No. 10 shaft. (b) Idealized west–east section across the No. 10 shaft lease area, showingunconformable relationships (Trewick, 1994).

nique generated a satisfactory range of offsets (0–1100 m) andazimuths within the bins to facilitate maximum fold at targetreflection times with adequate NMO, and it accommodatedthe requirements set for an optimum refraction statics solu-tion. The staggered brick pattern provided better coverage atshort offsets than a rectangular checkerboard layout, but it waslogistically more difficult to implement and thus incurred costpenalties. Ashton et al. (1994) summarize the merits of variousshooting patterns. Note that the area highlighted in yellow onFigure 6 represents the recording template and not the corearea displayed on Figure 5. Also note the omissions and com-pensation shots, which were necessary around the mine dumpin the northeast corner.

Statics.—The importance of good static control in the nearsurface to improve structural resolution at great depths wasappreciated at an early stage in AAC’s 3-D seismic program.To this end a multiline receiver template, restricted only bythe number of traces available and acquisition efficiency, wasadopted. This provided the capability of producing a moreequal distribution of the total fold between the in-line fold andthe cross-line fold, a factor important for an accurate solutionof refractor-based static algorithms.

Additionally, a low-velocity-layer (LVL) seismic refractionprogram on a 1-km grid over the survey area was conductedtogether with eight uphole surveys. The data obtained fromthese surveys were used to optimize the final static solutionderived from the generalized linear inversion (GLI) refractionstatic algorithm.


FIG. 4. A portion of 2-D seismic line OG-54 in the vicinityof No. 10 shaft.

FIG. 5. 3-D seismic coverage at Vaal Reefs No. 10 shaft.

SEISMIC DATA PROCESSING

Data processing operations at No. 10 shaft were divided intotwo phases:

1) field quality control processing conducted on site bySchlumberger/Geco-Prakla (SGP) using its Voyager sys-tem and

2) full 3-D processing undertaken by SGP at its BuckinghamGate (Gatwick) Processing Center in the UnitedKingdom.

The emphasis in the field center was on producing 2-D Brutestacks for quality control purposes, within a day of completingeach swath. This procedure ensured that potential data acqui-sition problems (e.g., geometry, statics) could be quickly de-tected and rectified in the field if necessary. GLI statics werealso derived in the field from the Vibroseis records, ensuring afinal check on the crucially important aspect of static controlbefore the seismic, LVL, and survey crews were demobilized.

The final processing route in the U.K. is summarized inTable 2. Most of the processing methods and parametersare normally accepted techniques adopted for most modernland-acquired 3-D seismic surveys. The refraction and residualstatic processes, together with DMO, proved to be the criticalfactors in improving the S/N ratio. The main problem reportedby the processing team was the elimination of air-blast andnear-surface noise trains without disturbing the relativeamplitudes of the shallow data. This was achieved using


Table 2. Vaal Reefs No. 10 shaft 3-D seismic survey final dataprocessing route.

Factor Description

Demultiplex SEGD 6250 bpi 9-track reels producedon Geoseis field crew

Geometry SEGD reels copied to Phoenix vectorformat 3480 cartridges. Line geometryand field statics appended to traceheaders. Traces dropped outside35–1064 m offset.

Statics Hampson-Russell GLI statics applied.Airwave ProMAX air blast attenuation 331 m/s.Gain T**1.0 spherical divergence correction.f -k filter Full off 0–5200 m/s fan taper 5200–9360 m/s.

500 ms recoverable AGC.D.B.S. Predictive 180-ms operator, 2 ms gap.Equalization Window: near # 300–1100 ms, far

# 400–1200 ms.Sort To CMP order, 20 m bin width.Autostatics Surface-consistent residual statics. Derived

from first-round NMO-corrected gathers.NMO Velocities picked from constant-velocity

stacks after residual statics and DMO.Mute 120% stretch mute.Equalization 500-ms sliding window, 250-ms move-up.DMO 8-ms/trace dip limits.Stack Square root compensation, 20-fold nominal.Phase Zero-phase conversion.Migration One-pass depth migration.Filter 10-20-80-90 Hz Band-pass filter

(time domain).

FIG. 6. Acquisition geometry used at Vaal Reefs No. 10 shaft.

Air-Blast attenuation software available within the Promaxprocessing system.

The means by which the velocity model for depth migrationwas constructed was, however, fundamentally different fromnormal accepted practices. With the restricted far offsets andhigh interval velocities associated with the hard-rock stratig-raphy, stacking velocities are highly inaccurate when used asan approximation of the true P-wave velocity field of thesubsurface. An alternative method was therefore conceivedand the model constructed in a three-phase manner.

First, based on borehole geophysical logs and prior 2-D seis-mic knowledge, an initial velocity approximation of 6000 m/sfor the entire section was used to migrate 2-D dip lines ex-tracted from the 3-D data set.

Second, the major velocity boundaries were interpreted onthese migrated sections, and time-based structure maps wereconstructed. Local average velocities for the main geologi-cal formations were then extracted from borehole geophysicallogs, and a 3-D velocity and time model was constructed. Fortu-nately, only two layers and one horizon (Ventersdorp contactreef) needed construction for the No. 10 shaft area, makingthis operation relatively simple. This provided the final inputmodel for a 3-D time migration of the seismic data volume.

Finally, the mine geologists and managers required their finaldata set in depth. A depth conversion was therefore carried outby an experienced interpretation geophysicist, familiar withthe Witwatersrand stratigraphy, and integrated into the finalmigration phase of the processing.


The output data set from the 3-D time migration was firstloaded onto an interpretation workstation and the major veloc-ity boundary at the Ventersdorp contact reef was interpreted.Borehole depth control together with extracted two-way trav-eltime values from this interpretation were then used to calcu-late the average velocity down to the Ventersdorp contact reefat each borehole position. These values were contoured to pro-duce a velocity map for the Klipriversberg lava which, togetherwith the time interpretation of the Ventersdorp contact reef,was used as an input model to the depth migration of the dataset. The quartzites and shales of the Witwatersrand Supergroupunderlying the lava were given a constant velocity of 5800 m/s.

The final depth-migrated data set was output in depth fordetailed interpretation on the mine. The accuracy of this finalvelocity model for migration is borne out by the accuracy ofthe predicted depth and position of the faults intersected todate on the mine, as discussed below.

INTERPRETATION

Interpretation of the No. 10 shaft seismic data com-menced on the depth-migrated data cube using Schlumberger-Geoquest’s IESX software with the Geoviz geovisualizationmodule. On the basis of this interpretation, the mine designhas been completed for the entire life of the mine.

One must appreciate that the data cube will still containsome processing artifacts and depth mismatches. The interpre-tation is therefore expected to require minor updates through-out the life of the mine. The important principle is to applythese updates dynamically to maintain the required predictivecapability ahead of development ends (200–300 m).

Figure 7 is an interpreted in-line (dip line) through the centerof the data cube, showing the Ventersdorp contact reef horizonand associated faulting. This is a direct screen dump from theworkstation. The stratigraphy is not described in detail becausethis has already been done for the 2-D type section (0G-54) inFigure 4. (Although 0G-54 does not fall within the 3-D surveyarea, the seismic stratigraphies are broadly similar.)

FIG. 7. Portion of an interpreted dip line through the centerof the 3-D depth-migrated seismic data cube.

Clearly, the Ventersdorp contact reef is well imaged on Fig-ure 7 and there are no indications of residual sideswipe, such asfeature fpr in Figure 4. Figure 8 is a 3-D representation of theVentersdorp contact reef and fault surfaces after computer-assisted tracking of the horizon and faults has taken placethrough approximately half of the data volume. For illustrativepurposes, the seismic cube has been peeled back to the cur-rent in-line section, revealing the orebody surface, displayedin yellow, with fault planes shown in blue. One of the surfacedrillholes (G 40) to which the interpretation has been tied isdisplayed in the northwest. Note how grid shading of the to-pographic surface in Geoviz, using a shallow sun angle fromthe north, helps highlight smaller faults such as f 5 (throw ofapproximately 20 m) by casting a significant shadow across thediscrete change in topography.

In Figure 9 the seismic data cube has been completely rolledback to reveal the interpreted Ventersdorp contact reef surface.As shown by the reference wireframe, the Ventersdorp contactreef depths range from 1000 m to about 3000 m within this area.The survey has accurately and spectacularly delineated faultswith throws in the 20 m ( f 2) to 1200 m ( f 3) range, with a lateralpositioning accuracy of better than 40 m. Note that the surfacedrillhole control has done little to define the fairly complex

FIG. 8. Interpreted VCR surface approximately midwaythrough the 3-D cube.

FIG.9. Grid-shaded 3-D perspective display of the VCR at VaalReefs No. 10 shaft.


faulting in the subsurface, particularly the smaller faults withthrows in the range of 20 to 100 m (e.g., 100-m fault f 4). Few ifany of these faults could be interpreted from surface drilling,and most will have a bearing on short- to medium-term mineplanning. The 3-D imaging of structures is very satisfying. Asshould be expected, the fault planes are generally nonlinear.Fault throws vary along strike, and there is evidence for transferstructures displacing faults such as f 4.

An unexpected result of this survey was the definition ofthe large upthrown block of Ventersdorp contact reef to thenorthwest of the 1200-m Mariendal fault, f 3. This structurehas since been tested by drillhole G40, west of current mineworkings and sited to intersect the block within an extrapo-lated Ventersdorp contact reef channel facies with good gradepotential. This hole intersected the Ventersdorp contact reefwithin 20 m of the interpreted depth based on the seismiccube.

Several of the smaller faults were intersected by mining op-erations between 1994 and 1996, generally confirming the pre-dicted structure. One example is illustrated in the enlarged dipsection displayed in Figure 10. On this section the Ventersdorpcontact reef pick is displaced by a 45-m fault ( f 6). Subsequentmining operations have confirmed the presence and throw ofthe fault, as shown by the underground survey pegs (p). Re-cent underground survey control indicates that fault throws assmall as 15 m can be resolved at shallower Ventersdorp contactreef depths within the 3-D data cube. These results are a re-markable confirmation of the seismic method and the velocitymodel when one considers a regional dip of up to 30◦ resultingin Ventersdorp contact reef event migration of over 500 m.

DYNAMIC MINE PLANNING AND DEVELOPMENT

Development of a reliable facies plan has greatly assistedwith the evaluation of Vaal Reefs No. 10 shaft. Together withthe accurate structural information provided by the 3-D seis-mic survey, this has enabled shaft geologists to develop a ro-bust geological model. Availability of the geological model, to-

FIG. 10. Mining confirmation of VCR seismic structure plan atVaal Reefs No. 10 shaft.

gether with the necessary computer technology in the form ofGeoquest, Microstation, and CADS-Mine, enabled the merg-ing of seismic, drillhole, underground survey, and sampling datainto an integrated mine modeling, reserve estimation, and minescheduling package. The new system is being used to generatean upgraded, accurate, and comprehensive planning databasewhich is sufficiently flexible to facilitate dynamic replanning inresponse to new information.

The planning database is currently being used to optimizethe positioning of the subshaft system, which will allow accessto the deeper portions of the Ventersdorp contact reef orebodyin the west and may extend the life of the mine to the year 2012.This subshaft must be optimally sited within Ventersdorp con-tact reef fault loss to minimize the sterilization of resourceswhile avoiding complex fault intersections that could compro-mise safety. The structure plan displayed in Figure 9 will greatlyhelp planners achieve this objective.

CONCLUSIONS

The 3-D seismic method has proven to be a very cost-effective technique for mapping structure on the Ventersdorpcontact reef at No. 10 shaft, delivering adequate resolution forshort-, medium-, and long-term mine planning and develop-ment. Figure 11 illustrates schematically just how cost effective

FIG. 11. Information economics at Vaal Reefs No. 10 shaft.


the survey has been, compared to surface drillholes, as a meansof providing structural information on a reflective geologicalhorizon. If each binned seismic trace is considered to be theequivalent of a structural drillhole, the survey has arguably de-livered spatial information equivalent to 12 000 surface drill-holes, on a 20 × 20 m grid, extending to Ventersdorp contactreef depths and beyond within the core area. Cross-sectionswithin the seismic cube can be viewed and interpreted in anyorientation, including horizontal time slices, as schematicallydisplayed in Figure 11. The total survey cost of US $1.07 million,in 1994 money, would fund only one 3000-m-deep drillhole, in-cluding deflections.

The ability of 3-D seismics to predict structure well aheadof the current stope faces and development ends has been ad-mirably demonstrated over the last three years. The seismicdata set is expected to play a key role in the modeling anddesign phases of mining operations as well as in the auditingphase, where compliance with the extraction plan is audited.In essence, the Mine Geology Dept. now manages the plan-ning function at No. 10 shaft, and 3-D seismics has played asignificant role in placing this responsibility firmly within thegeologists’ domain.

ACKNOWLEDGMENTS

The authors are grateful to the Anglo American Corp.and Vaal Reefs GM for their permission to publish this pa-per. We also extend our gratitude to our colleagues at AACand Vaal Reefs and our consultants at Schlumberger/Geco-

Prakla, Geoseis, Hydrosearch, Armstrong de Klerk, and IronsGeophysical for their suggestions and comments during thepreparation of the manuscript. A final special thank you toDesiree Hendriques for her patient and professional assistancein preparing the diagrams, under enormous time pressure.

REFERENCES

Ashton, C. P., Bacon, B., Deplante, C., Ireson, D., and Redekop, G.,1994, 3-D seismic survey design: Schlumberger Oilfield Review, 6,No. 2, 19–32.

de Wet, J. A. J., and Hall, D. A., 1994, Interpretation of the Oryx 3-Dseismic survey: 15th Congress, Council for Mining and MetallurgyInstitution, Geology 3, 259–270.

Milkereit, B., and Eaton, W. D., 1996, Towards 3-D seismic explo-ration technology for the crystalline crust: in Lawton, S. E., Ed.,Trends, technologies and case histories for the modern exploration-est: Prospectors and Developers Assn. of Canada short course notes,17–36.

Pretorius, C. C., Jamison, A. A., and Irons, C., 1987, Seismic explorationin the Witwatersrand basin, Republic of South Africa: Exploration87, Proceedings, 3, 241–253.

Pretorius, C. C., Steenkamp, W. H., and Smith, R. G., 1994, Devel-opments in data acquisition, processing and interpretation over tenyears of deep vibroseismic surveying in South Africa: 15th Congress,Council for Mining and Metallurgy Institution, Geology 3, 249–258.

Pretorius, D., 1986, Compilation of the geological map of theWitwatersrand basin, in Anhaueusser, C. R., and Maske, S., Mineraldeposits of southern Africa: Geol. Soc. South Africa, 1, 1019–1020.

Trewick, W. S. F., 1994, Exploration history and mining strategy for theVentersdorp contact reef at Vaal Reefs No. 10 shaft—Klerksdorpgoldfield: 15th Congress, Council for Mining and Metallurgy Insti-tution, Geology 3, 67–70.

Weder, E. E. W., 1994, Structure of the area south of the centralRand gold mines as derived from gravity and Vibroseis surveys: 15thCongress, Council for Mining and Metallurgy Institution, Geology3, 271–282.


Mineral exploration in the Thompson nickel belt, Manitoba, Canada,using seismic and controlled-source EM methods

Don White∗, David Boerner∗, Jianjun Wu∗, Steve Lucas∗,Eberhard Berrer‡, Jorma Hannila‡, and Rick Somerville‡

ABSTRACT

Seismic reflection and electromagnetic (EM) datawere acquired near Thompson, Manitoba, Canada, tomap the subsurface extent of the Paleoproterozoic,nickel ore-bearing Ospwagan Group. These data aresupplemented by surface and borehole geology andby laboratory measurements of density, seismic ve-locity, and electrical conductivity, which indicate thatOspwagan Group rocks are generally more seismicallyreflective and electrically conductive than the Archeanbasement rocks that envelop them. The combined seis-mic/EM interpretation suggests that the ThompsonNappe (cored by Ospwagan Group rocks) lies blindbeneath the Archean basement gneisses, to the eastof the subvertical Burntwood lineament, in a series oflate recumbent folds and/or southeast-dipping reversefaults. The EM data require that the shallowest of thesefold/fault structures occur within the basement gneissesor perhaps less conductive Ospwagan Group rocks. Theresults of this study demonstrate how seismic and deepsounding EM methods might be utilized as regional ex-ploration tools in the Thompson nickel belt.

INTRODUCTION

The Thompson nickel belt (TNB) in northern Manitoba,Canada (Figure 1) has been an important producer of nickelsince the early 1960s. Known ore deposits in the belt oc-cur without exception along distinct stratigraphic horizonswithin the Paleoproterozoic supracrustal Ospwagan Group.The supracrustal rocks and the ore deposits they host havebeen subjected to medium- to high-grade metamorphism andintense multiepisode deformation, resulting in a complex in-terference pattern of early recumbent folds overprinted bytight, upright, doubly plunging folds. In spite of the extensive

Manuscript received by the Editor February 1, 1999; revised manuscript received May 2, 2000.∗Geological Survey of Canada, 615 Booth St., Ottawa, Ontario K1A 0E9, Canada. E-mail: [email protected]; [email protected]; [email protected].‡Inco Limited, Exploration, Copper Cliff, Ontario P0M 1N0, Canada.c© 2000 Society of Exploration Geophysicists. All rights reserved.

postmineralization structural and metamorphic history, the sul-fide ore bodies have generally remained within their originalhost units. Thus, subsurface mapping of the structurally com-plex supracrustal unit is essential to future exploration withinthe TNB.

In an attempt to map the subsurface extent of the OspwaganGroup in this complex geological environment, 19 km of seis-mic reflection data were acquired in 1991 (Figure 1). In addi-tion, 20 km of deep-probing EM sounding data were acquiredto help constrain the interpretation of subsurface geologicalstructures. Coincident controlled-source EM data and seismicreflection data have proven to be complementary in previousstudies (e.g., Boerner et al., 1993, 1994). Both data sets wereacquired with joint funding from Inco Limited, Exploration;Lithoprobe; and the Geological Survey of Canada.

Seismic reflection methods have been applied within a va-riety of geological environments in various mining campsin Canada, e.g., low-grade greenstone belts of the ArcheanAbitibi subprovince (Selbaie area, Perron et al., 1997;Matagami area, Milkereit et al., 1992a; Ansil mine, Perronand Calvert, 1997), the Sudbury igneous complex (Milkereitet al., 1992b), and the Buchans mine in Newfoundland (Spenceret al., 1993). The TNB geological environment is distinct fromthese previous cases, representing a highly strained, moder-ately to steeply dipping, medium- to high-grade Proterozoiccollisional belt (see Weber, 1990; Bleeker, 1990b; and citationstherein). We present an interpretation of seismic reflection andEM sounding data from the immediate vicinity of the Birchtreemine (line 1A, Figure 1). Seismic data from a second profile tothe north (line 1B, Figure 1) are presented elsewhere (Whiteet al., 1999). The seismic and EM interpretation is constrainedby laboratory measurements of density, seismic velocity, andelectrical resistivity for representative TNB lithologies. Themodel is also constrained by surface and borehole geology.We conclude by summarizing how these geophysical mappingmethods might be successfully used as exploration tools in theTNB.

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1872 White et al.

GEOLOGICAL SETTING

The TNB (Figure 1) lies at the western margin of theArchean Superior province, where it is in fault contactwith the Kisseynew domain, a part of the PaleoproterozoicReindeer zone of the Trans-Hudson orogen (Hoffman, 1988),that is dominated by 1.86–1.84 Ga high-grade metaturbidites(Machado et al., 1999). Near Thompson, Manitoba (Figure 1),the TNB comprises variably reworked Archean gneisses inter-leaved and infolded with the remnants of a thin Paleoprotero-zoic cover sequence, the Ospwagan Group (Scoates et al., 1977;Bleeker, 1990b). All ore deposits in the vicinity occur withinthe Ospwagan Group and are associated with ultramafic sillsthat intrude the Pipe Formation of the Ospwagan Group andhave been variably dismembered (Bleeker, 1990a).

Bleeker (1990a) interprets the TNB geology in terms of arefolded nappe structure (the Thompson Nappe) as depictedin Figure 2. The overall geological history for the TNB, basedon Bleeker (1990a), is as follows:

1) deposition of Proterozoic Ospwagan sediments on Supe-rior Archean basement in a continental rift margin setting(ca. 2100–2000 Ma?);

FIG. 1. Map of the study area near Thompson, Manitoba,Canada. The Thompson nickel belt comprises all of the areato the east of the Superior boundary fault. The seismic andEM profiles are indicated, as are the mapped occurrences ofthe Ospwagan Group. The town of Thompson is indicated bythe T in the shaded rectangle. The base map is from Bleeker(1990a). The upper-left inset shows the approximate locationof Thompson (indicated by the star) near the boundary of theSuperior province and the Trans-Hudson orogen (THO).

2) formation of the Thompson Nappe during an early (F1)deformation event (>1880 Ma);

3) F2 recumbent folding coeval with peak metamorphism(ca. 1820 Ma);

4) refolding of the nappe structure into nearly upright, dou-bly plunging F3 folds;

5) late east-side-up ductile reverse faulting; and6) localized brittle-ductile strike-slip faulting along the

boundary with the Kisseynew domain.

Figure 2a shows the locations of mines around Thompson inrelation to the regional Thompson Nappe structure proposedby Bleeker (1990a). The Birchtree mine resides on the lowerlimb of the nappe structure, whereas the antiform that hoststhe Thompson mine and the Owl antiform are F3 structuresassociated with the overturned limb of the nappe. This modelallows for the possibility of Ospwagan metasediments in thecore of the nappe lying blind beneath the overturned basementgneisses.

SEISMIC IMAGING IN A COMPLEX ENVIRONMENT

The seismic reflection method is primarily suited to imagingstructures with shallow/moderate attitudes. Thus, the geolog-ical setting in the Thompson area, which comprises generallymoderately/steeply oriented rock fabric with layering as wellas fold and fault structures, presents a challenging environmentin which to obtain useful seismic images. This limitation mustbe respected during seismic interpretation. For example, con-sider the steep fold geometry from the Thompson area shownin Figure 3a (Bleeker, 1990a). The outline of a single lithologichorizon is reproduced in Figure 3b along with the calculatedseismic response before (Figure 3c), and after (Figure 3d) datamigration. The migrated seismic image fails to reveal the steepupright limbs of the folds, but the fold hinges and moderatelydipping limbs are well represented. The absence of the steepfold limbs in the seismic image makes interpretation of theimage nonunique. The moderately dipping fold limb that is im-aged might be interpreted as a fault plane. In this situation, sur-face and borehole constraints and the structural style assumedfor the area are critical in interpreting the seismic images.

EM IMAGING IN A COMPLEX ENVIRONMENT

Diffusive EM methods provide an image of the volume-averaged electrical resistivity of the subsurface. In contrast toseismic reflection methods, EM methods are ideally suited toimaging near-vertical structures. The EM data provide lowerspatial resolution than seismic reflection images because resis-tivity anomalies must have lateral dimensions roughly equiva-lent to their depth to be imaged properly. Intrinsically limitedresolution can be a benefit in a complicated environment suchas the TNB because minor parasitic folding or limited offsetfaulting should not disturb the EM image significantly. Fur-thermore, EM methods provide a means of mapping specificlithologic units in the subsurface. The major conductive rockunits in the Thompson area are generally limited to certainformations of the Ospwagan Group supracrustals, whereas theArchean gneisses are resistive (see following).

From limited rock property measurements (J. Katsube, per-sonal communication, 1993), it appears that the Pipe Formationis electrically conductive. Thus, a surface EM survey across thestructure illustrated in Figure 3 would provide evidence for

Exploration in the Thompson Nickel Belt 1873

steeply dipping elongated conductors wherever the Pipe For-mation crops out. In contrast, the same structure buried under1 km of resistive gneiss would appear as a single broad conduc-tive zone.

THE CAUSE OF REFLECTIVITY

Rock property measurements

Compressional-wave velocities (Vp) and density measure-ments have been made on a suite of rock samples from theThompson area to assess the reflectivity of the various litho-logic units. Seismic reflectivity is controlled by variations inseismic impedance (the product of velocity and density). Seis-mic waves are reflected from contacts between rock units withcontrasting impedances when the thicknesses of the adjacent

FIG. 2. (a) A 3-D sketch of the northern segment of the refolded Thompson Nappe structure as seen from the southeast (fromBleeker, 1990a). Mine locations are indicated by the small boxes and correspond to those shown in Figure 1. (b) Schematic transversesection through the Thompson Nappe structure (from Bleeker, 1990a), showing the structural positions of the mines from Figure 1.

rock units are at least 1/8 of a seismic wavelength (Widess,1973). For example, this corresponds to a thickness of 7.5 mfor a signal with a 100-Hz dominant frequency in rocks havingV p= 6000 m/s. Case studies from other areas indicate that re-flections can be caused by lithologic contacts (e.g., White et al.,1994), alteration zones, brittle fault zones, and shear (mylonite)zones. Both brittle and ductile shear zones are well documentedin the Thompson area (e.g., Fueten and Robin, 1989).

The Vp density measurements for a suite of rock samplesfrom the Thompson area are shown in Figure 4. The samplesinclude velocities measured under an applied uniaxial stressof 17 to 25 MPa or under a hydrostatic confining pressureof 30 MPa. Relatively high impedance (Z) values (Z = 18.5 ×106 − 21.5 × 106 kg/m2 · s) are observed for amphibolite, ironformation, pyroxenite, and sulfide ore, as compared with the

1874 White et al.

other lithologies (Z = 13.0 × 106 − 16.5 × 106 kg/m2 · s). Re-flection coefficients for juxtaposed high- and low-impedancelithologies range from 0.06 to 0.25, compared with 0.0 to 0.11for juxtaposed low-impedance lithologies. Thus, the largest am-plitude reflections will likely be associated with amphibolite,iron formation, pyroxenite, and sulfide ores embedded withinany of the low- to medium-impedance units. The geology logs inFigure 5 reveal that all of these high-impedance lithologies—with the exception of sulfide ore, which was not intersectedin any significant thickness in this borehole—are observed incontact with various low-impedance lithologies.

BOREHOLE LOGS

Knowledge of the spatial distribution of lithologies (i.e.,thickness, juxtaposition sequence, lateral extent) is required tocharacterize the reflectivity of the subsurface. This is best doneusing borehole velocity and density logs in conjunction withvertical seismic profiling (VSP). However, in the absence ofdownhole geophysical logging, the spatial distribution of seis-

FIG. 3. (a) Profile through the Thompson Nappe structure at the Thompson mine (from Bleeker, 1990a). (b) Trace of one of thelayers from (a). (c) Simulated seismic vertical incidence response (unmigrated) of the structure in (a). Response was calculated usinga forward modeling algorithm based on wave equation datuming (Berryhill, 1979). (d) Vertical incidence response of the structurein (a) after constant-velocity migration of the seismic data using an algorithm based on the frequency wavenumber phase-shiftmethod (Gazdag, 1978). The Pipe Formation is represented by the pelitic schist unit.

mic impedance down the borehole can be estimated by usingthe laboratory-measured velocities and densities in conjunc-tion with the borehole geological logs. The estimated variationof seismic impedance can then be used to calculate the syn-thetic normal incidence seismic response (Wuenschel, 1960).The reflectivity estimated in this manner is strictly applicablefor a layered medium where individual lithologic units are lat-erally continuous. To apply these results in the TNB geologicalenvironment, we assume that the interface between contrastinglithologic units is laterally continuous over a minimum lateraldistance on the order of a Fresnel zone (Sheriff and Geldart,1982). For example, this corresponds to a distance of ∼170 mat 1000 m depth and ∼300 m at 3000 m depth for a signal witha 100-Hz dominant frequency in rocks having V p= 6000 m/s.

The seismic response estimated in this manner for two bore-holes in the vicinity of the Birchtree mine are shown in Fig-ure 5. Significant reflections are predicted for (1) the amphi-bolite bodies both within the basement gneisses (900–1500 min borehole 86 250) and the Ospwagan supracrustals (700 and850 m in borehole 38 682-1), (2) iron formation within the


Ospwagan Group (1600 and 1800–2000 m in borehole 86 250),(3) pyroxenite (1050 and 1150 m in borehole 38 682-1), (4)serpentinite when in contact with some lithologies (1570 m inborehole 86 250; 1030 and 1150 m in borehole 38 682-1), (5)schist in contact with quartzite (<500 m in borehole 38 682-1),and (6) an ore zone (1720 m in borehole 86 250). A hypotheti-cal 10-m-thick massive sulfide zone placed within a schist hostrock produces a large-amplitude reflection (1300 and 2100 min boreholes 38 682-1 and 86 250, respectively).

Based on these results, we conclude the following:

1) the contact of the basement gneisses and the Ospwagansupracrustals should not be a significant reflector, region-ally, because the orthogneisses and paragneisses havesimilar impedances;

2) the reflectivity of the Ospwagan Group should allowdiscrimination from the more homogeneous basementgneisses;

3) where amphibolite is prevalent within the basementgneisses, they will be highly reflective, making them dif-ficult to distinguish from the Ospwagan supracrustals;

4) massive sulfide units will be reflective when they are>10 m thick; and

5) the reflectivity of serpentinite may provide a more use-ful marker for the location of massive sulfides becauseof the larger dimensions of the serpentinite bodies andthe genetic relation between the nickel ore deposits andultramafic rocks in the TNB.

FIG.4. Mean Vp and densities measured for a suite of rock sam-ples from the Thompson area. The solid curves indicate lines ofconstant acoustic impedance (product of Vp and density withunits of kg/m2 · s). Measurements were made on dry samplesunder hydrostatic pressure (30 MPa) or under an applied uni-axial stress (17–25 MPa). In the absence of measurements onmassive sulfides from the Thompson area, the Vp and densityvalues for massive sulfide are for a pyrhotite-rich sample fromSudbury measured at a confining pressure of 200 Mpa.

THE CAUSE OF CONDUCTIVITY

Surface observations of induced subsurface electrical cur-rents cannot distinguish between the two dominant uppercrustal conduction mechanisms: metallic conduction or ionictransport in fluids. However, knowledge of local geologicalstructure lets us infer the predominant conduction mode. Forexample, fluids tend to produce laterally continuous, subhor-izontal conductivity anomalies unless trapped by a sealingmechanism that prevents them from migrating. Thus, the pres-ence of fluids may be a reasonable interpretation where a per-vasive, subhorizontal conductive layer is observed. In contrast,solid semiconductors or conductors can attain any attitude ororientation; thus, local discrete conductors of variable attitudeand orientation are likely metallic conductors.

Electrical rock property measurements

Resistivity and porosity measurements made on samples ofvarious lithologies from the Thompson area are plotted in Fig-ure 6. The resistivity measurements were made using methodsdescribed by Katsube et al. (1991) with samples saturated withdistilled water. The serpentinites and peridotites are the mostconductive and porous. Moderately conductive rocks (i.e., be-tween 100 and 1000 ohm-m) include the dolomite and quartziteof the Ospwagan Group. Amphibolite, biotite schist, schist, andorthogneiss are all more resistive and show little porosity. Fig-ure 6 illustrates that any known conductive units are intimatelyrelated to the Ospwagan Group.

A useful means of characterizing electrical conductivity asdetermined by ionic transport is through a simplified form ofArchie’s law:

ρobserved = ρfluidϕ−m,

where m is the saturation exponent, φ is the porosity, ρobserved

is the measured in-situ resistivity, and ρfluid is the resistivity ofthe pore fluid. The straightline plotted on Figure 6 is a best fitof Archie’s law to all of the rock property data, regardless ofrock type. The exponentm= 2.33 is within the range of experi-mentally observed values (typically 1.3–3.0). The low porositiessuggest that most rocks in the Thompson area are electricallyresistive. A porosity of 3% would be required to generate a re-sistivity of 10 ohm-m. This porosity is at least an order of magni-tude greater than that observed for the Ospwagan Group andArchean gneisses, with the exception of some highly porousultramafic rocks.

These data indicate that, in general, Ospwagan Groupsupracrustal rocks and the Archean gneiss are characterizedby low porosity and that the dolomites and quartzites aremarginally conductive whereas the ultramafic rocks are highlyconductive. The resistive host rocks afford deep penetrationof the EM fields by virtue of limited attenuation. This obser-vation, coupled with the association of the conductive bodieswith the Ospwagan Group, means that EM mapping methodsshould be capable of delineating the subsurface extent of thesupracrustal rocks—a key objective of this study.

SEISMIC AND EM SURVEY DESIGN

Figure 7 shows the location of the coincident seismic and EMprofiles in relation to the Inco Birchtree mine and the outline ofthe Ospwagan Group supracrustals. We refer to this combined

1876 White et al.

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profile as line 1A. Also shown is the location of an orthogonalEM profile that is referred to in the text but is not shown. Theseismic/EM profile was intended to provide a complete cross-ing of the Ospwagan Group using existing roads. Starting to thenorthwest of the Birchtree mine, the profile crosses the F1 syn-form of the Thompson Nappe and continues southeast acrossF3 antiforms (Owl and basement antiforms) which lie alongstrike from the antiform that hosts the Thompson mine to thenortheast (see Figure 2). The seismic data were acquired usinga 240-channel telemetry acquisition system with in-field stack-ing, noise-rejection, and correlation capabilities. An asymmet-ric (80/160 channel) split-spread geophone array was used witha 20-m geophone group interval, 10-m source group interval,and geophones with a resonant frequency of 30 Hz. The seis-mic source consisted of two Heaviquip Hemi 50 vibrators, po-sitioned in-line with one directly behind the other, sweepingtwice at each shot station and using a 12-s linear upsweep from30 to 130 Hz. Data processing applied to these data includedcrooked-line binning, deconvolution, time-variant band-passfiltering, weathering static corrections, detailed velocity anal-ysis, trim static corrections, and constant-velocity (6000 m/s)f -k migration. A general description of these processes can befound in Yilmaz (1987). A hybrid of constant-velocity stackswas more effective in imaging the structures with highly vari-able dip than standard dip-moveout processing.

The EM survey consisted of two profiles, parallel and per-pendicular to strike of the Ospwagan Group, designed to con-strain the true structural dip. Source loop dimensions of ap-proximately 1000 × 1000 m (±1% precision) were designedto obtain reliable information about the diffusing current sys-tem to depths of approximately 4000 m. Data were collectedwith the UTEM system (West et al., 1984) in 20-channel modewith a base frequency of 31 Hz. Receiver profiles of at least3000 m in length and centered about the loop allowed a three-fold redundancy while allowing uniform resolution in the nearsurface along the entire profile. The EM data were processedusing the depth image processing method (e.g., Macnae and

FIG. 6. Rock-property results from a suite of samples providedby Inco Limited, Exploration. Plotted is the log of resistivityversus the log of porosity to facilitate the interpretation of con-ductivity in terms of ionic transport (fluids) versus electronictransport (metals and semiconductors).

Lamontagne, 1987) to convert step response data into esti-mates of conductivity versus depth. Discrete, steeply dippingconductors disrupt this processing, particularly if they are suffi-ciently conductive that the secondary EM field induced withinthem is characterized by a long time constant decay. The lo-cations of such bodies are indicated with black arrows on Fig-ure 7; the imaged conductivity section is not reliable in theseregions.

INTERPRETATION OF LINE 1A PROFILES

The unmigrated seismic data for line 1A are shown in Fig-ure 8, and the combined migrated seismic data and EM resis-tivity image are shown in Figure 9. Large, steeply dipping andhighly conductive bodies at the west end of the line near a broadunit of exposed iron formation make the EM image west of theBurntwood lineament unreliable. Further east, there is littledisturbance from near-surface conductors. An interpretationof the combined seismic/EM image is marked on Figure 9, anda perspective view of the interpretation is shown in Figure 10.

Seismic interpretation

1) Based on rock property studies, zones of high reflectiv-ity are interpreted as being associated primarily with theOspwagan Group. The highly reflective regions outlinethe F1 nappe folded by the upright F3 structures. TheF1 structure in general dips to the southeast, as do mostof the reflections. Since the reflection data are biased to-ward shallow dips (<60◦), the most prominent part of theoverall F1 structure imaged is the crest of the F3 antiform(Owl antiform), which is approximately along strikefrom the F3 antiformal culmination at the Thompsonmine.

2) The depth extent of the Ospwagan Group within theF1 fold structure is difficult to determine from the seis-mic data alone. In areas of favorable structural attitudes(e.g., the core of the F3 Owl antiform), the seismic im-age is dominated by shallowly southeast-dipping reflec-tions to at least 5000 m depth. Using the thickness of theOspwagan Group rocks in the vicinity of the Birchtreemine (∼1800 m, Figure 7) as a guide, the lower limb (F1)basement-cover contact must occur within this generallyreflective zone (dotted line).

3) The pattern of reflections observed to the southeast ofthe Owl antiform is interpreted as showing the over-turned basement-cover contact imbricated along steeplysoutheast-dipping reverse faults that necessarily post-date F3. This interpretation implies that the overturnedOspwagan Group stratigraphy (i.e., upper limb of thenappe) should be within 1000 m of the surface near thesoutheast end of the line. A mapped antiform within base-ment rocks southeast of the Grass River lineament (Fig-ure 7) appears to have been imaged within the hangingwall of the easternmost of the three major faults. Thefaults in this model can be projected to about 2000 to4000 m depth and appear to become listric, dipping moreshallowly to the southeast. These structures may accom-modate northeast-side-up displacement and thus are ex-amples of the late, ductile reverse faults found throughoutthe TNB (see Fueten and Robin, 1989).

1878 White et al.

4) The Burntwood lineament (mylonite zone) was not di-rectly imaged on the seismic profiles but is representedby a zone of truncated reflectivity west of the Owl an-tiform. The seismic signature changes dramatically acrossthe Burntwood lineament from highly reflective in theeast to nonreflective in the west as a result of the west-ward change across the structure to steep dips.

5) West of the Burntwood lineament, the geologic sections(from borehole information) show all of the lithologiccontacts dipping steeply to the northwest in contrast tothe seismic data, which are dominated by moderatelysoutheast-dipping reflections. These southeast-dippingreflectors are interpreted as being associated with knownsoutheast-dipping faults. The northwest-dipping litho-logic contacts were not imaged for two principal reasons:the steep dips are difficult to image with conventionalCMP processing and insufficient maximum offsets inthe downdip direction (northwest) were recorded duringacquisition.

FIG. 7. Location of seismic/EM profile in the vicinity of the Birchtree mine. The numbers annotated along the seismic line denotevibration-point stations (every 10 m) that increment by 5 and spatially coincident geophone group stations (every 20 m) thatincrement by 10. The dark arrows indicate the locations of strong conductors detected near the surface by the EM survey.

Interpretation of the EM data

The conductivity structure southeast of the Burntwood lin-earment is characterized by a resistive (103–105 ohm-m) near-surface layer that overlies a southeastward-dipping layer ofhigh conductivity (<100 ohm-m). The depth to the conduc-tive layer increases from approximately 1000 m near theBurntwood lineament to 2500 m at the southeast end of theline. Decay curves from adjacent loops on the southeast end ofthe line demonstrate a gradual northwestward slowing of thesecondary magnetic field diffusion along the profile, diagnosticof a conductive layer that shallows to the northwest. The EMdata from the perpendicular profile (not shown) suggest thatthe conductive layer also dips gently to the northeast, indicat-ing a north true dip. The other discrete conductors identifiedwith the surface EM survey (Figure 7) all parallel the strike ofthe Ospwagan supracrustal group and dip vertically with littleor no discernible plunge, although plunge is difficult to resolvewith these data.


The spatial association of surficial EM anomalies with themetasediments and the results of the laboratory resistivity mea-surements both suggest the most plausible source of the deepconductive layer is Ospwagan Group rocks. While these rocksare more porous than the basement gneiss, the conductive layer

FIG. 8. Unmigrated seismic data for line 1A. This stack was obtained by using low stacking velocities (6000 m/s). Vibration-pointstations are annotated along the top of the section. The location of geological features (from Figure 7) is also shown.

FIG. 9. Migrated seismic data for line 1A overlying the color resistivity versus depth image for the coincident EM profile. Vibration-point stations are annotated along the top of the section. The location of geological features (from Figure 7) is also shown.

dips markedly to the north, making fluids an unlikely conduc-tion mechanism. The observed anomaly is laterally continu-ous and thus may represent a conductive formation of theOspwagan Group. Alternatively, it may represent the cumu-lative response of several large, discrete conductors such as

1880 White et al.

peridotite/serpentinite bodies embedded within a mildly con-ductive matrix. The graphitic/sulfitic shale member of the PipeFormation probably contributes to the overall conductivity ofthe formation (Zablocki, 1966).

COMBINED INTERPRETATION

A 3-D perspective view of the surface geology and subsur-face interpretation is shown in Figure 10. The subsurface inter-pretation is based on the seismic/EM image in conjunction withsurface and borehole/mine geology. The Burntwood lineamentappears to be a subvertical structural discontinuity that disruptsthe inferred continuity of the Thompson Nappe structure (seeFigure 3) between the Birchtree mine and the Owl antiform.The significance of the Burntwood lineament is suggested bythe discontinuity in the seismic signature across the lineament,the disparity in conductivity associated with Ospwagan out-crops on either side of the lineament (i.e., Birchtree versus Owlantiform), and an increase in metamorphic grade from north-west to southeast across the fault, suggesting relative southeast-side-up movement.

Details of the interpretation model west of the Burntwoodlineament are based primarily on borehole and mine geology.The seismic and EM surveys failed to provide robust subsur-face images in this region because of the subvertical attitude ofthe Ospwagan Group and the presence of large, strong, near-surface conductors in this region (black arrows, Figure 7).

The Owl antiform containing Ospwagan supracrustalsplunges northeastward beneath the profile as suggested by sur-face geological observations. This is supported by spatial coinci-dence of (1) the antiform imaged seismically and its position onrudimentary downplunge projections of the surface fold tracedinto the seismic imaging plane and (2) the top of the conduc-tive zone with the crest of the antiform. This interpretationis consistent with the observation that the Ospwagan Groupis generally conductive, but it is in apparent contradiction tothe resistivity image from the perpendicular EM profile (notshown), which indicates that Ospwagan supracrustals withinthe Owl antiform have a lower conductivity than those westof the Burntwood lineament and a shallow rather than steepplunge of this conductive layer to the northeast. This questionremains unanswered.

FIG. 10. Perspective view (looking from the northeast) of thecombined seismic/EM interpretation.

Southeast of the Owl antiform, the seismic interpretationsuggests that the overturned basement-cover contact is imbri-cated along a series of late steeply southeast-dipping reversefaults carrying Ospwagan supracrustals to the shallow subsur-face. Although this structural interpretation is plausible, thehigh resistivities observed for this same zone require that thefolds/faults actually lie within reflective basement gneisses orresistive supracrustal rocks. Assuming the latter, it is possiblethat the reflective rocks represent Ospwagan metasedimentaryrocks from the overturned limb of the Thompson Nappe (gen-erally resistive) and that the deep conductive layer which dipsto the north actually represents metasedimentary rocks fromthe basal (upright) limb of the nappe. If this is the case, the twolimbs of supracrustals may not be physically connected and areperhaps separated by the Burntwood lineament.

CONCLUSIONS

The results of this combined seismic and EM study suggestthat these methods are effective in determining the subsurfaceconfiguration of the nickel ore-bearing Ospwagan supracrustalgroup. This conclusion is based on rock-property measurments,borehole logs, and correlation of the geophysical images withmapped surface geology. The Ospwagan supracrustals shouldgenerally be more seismically reflective and electrically con-ductive than the Archean gneisses that surround them in F1–F3fold interference structures in the Thompson area.

Further work is required to substantiate this hypothesis.In particular, more rigorous ground-truthing of geophysicalimages is essential. The only area along the existing profilehaving boreholes or mineworks to depths sufficient to cor-relate directly with the geophysical images is in the vicinityof the Birchtree mine. Unfortunately, this area is where theseismic and EM images are poorly constrained because of sub-vertical lithology and strong near-surface conductors, respec-tively. The in-situ reflectivity of the subsurface can be deter-mined by acquiring downhole sonic and density logs and VSPsto test the conclusions based on the existing rock-propertymeasurements.

The structural complexity of the Thompson Nappe dictatesthat no single method of subsurface imaging will be effectiveeverywhere. The existing mines in the Thompson area are lo-cated where steeply dipping nappe limbs breach the surface. Inthese areas, vertical seismic profiling (e.g., Eaton et al., 1996)may be a more effective means of subsurface imaging sincesurface seismic methods are hampered by the steep structuralattitudes. However, if the nappe model is valid, much of theOspwagan Group may lie blind beneath the Arohean gneisseswith structural attitudes that are amenable to imaging using sur-face seismics (e.g., east of the Burntwood lineament). In theseareas, surface seismics and EM methods may be strategicallyexploited. Careful survey design will be required to account for3-D structure; ultimately, 3-D surveys will be needed to obtainaccurate structural information.

ACKNOWLEDGMENTS

The seismic data were acquired by Enertec GeophysicalLtd. with original data processing by Western Geophysical.The UTEM data were acquired and processed by LamontagneGeophysics. An early version of the manuscript was reviewedby B. Roberts, W. Bleeker, D. Eaton, and an anonymous


reviewer. The authors thank Inco Limited, Exploration, forproviding access and allowing publication of the geological logsand rock-property measurement. This is Geological Survey ofCanada paper 2000021 and Lithoprobe contribution 1169.

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Machado, N., Zwanzig, H. V., and Parent, M., 1999, From basin tocontinent: The chronology of plutonism, sedimentation and meta-morphism of the southen Kisseynew metasedimentary belt, Trans-Hudson orgen (Manitoba, Canada): Can. J. Earth Sciences, 36, 1829–1842.

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Mikereit, B., Green, A., and the Sudbury Working Group, 1992b, Ge-ometry of the Sudbury structure from high-resolution seismic reflec-tion profiling: Geology, 20, 807–811.

Perron, G., and Calvert, A. J., 1997, Shallow, high-resolution seismicimaging of the Ansil mining camp in the Abitibi greenstone belt:Geophysics, 63, 379–391.

Perron, G., Milkereit, B., Reed, L. E., Salisbury, M., Adam, E., andWu, J., 1997, Integrated seismic reflection and borehole geophysicalstudies at Les Mines Selbaie, Quebec: CIM Bull., 90, 75–82.

Scoates, R. F. J., Macek, J. J., and Russell, J. K., 1977, Thompson nickelbelt project: Report of field activities, Manitoba, Min. Res. Div., 47–53.

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Spencer, C., Thurlow, G., Wright, J., White, D. J., Carroll, P., Milkereit,B., and Reed, L., 1993, A Vibroseis reflection seismic survey at theBuchans Mine in central Newfoundland: Geophysics, 58, 154–166.

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White, D. J., Milkereit, B., Wu, J., Salisbury, M. H., Berrer, E. K.,Mwenifumbo, J., Moon, W., and Lodha, G., 1994, Seismic reflectivityof the Sudbury structure North Range from borehold logs: Geophys.Res. Lett., 21, 935–938.

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Physical properties and seismic imaging of massive sulfides

Matthew H. Salisbury∗, Bernd Milkereit‡, Graham Ascough∗∗, Robin Adair∗∗,Larry Matthews§, Douglas R. Schmitt§§, Jonathan Mwenifumbo∇ ,David W. Eaton�, and Jianjun Wu≈

ABSTRACT

Laboratory studies show that the acoustic impedancesof massive sulfides can be predicted from the physicalproperties (Vp , density) and modal abundances of com-mon sulfide minerals using simple mixing relations. Mostsulfides have significantly higher impedances than sili-cate rocks, implying that seismic reflection techniquescan be used directly for base metals exploration, pro-vided the deposits meet the geometric constraints re-quired for detection. To test this concept, a series of 1-,2-, and 3-D seismic experiments were conducted to im-age known ore bodies in central and eastern Canada. Inone recent test, conducted at the Halfmile Lake copper-nickel deposit in the Bathurst camp, laboratory measure-ments on representative samples of ore and country rockdemonstrated that the ores should make strong reflectorsat the site, while velocity and density logging confirmedthat these reflectors should persist at formation scales.These predictions have been confirmed by the detectionof strong reflections from the deposit using vertical seis-mic profiling and 2-D multichannel seismic imaging tech-niques.

INTRODUCTION

In the early 1900s, the petroleum industry relied on surfacemapping, potential field techniques, and wildcat drilling for ex-ploration purposes, much as the mining industry still does to-

Manuscript received by the Editor January 27, 1999; revised manuscript received June 20, 2000.∗Geological Survey of Canada, Bedford Inst. of Oceanography, 1 Challenger Drive, Dartmouth, Nova Scotia B2Y 4A2, Canada. E-mail: [email protected].‡Institut fur Geophysik, Universitat zu Kiel, Olshausenstrasse 40, D-24098, Kiel, Germany. E-mail: [email protected].∗∗Noranda Mining and Exploration, Inc., 960 Alloy Drive, Thunder Bay, Ontario, P7B 6A4, Canada. E-mail: [email protected]; [email protected].§Noranda and Mining Exploration, Inc., 605 5th Avenue SW, Calgary, Alberta, T2P 3H5, Canada. E-mail: [email protected].§§University of Alberta, Department of Geology, Edmonton, Alberta, 2G6 2E3, Canada. E-mail: [email protected].∇Geological Survey of Canada, Mineral Resources Division, 601 Booth St., Ottawa, Ontario K1A 0E8, Canada. E-mail: [email protected].�University of Western Ontario, Department of Earth Sciences, London, Ontario N6A 5B7, Canada. E-mail: [email protected].≈Pulsonic Corporation, 301, 400-3rd Avenue SW, Calgary, Alberta, T2P 4H2, Canada. E-mail: [email protected]© 2000 Society of Exploration Geophysicists. All rights reserved.

day. Following the initial tests of the seismic reflection methodby Karcher more than 75 years ago, however, and a stringof oil discoveries during the first commercial reflection sur-veys several years later, the exploration methods employed bythe two industries rapidly diverged. Within a decade, reflec-tion seismology had become the principal exploration tool ofthe petroleum industry, a position maintained to the presentday through spectacular advances in acquisition, processing,and imaging technology (see Weatherby, 1940, and Enachescu,1993, for review).

Despite the remarkable success of the seismic reflectionmethod, the mining industry has been reluctant to embrace thistechnology because, until recently, its needs could be met bytraditional methods. With the known shallow reserves of cop-per and zinc declining, however, it has become obvious thatnew deep exploration tools must be developed if the industryis to remain viable in the future (Debicki, 1996). Given theoverall similarity of the exploration problems faced by the twoindustries, it is appropriate to ask whether or not this sophis-ticated technology might also be used for the direct detectionof massive sulfides.

Although no single relationship can be used to predict theresolution of seismic reflection techniques, it should be possi-ble to image a sulfide deposit using seismic reflection if threegeneral conditions are met:

1) The difference in acoustic impedance between the oreand the country rock must be sufficiently large to pro-duce strong reflections. If Zo and Zc are the acousticimpedances, or velocity-density products, of the ore andcountry rock respectively, the reflection coefficient R (the

1882

Properties and Imaging of Sulfides 1883

ratio of reflected to incident energy) can be calculated forthe case of vertical incidence from the relation,

R = Zo − Zc

Zo + Zc(1)

In practice, an impedance difference of 2.5 × 105 g/cm2 s(the contrast between mafic and felsic rocks) givesrise to a value of R = 0.06, the minimum value re-quired to produce a strong reflection in most geologicsettings.

2) To be imaged as a planar reflecting surface, the body musthave a diameter which is greater than the width of the firstFresnel zone, dF , defined from the relation

dF = (2zv/ f )1/2, (2)

where z is the depth of the deposit, v is the average for-mation velocity, and f is the dominant frequency used inthe survey. Smaller deposits with a diameter equal to onewavelength can be detected as point sources, but not im-aged. Since amplitudes are severely attenuated for bod-ies which are less than one wavelength across (Berryhill,1977), the diameter of the smallest body which can bedetected in practice, dmin, is

dmin = v/ f. (3)

3) The minimum thickness, tmin, that can be resolved us-ing seismic reflection can be estimated from the quarterwavelength criterion,

tmin = v/(4 f ). (4)

Thinner deposits can be detected, but their thickness can-not be determined, and the reflection amplitudes will bedecreased by destructive interference (Widess, 1973).

While it can be shown from these equations that many sul-fide deposits meet or exceed the size requirements for bothdetection and imaging (for example, a 500 m diameter × 15 mthick deposit could easily be imaged at a depth of 2 km, as-suming a peak frequency of 100 Hz and a formation velocity of6.0 km/s), early tests in mining camps were inconclusive (e.g.,Dahle et al., 1985; Reed, L., 1993, Mineral Industry Technol-ogy Council of Canada report), in part because the acousticproperties of the sulfide minerals themselves were poorly un-derstood, but also because significant differences in target size,structure, and acoustics between hard and soft rock environ-ments had not been fully taken into account. In particular, thesignal-to-noise (S/N) ratio is anomalously low in hard rock ter-rains, the targets are typically point sources or scatterers ratherthan continuous reflectors, and they are often steeply dipping(Eaton, 1999).

To assess and solve these problems, the Geological Surveyof Canada (GSC) recently embarked on a major collaborativeresearch program with INCO, Falconbridge, and Noranda in-volving integrated laboratory, logging, modeling, and seismictests at six mining camps in Canada, including Sudbury, KiddCreek, and Manitouwadge in Ontario, Matagami and Selbaiein Quebec, and Bathurst in New Brunswick. The objectives ofthe laboratory studies were first to determine the basic acousticproperties of the major sulfide minerals from measurements oftheir compressional wave velocities (Vp) and densities at ele-

vated pressures, and then to determine for each camp the pairsof lithologies which might be expected to produce strong re-flections. Following the laboratory studies, velocity and densitylogs were then run in selected boreholes in each camp to de-termine if the laboratory results persisted at formation scales.If the results of these tests were promising, seismic modelingbased on the laboratory and logging results plus the knowngeology in each camp was performed to guide the subsequentacquisition and interpretation of seismic reflection data. Verti-cal seismic profiling (VSP) and/ or 2-D and 3-D multichannelseismic (MCS) surveys were then conducted over known oredeposits and marker horizons in each camp using state-of-the-art technology to map the geology at depth and determineif the deposits themselves could actually be detected and im-aged. While the surveys were based on oil-field techniques,many changes were made to accomodate the unusual condi-tions encountered in hard rock terrains. For example, dynamitewas used as the source because of its high-frequency content,and particular care was taken to ensure high-fold coverageand good shot and receiver coupling to bedrock to compen-sate for low S/N ratios. In addition, large shot-receiver offsetsor VSP techniques were used where the targets were steeplydipping. Finally, since the targets were small, unconventionalprocessing sequences based on Born scattering were oftenused to process the data (Eaton et al., 1997; Milkereit et al.,1997).

The results of these studies have been spectacular. The labo-ratory studies have provided a quantitative basis for predictingthe reflectivity of an ore body of any composition in any setting(Salisbury et al., 1996), while the seismic studies successfullydetected all of the known deposits and identified several newtargets to be tested by drilling. Preliminary seismic results havealready been published for several of these studies (Milkereitet al., 1992, 1996; Eaton et al., 1996) and the results of many ofthe more recent surveys conducted during this program, includ-ing 2-, and 3-D surveys in Bathurst, Manitouwadge, Matagami,and Sudbury, were presented at Exploration ’97 (Adam et al.,1997; Eaton et al., 1997; Milkereit et al., 1997; Roberts et al.,1997; Salisbury et al., 1997). The purpose of the present paper isto outline the basic acoustic properties of sulfides and to show,through a case study involving laboratory, logging, and seis-mic imaging tests at the Halfmile Lake deposit in the Bathurstcamp, how these properties govern the reflectivity of massivesulfides.

ACOUSTIC PROPERTIES OF SULFIDES

While the acoustic properties of silicate rocks are wellknown from decades of laboratory studies (e.g., Birch, 1960;Christensen, 1982), until very recently the properties of sulfideswere so poorly known that it was difficult to predict whetheror not they should be reflectors. The principal difficulty wasthat while the velocities and densities of some sulfide miner-als, such as pyrite (py) and sphalerite (sph) were well known(Simmons and Wang, 1971), the properties of other volumet-rically and thus acoustically important minerals such as chal-copyrite (cpy) and pyrrhotite (po) had never been measured,making it difficult to estimate the properties of mixed sulfidedeposits.

To answer this question, we measured the densities and com-pressional wave velocities of a large suite of ore and host rocksamples of known composition in the laboratory at confining

1884 Salisbury et al.

pressures ranging from 0 to 600 MPa using the pulse transmis-sion technique of Birch (1960). In addition to ores of mixedcomposition, samples of pure py, sph, po, and cpy were mea-sured to establish the theoretical limits of the velocity-densityfield for common ores. The results, summarized in Figure 1from Salisbury et al. (1996) for data at a standard confiningpressure of 200 MPa (the crack closure pressure), show sev-eral important trends:

1) As predicted from earlier studies, the velocities of thehost rocks increase with density along the Nafe-Drakecurve for silicate rocks (Ludwig et al., 1971).

2) The sulfides, however, lie far to the right of the Nafe-Drake curve in a large velocity-density field controlled bythe properties of pyrite, which is fast and dense (8.0 km/s,5.0 g/cm3), pyrrhotite, which is very slow and dense(4.7 km/s, 4.6 g/cm3), and sphalerite and chalcopyritewhich have intermediate and very similar velocities anddensities (∼5.5 km/s, 4.1 g/cm3).

3) The properties of mixed and disseminated sulfides liealong simple mixing lines connecting the properties ofend-member sulfides and felsic or mafic gangue. Thus, forexample, velocities increase dramatically with increasingpyrite content, but they actually decrease with increas-ing sphalerite, chalcopyrite, or pyrrhotite content alongtrends which can be calculated using the time-averagerelationship of Wyllie et al. (1958).

4) If lines of constant acoustic impedance (Z) correspondingto the average impedances of mafic and felsic rocks aresuperimposed on the sulfide fields as in Figure 1, it is clearthat most sulfide ores have higher impedances than theirfelsic or mafic hosts. Since a reflection coefficient of 0.06

FIG. 1. Velocity (Vp)-density fields for common sulfide oresand silicate host rocks at 200 MPa. Ores: py = pyrite,cpy = chalcopyrite, sph = sphalerite, po = pyrrhotite. Silicaterocks along Nafe-Drake curve: SED = sediments, SERP = ser-pentinite, F = felsic, M = mafic, UM = ultramafic, g = gangue,c = carbonate. Dashed lines represent lines of constant acous-tic impedance (Z) for felsic and mafic rocks. Bar shows mini-mum impedance contrast required to give a strong reflection(R = 0.061).

is sufficient to give a strong reflection, then in principle,massive sulfides with the appropriate geometry should bestrong reflectors in many common geologic settings. Forexample, massive pyrrhotite should be readily detectablein felsic settings, and any combination of sphalerite, chal-copyrite, and pyrite should be a strong to brilliant reflec-tor in most mafic and felsic settings, depending on thepyrite content.

CASE STUDY: SEISMIC REFLECTIONS FROM THEHALFMILE LAKE DEPOSIT, BATHURST

While the laboratory results presented above show that mas-sive sulfides should often be strong reflectors, it is also clearthat the reflectivity of any given deposit will be strongly influ-enced by local conditions, such as the size and configurationof the deposit, its actual mineralogy, and the composition andmetamorphic grade of the country rock. Thus to evaluate themethod, it was necessary to study the actual seismic responseof several deposits at different scales of investigation.

The Bathurst mining camp in New Brunswick was selectedfor one of these studies because it provides an excellent oppor-tunity to examine the seismic response of volcanogenic mas-sive sulfide deposits in low-grade metamorphic settings. Theresults, which were obtained over the Halfmile Lake deposit,the largest undeveloped deposit in the camp, provide a par-ticularly clear example, not only of reflections from a massivesulfide deposit, but of the factors which must be taken intoaccount in imaging these bodies.

Geology of the Halfmile Lake deposit

The Halfmile Lake deposit is an extensive massive sulfidesheet which ranges from 1 to 45 m in thickness and contains26 million tons of total sulfides. Mineralization is characterizedby py-po-rich layered sulfides and po-rich breccia-matrix sul-fides with variable amounts of sphalerite, galena, and arsenopy-rite. The deposit is hosted by a thick sequence of turbidites,felsic volcanic and epiclastic rocks, argillites, and intermediatevolcanic rocks of Cambro-Ordovician age (Adair, 1992). Fourperiods of fold deformation are documented in the depositarea and are accompanied by faulting. The deposit occurs onthe overturned south limb of a large antiform (Figure 2). Thesulfide sheet extends 3 km along strike (Figure 2) and has beendrilled to a depth of 1.2 km. It is structurally overlain (strati-graphic footwall) by a stringer zone containing between 5 and30% stringer po and cpy (Figure 3). The overall dip of thestratigraphy is 45◦ to the north-northwest, but structural influ-ences locally steepen dips to near vertical. Metamorphism isgreenschist facies, and the sheet displays 100 m of topography.

Hosted within the sulfide sheet are two significant build-upsof massive sulfides in the Upper and Lower Zones (Figure 3),where the thickness of the sulfides ranges between 5 and 45 m.The most important and largest undeveloped deposit in thecamp is the Lower Zone, which contains 6.1 million tons of9.7% Zn, 3.34% Pb, 0.1% Cu, and 43 g/ton Ag. The HalfmileLake deposit and specifically the Lower Zone were selectedfor this study because it had been carefully mapped and de-lineated by drilling, core was available for laboratory stud-ies, drillholes were still open for logging and VSP tests, thesize and dip of the deposit seemed appropriate for detection


using 2-D surface seismic techniques and, since it had neverbeen mined, there would be no spurious reflections from mineworkings.

Laboratory impedance measurements

To determine which lithologies were potential reflectors atHalfmile Lake, velocities and densities were measured in thelaboratory at elevated pressures on minicores cut from 28 sur-face and drill core samples representing all of the major ore andhost rock lithologies along the seismic line. Since sedimentaryand metamorphic rocks are often anisotropic, velocities weremeasured parallel and perpendicular to bedding, banding, orfoliation in many samples, bringing the total number of mea-surements to 53. The results, presented at a confining pressureof 200 MPa in Figure 4, show that the host rocks all have verysimilar average velocities and densities (about 6.0 km/s and2.75 g/cm3). This initially surprising result is due to the fact thatthe felsic igneous rocks (rhyolite, quartz porphyry, tuff) and themetasediments all have very similar compositions, while themafic rocks are actually basaltic andesites of intermediate com-position in which the velocities have been depressed by thealteration of mafic minerals to chlorite by greenschist faciesmetamorphism. The Halfmile Lake ores, on the other hand,display a wide range of velocities (5.1–7.3 km/s) due to vary-

FIG. 2. Geologic map of Halfmile Lake deposit showing location of 2-D seismic line presented in Figure 7. Geologic cross-sectionshown in Figure 3 extends from station 1 to station 166 (10–1660 m along line). Geophysical logging was conducted in holes HN94-63 (Figure 5) and HN 94-65. Offset VSP shown in Figure 6 was conducted in hole HN 92-30. Dashed arrow shows axis and plungeof F1 antiform. Inset shows regional setting of Halfmile Lake deposit.

ing proportions of po and py. Interestingly, the iron formationsample plots in the velocity-density field for the sulfides due tothe high intrinsic velocity and density of magnetite (7.4 km/s,5.2 g/cm3). As a consequence, the impedance contrasts betweenthe various country rock lithologies at Halfmile Lake shouldbe small, while the contrast between any of the ores and thecountry rock will be very large. Since an impedance contrast of2.5 is sufficient to give a strong reflection, this implies that atleast in terms of their acoustic properties, the ores at HalfmileLake should be strong reflectors in a virtually transparent hostrock setting.

Geophysical logging

Once the laboratory measurements had been completed,two boreholes through the deposit (holes HN 94-63 and HN94-65 in Figure 3) were logged by the GSC Mineral ResourcesDivision using slim-hole sonic velocity and density tools to de-termine if the impedance differences measured in the labora-tory persist at formation scales. The results, presented in Fig-ure 5 for the deeper of the two holes, show that the densitiesrange narrowly from 2.6 to 2.75 g/cm3 throughout the silicaterocks, but increase erratically to values as high as 2.95 g/cm3 inthe stringer zone, then dramatically to about 3.4 g/cm3 in themassive sulfides, while negative excursions correspond to faults


marked by thin intervals of fault gouge or breccia. Similarly, thevelocity log varies from 5.5 to 6.0 km/s throughout most of thehole, with negative spikes corresponding again to narrow faults.Significantly, the velocity-density product, or impedance log,shows very little variation about a mean of about 15, except forthe ores, which reach values of about 20, and the faults, whichreach values as low as 11. While the absolute values of the ve-locity and impedance logs are lower than the laboratory resultsdue to differences in pressure, the impedance contrasts are sim-ilar to those predicted from laboratory studies, implying thatthe massive sulfides will be strong reflectors, while the countryrock will be transparent except possibly where cut by faults.

Vertical seismic profiling

While the laboratory and logging results were promising,they were not definitive because they were obtained at muchhigher frequencies than seismic surveys (1 MHz and 50 KHzversus 10–200 Hz) and the propagation paths were muchshorter (2–5 cm and 1–2 m versus a few kilometers). VSP pro-vides a more convincing test because the method can be usedto determine if reflections are actually generated at seismic fre-quencies and if they are sufficiently strong to be detected overpropagation paths of 1 km or more. To this end, an offset VSPsurvey was conducted in borehole HN 92-30 by the Universityof Alberta using 350 g Pentolite charges in a series of shallow

FIG. 3. Simplified geologic cross-section through Halfmile Lake deposit based on drilling results projected onto seismic line betweenstations 1 and 166. UZ = Upper Zone, LZ = Lower Zone. VSP survey (Figure 6) was conducted in hole HN 92-30 (shots at X) andlogging was conducted in holes 94-63 and 94-65. No vertical exaggeration.

FIG. 4. Average compressional wave velocity (Vp) at 200 MPaversus density for ore and host rock samples from the HalfmileLake deposit superimposed on velocity-density fields for sul-fides and silicate rocks shown in Figure 1. Also shown arelines of constant acoustic impedance for mafic rocks (Z = 20)and felsic rocks (Z = 17.5). Impedances of Halfmile Lake ores(ellipse) are much greater than those of their silicate hosts.


(3–6 m) holes drilled to the top of basement about 100 m northof hole HN 92-30 (X in Figure 3) and a 3-component boreholeseismometer clamped at 5 m intervals as the tool was raisedfrom a depth of 575 m to the surface.

Analysis of first arrival traveltimes shows that the crust hasan average P-wave velocity (5.55 km/s) consistent with thelogging data, while the reflection results, presented in Figure 6after routine processing (upgoing wavefield separation, high-pass 50–290 Hz filter, deconvolution, scaling, first break mute)and transformation to geometric coordinates using the CDPtransform method (Wyatt and Wyatt, 1984; Hardage, 1985;Kohler and Koenig, 1986), show a prominent, north-dippingreflector between 300 and 400 m depth which corresponds tothe massive sulfide deposit south of the borehole (Figure 3). In-terestingly, a deeper sulfide horizon was also imaged at the baseof the hole. As predicted from the laboratory and logging stud-ies, no other contacts in the immediate vicinity have sufficientlylarge impedance contrasts to produce reflections, and the faultsare too thin to detect. From the results of this test, it is thus clearthat the Halfmile Lake deposit generates strong reflections atseismic frequencies, these reflections propagate for significantdistances in basement, and the deposit can be readily imagedusing borehole VSP techniques and conventional sources.

FIG. 5. Density, velocity (Vp) and calculated impedance versusdepth in hole HN 94-63 from geophysical logging. Fault gougein core indicated by dashed lines. Note high impedance of mas-sive sulfides.

Multichannel seismic profiling

While obviously successful, the borehole VSP survey con-ducted at Halfmile Lake did not prove that 2-, or 3-D surveysconducted at the surface would be successful. Despite the manysimilarities between the two techniques, significant differencesstill remain: The source to receiver paths are shorter for VSPsurveys, the S/N ratio is improved by clamping the receiver inbasement rather than placing it at the surface, and the over-burden path is eliminated.

The definitive test of the seismic reflection method is thus toconduct a 2-, or 3-D survey over the deposit from the surface asif no boreholes were available. To this end, a 2-D multichannelsurvey was conducted over the deposit along a 5.85 km linewhich intersects the deposit at its southern end and extendsdowndip for a considerable distance to the north (Figure 2),with downhole control provided by the VSP and logging results.The survey was conducted by Enertec Geophysical Services un-der contract to Noranda using 340 g Pentolite charges in holesdrilled to basement every 40 m along the line; a portable, state-of-the-art, 480-channel, 24-bit recording system; and 14 Hz re-ceivers laid out along three parallel lines spaced 50 m apart:a center line with groups every 10 m, 9 phones/station, giving30 fold coverage, and a line to either side with a 20 m groupspacing, 9 phones/station, giving 15 fold coverage.

Although shooting conditions were difficult and the fieldrecords were very noisy, careful processing involving static cor-rections, scaling, the application of a high-pass filter, deconvo-lution, common-midpoint binning, stacking velocity analysis,noise suppression, and poststack scaling gave a clear image ofthe ore body (Figure 7) which coincides with the known lo-cation of the deposit after migration. As in the VSP survey,

FIG. 6. CDP transform of VSP survey in borehole HN 92-30showing reflection from Halfmile Lake deposit (arrow). X inFigure 3 shows location of shots. Reflections are also observedfrom a thin sulfide layer between 550 and 600 m (see Figure 3).No vertical exaggeration.


the country rock was weakly reflective to virtually transparentalong the rest of the line. The results at Halfmile Lake thusdemonstrate not only that massive sulfides can be detected us-ing surface seismic reflection techniques but that nothing elsein the immediate vicinity of the line causes strong reflections.Thus at Halfmile Lake, seismic reflection provided relativelylittle new information about deep structure, but the methodappears to be almost ideal for exploration.

CONCLUSIONS

From the results of this study, it is clear that massive sulfidescan make strong reflections in hard rock settings and that theycan be detected using state-of-the-art seismic reflection tech-niques. This has major implications, both for exploration in theBathurst camp itself and for the mining industry as a whole.

Since the first major discovery in Bathurst more than 40 yearsago, more than 100 base metal deposits, 35 with defined ton-nage, have been identified in the upper few hundred meters

FIG. 7. Unmigrated 2-D multichannel seismic image of the Halfmile Lake deposit. TWT = two-way traveltime.No vertical exaggeration.

of basement in the camp using surface mapping and potentialfield techniques. As in many camps, however, the known shal-low reserves are declining at Bathurst, forcing exploration todeeper levels. Statistically, there should be just as many de-posits per unit volume at greater depths in the camp, but thesehave proven difficult to find because the surface geology is toocomplex to project to depth and potential field methods losetheir resolution at depths greater than a few hundred meters.Seismic reflection, however, appears to be an ideal tool for deepexploration at Bathurst and elsewhere because it has high res-olution to the full depth limits of mining (∼3 km) and becauseit can scan large volumes of crust at a cost which is low com-pared to deep wildcat or delineation drilling (Pretorius et al.,1997). While conditions at Bathurst appear to be particularlysuitable for seismic exploration, recent experimental surveysover deposits in quite different settings such as Matagami, KiddCreek, and Sudbury (Adam et al., 1997; Eaton et al., 1997;Milkereit et al., 1997) are equally encouraging, suggesting thathigh-resolution seismic reflection techniques can be modified


to meet the deep exploration needs of the mining industry inthe new millennium.

ACKNOWLEDGMENTS

We thank R. Iuliucci from the Dalhousie/GSC High PressureLaboratory for his assistance in measuring the acoustic proper-ties of the Bathurst samples, the members of the GSC MineralResources Division logging crew for obtaining the boreholegeophysical logs presented in this study, and the members ofthe University of Alberta borehole geophysics field crew fortheir assistance in conducting the VSP survey. Special thanksare due to Kevin Coflin from GSC Atlantic for his assistancein the field during the VSP survey. Finally, the MCS survey atBathurst would not have been possible without the consider-able assistance of the Enertec Geophysical Services field teamand Noranda staff from Bathurst.

REFERENCES

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Adam, E., Arnold, G., Beaudry, C., Matthews, L., Milkereit, B., Perron,G., and Pineault, R., 1997, Seismic exploration for VMS deposits,Matagami, Quebec, in Gubins, A., Ed., Proc. Exploration 97: 4thDec. Internat. Conf. on Mineral Expl., 433–438.

Berryhill, J. R., 1977, Diffraction response for non-zero separation ofsource and receiver: Geophysics, 38, 1176–1180.

Birch, F., 1960, The velocity of compressional waves in rocks to 10 kilo-bars, 1: J. Geophys. Res., 65, 1083–1102.

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Dahle, A., Gjoystdal, H., Grammeltdveldt, G., and Hansen, T., 1985,Application of seismic reflection methods for ore prospecting in crys-talline rock: First Break, 3, 9–16.

Debicki, E., 1996, MITEC’s Exploration Technology Division: Helpingreverse the trend of declining mineral reserves in Canada: CanadianInst. Min. Metal. Bull., 89, 53–59.

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Eaton, D., Guest, S., Milkereit, B., Bleeker, W., Crick, D., Schmitt, D.,and Salisbury, M., 1996, Seismic imaging of massive sulfide deposits:Part III. Borehole seismic imaging of near-vertical structures: Econ.Geol., 91, 835–840.

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Enachescu, M., 1993, Amplitude interpretation of 3-D reflection data:The Leading Edge, 12, 678–683.

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Kohler, K., and Koenig, M., 1986, Reconstruction of reflecting struc-tures from vertical seismic profiles with a moving source: Geophysics,51, 1923–1938.

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Milkereit, B., Berrer, E. K., Watts, A., and Roberts, B., 1997, Devel-opment of 3-D seismic exploration technology for Ni-Cu deposits,Sudbury basin, in Gubins, A., Ed., Proc., Exploration 97: 4th Dec.Internat. Conf. on Mineral Expl., 439–448.

Milkereit, B., Eaton, D., Wu, J., Perron, G., Salisbury, M., Berrer,E., and Morrison, G., 1996, Seismic imaging of massive sulfide de-posits: Part II. Reflection seismic profiling: Econ. Geol., 91, 829–834.

Milkereit, B., Reed, L., and Cinq-Mars, A., 1992, High frequency re-flection profiling at Les Mines Selbaie, Quebec: Can. Geol. Surv.Curr. Res., 92-1E, 217–224.

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Roberts, B., Zaleski, E., Adam, E., Perron, G., Petrie, L., Darch, L.,Salisbury, M., Eaton, D., and Milkereit, B., 1997, Seismic explorationof the Manitouwadge greenstone belt, Ontario, in Gubins, A., Ed.,Proc. Exploration 97: 4th Dec. Internat. Conf. on Mineral Expl.,451–454.

Salisbury, M. H., Milkereit, B., Ascough, G. L., Adair, R., Schmitt, D.,and Matthews, L., 1997, Physical properties and seismic imaging ofmassive sulphides, in Gubins, A., Ed., Proc. Exploration 97: 4th Dec.Internat. Conf. on Mineral Expl., 383–390.

Salisbury, M. H., Milkereit, B., and Bleeker, W., 1996, Seismic imagingof massive sulfide deposits: Part I. Rock properties: Econ. Geol., 91,821–828.

Simmons, G., and Wang, H., 1971, Single crystal elastic constants andcalculated aggregate properties: M.I.T. Press.

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ture using the vertical seismic profile, in Toksoz, M. N., and Stewart,R. R., Eds., Vertical seismic profiling, part B: Advanced concepts:Geophysical Press, 148–176.

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Development of 3-D seismic exploration technology for deepnickel-copper deposits—A case historyfrom the Sudbury basin, Canada

Bernd Milkereit∗, E. K. Berrer‡, Alan R. King‡, Anthony H. Watts∗∗,B. Roberts§, Erick Adam§, David W. Eaton§§, Jianjun Wu∇ ,and Matthew H. Salisbury§

ABSTRACT

Following extensive petrophysical studies and presitesurveys, the Trill area of the Sudbury basin was selectedfor conducting the first 3-D seismic survey for mineral ex-ploration in North America. The 3-D seismic experimentconfirms that in a geological setting such as the SudburyIgneous Complex, massive sulfide bodies cause a char-acteristic seismic scattering response. This provides anexcellent basis for the direct detection of massive sul-fides by seismic methods. The feasibility study suggeststhat high-resolution seismic methods offer a large de-tection radius in the order of hundreds to thousands ofmeters, together with accurate depth estimates.

INTRODUCTION

The Sudbury structure (Figure 1) is located at the erosionalboundary between the Archean Superior province and theoverlying sequence of early Proterozoic continental margindeposits. The structure consists of the Sudbury Igneous Com-plex (SIC), a differentiated sequence of norite, gabbro, andgranophyre overlain by breccias and metasedimenary rocks.The Sudbury basin is thought to be associated with a largeimpact event (e.g., Boerner et al., 1994). The Sudbury basin isthe richest nickel-producing area in the world, with significantby-products in copper and precious metals. The basin hosts

Manuscript received by the Editor March 23, 1999; revised manuscript received June 8, 2000.∗Kiel University, Department of Geophysics, 24118 Kiel, Germany. E-mail: [email protected].‡Inco Ltd., Exploration, Copper Cliff, Ontario P0M 1N0, Canada. E-mail: [email protected]; [email protected].∗∗Falconbridge Ltd., Exploration, Toronto, Canada. E-mail: [email protected].§Geological Survey of Canada, Ottawa, Canada. E-mail: [email protected]; [email protected]; [email protected].§§Formerly Geological Survey of Canada, Ottawa, Canada; presently University of Western Ontario, Department of Earth Sciences, London,Ontario, N6A 5B7, Canada. E-mail: [email protected].∇Formerly Geological Survey of Canada, Ottawa, Canada; presently Veritas, 715 5th Avenue S.W., Calgary, Alberta T2P 5AZ, Canada. E-mail:jj [email protected]© 2000 Society of Exploration Geophysicists. All rights reserved.

numerous mines along the outer rim of the SIC. In 1995, themines produced 170,000 metric tons of nickel and 160,000metric tons of copper. This industry is a significant economicresource for Canada. The discovery of new deposits is vital forthe continued use of the infrastructure of this important miningdistrict. In the early part of the twentieth century, explorationwas based on surface prospecting and geology. Soon thereafter,exploration received assistance from magnetic surveys employ-ing dip-needle instruments. Starting in the late 1940s, groundelectromagnetic methods were used to locate conductivesulfide deposits which contained nickel ore. With these instru-ments, the highly-conductive sulfide bodies containing mainlypyrrhotite could be detected to a depth of about 100 m. Likethe oil and gas industry, the mining industry is looking progres-sively deeper for new ore deposits to replace declining reservesat existing mines. This requirement for deep exploration is driv-ing the search for new technology to reduce costs and increasedrilling success rates. With modern large-loop electromagneticsystems, the depth of exploration is extended up to 400 mdepending on the size of the target. It is highly likely that moresulfide nickel deposits will exist at greater depths, near the SICfootwall contact. Therefore, there is great interest in locatingthese deposits at depths up to about 2500 m, the depth limitto which modern mining methods are capable of extractingthe ore.

Adapting seismic reflection methods to hard-rock geologymay extend the depth range of exploration (e.g., Milkereitet al., 1996a). Although recognized as having both thedepth penetration and resolving power required for deeper

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3-D Seismic Exploration Technology 1891

exploration, seismic profiling has often been viewed by the min-ing industry as cost-ineffective, in part because of the paucityof strong, unambiguous reflections that typify a sedimentarybasin and in part because of a general lack of understanding ofhow observed signals correlate with subsurface features. In thispaper, we present a case history for the evaluation of 3-D seis-mic exploration technology. We review the physical rock prop-erties of massive sulfides, present borehole geophysical data,and discuss the seismic reflection response of deep-seated oredeposits. Examples illustrate various stages of the feasibilitystudy from survey design to heliportable data acqusition, andfrom special processing considerations to final interpretationof the 3-D seismic data volume.

Geological setting

The Sudbury Ni-Cu deposits occur at the moderately tosteeply dipping lower contact of the SIC. Figure 1b shows asimplified model of the crater floor derived from drilling and

FIG. 1. (a) Location map of Sudbury structure, Ontario,Canada, with borehole locations and 3-D seismic survey area.Norite, gabbro, and granophyre of the Sudbury Igneous Com-plex (SIC) indicated by gray pattern. Stars indicate boreholelocations. (b) Geological model of the crater floor from drilland mining operations with massive sulfide deposits locatedbeneath the lower contact of the SIC [cross-section modifiedafter Morrison (1984)] 1 = norite, 2 = sublayer, sulfides, andbreccia, 3 = footwall complex.

mining operations. At the base of the SIC, the sublayer is ahost for the ore. The sublayer consists of a mass of basic to ul-trabasic inclusions of varying size and frequency of occurrencein a matrix of norite and sulfides. When the sulfides are suffi-ciently concentrated, this zone constitutes the ore (Morrison,1984). Although results from 2-D reconnaissance seismic pro-files (Milkereit et al., 1992, 1996a) acquired across the Sudburybasin have provided important information on gross structureand regional geological setting, there have been problems in-tegrating this new mapping technique into normal explorationprocedures. For example, borehole geophysical logs (i.e., den-sity and sonic velocity), important for the interpretation andcalibration of reflection seismic data, have not been routinelygathered in existing slim diamond-drill holes.

Over the past five years, an extensive database of geolog-ical mapping information, geophysical logs of existing deepdrill holes, and core samples for physical rock property studieshas been assembled to support interpretation of the early 2-Dseismic data across the Sudbury structure. In-situ logging andborehole seismic experiments have been conducted at multiplelocations within the SIC using full-waveform sonic and densitylogging tools. Small-diameter holes (NQ, BQ) were drilled todepths of about 2000 m, providing superb control on the ori-gin of reflections and the basic velocity structure of the SIC.Figure 2 shows density and velocity logs aligned along the SIC-footwall contact. A narrow range of velocities characterize theSIC norite, ranging from 6200 to 6400 m/s, whereas the sublayerand footwall complex display more extreme velocity variations(6000–6700 m/s). Density logs follow the same trend as theP-wave velocities: relatively uniform densities between 2.75and 2.8 g/cm3 are associated with the norite, while sublayer andfootwall complex densities scatter between 2.75 and 3.0 g/cm3.There appears to be no variation of the physical properties lat-erally within the SIC. In terms of seismic exploration, the con-tact between the “transparent” SIC norite and the “reflective”footwall complex represents a clear regional marker, makingit possible to map the bottom of the SIC.

Physical rock properties of massive sulfides

Sedimentary basins, where seismic techniques have beenused most extensively, differ from the crystalline crust in a num-ber of characteristics that are significant for seismic acquisitionand processing (Milkereit and Eaton, 1998). Unlike most sed-imentary basins, the crystalline crust often lacks pronouncedhorizontal continuity of prominent seismic reflectors (reflec-tion coefficients rarely exceed 0.1). In addition, the velocitystructure of young sedimentary basins is characterized by lowcompressional-wave velocities and relatively strong velocitygradients. On the other hand, crystalline crust is characterizedby uniformly high compressional velocities, especially below200 m where fracturing is less prevalent. For the analysis ofseismic reflectivity, acoustic impedance becomes the mostimportant parameter. In-situ studies in deep boreholes con-firm that significant impedance contrasts exist at the contactsbetween major lithological units of the SIC and the footwallcomplex.

Reflected and scattered seismic events occur at the contactbetween rocks with contrasting elastic properties (e.g., com-pressional wave velocity, Poisson’s ratio and, density). Fig-ure 3 shows the velocity-density field for common silicate rocksranging from unconsolidated sediments to dense, high-velocity

1892 Milkereit et al.

ultramafics. The velocity-density field for sulfides (includ-ing the pyrrhotite, chalcopyrite, millerite, niccolite, and pent-landite assemblages typical for ores in Sudbury) is distinctlydifferent from that for common silicate rocks (such as norite inthe hanging wall and gneiss in the footwall). Composite ma-terial, containing various sulfides and silicate rocks, is gov-erned by simple linear mixing rules (Salisbury et al., 1996).Thus, the presence of sulfide minerals will tend to define aunique velocity-density field. Depending on grade and type,the Sudbury massive-sulfide deposits become strong seismic re-flectors based on impedance contrast with unmineralized hostrocks. For regional seismic exploration programs, the contactbetween the reflective footwall complex and the transparentnorite unit of the SIC represents the most prominent regionalseismic marker. For mineral exploration, high-density massivesulfides located at or close to embayment structures at thenorite-footwall contact will produce a strong seismic reflec-tion or scattering response (Milkereit et al., 1996a). Densitymeasurements from core samples confirm that sulfide samplesfrom the 3-D seismic study area occupy the distinct densityfield (Milkereit et al., 1997).

Seismic modeling of deep massive sulfide deposits

High-density mineralization introduces new problems formodeling, acquisition, processing, and interpretation of seis-mic data. In seismic modeling, ray tracing is most effective indiscrete layers that are separated by smooth interfaces. Raytheory breaks down on highly irregular interfaces, as might bethe case for high-impedance bodies and other geological fea-tures in the shallow crystalline crust. Other techniques, such asa direct solution of the elastic wave equation by finite differ-ences, are currently impractical for routine 3-D seismic mod-eling. In this study, we apply the 3-D Born-approximationmethod of Eaton (1996). The Born approximation method isdesigned to handle geological situations where short wave-length impedance anomalies (such as ore bodies) are super-

FIG. 2. Summary sketch of velocity and density logs from the Sudbury structure aligned at the contact between norite and footwallcomplex. Borehole locations are shown in Figure 1.

imposed onto a smoothly varying background medium. Con-sider the case of a dipping-lens model consisting of a 200 mwide, disk-shaped high-impedance unit at 1650 m depth. Thebackground velocity is 6000 m/s. Figure 4 shows the zero-offset scattering response from such a feature, inclined at45◦. The unmigrated data shows the region over which thediffracted energy is visible is considerably larger than theactual size of the scattering body. In the unmigrated data,scattered energy is concentrated primarily in the downdipdirection.

In contrast to reflections from continuous interfaces, the scat-tering response from local heterogeneities remains stationary(i.e., the traveltime response is symmetric with respect to the lo-cation of the scatterer in the 3-D cube) (Milkereit et al., 1996b).The symmetric scattering response is best detected in a seriesof time slices (Figures 4b, c). It is worth noting that for dipping

FIG. 3. Velocity-density fields for silicate rocks and sul-fides/oxides with lines of constant seismic impedance (modi-fied after Salisbury et al., 1996). SED = sediments, F = felsic,M = mafic, UM = ultramafic.


FIG. 4. 3-D modeling of scattering event caused by a localhigh-impedance contrast (a dipping lens at 1650 m depthand a background velocity of 6000 m/s). (a) In-line section.Note the asymmetric zero-offset response. (b, c) Time slicesof zero-offset response. Note enhanced amplitudes in thedowndip direction (M). A = apex.

impedance contrasts, the strongest reflection/scattering devel-ops at large offsets in the dip direction, away from the deposit.The directional characteristics of the scattering response areimportant for the acquisition, processing, and interpretation of3-D seismic data for massive sulfides. Because of the inherentlimitations of conventional seismic modeling methods, someof the strike- and dip-related characteristics of small seismicenergy scatterers may not have been fully appreciated inthe past (most conventional 2-D seismic reconnaissanceprofiles may have overlooked or underestimated the signif-icance of local bright seismic diffractions caused by densityanomalies).

Special problems posed by the hard-rock environment

In the past, many tests of high-resolution seismic imagingmethods using “off-the-shelf” technology from hydrocarbonexploration have had limited success in hard-rock environ-ments. High levels of source-generated noise in the 10–30 Hzrange can overwhelm the relatively weak signals (Milkereit andEaton, 1998). The presence of the low-frequency noise windowand the need to resolve small features at depth demand the useof a source spectrum that is shifted more towards higher fre-quencies than those commonly used for seismic profiling insedimentary basins. This fundamental difference must be con-sidered in the design of hard-rock seismic surveys in order toachieve success.

Until recently, the vast majority of seismic surveys for min-eral exploration have used 2-D profiling, where source and re-ceiver locations are colinear for the basic acquisition geometry.This approach, which essentially yields vertical cross-sectionalimages of the subsurface, is most effective where major struc-tural elements have a well-defined strike direction and the2-D seismic profiles can be oriented perpendicular to struc-tural trends. In cases where subsurface structures do not havea well-defined strike and dip direction, out-of-plane reflections(sideswipe) can seriously contaminate 2-D reflection images,producing false structural images. Why should 3-D reflectionseismic surveys be considered for mineral exploration? Thespatial dimensions of economically viable targets embeddedin complex crustal structures make them less likely to be de-tected in the course of reconnaissance 2-D seismic work. Inpractice, spatial aliasing and the number of recording channelspose some limitations on the design of 3-D seismic surveys.For example, the use of high seismic frequencies requires smallseparation of sensors to avoid spatial aliasing, the low signal-to-noise ratio in the crystalline environment requires digitalrecording equipment with large dynamic range and high stack-ing fold, and the need for large source-receiver offsets demandssimultaneous recording of hundreds of sensors. Although high-frequency seismic data promise improved resolution, there areimplicit complications that must be overcome in the hard-rockenvironment (Milkereit and Eaton, 1998). Special attentionmust be given to data acquisition and processing if the band-width and resolution of such data are to be preserved. Unfa-vorable near-surface conditions at the source or receiver mayattenuate the high frequencies. Thus, high frequencies of about100 Hz must be generated by the seismic sources, recorded bythe receiver grid, and be preserved throughout the processingsequence.


THE TRILL SITE 3-D SEISMIC EXPERIMENT

Results from 2-D profiling, physical rock property studies,and forward modeling provided strong support for the hypoth-esis that high-density mineralization (for example, a massivesulfide deposit) is detectable by its characteristic scattering re-sponse in a 3-D seismic data set. In the spring of 1995, it was de-cided to conduct the first 3-D seismic survey for deep mineralexploration in Canada. The relatively low-noise backgroundlevels (from man-made infrastructures) led to the selection ofthe Trill area (Figure 1a) for the 3-D seismic feasibility studyto detect and delineate deep massive sulfides. With the avail-ability of a well-defined target at about 1800 m depth and asubstantial borehole data base, enough information is avail-able to evaluate the performance of this new exploration tech-nology. In addition, the 3-D seismic data could provide newinformation about the deep geology and associated mineraldeposits of the Trill structure. The objectives of the 3-D sur-vey at Trill were threefold: (1) to image the steeply dippingcontact between norite and footwall complex, (2) to detect aseismic response of known massive sulfides at depth, and (3)to integrate physical rock property, seismic and drill hole data.

Survey design and field operations

Early 2-D seismic profiles were acquired with vibroseissources along existing roads or trails (Milkereit et al., 1992;1996a). The Trill area is characterized by steep hills, extensiveswamps, and limited access. For 3-D seismic surveys in such aremote area, it is not possible to rely on vibroseis sources. Thus,the 3-D seismic survey had to use small explosive charges. Bore-hole geophysical logging, vertical seismic profiling (VSP), andseismic source strength evaluations were conducted at the site(Milkereit et al., 1997). Extensive tests were required to opti-mize source parameters (i.e., bandwidth and dynamic range)for off-road 3-D seismic exploration programs. Small explo-sive charges (from 0.05 to 1.0 kg) in shallow boreholes wereused as the sources for VSP and noise spread surveys. Theseismic data recorded by stationary surface geophone spreadsand downhole three-component seismometers were analyzedfor bandwidth, dynamic range, and geometrical spreading ofthe signal. Source depth tests were required to evaluate wave-form consistency and to minimize source-generated noise (inparticular shear-wave energy).

The important contact between the norite of the SIC andthe footwall complex dips 30◦–60◦. From surface and boreholedata, it is difficult to define a single strike direction for thestudy area. Massive sulfides are located in an embayment inthe footwall. In map view, the top of the deep mineralization islocated at about 1800 m depth in the center of the survey area.A detailed 3-D forward modeling study was conducted to ad-dress the technological challenges of detecting and delineatingore in a complex geological setting between 1 and 3 km depth.At the same time, a geographic information system (GIS) wasbuilt for the study areas. Special emphasis was placed on theavailability of digital topography, surface geological maps, andborehole data (depth to footwall contact) to conduct a detailedsimulation of the proposed 3-D seismic experiment. In paral-lel, environmental assesssment/impact studies were conducted,and line cutting, drilling, and seismic data acquisition contrac-tors were selected.

The final 3-D survey design was based on detailed 3-D for-ward modeling of the Trill area and operational constraints. TheGIS data base supported the presurvey 3-D modeling of theTrill area through a simple 3-D geological subsurface model.We used the 3-D Born modeling approach to accommodatethe complex shape of the contact between SIC and footwallcomplex, known mineralization at 1800 m depth, and source-receiver geometries used for the actual seismic experiment. Theresulting synthetic data were processed and migrated to obtaina realistic approximation to the anticipated seismic processingsequence. Synthetic models indicated that steep dips and deeptargets required additional shot points at the southeastern cor-ner of the study area. In addition, modeling of the scatteringresponse of deep massive sulfides predicted a unique amplituderesponse (Figure 4). As a result, the design of the seismic dataacquisition program was modified. Additional shot points hadto be placed to compensate for irregular source and receiverline spacing due to swamps, lakes, and the rough topography.The Trill study area with 12 receiver lines and 14 source lines isshown in Figure 5a. The receiver lines are oriented east-west,perpendicular to the grid of shot lines. Details about the ac-quisition parameters are given in Table 1.The recording crewbegan the project in late October 1995. Due to heavy rain-fall, many of the receiver lines became difficult to walk, andwater levels in the swamps and creeks were rising. The sur-veying crews were forced to use hip waders, canoes, and boatsto a much greater extend then anticipated. During start-uptests, temperatures dropped, ground froze, and lakes becameice covered.

Receiver spacings of 30 m were used to give a nominal foldof 30, and marsh phones were used to guarantee good cou-pling under difficult near-surface conditions. Deployment andpickup of the recording equipment was made possible throughthe use of a helicopter. The data were collected using a 2000-channel telemetry acquisition system. It is worth noting that ittook nine days to deploy the recording spread. Production (i.e.,recording 1050 shot points), however, was completed within30 hours. The first snow storm hit the Sudbury area just hoursbefore completion of the equipment pickup, causing frozen ca-bles and geophones.

Table 1. Data acquisition parameters.

Original survey parameters

Survey area approximate by 30 km2

15 × 25 m sub bin size12 receiver lines30 m receiver station spacing300 m receiver line spacing14 source lines50 m source point spacing600 m source line spacing1050 shot points (0.25–0.5 kg in 5–10 m deep holes)

Instrumentation2000-channel, 24-bit telemetry system I/O System IIHigh cut filter: 270/188 HzPre-amp gain: 36 dBGeophone type: Mark L-210 (marsh phone)Geophone array: bunchedThree-component downhole geophone SIE T42 locked at

1070 m depthAll channels liveHelicopter assisted deployment and pick-up of equipment


After collating the data from all of the sources (1050), thedata were sorted into bins with dimensions of 15 × 30 m, defin-ing the ultimate resolution limits of the final migrated image.Note that the bin size was changed to 20 × 20 m during dataprocessing. The final fold diagram is shown in Figure 5b.

In the center of the study area, a three-component seis-mometer was placed in a deep borehole (at 1070 m) andrecorded all shots. The three-component borehole seismome-ter proved to be a valuable tool for quality control and dataediting (Milkereit et al., 1997). For example, small seismicsources in thick, dry overburden do not produce sufficient en-ergy for reflection seismic studies but small charges in bedrockdo. During data acquisition, it was very difficult to observeclear, high-quality reflections. In the hard-rock environment,we were dealing with source-generated noise (strong shearwaves) and low-frequency noise from flowing water. In dataprocessing, bandpass filtering and high stacking fold were re-quired to attenuate these types of noise.

Data processing

The data processing sequence had to take into account highlyvariable overburden conditions (deep swamps, thick eskers,basement outcrop, and rough topography) as well as steeply-dipping geological structures. First results were “blurred,”and it appeared that conventional 3-D seismic data process-ing strategies were not well suited to handle weak seismicsignals (from individual scatterers) in the crystalline crust.A time slice through the data cube is shown in Figure 6a.Various data processing options were evaluated and a ro-bust processing sequence was developed, which consisted ofcomputation of weathering static corrections, deconvolution,time-variant bandpass filtering, and stacking velocity analy-sis. Alternative 3-D data binning and stacking schemes weretested. Once strike and dip were determined, stacking with re-stricted azimuthal coverage was used for stacking velocity anal-ysis. The evaluation of various 3-D stacked sections revealedthat a limited azimuth stack provided the most robust results

FIG. 5. (a) Trill area geology with survey grid. (b) Fold diagram based on actual source and receiver locations.

(sufficient bandwidth and dynamic range). The final processed3-D data cube covers an area of 24 km2 and consists of 200 east-west lines and 301 north-south lines; the lines are separated by20 m. The processing sequence is summarized in the Table 2.A time slice from the final stacked volume is shown in Figure6b. The stacked and depth-migrated 3-D data volume cubes ofthe study area were interpreted using workstations.

3-D seismic data interpretation

The Trill 3-D seismic survey provides new insights intothe deep structure of the Sudbury North Range. Local high-impedance contrasts such as massive sulfides cause scatteringof seismic energy. Directly above the known mineralization,prominent circular and semicircular scattering events are ob-served in the 3-D seismic data. The strongest events occur atabout 0.65 s two-way reflection time (about 1800 m depth). Anin-line section through the 3-D cube shows the asymmetric

Table 2. Processing and interpretation.

Brute stack and parameter tests

GeometryEdit shots and receiversCompute static corrections (datum: 350 m above sea level)Select deconvolution, filtering, and scaling3-D binning (15 × 25 m bins)StackRevise strike and dip estimatesEvaluate 3-D dip moveout (DMO) processing

Final processing sequenceRevise geometry3-D binning (20 × 20 m bins)High-pass filter: 30 HzEvaluate azimuthal coverageStacking velocity analysisStackPoststack deconvolutionOne-pass 3-D phase-shift migration


diffraction-like response of the known mineralization (Fig-ure 7a). The high-impedance contrast between the massivesulfides and the steeply dipping norite causes the character-istic high amplitudes of the reflection response to be shiftedtowards larger offsets in the downdip direction. The strongdiffraction response of the known mineralization covers a dis-tance of about 1000 m, a distance which is considerably largerthan the actual size of the ore body. The prominent circularand semicircular scattering events are best observed in a se-quence of time slices through the 3-D stacked volume. Thestrongest events occur at about 0.6 s two-way reflection time(about 1800 m depth). This sequence of events in Figure 7 iscentered above the known mineralization and can be tracedfrom 590 ms to about 700 ms. The observed seismic data con-firms the forward modeling results shown in Figure 4. The in-terpretation of the scattered energy is complicated by the factthat structures located northeast and southwest of the knownmineralisation interfere at later times. In addition, phase andamplitude of the scattered wavefronts are not constant, causingsevere amplitude-versus-azimuth variations.

Important for the interpretation of the surface seismic data(and its time-to-depth conversion) is the observation that theaverage velocity of the hanging wall (above the target struc-ture) is about 6300 m/s. Migration of the 3-D stacked data vol-ume with its steeply dipping reflections provided robust results.Interpretation of the depth-migrated seismic data must honorthe available borehole data and geophysical logs. For the Trillarea, the data base consists of borehole locations and associ-ated “depth-to-footwall-contacts.” Migrated sections must beused for the interpretation of dipping lithological contacts. Aspredicted by the borehole geophysical logs shown in Figure 2,the lower contact of the SIC and associated footwall topogra-phy are well imaged between 1000 and 2500 m depth (Figure 8).Strike and dip of the footwall contact change laterally withinthe study area. Toward the east, the footwall contact forms alocal embayment structure at about 2000 and 2500 m depth.

FIG. 6. Scattering response as seen by a 3-D seismic time slice from the Trill area. The slice is from 790 ms into the data volume:(a) brute stack, (b) final stack.

In the east, the interpreted sublayer thickness reaches its max-imum. The main results of the Trill 3-D feasibility study aresummarized in Figure 9. The perspective view (from the east)shows the steeply dipping footwall contact, the location of theknown mineralization, and the associated scattering response( a time slice through the stacked volume at 612 ms) caused bythe known mineralization.

FOLLOW-UP STUDIES

The 3-D seismic experiment confirmed that in a geologi-cal setting such as the Sudbury North Range massive sulfidebodies cause a characteristic seismic scattering response. Thisprovides an excellent basis for the direct detection of massivesulfides by seismic methods. In addition, the seismic surveyprovided new information about strike and dip of the footwallcontact. The eastward extension of the footwall embaymentstructure led to a major revision of the geological explorationmodel. Toward the southeast, the thickness of the sublayer de-creases, and “reflectivity” in the footwall of the SIC increases.These deeper reflections within the footwall complex couldnot be explained by the available geological and geophysi-cal logs, and required further investigation. In order to bet-ter evaluate the performance of the 3-D seismic profiling inthe hard-rock environment, it was decided to initate a com-prehensive calibration and ground-truthing program to furtherguide the development of this new exploration technology. Themain objective of the follow-up project was to evaluate theresolving power (for example, accuracy of depth and dip es-timates in an area without borehole control) of 3-D seismicimages.

A prominent target from the eastern part of the study areais shown in Figure 10. The in-line section through the stackedcube shows prominent scattering events A and B (Figure 10a).The depth of the scatterers is well defined by the apex of thediffraction events (assuming an average velocity of 6300 m/s


for the hanging wall). The horizontal slice at 2040 m throughthe depth-migrated volume highlights the location of the fo-cused scatterer A. The predrilling interpretation of the mi-grated volume is shown in Figure 11. Depth and dip of thegranophyre (G), quartz-gabbro (QG), and norite (N) units ofthe Sudbury Igneous Complex, the sublayer (SL), and footwallcomplex (FW) are indicated. In the migrated data, events Aand B are characterized by local bright amplitudes at about2050 and 2400 m depth. The proposed borehole location is in-dicated. Drilling began in 1997 and verified the dip and depthestimates for the SIC and footwall contact. In addition, twolocal density anomalies were encountered at approximately

FIG. 7. Seismic response of a known mineralization located at about 1800 m depth. Cross-line section (a) andsequence of time slices through the stacked 3-D data cube (b–d). Top of the known mineralization is locatedat about 1800 m depth (about 590 ms, reflection time assuming an average velocity of 6300 m/s). At 590 ms,we observe a weak circular reflection event. The event expands and, at 640 ms, a high-amplitude semicircularreflection develop towards the east. Despite interference from structure and other events, the semicircular eventcontinues to grow (time slice at 670 ms). Arrows indicate diffraction. Compare with the synthetic response inFigure 4.

2050 and 2400 m depth. The density log (derived from coresamples) is shown in Figure 11; magnite-rich units with highdensities are responsible for the scattering response shown inFigure 10. As indicated by the velocity-density relationship inFigure 3, magnetite-rich units are characterized by high seismicimpedance values. These will tend to produce seismic scatter-ing/reflection anomalies at least as bright as massive sulfide orebodies.

In summary, postseismic drilling verified the depth and dipestimates of the Trill 3-D seismic data interpretation. How-ever, the interpretation of local seismic scatterering eventsis not free of pitfalls. The 3-D seismic data delineated local


scattering events and provided accurate depth estimates. Scat-tering shown in Figure 10 was caused by high densities. Atpresent, we cannot distinguish between scattering caused bymagnetite-rich and sulfide-rich units.

CONCLUSIONS

The 3-D seismic survey in Sudbury demonstrates for thefirst time that large massive sulfides generate a characteris-tic seismic reflection response. By adjusting acquisition andprocessing parameters, high-frequency seismic reflection pro-filing techniques can be tailored to (1) image important litho-

FIG. 8. Depth-migrated section with interpretation overlay(sublayer and footwall complex). The Sudbury Igneous Com-plex above the sublayer reflection is seismically transparent(compare with borehole logs in Figure 2).

FIG. 9. Composite perspective view of interpreted footwallcontact (green) derived from migrated data, known mineral-ization at 1800 m depth (red), and scattering event (indicatedby arrow) caused by the mineralization (time slice at 612 ms).

logical contacts and key geological structures such as embay-ments, and (2) identify and delineate deeply buried, large mas-sive sulfide deposits in the crystalline crust. The effective useof this new exploration technique requires an integrated ap-proach incorporating detailed knowledge of the geologicalsetting, comprehensive physical rock-property studies, state-of-the-art forward modeling techniques, and high-resolutionseismic data sets. The 3-D reflection seismic method can sup-port deeper exploration projects in existing and prospectivesites as well as prolonging the life of established mines in theSudbury basin. In addition, this new methodology could reju-venate deep exploration projects in mature base-metal miningcamps.

ACKNOWLEDGMENTS

Inco Ltd., Falconbridge Ltd., and the Geological Survey ofCanada funded the 3-D feasibility study. The 3-D surface seis-mic data were acquired by Solid State Geophysical of Calgaryand processed in-house.

FIG. 10. (a) In-line seismic section with scattering events Aand B. The apex of event A is located at 0.64 s two-way time(approximately 2050 m depth at 6300 m/s). (b) Migrated section(depth slice at 2040 m) with reflection from footwall contact(FWC) and local amplitude anomaly A.


FIG. 11. Depth-migrated section with borehole location and density log. Magnetite-rich units (A, B) with highdensities are responsible for the scattering response shown in Figure 10. G = granophyre, QG = quartz-gabbro,N = norite, SL = sublayer, and FW = footwall complex.

REFERENCES

Boerner, D. E., Milkereit, B., and Naldrett, A. J., 1994, Introductionto the Special Section on the Lithoprobe Sudbury: Geophys. Res.Lett., 21, 919–922.

Eaton, D. W., 1996, BMOD3D: A program for three-dimesional seis-mic modelling using the Born approximation: Geol. Surv. Can. OpenFile Report 3357.

Milkereit, B., Berrer, E. K., Watts, A., and Roberts, B., 1997, Devel-opment of 3-D seismic exploration technology for Ni-Cu deposits,Sudbury basin, in Grubbins, A., Ed., Proceedings of Exploration 97,439–448.

Milkereit, B., and Eaton, D., 1998, Imaging and interpreting the shallowcrystalline crust: Tectonophysics, 286, 5–18.

Milkereit, B., Eaton, D., and Berrer, E., 1996b, Towards 3-D seismic

exploration technology for the crystalline crust: 66th Ann. Internat.Mtg., Soc. Expl. Geophys., Expanded Abstracts, 638–641.

Milkereit, B., Eaton, D., Wu, J., Perron, G., Salisbury, M., Berrer, E.,and Morrison, G., 1996a, Seismic imaging of massive sulfide deposits,Part 2: Reflection seismic profiling: Econ. Geol., 91, 829–834.

Milkereit, B., Green, A., and Sudbury Working Group, 1992, Deepgeometry of the Sudbury structure from seismic reflection profiling:Geology, 20, 807–811.

Morrison, G. G., 1984, Morphological features of the Sudbury struc-ture in relation to an impact origin, in Pye, E. G., Naldrett, A. J.,and Giblin, P. E., Eds., Geology and ore deposits of the Sudburystructure: Ontario Geol. Surv., Special Volume 1, 513–520.

Salisbury, M. H., Milkereit, B., and Bleeker, W., 1996, Seismic imagingof massive sulfide deposits, Part 1: Rock properties, Econ. Geol., 91,821–828.


Crosshole seismic imaging for sulfide orebodydelineation near Sudbury, Ontario, Canada

Joe Wong∗

ABSTRACT

Crosshole seismic instrumentation based on a piezo-electric source and hydrophone detectors were usedto gather seismograms between boreholes at theMcConnell orebody near Sudbury, Ontario. High-frequency seismograms were recorded across rock sec-tions 50 to 100 m wide containing a continuous zoneof massive sulfide ore. First-arrival traveltimes obtainedfrom a detailed scan were used to create a P-wave ve-locity tomogram that clearly delineated the ore zone.Refraction ray tracing on a discrete layer model con-firmed the main features of the tomogram. The surveydemonstrated that it is possible to conduct cost-effective,high-resolution crosshole seismic surveys to delineateore bodies on a scale useful for planning mining oper-ations.

INTRODUCTION

There is growing interest in using geophysics to assist in plan-ning mining operations. In particular, borehole-to-boreholemethods are being recognized as having exceptional potentialin this regard. By using crosshole geophysics to provide greatergeometrical detail than is available from traditional delineationvia pattern drilling, mining of an economic deposit can be de-signed or modified to minimize costs (Williams, 1996). Thispaper presents a test survey in which high-resolution cross-hole seismic tomography is demonstrated to be effective fororebody delineation. Previous applications of seismic tomog-raphy used sparkers (Greenhalgh and Mason, 1997) and mi-croexplosions (Duncan et al., 1989) as seismic sources. I de-scribe a nondestructive piezoceramic vibrator source drivenby a pseudorandom waveform.

The McConnell deposit, owned by INCO Ltd., is locatednear Sudbury, Ontario, Canada. It has been used for testingvarious downhole and crosshole geophysical techniques. Mengand McGaughey (1996) applied crosshole seismic reflection to

Manuscript received by the Editor January 26, 1999; revised manuscript received April 18, 2000.∗JODEX Applied Geoscience Ltd., 9715 Horton Road S.W., Unit 101, Calgary, Alberta T2V 2X5, Canada. E-mail: jjodex @telusplanet.net.c© 2000 Society of Exploration Geophysicists. All rights reserved.

image the deposit. They used blasting caps as downhole seismicsources and recorded seismic frequencies in the 1- to 5-kHzrange. Livelybrooks et al. (1996) and Fullagar et al. (1996)describe crosshole electromagnetic surveying to map the orebody. The technique, sometimes referred to as the radio imag-ing method (RIM), uses electric dipoles operating at frequen-cies of 500 kHz and 5 MHz.

The target at the McConnell site is a near-surface, steeplydipping massive sulfide orebody residing in a host rock con-sisting chiefly of amphibolite and metasediments. A number ofBQ-sized (60 mm ID) boreholes have been drilled through theore zone (Figure 1). Mueller et al. (1997) give more details con-cerning the geological units intersected by the boreholes. Bore-hole velocity logging has indicated the P-wave velocities of thehost rock range from 5.8 to 6.5 km/s. In contrast, the velocitieswithin the massive sulfide zone are at least 20% lower—in therange of 4.2 to 4.6 km/s. The situation is almost ideal for eval-uating crosshole seismic methods for possible use to supportplanning mining activities.

METHODS

Instrumentation

The CORRSEIS borehole seismic system, shown schemat-ically on Figure 2, is designed for rapid and efficient acquisi-tion of high-frequency seismograms between boreholes. Thesystem includes a controlled piezoelectric vibrator and mul-tiple (8 or 16) high-sensitivity hydrophone detectors. Similarequipment has been described by Wong et al. (1983, 1987) andHarris et al. (1995). In the present case, the source was about53 mm in diameter and 75 cm in length. Each hydrophone wasabout 50 mm in diameter and 10 cm in length. Because boththe source and the detectors are coupled to the rock by wa-ter in the boreholes, the recorded seismic energy is primarilyP-waves. The dominant seismic frequencies generated by thesource are about 3.3 kHz, resulting in wavelengths of about2.0 m in the host rock with P-wave velocities of 6.5 km/s andabout 1.4 m in the massive sulfides with P-wave velocities ofabout 4.2 km/s. These wavelengths indicate the resolution that

1900

Crosshole Seismic Imaging 1901

can be expected from data acquired with the CORRSEIS sys-tem. The output power of the vibrator is adjustable and is nom-inally on the order of 10 to 30 W. The total source energies asso-ciated with field seismograms are typically in the range of 0.1 to2.0 kJ.

There are several practical advantages in using a controllablepiezoelectric vibrator as the source for crosshole seismic scan-ning. Once deployed with electrical power in a borehole, thevibrator can be operated for long periods of time without need-ing to be returned to the surface. The piezoelectric source ismuch more repeatable and safer than microexplosives or blast-ing caps, and there is very little risk of damage to boreholes.If 16 or more hydrophones are used in the detector array andthe equipment is configured with automatic winches, acquisi-tion rates >2500 crosshole seismic traces per hour are possibleunder low-noise conditions in low-loss rock.

Crosscorrelation used for data acquisition

The vibrator in the CORRSEIS system is driven by a pseu-dorandom binary sequence (PRBS) in continuous cycles, andimpulsive seismograms are obtained from the detected vi-brations by crosscorrelation with the PRBS pilot signal. Thelength of the PRBS used in acquisition is 2047, but the ba-sic concepts of crosscorrelation can be illustrated using thePRBS of length 63 shown on Figure 3a. When the vibratoris driven by a high-voltage replica of the sequence, its sys-tem response modifies the original sequence by convolving itwith a time-domain filter equal to, for example, the seismicsource wavelet shown on Figure 3b. The resulting vibration

FIG. 1. Cross-section of the McConnell ore body with ex-ploratory boreholes.

detected at some distance from the source is the filtered PRBSdelayed by the seismic traveltime between the source and thereceiver. The detected signal, shown on Figure 3c, is digitized,stacked, and then crosscorrelated with the original PRBS. Theresult is the trace in Figure 3d, which is the source wavelet(with some high-cut filtering) delayed by the seismic travel-time. This is the equivalent of a seismic trace from an impulsivesource.

The defining characteristic of a pseudorandom or pseudo-noise sequence is that its autocorrelation simulates a pure deltafunction in time domain. The autocorrelation of the PRBS oflength 63, shown in Figure 3e, has a triangular shape with noside lobes and is a good approximation to the delta function.The height of the triangle depends on the sequence length N ,being proportional to the sum of N individual pulses of unitamplitude. The absence of side lobes is important because it re-sults in cleaner seismograms after crosscorrelation. The cross-correlation method enhances S/N ratios by a factor equal tothe square root of the sequence length N . Additional signalenhancement is achieved by synchronously summing or stack-ing 20–500 repeated waveforms. In many cases, using long se-quences in conjunction with waveform stacking makes it possi-ble to record good-quality seismograms even in the presence ofdrilling noise. Hurley (1983) gives a more detailed discussionon PRBS.

FIG. 2. Schematic of the CORRSEIS borehole seismic system.

1902 Wong

RESULTS

Field seismograms

Seismograms were recorded with detectors in borehole (BH)78 929 and the source in BH-78 930 and BH-80 555. Sampleseismograms between BH-78 929 and BH-80 555 are shown onFigure 4 as a common detector gather. Each recorded seis-mogram consists of 512 or 1024 sixteen-bit samples with asampling interval of 50 µs. The columns of numbers to theright of the figure under the headings TX and RX are thesource and detectors depths in meters. The S/N ratios ofthe traces are very good, and the dominant frequencies arein the 3- to 4-kHz range. First-arrival times fall in the range of10 to 17 µs. These data indicate that piezoceramic-based equip-ment produces excellent crosshole seismograms for source-detector distances of over 100 m in the rocks at the McConnellsite.

For tomographic imaging, a complete seismic scan of therock panel between BH-78 929 and BH-78 930 was done. Thedetectors were located in BH-78 929 at depths from 30 to

FIG. 3. (a) PRBS of length 63. (b) Time-domain filter, i.e.,seismic wavelet. (c) Convolution of delayed PRBS and seis-mic wavelet. (d) Cross-correlation of trace (c) with the PRBS,recovering a delayed filtered version of the seismic wavelet.(e) Autocorrelation of PRBS simulating a delta function.

140 m on 2-m intervals. The source was located in BH-78 930at depths from 30 to 170 m on 0.62-m intervals. The scanningpattern is depicted schematically on Figure 5, where seismicraypaths joining source and detector positions are representedby straight lines (true raypaths would curve or bend accord-ing to Snell’s Law). The entire scan consisted of over 4200raypaths, but only every fifth raypath has been drawn on Fig-ure 5. Examples of seismograms from this scan are shown onFigure 8.

Tomographic imaging

First-arrival times for the scan of Figure 5 were picked us-ing an automatic routine and were checked and edited visuallyon a computer display. These first-arrival times were used toproduce a P-wave velocity tomogram via a straight-ray back-projection algorithm (Peterson et al., 1985). In essence, thealgorithm assigns seismic velocity values to an array of rectan-gular pixels overlying the rock section between the source anddetector boreholes. Seismic traveltimes for direct P-wave ar-rivals are calculated and compared with the observed times forall source and detector positions in the scan. The velocity val-ues are then automatically adjusted to reduce the differencesor residuals between observed and calculated arrival times.This is repeated until the residuals are reduced to an accept-able level, at which point we assume that the array of velocityvalues within the pixels is a good representation of the truevelocity distribution in the scanned rock section between theboreholes.

For the panel between BH-78 929 and BH-78 930 at theMcConnell site, the rms value of the 4202 observed arrival timeswas about 9.44 ms. Pixel dimensions used for tomographicimaging were 1.5 m high by 2.5 m wide. After 10 iterationsof the imaging algorithm, the rms value of the residual arrivaltimes was reduced to about 0.38 ms. The resulting P-wave ve-locity tomogram is shown on Figure 6. The ore zone is clearlyoutlined by the low-velocity (4.0–4.5 km/s) zone residing in thehigher velocity (5.9–6.5 km/s) host rock.

Tomographic analysis based on the straight-ray path assump-tion provides a rapid interpretation of the first-arrival times ina seismic scan like that shown in Figure 4. However, the ac-tual raypaths in a geological medium with large variations inseismic velocities will inevitably curve or bend from refraction.Therefore, while the gross aspects of the tomogram in Figure 6are likely to be reliable, the details may be considerably differ-ent from the actual geology. Also, velocity tomograms can bedegraded by the presence of false features or artifacts. Thesearise during image reconstruction because of the large numberof pixels and degrees of freedom if the process is not carefullyconstrained. In light of these shortcomings in the tomographicanalysis, the structure represented by the tomogram should bechecked against other, independent models.

Refraction ray tracing

One way to accomplish this is to construct a discrete layermodel using the tomogram (and other information such asborehole velocity logs) as a guide. The velocities are constantwithin each layer. The layers may vary in thickness and dip,and boundaries between layers may be curved. Refraction


FIG. 4. Crosshole seismograms between boreholes 78 929 and 80 555. First-arrival times are between 10 and 17 µs; source-detectorseparations are approximately 60–100 m. The columns of numbers to the right of the gathers are, respectively, the source anddetector depths in meters.

according to Snell’s Law at the seismic boundaries is used totrace seismic raypaths from one borehole to another.

On Figure 7, raypaths traced across one such model areshown for two common detector gathers with the detector in

the left-hand borehole. The ore zone is represented by the yel-low layer. Many raypaths are severely refracted at the bound-ary between the high-velocity host rock and the low-velocitysulfide ore body. In some cases, the rays are reversed in the

1904 Wong

direction normal to the rock–ore interface by postcritical (to-tal) reflection. Total reflection may result in shadow zones inwhich the seismograms for particular source–detector loca-tions have very weak first-arrival amplitudes.

As part of the ray tracing, the arrival time for each ray can becalculated. By adjusting the seismic velocities and the bound-ary positions and dips in the model, the calculated arrival timescan be made to closely fit observed arrival times. On Figure 8,two common detector gathers are displayed on an expandedtime scale. Arrival times calculated for corresponding raypathstraced across the velocity model on Figure 7 are plotted as redbars. The agreement between the observed and calculated first-arrival times is good, indicating that the velocity model is areasonable representation of the field data and the underlyinggeology. It is relatively easy to fit first-arrival times for individ-ual gathers separately. The difficulty lies in obtaining a singlemodel that closely match both (or more) gathers equally well.As more gathers of field seismograms are included for compar-ison, more layers may need to be added to obtain a good globalfit. In this event, the process must be automated in a nonlinearoptimization scheme (i.e., inversion).

An alternative to straight-ray imaging followed by refractionmodeling to confirm the tomogram is tomographic imagingusing curved raypaths. For example, Bishop et al. (1985) andBregman et al. (1989) describe imaging algorithms that ana-lytically trace refracted raypaths across triangular cells withinwhich velocities are linear. Jackson and Tweeton (1994) havepublished Migratom software, which creates tomograms using

FIG. 5. Schematic showing the raypaths of a detailed seismicscan between boreholes 78 929 and 78 930. In the completescan, there are over 4200 raypaths; only every fifth ray is shownhere.

refracted raypaths obtained by Huygens wavefront expansion.Tomographic image reconstruction using refracted raypaths isa nonlinear process. As such, it is slower than straight-ray imag-ing and is more prone to problems of nonuniqueness and imageartifacts.


The successful imaging of the McConnell ore body usingfirst-arrival times to create a P-wave velocity tomogram is at-tributable to the large, massive nature of the deposit and its rel-atively simple geometry. The velocity contrast between the oreand the host rock is high, and the ore zone is a single layer that isrelatively thick compared to both the seismic wavelengths usedand the hole-to-hole distances. If the ore zone had been in theform of thin stringers, first-arrival times would have been lessdiagnostic of the low-velocity sulfides since the first-arrivingraypaths would reside mostly in the high-velocity host rock.Also, if the velocity distribution had been more complex, ei-ther with much folding and faulting in the plane of the panel orwith significant out-of-plane 3-D structure, the velocity imagemight not have represented the true geology as well.

Nevertheless, in many situations, crosshole seismic imag-ing using high-resolution seismic data collected with a

FIG. 6. P-wave velocity tomogram for the scan of Figure 5.The tomogram was produced from observed first-arrival timesusing a straight-ray iterative back-projection algorithm.


FIG. 7. Refraction ray tracing between boreholes 78 929 and 78 930 across a discrete layer model. Common detector gather withdetector (a) below the orebody and (b) above the orebody.

piezoceramic source and hydrophone detectors appears to bea practical method for delineating ore bodies. Seismic imagingcomplements the information gained from pattern drilling byproviding more details about the geometric aspects of an eco-nomic deposit. The additional information gained by seismicimaging may assist in optimizing mining procedures to reducecosts.

The most critical factor in determining whether crossholeseismic imaging will become widely accepted by people respon-sible for planning mining operations is its cost effectiveness.In actual practice and for routine application, costs associatedwith field time for data acquisition are likely to be a major con-sideration in deciding whether the crosshole seismic methodis used. It is therefore important to have a technology that al-lows for the rapid recording of high-quality, high-resolutioncrosshole seismograms. The experience at the McConnell sitesupports the contention that piezoceramic-based equipmentsimilar to that described in this paper should have the neces-sary resolution, range, and operational efficiency to meet thepractical requirements.

ACKNOWLEDGMENTS

I thank INCO Technical Services Ltd. for providing accessto the boreholes at the McConnell site and especially GlennMcDowell of INCO Ltd. for logistical support during the fieldwork. I also thank Greg Turner and Lawrence Gochioco fortheir constructive review of this paper.

REFERENCES

Bishop, T. N., Bube, K. P., Cutler, R. T., Langan, R. T., Love, P. L.,Resnik, J. R., Shuey, R. T., Spindler, D. A., and Wyld, H. W., 1985,Tomographic determination of velocity and depth in laterally vary-ing media: Geophysics, 50, 903–923.

Bregman, N. D., Bailey, R. C., and Chapman, C. H., 1989, Crossholeseismic tomography: Geophysics, 54, 200–215.

Duncan, G., Downey, M., Leung, L., and Harman, P., 1989, The de-velopment of crosshole seismic techniques and case studies: Expl.Geophys., 20, 127–130.

Fullagar, P. K., Zhang, P., Yu, Y., and Bertrand, M. J., 1996, Applica-tion of radio-frequency tomography to delineation of nickel sulfidedeposits in the Sudbury basin: 66th Ann. Internat. Mtg., Soc. Expl.Geophys., Expanded Abstracts, 2065–2068.

Greenhalgh, S., and Mason, I., 1997, Seismic imaging with applicationto mine layout and development: 4th Decennial Internat. Conf. onMin. Expl., Exploration 97, Proceedings, 585–598.

Harris, J. M., Hoeksma, R. C., Langan, R. T., Van Schaak, M., Lazaratos,S. K., and Rector, J. W., 1995, High-resolution crosswell imagingof a West Texas carbonate reservoir: Part I—Project summary andinterpretation: Geophysics, 60, 667–681.

Hurley, P. M., 1983, The development and evaluation of a crossholeseismic system for crystalline rock environments: M.S. thesis, Univ.of Toronto.

Jackson, M. J., and Tweeton, D. R., 1994, MIGRATOM—Geophysicaltomography using wavefront migration and fuzzy constraints U.S.Dept of Interior Bureau of Mines Report RI 9497.

Livelybrooks, D., Chouteau, M., Zhang, P., Stevens, K., and Fullagar, P.,1996, Borehole radar and radio imaging to delineate the McConnellore body near Sudbury, Ontario: 66th Ann. Internat. Mtg., Soc. Expl.Geophys., Expanded Abstracts, 2060–2063.

Meng, F., and McGaughey, J., 1996, Ore body hunting with crosswellimaging methods: 66th Ann. Internat. Mtg., Soc. Expl. Geophys.,Expanded Abstracts, 626–629.

Mueller, E. L., Morris, W. A., Killeen, P. G., and Balch, S., 1997,Combined 3-D interpretations of airborne, surface, and borehole

1906 Wong

FIG. 8. Arrival times calculated from the rays in Figure 7 plotted on the corresponding field seismograms. The calculated arrivaltimes compare very well with the first-arrival times on the recorded traces.


vector magnetics at the McConnell nickel deposit: 4th DecennialInternat. Conf. on Min. Expl., Exploration 97, Proceedings, 657–665.

Peterson, J. E., Paulsson, B. N. P., and McEvilly, T. V., 1985, Appli-cation of algebraic reconstruction techniques to crosshole seismicdata: Geophysics, 50, 1566–1580.

Williams, P., 1996, Using geophysics in underground hard rock mining:

A question of value and vision: 66th Ann. Internat. Mtg., Soc. Expl.Geophys., Expanded Abstracts, 2046–2047.

Wong, J., Bregman, N., Hurley, P., and West, G. F., 1987, Crossholeseismic scanning and tomography: The Leading Edge, 6, 36–41.

Wong, J., Hurley, P., and West, G. F., 1983, Crosshole seismology andseismic imaging in crystalline rocks: Geophys. Res. Lett., 10, 686–689.


In-mine seismic delineation of mineralization and rock structure

Stewart A. Greenhalgh∗, Iain M. Mason‡, and Cvetan Sinadinovski∗∗

ABSTRACT

Significant progress has been made towards the goalof generating detailed seismic images as an aid to mineplanning and exploration at the Kambalda nickel minesof Western Australia. Crosshole and vertical-seismic-profiling instrumentation, including a slimline multi-element hydrophone array, three-component geophonesensors, and a multishot detonator sound source, havebeen developed along with special seismic imaging soft-ware to map rock structure.

Seismic trials at the Hunt underground mine estab-lished that high frequency (>1 kHz) signals can be prop-agated over distances of tens of meters. Tomographicas well as novel 3-D multicomponent reflection imagingprocedures have been applied to the data to produceuseful pictures of the ore-stope geometry and host rock.Tomogram interpretation remains problematic because

velocity changes not only relate to differing rock typesand/or the presence of mineralisation, but can also becaused by alteration/weathering and other rock condi-tion variations. Ultrasonic measurements on rock coresamples help in assigning velocity values to lithology,but geological assessment of tomograms remains am-biguous. Reflection imaging is complicated by the pres-ence of strong tube-wave to body-wave mode conversionevents present in the records, which obscure the weakreflection signatures. Three-dimensional reflection dataprocessing, especially three-component analysis, is timeconsuming and difficult to perform. Notwithstanding thedifficulties, the seismic migrations at the Hunt mine showa striking correlation with the known geology. Combinedseismic and radar surveying from available undergroundboreholes and mine drivages is probably needed in thefuture to more confidently delineate mineralisation.

INTRODUCTIONThere are potentially great economic benefits to be obtained

from an ability to seismically probe ahead of a mine face,or to image with a reasonable clarity a volume of rock sur-rounding an exploration or underground development bore-hole. The sort of geological information that a high-frequency(200–2000 Hz) seismic technique can, in principle, provide in-cludes existence of ore, changes of rock type, offset of mineral-ization, location of structures such as faults and troughs, extentof shear zones, etc. The seismic technique offers good resolvingpower (a few meters) coupled with a sufficient degree of pen-etration (probing distance tens to hundreds of meters), com-pared to other geophysical techniques like ground-penetratingradar, which suffers from severe attenuation.

Until recently, relatively few successful hard-rock seismicsurveys had been carried out in metalliferous mining areas.Surface reflection surveys for mapping mineralization havebeen reported by Nelson (1984); Gal’perin (1984); Dahle et al.

Manuscript received by the Editor April 20, 1999; revised manuscript received May 15, 2000.∗The University of Adelaide, Dept. of Geology and Geophysics, Adelaide, South Australia 5005, Australia. E-mail: [email protected].‡University of Sydney, School of Geoscience, Sydney, New South Wales 2006, Australia. E-mail: [email protected].∗∗Australian Geological Survey Organisation, GPO Box 378, Canberra, ACT 0206, Australia. E-mail: [email protected]© 2000 Society of Exploration Geophysicists. All rights reserved.

(1985), Spencer et al. (1993); Wright et al. (1994), Adam et al.(1996), and Pretorius et al. (1997). Underground reflection andtransmission measurements in hard rock mines and crossholetomography surveys in crystalline rock have been carried outin Germany (Schmidt, 1959; Wachsmuth and Schmidt, 1962),southern Africa (Reid et al., 1979; Mutyorauta, 1987; Carneiroand Gendzwill, 1996), the U.S.A. (Ruskey, 1981; Peterson et al.,1985; Freidel et al., 1995), Canada (Wong et al., 1984; Younget al., 1989; Gendzwill and Brehm, 1993), Sweden (Gustavsonet al., 1984), and Australia (Harman et al., 1987; Duncan et al.,1989; Sinadinovski et al., 1995; Cao and Greenhalgh, 1997;Greenhalgh and Mason, 1997).

In the 1990s, a major research program was initiated byWestern Mining Corporation at Kambalda, Western Australia,to develop high-resolution seismic and radar techniques toaid mining of and exploration for nickel sulfide orebodies.The economic justification is improved cost-effectiveness (andsafety) of underground development, orebody delineation, and

1908

In-Mine Seismic Delineation 1909

surface exploration. This paper is concerned with the seismicwork done and imaging results obtained at just one mine site us-ing both seismic tomography and seismic reflection techniques.The purpose was to assess the usefulness of underground (in-mine) methods in mapping stope geometry and hard rock ge-ological structure. Instrumentation developments and resultsfrom other sites are described by Cao and Greenhalgh (1997),Bierbaum and Greenhalgh (1997), Greenhalgh and Mason(1997), and Greenhalgh and Bierbaum (2000).

GEOLOGICAL AND GEOPHYSICAL TARGETS

The Kambalda nickel sulfide environment

The Kambalda region is situated within the south-centralpart of the Norseman-Wiluna greestone belt, in the EasternGoldfields province of the Achaean Yilgarn craton. The townof Kambalda is located 560 km east of Perth.

The volcanic sedimentary sequence at Kambalda, with itsenclosed magmatic nickel sulfide deposits, has undergone de-formation, metamorphism, intrusive activity, and late stagegold mineralization (Cowden and Roberts, 1990). Many rocksin the region have undergone upper greenschist to loweramphibolite-facies metamorphism. There has been a series ofintrusions of mafic to felsic stocks, dikes, and sills.

The nickel ores at Kambalda occur in 24 shoot complexes,each containing multiple ore surfaces (pods) caused by struc-tural dislocation of a few, grouped, north-northwest trendingribbonlike orebodies. Most ore occurs at the base of the low-est ultramafic flow, in elongate, shallow (100 × 1500 × 10 m)troughs, which are thought to have been original volcanic de-pressions at the base of the komatiite lava. The ore is ap-proached from the rear through stopes which lead off drivagesdriven through the Lunnon basalt. The contact zone undulates.The morphology of the ore zone is highly variable with massivesulfide layers at the base being irregular in thickness (generallyless than 5 m). Matrix and disseminated ore layers are moreregular in distribution and generally comprise 60–80% of thetotal thickness of the sulfide zone.

Velocities and densities of ores and host rocks

Refraction surveying and downhole velocity measurementswere carried out at Kambalda in the 1970s to determinewhether major rock boundaries (which control mineralization)could be mapped seismically (C. Porter, personal communica-tion, 1978). Five refraction spreads established that the P-wavevelocity of the ultramafics (5600–6500 m/s) is significantly lowerthan that of the hanging-wall basalt (6000–6500 m/s) and thatof the footwall basalt (6800 m/s). Four well-shoot experimentsgave slightly lower velocities (ultramafics, 4600–5400 m/s; foot-wall basalt, 6300–6800 m/s; porphyry intrusion, 4600 m/s). Notethat the ultramafics (comprising talc carbonate, talc chlorite,and amphibole-chlorite rocks) have been serpentinised and,therefore, exhibit lower velocities that one would expect forsuch rocks (which in an unaltered state show higher velocitythan mafic rocks).

Sonic logging carried out in just one hole in the late 1980sgave an average velocity for basalt of 6500 m/s, which droppedto 5500 m/s within a 2-m brecciated zone. The correspondingdensities were 2.9 and 2.65 g/cm3.

At the beginning of the new seismic research program, ad-ditional elastic property data were obtained. In-mine refrac-tion measurements at the Foster mine yielded high seismicwavespeeds for basalt: 6700 m/s for P-waves and 4000 m/sfor S-waves. Laboratory determinations of ultrasonic velocityand density were made for several hundred rock core sam-ples collected from various mine sites. An analysis is given byGreenhalgh and Mason (1997), who plotted velocity versusdensity for the various rock and ore samples. Compressionalwave velocity shows a clear inverse relationship with ore gradeor density. The ultramafics gave an average P-wave velocityof 5500 m/s and an average density of 2.87 g/cm3. The maficsamples showed slightly higher velocity (6050 m/s) and den-sity (2.93 g/cm3). Massive ore was characterised by low veloc-ity (4900 m/s) and high density (4.3 g/cm3). For matrix ore,the velocity was somewhat higher (5200 m/s) and the densitylower (3.52 g/cm3). Plane-wave reflection coefficients calcu-lated from these physical property data are marginal (<0.05)for disseminated-matrix ore/rock boundaries, but detectablefor the ultramafic/mafic boundary and the massive ore/maficand massive ore/ultramafic boundaries (0.08–0.15).

UNDERGROUND SEISMIC EXPERIMENTS

The seismic experiment was carried out at the Hunt mine,D zone decline, 16 level, below the B05 ore surface. A com-panion borehole radar survey was conducted at the same site.The northern portion of the site had also been the test area fora previous radar profiling experiment in which the bottom ofthe ore surface was partially imaged from a single scan alongthe mine decline. This experiment suffered from electromag-netic (EM) reverberation in the tunnel (noise which obscuredreflected signals) and lack of dimensionality. With only two di-mensions (time and horizontal distance along the scan line),only target range and horizontal position can be recovered.Angular elevation to the target requires that radar data beacquired along at least two directions—the decline as well asone or more in-mine boreholes. Similar considerations applyto seismic imaging.

The only previous seismic work attempted at Kambalda hadbeen at the Foster mine. In that work triaxial geophones weremounted along the decline wall, and shots fired in shallow holesdrilled into the tunnel wall. The survey clearly showed that itwas possible to generate high frequency (>1 kHz) signals andthat they propagated over significant distance (>100 m), butattempts at imaging the geology were frustrated by receiver-mount resonance problems, insufficient receiver array aper-ture, spatial aliasing, poor angular coverage, and the lack ofany well-defined target (the orebody had been mined out). TheHunt mine seismic experiment was thus designed to overcomemany of the Foster mine seismic survey limitations by operat-ing primarily in boreholes at a close shot and receiver spac-ing, using both crosshole and vertical-seismic-profiling (VSP)recording geometries. The anchors to the triaxial geophonesused along the decline were permanently grouted into the rockto ensure good coupling.

The Hunt seismic experiment was the first undergroundtest of the new borehole hydrophone streamer and the newexplosive borehole seismic source (Greenhalgh and Mason,1997). Two transportation/navigation systems for boreholetransducer deployment, (a pushrod system and a hydraulic

1910 Greenhalgh et al.

injection system) were also tested during this experiment. Inthis paper, we document the technical aspects of the work, aswell as present as results on the images obtained.

The test site

Figure 1 is a plan view of the test site. It shows the locationsof seismic boreholes DDH 16-18 and DDH 16-20 in relationto the D zone decline at the 16 level, and the overlying stopes.The stations G1–G8 are the collar positions of the triaxial geo-phones along the tunnel as well as being hammer impact points(see next section). The boreholes 16-18 and 16-20 were drilleddownwards at an angle of 30◦ from the horizontal (so as to holdwater) to a depth of 45 m. The distance between borehole col-lars is approximately 55 m. The volume of rock to be imagedwas largely bounded by the decline, the two boreholes, and hor-

FIG. 1. Seismic test site at the Hunt mine showing source-receiver boreholes, geophone array along decline, andstope geometry 20 m above decline.

izontal planes located 50 m above and below the level of thedecline. The stopes of the B05 ore surface are a height of about20 m above the borehole collars. This can be seen in Figure 2which is a 3-D perspective view of the site. The additional pairof upward-inclined boreholes shown, DDH 16-19 and DDH16-08, were used for the radar experiment. Most of the ore inthe vicinity of the seismic/radar test site has been mined out.The 16 level represents the deepest development of the mine.Extraction is continuing to the immediate south of the surveyarea. In fact, mining activity dictated that the geophysical workbe undertaken on the afternoon and night shifts, when vehicleswere not operating in the decline.

Figure 3 gives two geological cross-sections in the verticalplanes of boreholes DDH 16-18 and DDH 16-20. The originalore shoot (premined) is about 2 m thick and 20–30 m wide, andoccupies the base of several troughs within the contact surface


between the footwall basalt and the hanging wall ultramafics.There is remnant ore in the stopes where the surface has thick-ened in the form of local “jags.” The ore has about 1.5:1 densitycontrast with the enclosing rock, and possibly presents a slight

FIG. 2. 3-D perspective view of the Hunt D zone decline andB05 stopes at the seismic test site. Boreholes used in the exper-iment are shown.

FIG. 3. Cross-section at boreholes H16-18 and H16-20 showingsimplified stope geometry and ore troughs.

velocity decrease (say, 5000 m/s versus 6000 m/s). The com-puted reflection coefficient for the idealised case of a plane, lat-erally continuous mineralised surface is about −0.11. In prac-tice, the finite width of the ore, its discontinuous nature, and theroughness of the surface would greatly diminish its reflectivity,especially for wavelengths of comparable size to the structure.The ore would behave more as a diffuse diffractor than as a mir-ror. Furthermore, the target will only “reflect” sound back tothe receiver array if illuminated from the right range of angles.The stope cavities themselves, rather than the small remnantmineralization within them, are the more likely reflectors inthis survey. The air/rock interface presents a perfect reflectingsurface (reflection coefficient of −1), but once again the re-ceived signal will be downgraded by the finite extent (lengthand width) of the tunnels and the finite wavelength. For a 1000-Hz seismic signal, the wavelength is about 6 m (not unlike theradar wavelength), which is of the same order as the width ofeach stope.

The host medium is hardly homogeneous. Even within thefootwall basalt there is considerable variability in compositionand alteration, and hence elastic properties, as revealed by thetwo drill cores. There are several faults, fractures, and quartzveins, as well as a porphyry sheet (see Figure 3), which disturbthe host rock.

Data acquisition geometry

Three separate experiments (crosshole survey, hammer-source VSP survey, and explosive-source reversed VSP) wereconducted at the site shown in Figure 1.

Crosshole survey.—The crosshole survey entailed shots inhole H16-20 at 1-m depth spacing, from 3.2 to 42.3 m, andrecording on a hydrophone array in hole H16-19, at 1-m spac-ing, over the depth range 0 to 41 m. Since the hydrophone arrayhas its detectors at 2-m intervals and is limited to just 24 chan-nels, the 1-m receiver array was synthesised by shooting twiceat each depth level in hole H16-20, once for hydrophones atpositions 0 m, 2 m, . . . , 40 m, and again after displacing the eelby 1 m in the hole such that the elements were as positions1 m, 3 m, . . . , 41 m. It was not possible to push the hydrophonestreamer into the hole any deeper than 43 m. The number ofuseable recording channels was 21 per shot, yielding a total of40 × 2 × 21 = 1680 seismic traces.

It would have been desirable to have collected the seismicdata at a closer spacing (shot and receiver interval of 0.5 mor less), but this would have greatly increased the acquisitiontime. We were limited largely by the time to load and placeeach shot.

Hammer VSP experiment.—The hammer VSP experimentwas the first phase of the work undertaken. It involved plac-ing the hydrophone array in hole H16-18, with detectors at2-m intervals over the depth range 1–41 m, and impacting thetunnel wall with a 5-kg sledge hammer along the decline atone-quarter station positions from G1 to G8.5 (1.25-m inter-vals). This is a multilevel, walkaway VSP recording geometry.The number of seismograms collected was 31 × 21 = 651.

Triaxial VSP experiment.—This experiment was a reversedmultilevel walkaway reversed VSP. It entailed having shots at1-m spacing in both holes H16-18 (depth range 3.2–42.3 m)and H16-20 (depth range 3.6–42.6 m) and recording on eight


three-component geophones along the decline wall (G1–G8;see Figure 1) at a nominal spacing of 5 m. The sensors wereplaced in 2-m deep horizontal holes drilled perpendicular tothe tunnel axis, oriented into position, and then grouted per-manently into each hole. The orientation system was such thatthe X component was parallel to the decline, with its positiveterminal pointing northwest (up the decline), the Y componentpointing along and into the axis of the hole (northeast), andthe Z component vertically downwards. The purpose of usingthree-component receivers was to achieve directional sensitiv-ity in the response, which is of potential benefit in 3-D imagingwith a limited number of sensors arranged in a fairly short lin-ear array.

The number of traces acquired in the triaxial VSP experi-ment was 40 × 2 × 8 × 3 = 1920.

Instrumentation

A block diagram of the recording instrumentation is shownin Figure 4.

Seismic source.—The mechanical impact source was a stan-dard 5-kg sledgehammer, swung to strike the turned wall atabout waist height and at 90◦ to the decline axis. A piezoelectriccrystal sensor on the base of the hammer senses the moment ofimpact and transmits a trigger pulse back to the seismographvia a hammer cable.

The explosive source uses a single seismic detonator fired ina water-filled hole. Various detonator holding bars, including amultishot gun, were tested. The firing line was attached toa special control unit at the head of the tool and connectedat its other end to a Nissan capacitor discharge blaster that iscontrolled by a synchronous timing unit. The blaster delivers

FIG. 4. Schematic illustration of underground seismic instrumentation.

over 10 A of current to ensure quick burning of the detonatorbridge wire. The detonator gun was deployed by means of a setof 14 pushrods, each 3-m long and fitted with special couplersto allow quick joining and unjoining of sections. The rods weredistance marked to facilitate depth recovery to within a fewcentimeters.

Receivers.—A hydrophone array, or borehole eel, was de-signed specifically for this and similar experiments and wasconstructed by Australian Sonar Systems. It is a 24-channel ar-ray of 46-m aperture, connected to a 150-m leader cable. At adiameter of 30 mm, the eel was sufficiently narrow and rigid topush into a horizontal or declined borehole of 45-mm diame-ter to a depth of almost 200 m. The hydrophones are equippedwith preamplifiers and have a frequency range of 50–20 00 Hzand a dynamic range of 80 dB.

The array is provided with a front-end conditioner (FEC)which provides power to the array preamplifies, buffers thearray channels, and enables the level of the output to matchthe input of the recording device.

The three-component geophones, assembled at FlindersUniversity, use high frequency (100-Hz) moving-coil elements.The three pairs of leads, X , Y , and Z , connect to the takeouts ofa standard geophone cable. Each detector assembly is screwedonto a PVC anchor which is grouted into the borehole. A longwrench enables rotation of the package to align Z in the verticaldirection.

Seismograph.—A Bison 7024 digital IFP 24-channel seismo-graph, coupled to a Toshiba 1600 laptop computer, was usedfor data acquistion. Both are powered by 12-V batteries. Asynchronous clock was used for instrument triggering with theexplosions.


All seismic records were obtained using a sample rate of0.1 ms, which is the maximum possible with the Bison. Therecord length was 100 ms, more than adequate to capture anyreflected waves originating from within several hundred metersof the boreholes. The analog band-pass filter settings used onthe Bison were 32 Hz low cut and 2000 Hz high cut (antialias).

SIGNAL CHARACTERISTICS AND NOISE

Figure 5 is a sample hammer blow record from an impactposition at station G8. The record shows the direct P-, direct

FIG. 5. Sample hammer blow recording on hydrophone array,from impact at G8. Trace separation is 2 m.

FIG. 6. Set of shot-gather crosswell hydrophone records for shots in hole H16-20 and receivers in hole H16-18. Shot-depth spacingis 1 m, receiver spacing is 2 m.

S-, and tube-wave arrivals. The waves arrive first at the closesthydrophone station, and move out in time in accordance withthe source-receiver separation. The corresponding velocitiesare 6, 3.5, and 1.2 km/s. The signals are quite strong and displayfairly high frequency (up to 1000 Hz).

Figure 6 is a sample set of shot-gather records from the cross-hole experiment. Each panel represents a separate shot (depth22.2 m, 23.2 m, etc.) recorded on 21 hydrophone receivers atdepths of 1 m, 3 m, . . . , 41 m. The first arrival is the directP-wave. The record is full of other events which seem to moveout in both directions with the approximate speed of sound inwater (1400 m/s). This is best seen on receiver gathers (Fig-ure 7), which reveal the origin of these events as the top, bot-tom, and middle of the shothole. These secondary events aretube wave to P and S conversions at each end of the hole,and also from an impedance boundary (fracture zone) mid-way along the hole. The top and bottom of the hole presentimpedance barriers to the shock front set up from the actualsource in the hole. They behave as secondary “sources” ofbody waves, which then radiate out into the medium and arerecorded on the receiver array, superimposed on the direct andreflected arrivals from the actual shot.

There is a family of tube waves in the shothole (velocities 1.2–1.5 km/s), each of which undergoes mode conversion. The bodywaves (from true and secondary sources) can then excite tube-wave arrivals in the receiver hole from both ends as well as fromany fractures or other discontinuities in the hole. This makesfor a very cluttered record and represents a severe form ofcoherent noise contamination. It is difficult to remove. Velocityfiltering is partially effective (due to low water-wave velocityin the receiver-gather domain), but it can only be done if thewaves are not velocity aliased, which means a very close shotspacing (say, 0.25 m). Unfortunately this was not the case withthe Hunt experiment. The tube-wave problem arises largelyfrom shooting in a water-filled hole. Explosions in dry holesdon’t generate tube waves, but they don’t generate much body-wave energy either, because of the very poor coupling.


The detonator source produced high-amplitude, high-fre-quency (up to 3000-Hz) signals. The high frequencies areimportant for getting reflections from small objects like oreshoots, and for having the requisite resolution.

The 1-m shot and receiver interval used on the crossholesurvey, the 5-m geophone spacing used on the triaxial VSP, andthe 2-m hydrophone interval/1.25-m impact interval used onthe hammer VSP were inadequate to combat the low-velocitycoherent noise, which became aliased at the high (kilohertz)frequencies involved.

The seismic work at the Hunt mine was conducted mainly atnight. Background noise was plentiful, and included vibrationsfrom ventilation fans, pumps, surface drilling above, inductivepickup, and vehicles operating nearby. On some occasions, thenoise was so high that the recording had to be discontinued.Apart from the noise, the seismic survey was interrupted byseveral instrument malfunctions.

TOMOGRAPHIC IMAGING

Procedure

Tomography is a 2-D image reconstruction procedure. Theimage plane in the present context was formed by the two dip-ping boreholes and the section of decline joining the two holecollars. These three observation lines are not actually straightnor perfectly coplanar because of the slight irregularity of thetunnel and the drillholes. But as a starting point for tomo-graphic inversion, we computed the best-fit plane to the threelines and projected the actual shot-receiver positions onto thisplane.

The input data comprised the first arrival P-wave times,which were picked interactively for every shot gather usingSeismic Unix software. Appropriate static corrections wereapplied. Only the hydrophone data (crosshole and hammerVSP records) were used in the reconstruction. More than 2300

FIG. 7. Set of receiver gathers for crosswell experiment, showing direct P- and S-waves plus tube-wave generated events at top,bottom, and middle of shothole. Trace spacing is 1 m.

raypaths crisscross the image plane to yield good angular andspatial coverage. The source and receiver positions along thethree arrays used to form the “U” geometry are at roughly 1-mspacing. The medium was divided up into cells of dimension0.25 × 0.25 m.

The tomography procedure begins with a uniform (average)velocity model and then iterates through using a multistage,subspace method (Cao and Greenhalgh, 1995) in which thelong spatial wavelength velocity variations are solved for first,progressively adding in the shorter wavelength features. A con-jugate gradient–type algorithm is at the heart of the model up-date procedure, details of which are given by Zhou et al. (1992).Ray tracing is accomplished using the minimum traveltime treealgorithm (Cao and Greenhalgh, 1993).

Modeling results

Figure 8 shows a simplified model, containing a faulted ore-body sitting between two horizontal underground boreholes60 m apart. The ore has low velocity compared to the rock. Aset of 24 shots and 24 geophones are placed in the boreholesto simulate a seismic tomography experiment. The dimensionsand layout were not chosen to match the actual experiment, butmerely to illustrate certain features. Shot and detector spacingis 5 m. Raypaths are shown for first arrivals. The synthetic ar-rival times were then inverted by a DMLS tomography algo-rithm (Zhou et al., 1992) to produce the tomogram of Figure 9.The ore zones have been recovered, albeit with some blurringand distortion due to the limited angular coverage. This modelclearly demonstrates the possibilities of tomography in cross-well exploration.

Hunt Mine tomogram features

Figure 10 shows the final tomogram obtained at the Huntmine. The color code at the side of the map defines the various


velocity values. The boreholes have been superimposed on thetomogram to give the proper perspective (the crosses “+” de-note collar positions in the decline).

The velocities range from about 5300 to 6400 m/s. From thegeological log of the holes we know that the rock type is entirelybasalt, but the velocity field is anything but homogeneous. Theinhomogeneity is presumably due to changes in alteration andgeochemistry of the basalt. The presence of quartz veins andfractures will also affect the measured velocity values.

Ultrasonic velocity measurements were carried out on eightcore samples taken from the two holes to aid tomogram inter-pretation. The samples included the major change in basalt typeand condition. Velocity results are given in Table 1. The massivebasalt has a high velocity of 5900 m/s. Abundant disseminatedmagnetite yields a lower velocity, as expected, through the ef-fect of increased density. The ankerite-albite altered basalt hasa much lower velocity of just 2700 m/s. The massive quartz veinhas a velocity of 4700 m/s, similar to that of the amphibole-richbasalt. These velocities are all lower than those obtained in situfrom the seismic tomography experiment. This can be largelyexplained by the fact that the rock in situ is under considerablelithostatic pressure. It is well known that pressure increases ve-locity due to the closing of fractures and reduction of porosity.

FIG. 8. Synthetic 2-D crosshole seismic experiment showingthe orebody model and raypaths.

FIG. 9. Velocity tomogram reconstructed for the orebodymodel of Figure 8.

FIG. 10. Tomographic reconstruction of the first arrival VSPand crosshole data at the Hunt mine.


If the relative velocity variations rather than the absolutevariations are considered along borehole H16-18, then the to-mogram velocity variation is well matched by the ultrasonicvelocities. The high-, medium-, and low-velocity zones in thetomogram seem to correlate well with the mineralogy and al-teration of the basalt. This is an encouraging result, especiallygiven the fact that only single sample determinations for eachrock “type” were made. Besides random errors on single sam-ples, which can be smoothed out by multiple sample measure-ments, the laboratory sample may not be representative of therock behavior in situ because of the absence of macrofeaturessuch as fractures. Alternatively, macrofeatures (such as 1-mwide quartz veins) may have little effect on 6-m-long seismicwavelengths in the field, but multiple millimeter-wide fracturesin a laboratory sample may dramatically affect the velocity ob-tained using a 1-cm-long ultrasonic wave.

It thus appears that seismic tomography at this location of-fers a useful lithologic mapping tool. It is unfortunate that nomineralization lies within the image plane. The boreholes dipdownwards, whereas the ore is in the stopes above the decline.

3-D REFECTOR IMAGING

Kirchhoff modeling of stope diffractions

The principal targets of the crosswell/VSP reflection experi-ments were the stope tunnels and residual ore, lying about 20 mabove the decline, as explained earlier (see Figures 2 and 3).In order to assess the likely diffracting/reflecting response ofsuch a structure, to help identify any reflections in the field data,and as a test of our 3-D migration algorithm, we computed theacoustic response for the idealised model shown in Figure 11.The model has very similar geometry to the Hunt B05 site.Figure 11 shows the seismic observation plane defined by thetwo boreholes and the decline. The stope tunnels have beenapproximated by three bounded rectangles of width 4 or 5 mand of fixed length of 60 m. They have been assigned refectioncoefficients of −1.

The modeling was done using a modified Kirchhoff sur-face integral formulation (Sinadinovski, 1994). For noncoin-cident source-receiver geometry, it yields the complete reflec-tion/diffraction response of the structure:specular returns, aswell as corner and edge diffractions.

Figure 12 show a portion of a set of shot gathers for shotsat 2-m intervals in borehole H16-20, starting at 2-m depth,and detectors every 2-m in hole H16-18 over the depth range

Table 1. Ultrasonic velocities of core samples from HuntH16-18.

LongitudinalSample no. Rock description velocity (m/s)

1 Amphibolite rich basalt 46402 Massive basalt 59303 Massive basalt with abundant

disseminated magnetite 39904 Massive quartz vein 46905 Awaiting X-ray diffraction

results 57106 Epidote altered basalt 61307 Biotite-rich basalt and

disseminate pyrite 52408 Alteration zone: ankerite-

albite altered basalt 2680

2–40 m. The plotted quantity is

sgn(Z)√

X 2 + Y 2 + Z 2,

where X , Y , Z are the three components of particle displace-ment, in order to match the pressure waveforms of the cross-hole hydrophone data. The synthetic seismograms, computedfor a source center frequency of 2 kHz, show the complexdiffraction response arising from the different parts of thestructure. The seismograms do not include the direct-wave ar-rivals or the mode conversions. The P-wave reflections exhibitarrival times in the range 13–20 ms. The amplitudes vary withreceiver position, and interference of the different diffractionsignals can also be observed.

Figure 13 is a comparison of the model results with the exper-imental data for a source depth of 30 m. The reflection signalsare visible in the observed data, but they are obscured by shearwaves and by tube-wave generated secondary arrivals.

Migration procedure

The reflection imaging procedure is described bySinadinovski (1994). In simple terms, it is a prestack,vector Huygens-Kirchhoff migration method.

A reflector may be visualized as an ensemble of closelyspaced point scatterers in the subsurface. The impulse responseof a point scatterer is a hyperbola, whose curvature is a func-tion of the medium velocity. Therefore, summation (or inte-gration) over the numerous hyperbolas appropriate for eachtime sample of the input data set will add in-phase all diffrac-tion signals and image all reflectors. The summation opera-tor must be properly weighted to honor the wave equation.

FIG. 11. Simplified model of the Hunt reflection experiment,showing the source-receiver observation plane (black) and theoverlying stope diffractors (approximated by rectangular scat-terers).


FIG. 12. Set of synthetic shot gathers for the model of Figure 11 obtained by the Kirchhoffintegral method. Shots at 2-m intervals in hole H16-20; receivers at 2-m spacing in hole H16-18.Note the complicated diffraction/reflection response.

FIG. 13. Comparison of the synthetic diffraction response for model of Figure 11 (with a shotat depth of 30 m) and the real crosswell seismograms for approximately the same recordinggeometry.


The procedure involves breaking up the volume into smallcells and calculating the probability that a diffractor/reflectorexists at each grid point by summing amplitudes over all shotsand geophones. The raypaths are first traced in three dimen-sions from every shot to every voxel and back to every receiver.The corresponding traveltimes (for the assumed wave type: PP,PS, SP, SS) define the search trajectory through the traces. Forthree-component receiver data, the azimuth and inclination ofeach computed ray permit the additional projection of the sig-nal vector into the plane of the wavefront (S) or parallel tothe ray (P). Constructive interference will occur only for thosevoxels at which seismic scattering take place.

We modified the procedure for the Hunt experiment in twoways. First, we used a semblance-based measure of trace energyalignment by forming all of the possible cross products of sam-ple amplitude over a 10–20 point sliding time window for eachgather. Second, we performed a partially coherent migrationby adding only the absolute values for each shot gather, whichwere coherently summed individually over time, receiver po-sition, and vector component. The reason for doing a partiallyrather than a fully coherent migration is to minimize the effectof shot static-type timing errors and other random fluctuations.

The cell size used in the volume interrogation procedurewas a 1-m cube. The grid size was 60 × 60 × 15, or 54000 voxels.Larger cells were also explored, but computer memory limita-tion precluded an increased 3-D grid.

The data to be migrated must undergo some pretreatment,such as three-component rotation, muting of direct P- andS-wave arrivals (10–20 sample suppression window), and gainfunction application to boost the later arrivals. The input datacomprise the velocity field, the source receiver coordinates, andthe field seismograms. The process is repeated for the variousmodes (PP, SS, etc.), if desired.

The output from the migration is a set of positive numbers ateach point in the 3-D grid, which represent the scatterer prob-abilities. These numbers are color coded according to someappropriate nonlinear scale (e.g., logarithmic), and the imageplanes (horizontal slices through the volume) are then dis-played at various depths.

Results

The data subsets used for the migration were the staticallycorrected and pre-processed records from the triaxial VSP ex-periment (80 shots, 8 receivers) and the crosswell experiment(40 shots, 2 × 21 receivers). The individual data sets were mi-grated separately, as well as being combined into a single mi-gration. The attraction of a combined geometry is that the re-flectors are illuminated from a greater range of angles.

Figure 14 is a horizontal slice through the final P-wave migra-tion 3-D volume at the −234 m elevation level. This elevationcorresponds to the approximate average position of the stopefloor across the survey area. The southwest stope tunnel actu-ally drops in depth by about 4 m over a horizontal distance of55 m between the two boreholes, from an elevation of −231 mat the northern end to an elevation of −235 m in the south.The northeastern stope floor maintains an elevation of −234 mover its northern extent.

The warm colors (brown, red, yellow) in Figure 14, indicatethe strongest scatterer positions, whereas the cold colors (blue,purple) indicate the absence of any reflectors. Geographical

coordinates, as well as the projected position of the D zonedecline (20 m below the elevation shown), are given on themap to aid location and identification of features. The migra-tion result show a striking correlation with the known mineworkings (see Figure 1). The two green elongated zones ofmoderate reflectivity, which subparallel the decline, delineatethe stopes. The southwestern ridge bends in close to the declineat its northern end, following the actual stope cavity. The re-gion of strong reflectivity, colored red, seems to coincide withthe far boundary of the northeastern tunnel. The reflectivitymay well be enhanced by the curvature of the ore trough and

FIG. 14. P-wave migration result for VSP/crosshole data. Themap is a horizontal slice of the 3-D volume at the −234 m level.The scatterers parallel the stope geometry.

FIG. 15. Migration map for horizontal slice of −230 m level.The response is less intense than Figure 14 as the level risesabove the stopes.


the sharp lip of the structure, as seen in Figure 3. This wouldcause a focussing of back-scattered energy, as observed in thedecline radar profile. The presence of residual, unmined ore onthis side of the stope (see Figure 3) may also be contributingto the strong reflection signal.

It is interesting to observe on the computer how the re-flection (migration) pattern changes with elevation. Migrationslices at the −230 m (Figure 15) and the −240 m levels (notshown), which are just above and below the stope floor androof elevations, respectively, yield reduced reflectivity fromthe maximum which occurs within the −237 to −236 m depthrange.

The finite time window used in forming the semblance, aswell as the finite wavelength of the signal (say, 6 m) means thatresolution is limited, and features get averaged, or smoothed inthe migration, over a spatial distance of at least several meters.Migration images obtained at elevations of −226 and −246 m(not shown) exhibit almost no structure, confirming that thereflectors are confined in depth to the vicinity of the stopes.

CONCLUSIONS

A high-resolution seismic hardware-software system hasbeen developed for crosshole/VSP imaging of orebodies andassociated hardrock geological structure, either within minesor from the earth’s surface using available boreholes.

Although the seismic signals are of high frequency and highquality, records are strongly contaminated by tube-wave andmode-conversion noise. Dense arrays of sources and/or re-ceivers are needed to enable proper preprocessing of the dataprior to imaging.

An example of an underground survey at the Hunt mine,Kambalda, Western Australia, demonstrates the possibility ofboth tomographic imaging to map velocity inhomogeneities,and reflection imaging to delineate stope geometry. The seismicmaps show a striking correlation with the known geology.

The seismic images (especially reflection) are difficult to ob-tain and not easy to relate to mineralization. Companion elec-trical images such as radar, radiowave imaging method (RIM),and applied potential, are needed to aid interpretation, and arelikely to see increased use in mine development over the nextfew years. The seismic method offers great promise in prospectexploration. We are working on improvements to boreholegeophysical instrumentation and at developing more versatileimage reconstruction procedures.

ACKNOWLEDGMENTS

We acknowledge the support and cooperation of PeterFullagar, Peter Williams, and Greg Turner from Western Min-ing Corporation in the conduct of this work.

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Radio tomography and borehole radar delineation of the McConnellnickel sulfide deposit, Sudbury, Ontario, Canada

Peter K. Fullagar∗, Dean W. Livelybrooks‡, Ping Zhang∗∗, Andrew J. Calvert§,and Yiren Wu§§

ABSTRACT

In an effort to reduce costs and increase revenuesat mines, there is a strong incentive to develop high-resolution techniques both for near-mine explorationand for delineation of known orebodies. To investi-gate the potential of high-frequency EM techniquesfor exploration and delineation of massive sulfide ore-bodies, radio frequency electromagnetic (RFEM) andground-penetrating radar (GPR) surveys were con-ducted in boreholes through the McConnell massivenickel–copper sulfide body near Sudbury, Ontario, from1993–1996.

Crosshole RFEM data were acquired with a JW-4 elec-tric dipole system between two boreholes on section2720W. Ten frequencies between 0.5 and 5.0 MHz wererecorded. Radio signals propagated through the SudburyBreccia over ranges of at least 150 m at all frequen-cies. The resulting radio absorption tomogram clearlyimaged the McConnell deposit over 110 m downdip. Sig-nal was extinguished when either antenna entered thesulfide body. However, the expected radio shadow didnot eventuate when transmitter and receiver were onopposite sides of the deposit. Two-dimensional model-ing suggested that diffraction around the edges of thesulfide body could not account for the observed fieldamplitudes. It was concluded at the time that the sulfidebody is discontinuous; according to modeling, a gap as

small as 5 m could have explained the observations. Sub-sequent investigations by INCO established that pick-upin the metal-cored downhole cables was actually respon-sible for the elevated signal levels.

Both single-hole reflection profiles and crosshole mea-surements were acquired using RAMAC borehole radarsystems, operating at 60 MHz. Detection of radar reflec-tions from the sulfide contact was problematic. One co-herent reflection was observed from the hanging-wallcontact in single-hole reflection mode. This reflectioncould be traced about 25 m uphole from the contact.In addition to unfavorable survey geometry, factorswhich may have suppressed reflections included hostrock heterogeneity, disseminated sulfides, and contactirregularity.

Velocity and absorption tomograms were generated inthe Sudbury Breccia host rock from the crosshole radar.Radar velocity was variable, averaging 125 m/µs, whileabsorption was typically 0.8 dB/m at 60 MHz. Kirchhoff-style 2-D migration of later arrivals in the crossholeradargrams defined reflective zones that roughly parallelthe inferred edge of the sulfide body.

The McConnell high-frequency EM surveys es-tablished that radio tomography and simple radioshadowing are potentially valuable for near- and in-mineexploration and orebody delineation in the SudburyBreccia. The effectiveness of borehole radar in this par-ticular environment is less certain.

INTRODUCTION

Borehole EM methods are being applied increasingly in andnear metalliferous mines for both exploration and orebody

Manuscript received by the Editor May 10, 1999; revised manuscript received June 16, 2000.∗Fullagar Geophysics Pty. Ltd., Level 1, 1 Swann Road, Taringa, Queensland 4068, Australia. E-mail: [email protected].‡University of Oregon 1274, Department of Physics, Eugene, Oregon 97403-1274. E-mail: [email protected].∗∗Electromagnetics Instruments, Inc., P.O. Box 463, El Cerrito, California 94530-0463.§Simon Fraser University, Department of Earth Sciences, 8888 University Drive, Burnaby, British Columbia V5A 1S6, Canada.§§Beijing Xin Yi Hightech Research Institute 70 Bei Lishi Road, Beijing 100037, China.c© 2000 Society of Exploration Geophysicists. All rights reserved.

delineation. In the context of near-mine exploration for basemetals, downhole transient EM methods such as UTEM haveproved effective for detecting deep orebodies at ranges of hun-dreds of meters from boreholes (e.g., King, 1996). At the same

1920

Radio and Radar Delineation 1921

time the drive for more efficient mine production techniquesto reduce costs and increase revenues has stimulated inter-est in high-frequency borehole EM techniques. Radio imagingand ground-penetrating radar (GPR) methods show particularpromise for orebody delineation.

The use of radio waves to define geological features be-tween boreholes can be traced back to a 1910 German patent(Thomson and Hinde, 1993). Radio tomography (RT) per sewas pioneered by Lager and Lytle (1977) and has since beenapplied extensively to detect faults and other disruptions inthe continuity of seams of coal or potash (McGaughey andStolarczyk, 1991; Vozoff et al., 1993). The utility of radio imag-ing in metalliferous exploration and mining is currently un-der investigation. Radio frequency methods in the 10 kHz to1 MHz band have already enjoyed some success in this context,e.g., Nickel and Cerny (1989), Anderson and Logan (1992),Thomson et al. (1992), Wedepohl (1993), Zhou et al. (1998),and Stevens and Redko (2000). RT can deliver higher resolu-tion than traditional borehole EM systems, both by virtue ofits higher frequencies and because the radio transmitter can belowered down a borehole, closer to the target.

GPR detects changes in permittivity and conductivity us-ing high-frequency electromagnetic pulses (10 MHz–1 GHz).Conventional GPR has found application in underground coalmines (Coon et al., 1981; Yelf et al., 1990) and is used routinelyto define auriferous zones in the Witwatersrand (Campbell,1994) and at the Sixteen to One mine in California (Raadsma,1994). Other mining applications include mapping conductivesalt structures (Stewart and Unterberger, 1976), exploring forplacer deposits (Davis et al., 1985), detecting geotechnical haz-ards (Fullagar and Livelybrooks, 1994a), and delineating lat-eritic deposits. Borehole radar is well established in salt andpotash mines (Mundry et al., 1983; Eisenburger et al., 1993;Thierbach, 1994), but application in nonevaporite mines is stillrelatively uncommon, notwithstanding the strong commercialincentive to accurately define ore boundaries and structures.Encouraging experimental applications of borehole reflectionradar have been reported from coal mines (Murray et al.,1998), Witwatersrand gold mines (Wedepohl et al., 1998), andbase metal sulfide mines (Liu et al., 1998; Zhou and Fullagar,2000).

This article summarizes activities undertaken at theMcConnell nickel sulfide deposit from 1993–1996 by a re-search group formed under the NSERC/TVX Gold/GoldenKnight Chair in Borehole Geophysics for Mineral Explorationat Ecole Polytechnique, Montreal. Borehole radar trials wereperformed with a 22-MHz RAMAC I system in December1993 (Fullagar and Livelybrooks, 1994b; Stevens and Lodha,1994) and with a 60-MHz RAMAC LI system in June 1996(Calvert and Livelybrooks, 1997). Both single-hole reflectionand crosshole data were recorded. A radio imaging survey wasundertaken in April 1994 with a JW-4 electric dipole system(Fullagar et al., 1996). Data were recorded at ten frequenciesbetween 0.5 and 5.0 MHz.

The McConnell deposit, owned by INCO Ltd., is locatednear Garson mine on the southeastern rim of the Sudburybasin, Ontario, Canada. It is a tabular body of massive sul-fides (pentlandite–pyrrhotite), about 200 m along strike, 300 mdowndip, and averaging 15 m true thickness (Figure 1). Themineralization is hosted by a quartz diorite dyke, intruded intoheterogeneous Sudbury Breccia country rock, comprised of

blocks of metavolcanics and metasediments. The dyke strikes85◦ east and dips 70◦ south.

The McConnell sulfide body is shallow, extensively drilled,and close to an operating mine, rendering it an excel-lent geophysical test site. Prior to the work described here,the Geological Survey of Canada (GSC) had recordeda comprehensive suite of downhole geophysical logs atMcConnell (Mwenifumbo et al., 1993; Killeen et al., 1996). Thedc resistivities varied over four orders of magnitude, from nearzero in the massive sulfides to over 30 000 ohm-m within thebreccia. The high resistivities of the host rocks rendered themfavorable for propagation of EM waves, while the high con-ductivity of the massive sulfide body was expected to produce

FIG. 1. McConnell deposit shown (a) in plan and (b) on section2720W. Top of the sulfide body has been projected to the sur-face in (a). Transmitter locations for the RT survey are markedwith dots in (b); thick black lines mark massive sulfide inter-sections. Westings in (a) are in feet; depths in (b) are in meters.

1922 Fullagar et al.

strong radar reflections and to create radio shadow zones. TheMcConnell site therefore represented an ideal location to as-sess the efficacy of radio tomography and borehole radar fordelineating a known massive sulfide body.

THEORY

For EM waves propagating at angular frequency ω through ahomogeneous isotropic medium with conductivity σ , real per-mittivity εr , and permeability µ, the real and imaginary parts, βand α, of the complex wavenumber k can be written as (Cook,1975)

β = ω

√µεr

2(√

1 + Q−2 + 1)1/2 (1)

and

α = ω

√µεr

2(√

1 + Q−2 − 1)1/2. (2)

The value Q is the ratio of displacement current to conductioncurrent,

Q = ωεr

σe, (3)

where σe is the effective conductivity (Turner and Siggins,1994),

σe = σ + ωεi , (4)

with εi denoting the imaginary part of permittivity.The scalar wavenumber β determines the wavelength, while

the absorption coefficient α controls attenuation. For both ra-dio imaging and GPR surveys, the effective range increaseswith decreasing frequency, but at the expense of resolution.Thus, any survey involves a trade-off between resolution andrange.

Radio imaging involves propagation of monofrequency EMsignals in the 1 kHz to 10 MHz band between boreholes or mineaccessways. In a typical crosshole RT survey, the transmitter isfixed in one hole while signal amplitude (and perhaps phase)is recorded at successive receiver stations in one or more otherholes. Signals which are weak and retarded in phase are indica-tive of more conductive material between the transmitter andreceiver. Tomographic reconstruction of the signal amplitudesand phases at all receiver stations provides an image of theconductivity distribution between the boreholes.

GPR involves emission, propagation, and detection of EMpulses, with center frequencies usually between 10 MHz and1 GHz. GPR can be deployed in transmission mode, like radioimaging, but it is more commonly applied in reflection mode.In the latter case it is closely analogous to the seismic reflectionmethod. GPR can map contrasts in electrical properties or fluidcontent which usually coincide with geological contacts andstructures. Radar reflection coefficients depend on permittivity,conductivity, and permeability contrasts.

Most radar surveys are carried out under low-loss conditions(Q� 1), for which the propagation involves polarization of themedium. Typical rocks are characterized by relative dielectricconstants between 3 and 30. The presence of water, with itslarge relative dielectric constant (∼80), serves to dramaticallyretard propagating waves.

The demarkation between GPR and RFEM applications isnot sharp, but generally radio imaging is used at frequenciesbetween 1 kHz and 10 MHz, often in lossy (low Q) environ-ments. RFEM equipment can operate effectively in the dif-fusion domain Q≤ 1, where conduction currents are at leastcomparable to, and often much larger than, displacement cur-rents. In the limit of low Q, α and β assume the same functionalform:

α = β =√

ωµσe

2. (5)

McCONNELL RADIO FREQUENCY EM SURVEY

RFEM data acquisition

Tomographic radio frequency surveys were undertaken be-tween boreholes 80 578 and 78 930 in April 1994 using a JW-4borehole radio imaging system developed by the Chinese Min-istry of Geology and Mineral Resources (MGMR). The JW-4 isan electric field system, capable of recording axial componentamplitude over a programmable sweep of frequencies between0.5 and 32 MHz (Qu et al., 1991). At McConnell, a suite of tenfrequencies was read between 0.5 and 5 MHz at 0.5-MHz in-tervals.

Tomographic coverage was 90–170 m in hole 78 930 and 130–310 m in hole 80 578, i.e., in both the hanging wall and footwallof the sulfide deposit (Figure 1b). The transmitter sites were10 m apart, and the receiver station interval was 2 m.

A center-fed half-wave dipole transmitter with 18-m armswas deployed in hole 78 930 on the first day, but a monopoleantenna with a 7-m arm was used in hole 80 578 on the secondday. Water infiltration into the transmitter filter pod promptedthe change. Signal strength was satisfactory at all frequencies atranges up to 150 m. The receiver was a monopole, with a 7-marm, throughout. Analog to digital conversion was affecteddownhole in the receiver electronics pod, which is 1.2 m longand has an outside diameter of 40 mm.

A limited repeatability test was conducted during ascent ofhole 80 578 at the McConnell site, with the transmitter at 30 mdepth in hole 78 930 (Figure 1b). The greatest difference inrepeat readings was only 0.6 dB at 0.5 MHz. In conjunctionwith a similar repeatability test performed at other sites, thesecomparisons engendered confidence in data precision.

The transmitter was operated from each hole in turn to per-mit a reciprocity test. However, given the change in transmit-ter and in view of the gradual infiltration of water into thetransmitter on the first day, there is no reason to expect reci-procity to apply exactly. The amplitude differences when thepositions of the transmitter and receiver were interchangedwere indeed substantial, typically between −4 and +4 dB at5 MHz.

RFEM data reduction and image construction

In RFEM surveys, the amplitude of a single component of ei-ther electric or magnetic field is measured at the receiver. To to-mographically reconstruct the conductivity structure betweenboreholes, far-field conditions are assumed. For an electricdipole in a homogeneous and isotropic medium, the receivedelectric field strength in the far-field is given by (Ward and


Hohmann, 1988, p. 173)

|E f (r)| = A0eαγ sin θt sin θr

r, (6)

where r is the transmitter-receiver separation, A0 is the sourcestrength, and θt and θr are the polar angles of the ray withrespect to the transmitter and receiver axes (Figure 2). Treatingabsorption as an analog for slowness, the apparent attenuationor traveltime, τn , for the nth ray is defined by

τn =∫Cn

α(x, y, z) dl, (7)

where integration is along the raypath Cn . Taking logarithmsof equation (6), it follows from equation (7) that τ can be ex-pressed as

τ = αar = −20 log10 |E f | + 20 log10 A0

+ 20 log10

(sin θt sin θr

r

), (8)

where αa is the apparent absorption coefficient in decibels permeter.

The three main sources of error for τ were (1) observationalerror, (2) uncertainty in A0, and (3) near-field effects. Fromour limited repeatability tests, described above, we concludedthat observational error is small (∼1 dB). This established dataprecision but not accuracy.

Generally, A0 is estimated via analysis of data collected inhomogeneous host rocks. After accounting for geometrical ef-fects, A0 is the intercept on the amplitude versus distance plot.Transmitter sites at depths 130, 140, and 150 m in hole 80 578,far from the massive sulfide contact, were selected for estima-tion of A0. After minimum l1-norm regression (Fullagar et al.,1994), A0 values of 0.38, 0.38, and 0.40 were obtained for thethree transmitter gathers. A starting estimate for the apparentabsorption coefficient was computed assuming σe was 10−4 S/mand εr was 6.5 ε0, in keeping with the best available informa-tion for Sudbury resistivities (Mwenifumbo et al., 1993) andpermittivities (Fullagar and Livelybrooks, 1994a). The details

FIG. 2. Crosshole survey schematic, where θt and θr are thepolar ray angles at the transmitter and receiver, respectively.

of the l1-norm minimization are given by Fullagar et al. (1994).An A0 value of 0.38 was ultimately adopted.

RFEM transmitter performance is strongly affected in ornear sulfide mineralization. Since the transmitter strength isnot identical at all locations, its output ideally should be contin-uously monitored. The JW-4 usually monitors transmitter inputcurrent, but this facility was inoperable for the McConnell sur-vey because no suitable three-conductor cables were availableon site.

If the source strength is assumed constant, a static shift erroris introduced in the tau values at each transmitter site if sourcestrength varies. McGaughey (1990) suggests a simple approachfor estimation of such static shifts by enforcing reciprocity. Animplementation of this approach is described by Fullagar et al.(1994). It was not applicable at McConnell because reciprocitydid not apply. More recently, Cao et al. (1998) proposed fre-quency differencing as a means to suppress variations in sourcestrength caused by large changes in conductivity near theboreholes. This approach may be suitable for the McConnelldata.

Near-field effects can become important as frequency de-creases, resistivity increases, path length decreases, or trans-mitter polar angle decreases (Pears, 1997). To assess the va-lidity of the far-field assumption for the McConnell data, thedifferences between the full analytic solution and the far-fieldapproximation for an electric dipole in a whole space werecompared for transmitters and receivers distributed in spaceas at McConnell. In a medium with effective of conductivityof 10−4 S/m and dielectric constant of 6.5 ε0, the wavelength at0.5 MHz is 235 m, larger than the longest raypath at McConnell.Therefore, far-field conditions did not apply at the lower fre-quencies.

The relative error between far-field and exact amplitudeswas computed for an electric dipole in a homogeneous medium,where

= |E | − |E f ||E | . (9)

For the McConnell survey geometry, the maximum relative er-ror in homogeneous host was estimated to be 5% at 500 kHz.This corresponds to an error of less than 1 dB. The McConnelltomogram presented here corresponds to a frequency of5 MHz, for which far-field conditions certainly apply.

To invert equation (7), the image plane is discretized andthe line integral on the rightside is represented as a summa-tion (e.g., Stewart, 1991). Tomographic reconstruction then re-duces to the solution of a system of linear equations for theunknown absorption coefficients within each of the grid cells.Migratom, a SIRT algorithm developed by the U.S. Bureauof Mines (Jackson and Tweeton, 1994), has been used here,assuming straight rays.

To perform a ray-based tomographic reconstruction, it is as-sumed that the amplitude at the receiver is governed solelyby absorption of the signal and that reflection, scattering, anddiffraction effects can be ignored. Thus, there is an implicit low-contrast assumption. If a highly conductive body intervenesbetween the transmitter and receiver, there will be no directtransmission and the inferred absorption coefficient after to-mographic reconstruction will not be an accurate indication ofthe actual absorption coefficient of the conductor.


RFEM interpretation

The absorption tomogram generated from the 5-MHzMcConnell data using a modified form of Migratom is pre-sented as Figure 3. The starting model was homogeneous, andclamping weights (Pears and Fullagar, 1998) were applied. De-spite the fact that reciprocity is assumed implicitly during to-mographic reconstruction, the tomogram based on the full dataset was superior (insofar as it provided better definition of thesulfide body) than that obtained when the data acquired withthe symmetric transmitter in hole 78 930 were disregarded.

The tomographic reconstruction clearly defines the contin-uation of the sulfide body over a distance of 110 m betweenintersections in the two boreholes. However, the inferred ab-sorption is not uniform along the conductor, and irregularitiesin the contacts appear to be related to the discretization of theimage plane.

When the transmitter and receiver are on opposite sides ofthe sulfide body, the signal amplitude is low but measurable.Propagation of radio signals through a 15-m-thick massive sul-fide body is impossible at megahertz frequencies. Nevertheless,the expected radio shadow at McConnell did not eventuate. Itwas concluded at the time that the signals received on the otherside of the conductor either travelled around the edges of thedeposit or passed through a gap in the sulfides.

Diffraction around the edges of the sulfide body is a plausi-ble explanation for the relatively high signals since the dimen-sions of the McConnell deposit (approximately 300 × 200 m)are comparable to the estimated radio wavelength (230 m) at500 kHz in the Sudbury Breccia and much larger than the es-timated wavelength (22 m) at 5 MHz. To determine whether

FIG. 3. Radio absorption tomogram at 5 MHz between holes78 930 and 80 578. Transmitter locations for the RT survey aremarked with dots.

diffraction alone could explain the observations, 2.5-D finite-difference modeling was performed using the EMSUN pro-gram (Smith et al., 1990). The model used to characterizediffraction around the bottom of the slab is depicted in Fig-ure 4. The model conductor is two dimensional (infinite in strikelength), but the source is finite (vertical electric dipole). Exceptfor the 2-D limitation, the cross-sectional geometry of the Mc-Connell experiment was replicated as closely as possible fortwo transmitter positions, T1 and T2, at depths 170 and 190 min hole 80 578 and two receiver positions, R1 and R2, at depths108 and 168 m in hole 78 930.

The orebody model was assigned a conductivity of 100 S/mat its core, enclosed within an inner layer of conductivity 1 S/mand an outer layer of conductivity 0.01 S/m. The transitionallayers were introduced for numerical stability, since a contrastof 1:100 was regarded as the greatest which could be handledaccurately. Geologically, such a transition zone often exists inthe form of a disseminated sulfide halo; indeed, at McConnellsulfide is known to occur as blebs in the quartz diorite andas disseminations in the breccia immediately adjacent to thequartz diorite (Grant and Bite, 1984).

The observed and modeled ratios of electric field amplitudeat the two receiver sites for the two transmitter positions at500 kHz are recorded in decibels on Figure 4. For a contin-uous conductor (no gap), modeling results indicate that afterdiffraction around the bottom of the slab the signal amplitudeis 61 dB lower at R2 on the far side of the conductor than atR1 on the near side for the transmitter at T1 and 54 dB lower

FIG. 4. A 2-D sectional model adopted to investigate diffrac-tion around the bottom of the McConnell deposit.


when the transmitter is at position T2. Even if the signal atR2 is increased by a factor of three to allow for diffraction lat-erally around the ends of the tabular body, the signal at R2would still be 51 or 44 dB down relative to that at R1 (notethe 15 dB or 16 dB differences observed). Therefore, diffrac-tion around the edges of the deposit cannot fully explain therelatively high field strengths recorded when the transmitterand receiver are on opposite sides of the conductor. Numeri-cal instability precluded 5-MHz modeling; however, at higherfrequencies diffraction would be a less effective mechanism.

In light of the modeling, a break (or hole) in the sulfide bodywas proposed as the explanation for the absence of a shadow(Fullagar et al., 1996). Geologically, there is a hint of a break incontinuity of the sulfide in hole 80 555, which passed through asmall sulfide interval (188–190 m) and then back into SudburyBreccia (190–197 m) before passing through the main sulfideinterval (Figure 1b). By introducing a 5-m-wide break in the2-D model in approximately this position, the predicted ra-tios of near-side to far-side signal amplitudes increase to levelscomparable with those observed, even though the gap was asmall fraction (∼2%) of a wavelength in the breccia at 500 kHz(Figure 4).

More recently, INCO geophysicists have demonstrated thatthe radio signal strength drops substantially if the metal-coredreceiver cable is replaced with reinforced fiber-optic cable(McDowell and Verlaan, 1997). Thus, we now conclude thatthe JW-4 in-line filters were inadequate to prevent cable pick-up, i.e., that the receiver and transmitter cables were acting asparasitic antennae during the 1994 survey. It is therefore notnecessary to invoke a gap or hole in the sulfide body to explainthe anomalously high amplitudes in the RT data.

McCONNELL BOREHOLE RADAR

Radar data acquisition and processing

The borehole radar data described here were recordedin 60-mm-diameter boreholes in June 1996 with the EcolePolytechnique omnidirectional 60-MHz RAMAC LI system.Both single-hole and crosshole configurations were used. Thetransmitter was positioned automatically using a computer-controlled electric winch.

The RAMAC system is comprised of a control unit con-nected via optical fibers to a transmitter pulser, amplifier, andantenna and to a receiver antenna, amplifier, trigger, and A/Dconverter (Olsson et al., 1992). The transmitter antenna emitsa short pulse of approximately three half-cycles duration. Thereceived wavetrain for each transmitter–receiver pair is con-structed over a window, with only one time sample recordedeach time the transmitter fires. For each of 512 time lags, 128measurements are stacked.

Although the radar data are single fold, the processing hasnevertheless been adapted from seismic processing techniquesdrawn from a number of sources, including Yilmaz (1987),Fisher et al. (1992), Liner and Liner (1995), and Young et al.(1995). Some care must be exercised when borrowing from seis-mic technology since radar signals exhibit greater attenuationand dispersion than seismic waves.

A dc drift correction, entailing subtraction of the averagevalue for each record prior to the first arrival, was performedon all traces. Subsequent processing steps included spreadingand exponential compensatory (SEC) gaining (Annan, 1993),

frequency-domain band-pass filtering, and spectral equaliza-tion. Processed crosshole reflection radargrams were migratedin an effort to delineate radar reflectors.

Radar tomography

Crosshole 60-MHz radar data were acquired between bore-holes 78 930 and 78 929 in 1996. For each of 18 transmitter posi-tions in hole 78 929, data were recorded at 25 receiver depths inhole 78 930. Transmitter and receiver spacings were both 2.5 m,comparable to the expected radar wavelength. The crossholesurvey was originally designed to image the radar velocity andattenuation above the ore deposit, primarily to gauge the ab-sorption and velocity heterogeneity of the Sudbury Breccia.The scanning pattern is not suitable for imaging the ore bound-aries.

Straight-ray tomographic inversion of first-arrival times wasperformed using Migratom. The velocity tomogram (Figure 5a)reveals a fairly high degree of heterogeneity within the breccia.A lobe of low velocity extending down from hole 78 929 towardhole 78 930 may correspond to the low-resistivity zones definedin the borehole logs: 88–95 m in 78 929 and 114–118 m in 78 930(Figure 6). The low velocity and low resistivity might indicatewater-filled fractures.

FIG. 5. (upper) Hanging-wall radar velocity tomogram be-tween boreholes 78 929 and 78 930. Text gives details regardingthe tomographic inversion process. Radar velocities are in me-ters per microsecond. (lower) Radar absorption tomogram forthe same region. Absorption coefficient in decibles per meter.


First-arrival amplitudes were corrected for spherical spread-ing and for a dipolar radiation pattern, as per equation (8),prior to SIRT reconstruction. The resulting absorption to-mogram is shown in Figure 5b. The radar absorption co-efficient in the breccia is typically 1 dB/m at 60 MHz. Ingeneral, velocity (ω/β) and absorption are anticorrelated, asexpected from equations (1) and (2), when attenuation isappreciable.

Single-hole reflection GPR imaging

The 1996 single-hole reflection surveys were undertaken inholes 78 929 and 78 930. The transmitter and receiver were sep-arated by 4.7 m. Coverage was from 75 to 130 m in hole 78 929and from 75 to 170 m in hole 78 930 at a station spacing of0.25 m. The center frequency of the received radar signals wasapproximately 48 MHz.

An SEC gain was applied to the radargrams. A velocity of125 m/µs (the average from the velocity tomogram) and an ab-sorption coefficient of 0.75 dB/m (based on the attenuation rateof trace envelopes) were adopted to specify the gain function.A band-pass filter with corner frequencies of 30 and 75 MHzwas applied to all traces to suppress long-period fluctuationsand late-time drift. Spectral equalization (Young et al., 1995),which flattens the spectrum in a manner akin to deconvolution,was also applied.

Reflection data were recorded above, within, and belowthe massive sulfide mineralization, and data amplitudes varymarkedly. The resulting radargrams for boreholes 78 929 and78 930 are presented in raw and processed form in Figure 6, withgeology and resistivity logs (Mwenifumbo et al., 1993). The re-sistivity logs are clipped at around 30 000 ohm-m because ofloss of sensitivity (negligible current).

A clear radar reflection from the massive sulfide hanging-wall contact was observed in hole 78 929 (Figure 6a). How-ever, there is no obvious reflection from the footwall contactin 78 929 nor from either contact in hole 78 930 (Figure 6b). Thiswas a surprising and, from a practical viewpoint, disappointingobservation. The main factors responsible for the absence ofreliable reflections are probably (1) unfavorable endfire ori-entation of the antennae with respect to the contact, (2) het-erogeneity of the breccia, (3) local irregularity of the sulfidecontact, and (4) minor concentrations of disseminated sulfidein and immediately adjacent to the quartz diorite.

There is total extinction of signal when either antenna iswithin the sulfide body. In addition, there are highly absorptiveintervals and zones of low resistivity in the hanging wall inboth holes. The correlation between low resistivity and lowradar amplitude is by no means perfect as plotted in Figure 6,but a depth shift of approximately 5 m would achieve a closecorrespondence. A systematic depth error in one or the otherdata set is suspected.

The combination of SEC gain and spectral equalization re-covered interpretable signal in the low-amplitude intervals.The intervals of high absorption within the breccia are notcorrelated with lithology and may signify fluid-filled fracturezones. Weak reflections, coherent for tens of meters, can bediscerned elsewhere in the breccia. These are presumably fromfractures or lithological boundaries.

Crosshole radar reflection

The crosshole first arrivals carry little or no information onthe deposit geometry. However, an attempt has been madeto focus crosshole reflections from the sulfide contact into animage via migration (Calvert and Livelybrooks, 1997).

The survey geometry is unfavorable for reception of reflec-tions from the sulfide, given that the dipole antennae are inalmost endfire orientation with respect to the hanging-wallcontact (Figure 1b). Reflection amplitudes would be furtherreduced by attenuation when one or both of the antennae arefar from the sulfide contact. When both transmitter and re-ceiver antennae are close to the sulfides, the reflection fromthe contact would be difficult to distinguish from the directarrival. Survey geometry also dictates that any crosshole re-flections from the contact will originate from a relatively smallarea straddling the point where hole 78 929 pierces the sul-fide body (Bellefleur and Chouteau, 1998). From a practicalviewpoint, therefore, the analysis of crosshole reflections is des-tined to provide limited new information about the shape of thecontact.

After suppression of the direct arrivals and application ofgeometric spreading correction, the data were migrated using a2-D Kirchhoff-style algorithm. A constant velocity of 105 m/µswas assumed. The migrated data are depicted in Figure 7.

Reflective zones between the boreholes have been identi-fied. Those in midpanel, near true depth 60 m, could be fromlithological contrasts within the breccia. Of greater interest isthe inferred reflectivity trend paralleling the sulfide contact.Given the assumptions underlying the migration (2-D geome-try, constant velocity, spherical radiation pattern), this resultis reasonably encouraging. The recording window was only623 ns; some improvement could be expected if data wererecorded over a longer interval.

Bellefleur and Chouteau (1998) have achieved an improvedresult migrating these same data by (1) restricting reflector dipto a particular range, (2) suppressing down-going waves, and(3) adopting the velocity tomogram as the migration velocitymodel. In their migrated image, the strongest reflection is co-incident with the sulfide contact.

DISCUSSION

Investigation of the efficacy of radio tomography and bore-hole radar for delineation of massive sulfide bodies com-menced at McConnell in 1993. The impetus for the researchwas the need for accurate geophysical definition of ore con-tacts, both during near- and in-mine exploration and for ore-body delineation.

Radio frequency tomography with a JW-4 electric dipole sys-tem successfully imaged the McConnell sulfide deposit over adown-dip distance of 110 m. Data were recorded at ten fre-quencies between 0.5 and 5.0 MHz. Signals propagated over150 m through Sudbury Breccia host rock, even at 5 MHz. Thesulfide deposit shape was well defined on the resulting tomo-gram as a zone of high absorption. Thus, the first RT surveyat McConnell demonstrated the potential of radio imaging tomap sulfide bodies in highly resistive host rocks. The resultswere sufficiently encouraging to prompt further RT investiga-tions in the Sudbury area by INCO (McDowell and Verlaan,1997) and, more recently, by Falconbridge (Stevens and Redko,2000).


There was complete extinction of radio signal when eitherantenna was located within the massive sulfides. However, sig-nal amplitudes were considerably stronger than expected whenthe transmitter and receiver were on opposite sides of the sul-fide body. The 2.5-D modeling indicated that diffraction aroundthe edges of the conductor was inadequate to explain the ob-served signal levels, and the conclusion at the time pointed toa break in continuity of the sulfides. According to the model-ing, a 5-m-wide gap is compatible with the observed data. Abreak or hole in the mineralization had not been previously

FIG. 6. Two 60-MHz single-hole reflection radargrams, without and with AGC, for boreholes (a) 78 929 and (b) 78 930 (after Calvertand Livelybrooks, 1997). Resistivity logs (Mwenifumbo et al., 1993) shaded according to lithology, shown at left.

interpreted geologically, though two mineralized intervals hadbeen intersected in hole 80 555 and interpreted as a reentrant(Figure 1b).

The transmitter and receiver cables were copper cored (two-conductor) and could therefore have served as parasitic anten-nae. Although the JW-4 system included a filter to isolate thereceiver, investigations by INCO since the original experimenthave confirmed that the 1994 data were affected by cross-cablepick-up. The potential of radio imaging to detect narrow gapsin conductors remains an interesting possibility, but there is no


longer any reason to invoke the existence of a gap to explainthe 1994 McConnell RT data.

GPR, especially in reflection mode, offers the potential forhigh-resolution mapping of geological contacts and structures.The single-hole borehole radar reflection surveys at McConnellin 1993 and 1996 demonstrated that the sulfide contacts onlysometimes produce strong coherent reflections. The main fac-tors responsible for the erratic occurrence of clear reflectionsare probably (1) unfavorable endfire geometry, which reducesthe S/N ratio, (2) heterogeneity of the host breccia, (3) mi-nor concentrations of disseminated sulfide in or immediatelyadjacent to the quartz diorite, and (4) local irregularity of thesulfide contact. Of these, only the first can be altered. Boreholeradar would be far more likely to succeed in holes drilled par-allel to the sulfide contact, permitting broadside illuminationof the target. This exposes the contradistinction between thegeophysical view of drillholes as accessways for instrumentsand the geological view of drillholes, as voids created duringsampling. Purpose-drilled holes for borehole radar representan additional survey cost; on the basis of the results obtainedto date, it is unclear whether the additional expense would bejustified at McConnell.

The crosshole GPR survey between boreholes 78 929 and78 930 was suitable for tomographically imaging velocity andabsorption coefficient in the breccia above the orebody. Thevelocity within the breccia exhibited significant variation, rang-ing between 95 and 145 m/µs and averaging about 125 m/µs.The absorption coefficient ranged up to 2.5 dB/m, averaging1.2 dB/m. The crosshole tomography thus confirmed that thebreccia is appreciably heterogeneous and hence less favorable

FIG. 7. Kirchhoff-style migration of crosshole radar data forthe region between boreholes 78 929 and 78 930. Text gives de-tails regarding data processing and migration. The geologicallyinferred location of the McConnell orebody is also outlined forreference (after Calvert and Livelybrooks, 1997).

for radar propagation than its high resistivity might suggest.For α = 1.2 dB/m = 0.14 neper/m and ν = 125 m/µs, it followsthat Q≈ 11 in the Sudbury Breccia. Although derived from60-MHz data, this Q value could be reasonably characteris-tic of the breccia at all radar frequencies since conductivity isapproximately proportional to frequency for many materials(Johnscher, 1977), while the real part of permittivity is oftenonly weakly dependent on frequency (Collett and Katsube,1973).

Migration of crosshole reflection data has defined a reflec-tivity feature which lies above and parallel to the contact.While a worthwhile technical achievement, the migration ofMcConnell data was destined never to greatly advance theknowledge of the contact geometry, given that reflections orig-inate near where hole 78 929 pierces the contact. However, inanother situation, between parallel boreholes drilled orthogo-nal to a contact, migration of crosshole reflection could yieldvaluable information.

CONCLUSIONS

Radio imaging is a potentially effective means for imagingmassive sulfide orebodies in resistive environments. Specifi-cally,

1) RT surveys depict orebodies as zones of enhanced atten-uation;

2) radio imaging uses lower frequencies than GPR, therebyachieving greater range but lower resolution;

3) RFEM is less sensitive to the presence of disseminatedsulfide;

4) the absence of a physical link between transmitter andreceiver is a significant logistical advantage in mining en-vironments; and

5) typically, RFEM data can be collected more rapidly thanGPR, since finer spatial sampling is required for GPR.Recording times for both RFEM and GPR could beslashed using multichannel acquisition systems.

The advantages and disadvantages of single-hole reflectionradar can be summarized as follows:

1) radar reflection surveys offer greater resolution butsmaller range than either radio imaging or crossholeradar;

2) both transmitter and receiver are in the same borehole,so data acquisition is uncomplicated and efficient;

3) radar reflection data are amenable to seismic processing,a highly developed technology;

4) reflection radar is sensitive to heterogeneity in the hostrock and to the presence of even minor concentrationsof disseminated sulfide; and

5) unless dip and strike are known a priori, or unless direc-tional antennae are deployed, surveys in more than onehole are required to overcome rotational ambiguity.

Crosshole GPR surveying shows some promise for imagingore boundaries. Its strengths and weaknesses are

1) times and amplitudes of first arrivals can be used to imageinterhole velocity and absorption;

2) direct arrivals can be distinguished from reflections anddiffractions;


3) later reflections can potentially be migrated to definestrong reflectors;

4) larger data volumes and more elaborate processing addto the cost of crosshole GPR; and

5) the need for a physical connection between the trans-mitter and receiver limits applicability, especially under-ground.

ACKNOWLEDGMENTS

This work was completed under the auspices of theNSERC/TVX Gold/Golden Knight Chair in Borehole Geo-physics for Mineral Exploration, Department of Mineral En-gineering, Ecole Polytechnique, Montreal, Canada.

The McConnell surveys were jointly funded by INCO Ltd.and Noranda; their support is gratefully acknowledged. Dataacquisition was expedited by the efforts of many INCO person-nel, while Alan King and Gord Morrison assisted with planningand interpretation. The data have been released for publica-tion by the kind permission of Larry Cochrane of INCO Ltd.,Ontario.

The Migratom tomographic software was provided by DarylTweeton and Michael Jackson (U.S. Bureau of Mines, Min-neapolis). The EMSUN modeling program was made freelyavailable by Keeva Vozoff (Harbourdom, Sydney, Australia).Geoff Smith (Univ. of Technology, Sydney, Australia) assistedwith the EMSUN modeling.

The 1993 borehole radar data were recorded by an AECLcrew, led by Kevin Stevens. The 1994 radio imaging datawere recorded by Wu Yiren, Qu Xingchang, and Zhang Zilingfrom the Chinese Ministry of Geology and Mineral Resources(MGMR), Beijing. Maree-Josee Bertrand, Noranda, assistedwith interpretation of the radio imaging data.

The paper benefitted from comments by Greg Turner andan anonymous reviewer. Mike Asten is to be commended forhis editorial persistence and patience.

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3-D inversion of induced polarization data

Yaoguo Li∗ and Douglas W. Oldenburg‡

ABSTRACT

We present an algorithm for inverting induced polar-ization (IP) data acquired in a 3-D environment. Thealgorithm is based upon the linearized equation for theIP response, and the inverse problem is solved by mini-mizing an objective function of the chargeability modelsubject to data and bound constraints. The minimizationis carried out using an interior-point method in which thebounds are incorporated by using a logarithmic barrierand the solution of the linear equations is acceleratedusing wavelet transforms. Inversion of IP data requiresknowledge of the background conductivity. We studythe effect of different approximations to the backgroundconductivity by comparing IP inversions performed us-ing different conductivity models, including a uniformhalf-space and conductivities recovered from one-pass3-D inversions, composite 2-D inversions, limited AIMupdates, and full 3-D nonlinear inversions of the dc resis-tivity data. We demonstrate that, when the backgroundconductivity is simple, reasonable IP results are obtain-able without using the best conductivity estimate derivedfrom full 3-D inversion of the dc resistivity data. As a fi-nal area of investigation, we study the joint use of surfaceand borehole data to improve the resolution of the re-covered chargeability models. We demonstrate that thejoint inversion of surface and crosshole data produceschargeability models superior to those obtained frominversions of individual data sets.

INTRODUCTION

In recent years, there has been much progress in rigorous in-version of induced polarization (IP) data assuming a 2-D earthstructure. Published work on 2-D inversions has demonstratedthat inversion can help extract information that is otherwiseunavailable from direct interpretation of the pseudosections.

Manuscript received by the Editor February 23, 1999; revised manuscript received June 2, 2000.∗Formerly University of British Columbia, Department of Earth and Ocean Sciences; presently Colorado School of Mines, Department of Geo-physics,1500 Illinois St., Golden, Colorado 80401. E-mail: [email protected].‡University of British Columbia, Department of Earth and Ocean Sciences, 2219 Main Mall, Vancouver, B.C. V6T1Z4, Canada. E-mail: [email protected]© 2000 Society of Exploration Geophysicists. All rights reserved.

In application, the technique has matured sufficiently that it isnow routinely applied to data sets acquired in mineral explo-ration projects and in environmental problems.

The 2-D IP data are commonly inverted using a linearizedapproach (LaBrecque, 1991; Oldenburg and Li, 1994), in whichthe chargeability is assumed to be relatively small and the ap-parent chargeability data are expressed as a linear functional ofthe intrinsic chargeability. A linear inverse problem is solved toobtain the chargeability model. In addition to the linearized ap-proach, Oldenburg and Li (1994) also propose two other meth-ods. The second obtains the chargeability by performing twoseparate dc resistivity inversions and then taking the relativedifference of the recovered conductivities. The third methodmakes no assumption about the magnitude of the chargeabilityand performs a full nonlinear inversion to construct its distri-bution.

The effectiveness of IP inversions has been documented inseveral case histories (e.g., Oldenburg et al., 1997; Kowalczyket al., 1997; Mutton, 1997). When the data set is acquired in atruly 2-D environment, the inversion algorithm has performedwell. However, 2-D inversions face difficulties when the basic2-D assumption is violated because of the use of 3-D acquisitiongeometry or the presence of a 3-D geoelectrical structure suchas severe 3-D topography or 3-D variation of conductivity andchargeability. Under these circumstances, a 3-D algorithm isrequired.

The methods developed in 2-D are general and applicable to3-D problems. For instance, recovering chargeability by com-puting the difference between two conductivity inversions isdemonstrated by Ellis and Oldenburg (1994) using pole–poledata. The implementation of the linearized approach is alsostraightforward in principle; however, numerical and compu-tational challenges require specific treatment. The foremostchallenge is the computational complexity related to generat-ing background conductivity in three dimensions and the so-lution of the large-scale constrained minimization problem toconstruct the 3-D chargeability model. This paper concentrateson these associated computational issues. We assume that the

1931

1932 Li and Oldenburg

chargeability is small and that the data are not affected by EMcoupling effect. Therefore, we adopt the linearized representa-tion of the IP response and develop the inversion methodologyapplicable for general electrode configurations, including sur-face arrays, downhole arrays, and crosshole electrode configu-rations. We present a detailed algorithm that solves large-scaleproblems.

Our paper begins with a summary of the basics of IP in-version and the formulation of the inverse solution. The useof approximate conductivity models in the 3-D IP inversionis discussed next to demonstrate how an efficient IP solutioncan be obtained in practice. We then study the joint inversionof surface and crosshole data and its improvement in modelresolution. We conclude with an application to a field data setand a discussion.

BACKGROUND

The commonly used electrode configurations in most explo-ration work include the pole–pole, pole–dipole, dipole–dipole,and gradient arrays. These arrays are usually arranged in a co-linear configuration, and the source and potential electrodesare generally aligned parallel to the traverse direction. How-ever, to image a 3-D structure, truly 3-D data are often needed.This requires that off-line or cross-line data be acquired andthat the orientation of the current electrodes be varied. In addi-tion, high-resolution surveys carried out in ore delineation andgeotechnical investigations often acquire surface-to-boreholeand crosshole data in three dimensions. Thus, a generally ap-plicable inversion algorithm must be able to work with arbi-trary electrode configurations. In this paper, we assume that thetime-domain IP measurements are acquired using an arbitraryelectrode geometry over a 3-D structure. The current sourcecan be a single pole, dipole, or widely separated bipole eitheron the earth’s surface or in boreholes. The resulting potentialor potential difference can be measured as data anywhere onthe surface or in the borehole. The commonly used pole–pole,pole–dipole, and dipole–dipole arrays on the surface or in theborehole constitute only a small number of possible configura-tions.

Let σ (r) be the conductivity as a function of position in threedimensions beneath the earth’s surface and η(r) be the charge-ability as defined by Seigel (1959). The dc potential producedby a current of unit strength placed at rs is governed by thepartial differential equation

∇ · (σ∇φσ ) = −δ(r − rs), (1)

where φσ denotes the potential in the absence of IP effect.When the chargeability is nonzero, it effectively decreases theelectrical conductivity of the media by a factor of (1−η)(Seigel,1959). The corresponding total potential φη is given by

∇ · (σ (1 − η)∇φη) = −δ(r − rs). (2)

Thus, the secondary potential measured in an IP survey is givenby the difference

φs = φη − φσ , (3)

while the apparent chargeability is defined

ηa = φη − φσ

φη

. (4)

The apparent chargeability is the preferred form of IP data,and it is well defined in some surface and downhole surveys.

However, in crosshole experiments using dipole sources orreceivers, the electric field often reverses direction along theborehole, and the measured total potential differences can ap-proach zero in the vicinity of the zero crossing. The zero cross-ing can also occur with noncolinear arrays on the surface. Thesenear-zero potentials cause the apparent chargeability to be un-defined. It is therefore necessary to use the secondary potentialas data when these conditions occur.

When the magnitude of the chargeability is moderate, thesecondary potential φs measured in an IP experiment is wellapproximated by a linear relationship with the intrinsic charge-ability. Applying a Taylor expansion to equation (3), neglect-ing higher order terms, and discretizing the earth into cellsof constant conductivity σ j and chargeability η j results inthe following equation (e.g., Seigel, 1959; Oldenburg and Li,1994):

φsi =M∑j=1

−η j∂φηi

∂ ln σ j≡

M∑j=1

η j Jφ

i j , (5)

where J φ

i j is the sensitivity of the secondary potential φsi andφηi is the corresponding total potential. If the total potentialsdo not approach zero, the linearized equation for apparentchargeability ηa is given by

ηai =M∑j=1

−η j∂ ln φηi

∂ ln σ j≡

M∑j=1

η j Jη

i j , (6)

where J η

i j is the corresponding sensitivity. Note that J η

i j is un-defined when φηi approaches zero.

Given a set of measured IP data, inversion of either equa-tion (5) or (6) allows the recovery of the intrinsic chargeabil-ity model. Since the true conductivity structure is unknown inpractical applications, an approximation to it is substituted incalculating the sensitivities. This approximation is usually ob-tained by inverting the accompanying dc potential data. Thus,the IP inverse problem is a two-stage process. In the first stage,an inverse problem is solved to recover a background conduc-tivity from the dc resistivity data. This conductivity is then usedto generate the sensitivity for the IP inversion, and a linear in-verse problem is solved to obtain the chargeability.

FORMULATING THE INVERSION

Assume we have a set of N IP data, which can be apparentchargeabilities or secondary potentials. Further assume that adc resistivity inversion has been performed (see next section)to obtain a reasonable approximation to the true conductivity;the IP sensitivity is calculated from it. To invert these IP datafor a 3-D model of chargeability, we first use the same mesh asin the dc resistivity inversion to divide the model region intoM cells and assume a constant chargeability value in each cell.The data are formally related to the chargeabilities in the cellsby the relation in equations (5) and (6),

d = Jη, (7)

where the data vector d = (d1, . . . , dN )T and the model vectorη = (η1, . . . , ηM)T . J is the sensitivity matrix corresponding tothe data, whose elements Ji j are calculated from the assumedapproximation to the background conductivity by using anadjoint equation approach (McGillivary and Oldenburg, 1990).For this calculation and all the numerical simulations in this

3-D Inversion of IP Data 1933

paper, we use the finite-volume method (Dey and Morrison,1979) to solve equation (1) to obtain the electrical potentials.

The number of model cells is generally far greater than thenumber of data available; thus, an underdetermined problemis solved. To obtain a particular solution, we minimize a modelobjective function, subject to the data constraints in equa-tion (7). We used a model objective function that is similarto that for the 2-D case but that has an extra derivative term inthe third dimension. Let m = η generically denote the model.The objective function is given by

ψm = αs

∫V

(m − m0)2 dv + αx

∫V

{∂(m − m0)

∂x

}2

dv

+ αy

∫V

{∂(m − m0)

∂y

}2

dv + αz

∫V

{∂(m − m0)

∂z

}2

dv,

(8)where m0 is a reference model. The positive scalars αs , αx , αy ,and αz are coefficients that affect the relative importance of thedifferent components. We usually choose αs to be much smallerthan the other three coefficients, so the recovered model be-comes smoother as the ratios αx/αs , αy/αs , and αz/αs increase.For numerical solutions, equation (8) is discretized using thefinite-difference approximation. The resulting matrix equationhas the following form;

ψm = (m − m0)T(αsWT

s Ws + αxWTx Wx + αyWT

y Wy

+ αzWTz Wz

)(m − m0)

≡ ‖Wm(m − m0)‖2. (9)

The data constraints are satisfied by requiring that the totalmisfit between the observed and predicted data be equal to atarget value. We measure the data misfit using the function

ψd = ∥∥Wd(dpre − dobs

)∥∥2, (10)

where dpre and dobs are, respectively, predicted and observeddata and where Wd is a diagonal matrix whose elements arethe inverse of the standard deviation of the estimated error ofeach datum: Wd = diag{1/ε1, . . . , 1/εN }. If we assume that thecontaminating noise is independent Gaussian noise with zeromean, then ψd has X 2 distribution with N degrees of freedom,and its expected value is equal to N . Thus, a reasonable targetvalue is ψ

d = N .In addition to the data constraints, we also need to impose a

lower and an upper bound on the recovered chargeability. Thebounds are required because the chargeability is defined in therange [0,1). The bound constraints ensure that the recoveredmodel is physically plausible. For numerical implementation,the lower bound must be zero since the chargeability of thegeneral background is zero. The upper bound, denoted by u,can take on the theoretical value of unity or can be smaller ifa better estimate of the upper bound is known.

Having defined the model objective function, the data misfitand its expected value, and the appropriate bounds, we nowsolve the inverse problem of constructing the 3-D chargeabilitymodel by the Tikhonov regularization method (Tikhonov andArsenin, 1977) with additional bound constraints:

minimize ψ = ψd + µψm

subject to 0 ≤ m < u, (11)

where µ is the regularization parameter that controls the trade-off between the model norm and misfit. Ultimately, we wantto choose µ such that the data misfit function is equal to a pre-scribed target value ψ

d . The minimization is solved when a min-imizer m is found whose elements are all within the bounds.

This is a quadratic programming problem, and the main dif-ficulties arise from the presence of the bound constraints. Weuse an interior-point method to perform the minimization. Theoriginal problem in equation (11) is solved by a sequence ofnonlinear minimizations in which the bound constraints areimplemented by including a logarithmic barrier term in theobjective function (e.g., Gill et al., 1991; Saunders, 1995):

B(m, λ) = ψd + µψm − 2λ

{M∑j=1

ln(m j

u

)

+M∑j=1

ln(

1 − m j

u

)}, (12)

where λ is the barrier parameter and the regularization pa-rameter µ is fixed during the minimization. The minimizationstarts with a large λ and an initial model whose elements arewell within the lower and upper bounds. It then iterates to thefinal solution as λ is decreased toward zero. As λ approacheszero, the sequence of solutions approaches the model that min-imizes the original total objective function ψ in equation (11).Since we are only interested in the final solution, we do notcarry out the minimization completely for each value of λ inthe decreasing sequence. Instead, we take only one Newtonstep and limit the step length during the model update to keepthe model within the bounds throughout the minimization. Thesteps of the algorithm are as follows:

1) Set the initial model m and the µ, and calculate the start-ing value of the barrier parameter by

λ = ψd + µψm

−2M∑j=1

[ln

(m j

u

)+ ln

(1 − m j

u

)] . (13)

2) Take one Newton step for each value of λ by solving thefollowing equation for a model perturbation �m:(

JTWTd WdJ + µWT

mWm + λX−2 + λY−2)�m

= −JTWTd Wd(d − dobs) − µWT

mWm(m − m0)

+ λ(X−1 − Y−1)e, (14)

where X = diag{m1, . . . ,mM }, Y = uI − X, ande = (1, . . . , 1)T .

3) Determine the maximum step length of the model updatethat satisfies the bounds:

ρ− = min�m j<0

m j

|�m j | ,

ρ+ = min�m j>0

u − m j

�m j,

ρ = min(ρ−, ρ+). (15)


4) Update the model and barrier parameter by the limitedstep length:

m ← m + γρ�m,

λ ← [1 − min(γ, ρ)]λ. (16)

5) Return to step 2 and iterate until convergence accordingto the criteria that ψd + µψm has reached a plateau andthe barrier term is much smaller than this quantity. Theparameter γ is usually prescribed to be a value close tounity. Its role is to prevent model elements from reach-ing the bounds exactly so that the logarithmic barrieriteration can continue. Values between (0.99, 0.999) havebeen commonly used in literature (e.g., Gill et al., 1991).Our experience with IP inversions suggests that a slightlysmaller value works better, so we have typically used0.925 in our algorithm.

The central task of the algorithm is solving the linear sys-tem in equation (14). We obtain the solution by using the con-jugate gradient (CG) technique. The reasons for using a CGsolver are twofold. First, any practical application will require alarge number of cells (at least on the order of 104) to representthe geology reasonably. As a result, the linear system in equa-tion (14) is large, and explicit formation of JTWT

d WdJ is im-practical. This precludes the use of any direct solver. The CGtechnique is the obvious choice for an iterative solver since thematrix (JTWT

d WdJ + µWTmWm) is symmetric. Also, each sub-

problem at a given value of barrier parameter only generatesone step among a sequence that leads to the final solution. It isunnecessary to solve equation (14) precisely. Instead, it is com-mon to solve the central equation approximately to producea partial solution. The resulting update is called a truncatedNewton step. This is designed to reduce the required amountof computation without compromising the quality of the finalsolution. The CG technique can be terminated at an early stageby supplying it with a relaxed stopping criterion. We have typ-ically used a criterion that the ratio of the norm of the residualand the norm of the right-hand side in equation (14) be lessthan 10−2. This has led to large computational savings.

CG iterations require the repeated multiplication of the ma-trix JTWT

d WdJ and WTmWm to vectors. The matrix WT

mWm isextremely sparse, and the matrix–vector multiplication is eas-ily obtained. However, applying JTWT

d WdJ is computationallyintensive since J is dense. We perform a fast matrix–vectormultiplication by using wavelet transforms (Li and Oldenburg,1999a), in which a sparse representation of the dense matrix isformed in the wavelet domain and matrix–vector multiplica-tion is carried out by sparse multiplications.

The remaining issue is how to determine the optimal valueµ so that the data constraint ψd = ψ

d is satisfied. There aretwo situations that require different treatments. In the first,a reliable estimate is available for the standard deviation ofthe errors that have contaminated the data and, therefore, thevalue ofψ

d is known. We then need to find the value of µ thatyields this target misfit. This is achieved by an efficient line-search technique that uses a number of approximate solutionsto the minimization problem. In the second case, the standarddeviations of errors are unknown; hence, the optimal valueof µ must be estimated independently. We achieve this withthe generalized cross-validation technique (Golub et al., 1979;

Wahba, 1990; Haber and Oldenburg, 2000). The use of thesetechniques in large-scale 3-D inversions is detailed in Li andOldenburg (1999a).

We now illustrate our algorithm using a test model com-posed of five anomalous rectangular prisms embedded in a uni-form half-space (Figure 1). Three surface prisms simulate near-surface distortions, and two buried prisms simulate deeper tar-gets. The conductivity and chargeability of the prisms are listedin Table 1. The dc resistivity and IP data from both surface andcrosshole experiments have been computed.

The surface experiment is carried out using a pole–dipolearray with a= 50 m and n= 1 to 6. There are seven traversesspaced 100 m apart in both east–west and north–south direc-tions. There are 1384 observations, and these have been con-taminated with uncorrelated Gaussian noise whose standarddeviation is equal to 2% of the datum value. Figures 2 and 3show apparent conductivity pseudosections and apparentchargeability pseudosections at three selected east–west tra-verses. The pseudosections are dominated by the responses tothe near-surface prisms, and there are only subtle indicationsof the buried conductive prism.

We first inverted the dc resistivity data using a Gauss-Newtonapproach that constructs a minimum structure model using amodel objective function similar to that in equation (8) butapplied to the logarithmic conductivity as the model. We setthe coefficients to αs = 0.0001 and αx = αy = αz = 1 and used areference conductivity model of 1 mS/m. The recovered con-ductivity model is shown by two plan sections and one crosssection in Figure 4. It is a reasonably good representation ofthe true conductivity model. All three surface prisms and theburied conductive prism are clearly imaged, and there is indi-cation of a resistive prism at depth. This conductivity model isthen used to calculate the sensitivity for the subsequent IP in-version. The inverted chargeability model is shown in Figure 5in the same plan and cross-sections. The surface prisms areclearly imaged, and the chargeability at depth is concentratedat the location of the two buried targets. The separation of thesebodies is not clearly defined, but this decrease of anomaly def-inition with increasing depth is expected when surface dataare inverted. Overall, the model is a good representation ofthe true anomalous chargeability zone. The contrast betweenthe pseudosections shown in Figure 3 and the cross-section ofthe recovered model in Figure 5 illustrates the improvementgained by performing the 3-D inversion.

CONSTRUCTION OF APPROXIMATE CONDUCTIVITIES

As discussed in the preceding section, the inversion of IPdata requires a background conductivity model for calculatingthe sensitivity. IP inversion is therefore a two-stage process,and its success depends upon the availability of a conductivitymodel that is a reasonable approximation to the true conduc-tivity. The usual approach to generating such a conductivitymodel is to invert the dc resistivity data that accompany theIP data. Numerous papers have been published on 3-D dc re-sistivity inversions, and different approaches have been pro-posed. For example, Park and Van (1991), Ellis and Oldenburg(1994a), Sasaki (1994), Zhang et al. (1995), and LaBrecque andMorelli (1996) all perform linearized inversion to construct aconductivity model from the dc resistivity data, although de-tails of their algorithms and implementations may vary greatly.


Li and Oldenburg (1994), on the other hand, apply approxi-mate inverse mapping (AIM) formalism to construct a modelthat reproduces the data. For the current study, we implementa regularized inversion and use Gauss-Newton minimizationto accomplish this (Li and Oldenburg, 1999b). The basics ofthat algorithm are summarized here.

Let m= ln σ define the model used in the conductivity in-version, and let dobs be the dc potential data. As in the IP inver-

FIG. 1. Perspective view of the five-prism model. Seven surfacetraverses in the east–west direction and four boreholes are alsoshown. For clarity, seven traverses in the north–south directionare not shown. The physical property values of the prisms arelisted in Table 1.

FIG. 2. Examples of the apparent conductivity pseudosectionsat three east–west traverses. The data are simulated for apole–dipole array, and they have been contaminated by in-dependent Gaussian noise with a standard deviation equal to2% of the accurate datum magnitude. The pseudosections aredominated by the surface responses. The grayscale shows theapparent conductivity in mS/m.

sion, we minimize the model objective function in equation (8)subject to fitting the data to the degree determined by the es-timated error. Thus, the desired conductivity model solves thefollowing minimization problem:

minimize ψ = ψd + µψm

subject to ψd = ψ∗d , (17)

where ψd and ψm are the same as those defined in equations (8)and (10), µ is the regularization parameter, and ψ∗

d is the targetmisfit value for the dc resistivity problem.

The potential data depend nonlinearly upon the conductiv-ity; hence, minimization (17) must be solved iteratively. Let mbe the current model, d its predicted data, and �m a model per-turbation. Performing a first-order Taylor series expansion ofthe predicted data as a functional of the new model m+�m andsubstituting into the total objective function in equation (17),we obtain

ψ(m + �m) ≈ ∥∥Wd(d + J�m − dobs)∥∥2

+ µ‖Wm(m + �m − m0)‖2. (18)

Table 1. Conductivity and chargeability of the prisms. Thehalf-space has a conductivity of 1 mS/m and zero chargeability.

Prism Conductivity (mS/m) Chargeability (%)

S1 10 5S2 5 5S3 0.5 5B1 0.5 15B2 10 15

FIG. 3. Examples of the apparent chargeability pseudosectionsalong three east–west traverses. The data have been contami-nated by independent Gaussian noise with a standard deviationequal to 2% of the accurate datum magnitude plus a minimumof 0.001. The same masking effect of near-surface prisms ob-served in apparent-conductivity pseudosections is also presenthere. The grayscale shows the apparent chargeability multi-plied by 100.


The value J is the sensitivity matrix of the potential data. Itselements are given by

Ji j = ∂φi

∂ ln σ j(19)

and are evaluated at the current model. Differentiating withrespect to �m and setting the derivative to zero yields theequation for the model perturbation:

(JTWT

d WdJ + µWmWm)�m

= −JTWTd Wd(d − dobs) − µWT

mWm(m − m0). (20)

FIG. 4. The conductivity model recovered from inversion ofsurface data using a Gauss-Newton method. The model isshown in one cross-section and two plan sections. The posi-tions of the true prisms are indicated by the white lines.

This is the basic equation solved for a Gauss-Newton step.The new model is then formed by updating the current model:m ← m + �m. This process is repeated iteratively until theminimization converges and an optimal value of µ is found toproduce the desired data misfit in equation (17).

The nonlinear inversion of 3-D dc resistivity data providesthe best approximation to the actual conductivity distribution,but it is a costly undertaking. One may not always want toexpend that amount of computation, especially when the re-covery of the conductivity model is but an intermediate step to-ward the end goal of constructing a chargeability model. Moreimportantly, good IP inversion results are often obtained byusing less rigorous approximations to the conductivity. Our

FIG. 5. The chargeability model recovered from inversion ofsurface data. The conductivity from full 3-D dc inversion isused to calculate sensitivities. The positions of the true prismsare indicated by the white lines.


experience with 2-D inversions (Oldenburg and Li, 1994) hasshown that good first-order results concerning the chargeabilitydistribution can often be obtained by approximating the earthusing a homogeneous conductive half-space. This suggests thata reasonable recovery of a 3-D chargeability model might beachieved by using intermediate approximations between thetwo end members corresponding to a uniform half-space andthe conductivity model recovered from a full nonlinear 3-D dcinversion.

To explore this, we compare five options for generating aconductivity model to be used in the IP inversion. The firstfour require much less computation than does the full 3-D in-version:

1) A uniform half-space: This is the simplest approximation,and no inversion of dc data is involved. When invert-ing apparent chargeability data, the actual value of thehalf-space conductivity is arbitrary since the sensitivity isindependent of it. However, when the secondary poten-tials are inverted, the best fitting half-space from the dcresistivity data should be used.

2) One-pass approximate 3-D inversion: This conductivitymodel is obtained from a linear inversion of the dcdata assuming that the actual conductivity consists ofweak perturbations of a uniform half-space (e.g., Li andOldenburg, 1994; Mø/ller et al., 1996). Such a modelcaptures the gross features in the conductivity structureand demands the least amount of computation. We haveimplemented the approximate inversion in the spatialdomain, in which the model objective function in equa-tion (8) is minimized explicitly so that a minimum struc-ture model is obtained. This is identical to performing thefirst iteration of the Gauss-Newton inversion with boththe initial and reference model being equal to the chosenbackground, mb. The equation to be solved is(

JTWTd WdJ + µWmWm

)�m

= −JTWTd Wd(d − dobs), (21)

where d is the predicted data. The approximate conduc-tivity is given by m = mb + �m.

3) Composite 2-D inversions: When surface data along par-allel traverses are available, independent 2-D inversionscan be carried out along each line so that a 2-D model thatreproduces the observations is generated (Oldenburget al., 1993; Loke and Barker, 1996). The series of 2-Dmodels are then combined to form a 3-D representationof the true conductivity structure. Such a model shouldperform well when there are strong 2-D features in thedata.

4) Limited 3-D AIM updates: Using the one-pass 3-D inver-sion as an AIM, we can iteratively update the conductivitymodel by the AIM algorithm (Oldenburg and Ellis, 1991)

Table 2. List of the dc and IP misfit for different approximations to the background conductivity. The dc misfit is calculated betweenthe observed dc resistivity data and the predicted data obtained from 3-D forward modeling of the approximate conductivities. TheIP misfit is the value achieved by the IP inversion when an approximate conductivity is used to calculate the sensitivity.

Misfit Half-space One-pass 3-D approximation 2-D composite Limited AIM updates Nonlinear inversion

DC 2.96 × 106 6.44 × 104 1.22 × 105 6.96 × 103 1.35 × 103

IP 2184 1720 1637 1634 1510

such that a final model reproducing the 3-D observationsis constructed. The greatest misfit reduction is achievedwithin the first two or three iterations (Li and Oldenburg,1994). Thus, by performing only a limited number of AIMupdates, we can obtain a conductivity model for the IPinversion. Let F−1 denote the one-pass approximate in-version and m be the current model. The model pertur-bation is defined by the difference between models gen-erated by applying the approximate inverse mapping tothe observed and predicted data, respectively:

m ← m + F−1[dobs] − F−1[d], (22)

where d is the predicted data from the current model.The iteration starts with an initial model which can besupplied by m = F−1[dobs].

5) Full 3-D nonlinear inversion: We use the Gauss-Newtoninversion discussed at the beginning of this section. Thisapproach provides the best approximation to the conduc-tivity, but it is the most computationally intensive. Eachiteration requires calculation of the sensitivity and sev-eral additional forward modelings.

The relative merits of these five methods will probably de-pend upon the complexity of the actual conductivity distribu-tion. A general statement may therefore be difficult to make,but insight can be obtained from applications to specific datasets. We have applied these five methods to the inversion ofour synthetic test data set shown in Figures 2 and 3.

We first focus upon generating the approximate conductiv-ities. The composite 2-D conductivity was obtained by invert-ing the data from the seven east–west lines using a 2-D algo-rithm and stitching together the resulting 2-D conductivities toform a 3-D model. The one-pass approximate 3-D inversionwas carried out using a uniform background of 1 mS/m and amodel objective function with αs = 0.0001 and αx = αy = αz = 1;we chose an optimal regularization parameter by the L-curvecriterion (Hansen, 1992) to account for both the linearizationerror and the added random errors. The selected regularizationparameter, together with the objective function, also definedthe approximate inverse mapping. We performed two itera-tions of AIM updates to produce the AIM approximation ofthe conductivity. Last, we had the conductivity model from afull 3-D inversion (Figure 4). For comparison, we have listedin Table 2 the data misfit between the observed dc potentialdata and the predicted data obtained by applying 3-D forwardmodeling to each of the five approximate conductivity mod-els. A comparison of these models with the true conductivitymodel is shown in Figure 6. We selected the cross-section atnorthing = 475 m, which passes through four of the five prismsin the model. The four models from different inversions showdifferent levels of detail about the conductivity anomaly, andthey present a general progression toward better representa-tions of the true model. However, the improvement diminishes


as the approximation approaches the best model that is ob-tained from the full 3-D inversion. Although the inverted con-ductivity models are similar, there is a substantial differencebetween the true model and any one of these approximations.

Using these five approximations to calculate the sensitivities,we performed five different inversions of the IP data. Sincesome of the conductivity models are poorer approximations,the corresponding IP inversions are not expected to achievethe expected data misfit. Instead, we chose an optimal regular-ization parameter for each inversion according to a generalizedcross-validation criterion. The result is that different inversionsmisfit the observed IP data by different amounts (Table 2).The resulting chargeability models are compared with the truemodel in Figure 7. Each panel in that figure is the cross-sectionof the recovered chargeability model at northing 475 m. Allfive models recover the essential features of the true model,and they present a general trend of improvement as the ap-proximation to the background conductivity improves. How-ever, the improvement in the recovered chargeability model isnot proportional to the increased computational cost involvedin constructing a better conductivity approximation. Less rig-orous approximations of conductivity which require much lesscomputation have produced good representations of the truechargeability model.

FIG. 6. Comparison between the five approximate conductivity models with the true conductivity. All sections are atnorthing = 475 m, which passes through four of the five prisms. The positions of the true prisms are outlined by the white boxes. Asthe approximation improves, the inverted conductivity model is a better representation of the true model.

JOINT INVERSION OF SURFACE AND CROSSHOLE DATACrosshole data have been used to achieve higher resolution

image of the subsurface structure obtained from dc resistivityand IP experiments (e.g., Spies and Ellis, 1995; LaBrecque andMorelli, 1996). However, although crosshole data are sensitiveto the vertical variation of conductivity and chargeability, theyhave rather poor sensitivity to the lateral variation because thedata have limited spatial distribution and the array separationis restricted to a small range. Surface data, however, usuallyhave good areal coverage and therefore possess better resolv-ing power for determining lateral variations in the subsurfacestructure. Surface data can provide good complementary in-formation to the crosshole data if the targets are within thedepth of penetration of the surface arrays. Joint inversion ofthese two data sets was expected to improve the resolution ofthe recovered chargeability model.

We placed four vertical boreholes around the anomalous re-gion in the test model. The locations are shown in Figure 1. Wesimulated crosshole data from a pole–dipole tomographic ex-periment. Current sources were placed along the source holefrom 0 to 400 m depth at an interval of 25 m. For each currentlocation, potentials in another borehole (receiver hole) weremeasured with a 50-m dipole at an interval of 25 m betweenz= 0 and 400 m. Figure 8 illustrates the electrode configuration


between two holes. Only one borehole in any pair of boreholeswas used as the source hole, and the reverse configuration ofswitching the source and receiver holes was not used. This re-sulted in six independent pairs of source–receiver holes.

A total of 1530 observations were generated for both dc andIP experiments using this configuration. Because of the pres-ence of zero crossings in the measured total potentials, we usedthe secondary potential, instead of apparent chargeability, asthe IP data. The data were contaminated with independentGaussian noise. The standard deviation for dc potentials wasequal to 2% of each accurate datum; for secondary potentialsit was equal to 5% of each accurate datum plus a minimumof 0.1 mV. (All the potentials were normalized to unit currentstrength.) Figure 9 displays the crosshole dc data as the appar-ent conductivity between two pairs of boreholes. The verticalaxis of the plot indicates the position of the current electrode inthe source hole, and the horizontal axis indicates the midpointof the potential dipole in the receiver hole. The data plots areremarkably featureless, and identification of individual prismsin the true model is impossible. Figure 10 displays the secondarypotentials in the same two pairs of boreholes. Again, there isno distinct feature in the secondary potential plots. The lackof distinct features in the borehole data is a direct indicationof the data’s poor sensitivities to the lateral variations in theconductivity and chargeability distributions.

FIG. 7. Comparison of chargeability models recovered from the 3-D inversion of surface IP data using five different approximationsto the background conductivity. The process by which each conductivity approximation is obtained is shown in each panel. Thelower-right panel is the true chargeability model.

Next, we compared the chargeability models obtained frominverting the crosshole data alone with the model obtainedby jointly inverting the surface and crosshole data. For thisstudy, we inverted the dc resistivity data using the full nonlinearinversion so that the best conductivity approximation at ourdisposal was used for the sensitivity calculation. In both dc andIP inversions, we chose a model objective function by settingthe coefficients to αs = 0.0001 and αx = αy = αz = 1. A uniformhalf-space of 1 mS/m was used as the reference model for dcresistivity inversion, and a zero reference model was used forthe IP inversion. In the inversions, the known values of the errorstandard deviations were used, and the target misfit value wasset to the number of data points being inverted. All inversionsconverged to the expected misfit value.

Figure 11 shows the conductivity model obtained from in-verting the crosshole dc resistivity data, displayed in two plansections and one cross-section. This model is a crude represen-tation of the true conductivity. Only the large surface conduc-tor and the buried conductor are identified, and the recoveredanomaly amplitude is very low. We used this conductivity in theinversion of crosshole IP data and the recovered chargeabilitymodel (Figure 12). This model is a poor representation of thetrue chargeability. Anomalies are recovered near the surface,but they do not correspond to the locations of the true prisms.The two deep prisms are marginally identified. The vertical


FIG. 8. Crosshole electrode configuration for collecting tomo-graphic data. The current source A in hole B moves at an in-terval of 25 m from z = 0 m to z = 400 m. For each currentlocation, the potential electrodes M and N , separated by 50 m,measure the potential differences in hole D. The midpoint ofthe potential dipole moves at an interval of 25 m from z = 25 mto z = 375 m. For a given pair of holes, no data are collectedby interchanging the current and potential holes.

FIG. 9. The crosshole plot of apparent-conductivity data. The vertical axis is the location of the current source in the source hole,and the horizontal axis is the midpoint of the potential dipole in the receiver hole. The left panel is for data between holes C andA, and the right panel is for data between holes B and D, as shown in Figure 1.

FIG. 10. The crosshole plot of secondary potential data in the same format as the crosshole apparent conductivity plots in Figure 8.The potentials are normalized to unit current strength, and the grayscale indicates the value in millivolts.

extent is well imaged, as would be expected from boreholedata, but the orientations and horizontal boundaries of therecovered anomalies differ from those of the true model. Inaddition, there is excessive structure in the region immediatelysurrounding the boreholes. This is typical when crosshole dataare inverted unless special weighting in the objective functionis included to counter it.

We next jointly inverted the surface data in Figures 2 and 3and the crosshole data in Figures 8 and 9. Figure 13 displaysthe recovered conductivity model from the joint inversion ofsurface and borehole data. This model is dominated by the fea-tures recovered in the surface data inversion. The minor im-provements are the increased amplitude and the slightly betterdefinition of the depth extent of the conductivity prism. Usingthis model we calculated the sensitivity and then performed thejoint inversion of the two IP data sets to recover the chargeabil-ity model shown in Figure 14. It shows dramatic improvementcompared with the models from individual inversions in Fig-ures 5 and 12. All five prisms are well resolved, and artifactssurrounding the boreholes are minimized. The most noticeableimprovement is the clear image of the two separate buried tar-gets. The recovered amplitudes, positions, and orientations ofthe two anomalies all correspond well with the true model.


FIELD EXAMPLE

As our last example, we illustrate the 3-D inversion algo-rithm using a set of pole–dipole data from the Mt. Milligancopper–gold porphyry deposit in central British Columbia,Canada. These data were first analyzed by Oldenburg et al.(1997) using a 2-D algorithm. We invert them using the 3-Dalgorithm, which illustrates the 3-D inversion in a mineral ex-ploration setting and provides a comparison with the resultfrom a series of 2-D inversions.

The Mt. Milligan deposit lies within the Early MesozoicQuesnel terrane, which hosts a number of Cu-Au porphyry de-posits, and it occurs within porphyritic monzonite stocks andadjacent volcanic rocks. The initial deposit model consists of avertical monzonitic stock, known as the MBX stock, intruded

FIG. 11. Conductivity model recovered from the crosshole dataalone. This model shows an elongated conductor on the sur-face and a broad conductor at depth. Both conductors are sur-rounded by resistive halos, and the amplitude is small.

into volcanic host rocks. Dykes extend from the stock andcut through the porous trachytic units in the host. Emplace-ment of the monzonite intrusive is accompanied by intensivehydrothermal alteration primarily near the boundaries of thestock and in and around the porous trachytic units cross-cutby monzonite dykes. Potassic alteration, which produced chal-copyrite, occurs in a region surrounding the initial stock, and itsintensity decreases away from the boundary. Propylitic alter-ation, which produces pyrite, exists outward from the potassicalteration zone. Strong IP effects are produced by these alter-ation products, and the IP survey is well suited for mappingthe alteration zones. The pole–dipole dc resistivity and IP sur-veys over Mt. Milligan were carried out along east–west linesspaced 100 m apart. The dipole length was 50 m, and n-spacingwas from 1 to 4. This yielded 946 data points along 11 lines in

FIG. 12. Chargeability model recovered from the crossholedata alone. This model shows little resolution near the sur-face where the anomalies are confined to small volumes. Thedeeper anomalies are identified but not well resolved. As ex-pected, the depth extent of the buried chargeable bodies is welldelineated.


our study area of 1.2 × 1.0 km. This area, directly above theMBX stoc, has a gentle surface topography, and the total reliefis about 100 m. Figure 15 displays the apparent chargeabilitydata in plan maps of constant n-spacings. For brevity, we havenot shown the dc resistivity data here. The apparent chargeabil-ity data show large anomalies toward the western and southernregions. The north-central region of low apparent chargeabilityis related to the intrusive stock that has significantly less sulfidefrom the alteration processes.

To invert these data, we used a mesh that consisted of cells25 m wide in both horizontal directions and 12.5 m thick inthe region of interest. The mesh was extended horizontally anddownward by cells of increasing sizes. The total number of cellsin the inversion was about 72 000. We first performed the fullnonlinear inversion of the DC resistivity data and then used it to

FIG. 13. Conductivity model recovered from the joint inversionof surface and crosshole data. This model is similar to thatobtained from surface data alone, but it has a slightly higheramplitude for the buried conductive anomaly. The depth extentof the anomaly is also slightly better defined.

carry out the IP inversion. The resulting model is shown in Fig-ure 16. For comparison, we also plotted the chargeability modelcreated by combining the 2-D sections obtained from invertingthe 11 lines of data separately using a 2-D algorithm. The re-covered 3-D chargeability models from these two approacheswere consistent, and they both imaged the large-scale anoma-lies reasonably well. This was not surprising since the limitedarray length meant there was little redundant information inthe data from adjacent lines. The model recovered from the 3-D inversion was somewhat smoother and showed less spuriousstructure than the composite 2-D model. It also showed a well-defined central zone of low chargeability at depth. This was aclearer image of the monzonite stock than what was imaged inthe 2-D inversions.

FIG. 14. Chargeability model recovered from the joint inver-sion of surface and crosshole data. This model shows the greatimprovement achieved by joint inversion of the two comple-mentary data sets. All five anomalies are well resolved. Es-pecially noticeable is that the boundaries of the two buriedchargeable bodies are well delineated.


DISCUSSION

We developed a 3-D IP inversion algorithm that applies todata acquired using arbitrary electrode configurations on a to-pographically variable earth surface or in boreholes. We as-sumed that the chargeability is small and formulated the in-version as a two-step process. First, the dc resistivity data areinverted to generate a background conductivity. That conduc-tivity is used to generate the sensitivity matrix for the IP equa-tions. The 3-D chargeability model is then generated by solvingthe system of equations, subject to a restriction that the charge-ability is everywhere positive and smaller than an upper bound.

The analysis of large IP data sets often takes place in stages.The first goal is to obtain an image that reveals the major sub-surface structures, answers questions about the existence ofburied targets, and supplies approximate details about size andlocation. Major components affecting this image are the choiceof model objective function, the amount and type of errors onthe data, and the degree to which the data are fit. For our two-step process, we must also pay attention to how valid the lin-earization process is and how close the recovered conductivityis to the true conductivity.

The question of how close the recovered conductivity needsto be to the true conductivity so that sensitivity J is a good esti-mate of the true sensitivities is not addressed quantitatively inthis paper. We have, however, carried out an empirical test ina single example. Four approximate conductivities were usedto generate the sensitivities. In general, higher quality dc in-

FIG. 15. The IP data from an area above the MBX stock of the Mt. Milligan copper–gold porphyry deposit in central BritishColumbia. The data were acquired using a pole–dipole array with a dipole length of 50 m and n-spacing from 1 to 4. The four panelsare plan maps of the data corresponding to different n-spacings.

versions yielded better IP results, with the half-space conduc-tivity, a one-pass linearized inversion, a few passes of an AIMapproach, and the Gauss-Newton inversion giving progressiveimprovement. However, the differences in the final IP inver-sions from these various approximations were fairly subtle (seeFigure 7). In fact, these differences were smaller than changesin the image obtained by adjusting the degree to which the dataare misfit or by slightly altering the model objective functionbeing minimized. Yet the various approximations to the con-ductivity can be produced with substantially fewer computa-tions than the full Gauss-Newton solution. This allows the userto carry out a number of first-pass inversions with a data setto achieve insight about the gross distribution of earth charge-ability. If the conductivity structure is not overly complicated,then this result may be satisfactory for final interpretation. Thequestion of how well the conductivity must be known is a po-tential area for further research.

Another approximate conductivity model is that generatedby combining results from 2-D inversions. The prevalence of2-D inversion algorithms means that this information is gen-erally available when data have been collected along parallellines. We know that off-line anomalies and 3-D topography willcause distortions in the recovered 2-D conductivity models, sosome degree of caution is required. In the synthetic modelingpresented here and in the Mt. Milligan example, the 2-D anal-ysis for conductivity worked satisfactorily. Further research isrequired to provide more detailed rules about when 2-D is ap-plicable.


An important aspect of our IP inversion is the incorporationof upper and lower bounds on the chargeability. The lowerbound is physical since chargeability is positive. The upperbound might be (1) assignable from a priori knowledge aboutthe nature of the mineralization or (2) assigned to generate amodel consistent with the linearized formulation of the equa-tions. Linearization requires that the chargeability be small.The positivity and upper bounds are implemented through aprimal logarithmic barrier method. This increases the complex-ity of the algorithm, but the method is well established in theliterature and we provide an explicit algorithm for its imple-mentation.

Another major component of our algorithm is the introduc-tion of the wavelet transform to perform the matrix–vectormultiplications. The sensitivity matrix can be compressed by afactor of at least 10. This leads to substantial savings in both re-

FIG. 16. Comparison of the chargeability models obtained from 2-D and 3-D inversions of the data from Mt. Milligan shown inFigure 15. The column on the left shows one cross-section and two plan sections of the 3-D model obtained by combining eleven 2-Dsections recovered from 2-D inversions. The column on the right shows the model obtained by performing a single 3-D inversionof all the data. The two results are generally consistent. However, less spurious structure is present in the model from the 3-Dinversion, and the central zone of low chargeability corresponding to the MBX stock is imaged better.

quired memory and CPU time. This has made the algorithm atleast ten times faster than a direct approach and consequentlyhas allowed us to routinely handle problems that have a fewthousand data and a hundred thousand cells with relative effi-ciency.

Last, the application of our algorithm to joint surface andcrosshole data has demonstrated that the inversion of these twocomplementary data sets can greatly improve the resolutionof the inverted chargeability model. The noticeable gains arein the enhanced definition of both horizontal boundary andvertical extent of buried chargeable zones.

ACKNOWLEDGMENTS

We thank Roman Shekhtman for his valuable assistancein programming the code and in running numerical exam-ples. This work has been supported by an NSERC IOR grant


and an industry consortium, 3-D Inversion of DC resistivityand Induced Polarization Data (INDI). Participating compa-nies are Placer Dome, BHP Minerals, Cominco Exploration,Falconbridge, INCO Exploration & Technical Services, New-mont Gold Company, and Rio Tinto Exploration.

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The application of geophysics during evaluationof the Century zinc deposit

Andrew J. Mutton∗

ABSTRACT

During the period 1990 to 1995, experimental pro-grams using high-resolution geophysics at severalAustralian operating mines and advanced evaluationprojects were undertaken. The primary aim of those pro-grams was to investigate the application of geophysicaltechnology to improving the precision and economics ofthe ore evaluation and extraction processes. Geophysicalmethods used for this purpose include:

1) borehole geophysical logging to characterize oreand rock properties more accurately for improvedcorrelations between drill holes, quantification ofresource quality, and geotechnical information

2) imaging techniques between drill holes to mapstructure directly or to locate geotechnical prob-lems ahead of mining

3) high-resolution surface methods to map ore con-tacts and variations in ore quality, or for geotech-nical requirements

In particular, the use of geophysics during evalua-tion of the Century zinc deposit in northern Australiademonstrated the potential value of these methods tothe problems of defining the lateral and vertical ex-tent of ore, quantitative density determination, predic-tion of structure between drill holes, and geotechnicalcharacterization of the deposit. An analysis of the po-

tential benefit of using a combination of borehole geo-physical logging and imaging suggested that a moreprecise structural evaluation of the deposit could beachieved at a cost of several million dollars less thanthe conventional evaluation approach based on anal-ysis from diamond drill-hole logging and interpolationalone.

The use of geophysics for the Century evaluation alsoprovided substance to the possibility of using systematicgeophysical logging of blast holes as an integral part ofthe ore extraction process. Preliminary tests indicate thatore boundaries can be determined to a resolution of sev-eral centimeters, and ore grade can be estimated directlyto a usable accuracy. Applying this approach routinelyto production blast holes would yield potential benefitsof millions of dollars annually through improved time-liness and accuracy of ore boundary and quality data,decreased dilution, and improved mill performance.

Although the indications of substantial benefits result-ing from the appropriate and timely use of geophysics atRio Tinto’s mining operations are positive, some chal-lenges remain. These relate largely to the appropriateintegration of the technology with the mining process,and acceptance by the mine operators of the economicvalue of such work. Until the benefits are demonstratedclearly over time, the use of geophysics as a routine com-ponent of evaluation and mining is likely to remain at alow level.

INTRODUCTIONGeophysics has been used by the metalliferous mining in-

dustry primarily as a tool for exploration. By contrast, thepetroleum and coal industries have exploited geophysics toa far greater extent in terms of using the data to quantify thevalue, size, or production capabilities of their resource. The ap-plication of geophysics to mineral-deposit evaluation and min-ing is not developed well in general. Some possible reasons forthis are:

Manuscript received by the Editor February 5, 1999; revised manuscript received May 12, 2000.∗Rio Tinto Technical Services, P.O. Box 2207, Milton, Queensland 4064, Australia. E-mail: [email protected]© 2000 Society of Exploration Geophysicists. All rights reserved.

1) insufficient knowledge by mining managers, engineers,and operators of the existence of geophysical method-ologies, coupled with a “high tech–high cost” perceptionof geophysics

2) perceived infrastructure and logistical difficulties in usinggeophysics in a working mine environment

3) lack of knowledge of the physical properties of the oreand host rocks which may be exploitable in a miningenvironment

1946

Geophysical Applications at Century Zinc Deposit 1947

4) limited research effort applied to standard geophysi-cal methods to develop higher-resolution acquisition orinterpretation tools applicable to mining applications,rather than tools applicable to exploration

5) scarcity of geophysicists with access to mine problems orwith sufficient knowledge of mining “culture” or opera-tional requirements to champion the use of geophysics inthis area

During the period 1990 to 1995, CRA Limited (nowRio Tinto) undertook experimental programs using high-resolution geophysics at several Australian operating minesand advanced evaluation projects, to investigate opportunitiesto develop and test geophysical applications related directly toresource definition and mining problems. Using some exam-ples of results obtained during the evaluation by CRA of theCentury zinc deposit in northern Australia, this paper describesthe opportunities available and the rationale for applying geo-physics beyond the exploration stage of mineral development.

WORK UNDERTAKEN

A range of geophysical surveys, some conventional and someadapted for specific problems, has been tested in recent yearsat several CRA evaluation projects and mining operations toenhance ore-boundary definition and recoveries. The surveysfall into several categories, including:

1) High-resolution surface surveys to map “remotely” thelateral and vertical limits of ore or other geotechnicalparameters required for mine design. Successful applica-tion of such methods potentially can reduce the amountof closely spaced expensive drilling generally needed forthese requirements, and can enable improved reservescalculation and mine design. Examples of such workincorporate the use of standard methods such as mag-netics, electromagnetics (EM), and IP/resistivity usingimproved modeling and data enhancements, or “new-generation” methods such as borehole EM imaging andground-penetrating radar (GPR).

2) Geophysical logging of evaluation drill holes to quan-tify mineralogical, geotechnical, and structural parame-ters of ore and host rocks in situ. Many of the standardpetroleum logging tools, such as density, sonic, and dip-meter, have been adapted successfully to the hard-rockenvironment, along with developments in new tool tech-nology such as magnetic susceptibility and conductivity.Improved confidence and demonstrated success with thisapproach is aimed mainly at providing significant costsavings by the future routine use of noncore rather thancore drilling.

3) Interborehole imaging methods using electromagnetic orseismic sources. Examples of such methods include radio-imaging (RIM) surveys between holes to map ore con-tinuity and structure. The use of these methods is de-signed to improve the knowledge and reliability of theore boundaries at an earlier stage of evaluation, whichshould reduce the time and cost of the evaluation pro-cess.

4) Geophysical logging of blast holes to define ore-wastecontacts to a resolution of a few centimeters and, in somecases, to estimate ore grade directly. Applying this ap-

proach routinely to production blast holes is designed toyield substantial cost benefits through improved timeli-ness and accuracy of resource information, less dilution,and improved mill performance.

Much of CRA’s investigation of the use of geophysics formining and mine evaluation was carried out at the site ofthe Century zinc deposit in northern Australia, discovered byCRA Exploration in 1990. Since 1997, it has been owned andoperated by Pasminco Pty Limited. The discussion and exam-ples presented below will focus on the work at Century and therole geophysics played in the various phases of the evaluationof the deposit prior to 1995.

RESOURCE DEFINITION GEOPHYSICS AT THE CENTURYZINC DEPOSIT

Geology

The Century Zn-Pb-Ag deposit is located about 250 kmnorth-northwest of Mount Isa, Queensland. Substantialdrilling (about 500 diamond core holes) from 1990 to 1995 out-lined a geologic resource containing about 120 million metrictons of 10.5% zinc, 1.5% lead, and 35 g/ton silver.

The Century deposit occurs within sediments of theProterozoic Mount Isa inlier and is locally unconformably over-lain by Cambrian rocks comprising dolomitic limestone, chert,and chert breccia (Figure 1). Dolomitic siltstones and carbona-ceous shales host the deposit. The mineralized sequence is

FIG. 1. Century deposit location and geology.

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about 40 m thick and consists of four laterally continuous sub-divisions (units 1 to 4). The bulk of the mineralization occursas strata-bound, banded sphalerite, galena, and pyrite withinblack carbonaceous shales of units 2 and 4. A barren dolomiticsiltstone bed within unit 3 represents a semicontinuous markerbed throughout much of the deposit. The mineralized sequencehas excellent grade continuity but is disrupted by late faulting.A detailed description of the geology and mineralogy of thedeposit is included in Waltho et al., 1993.

Figure 2 shows a schematic north-south cross-sectionthrough the deposit. The deposit consists of a smaller, shallowsouthern block which subcrops in the southwestern margin ofthe orebody, and a larger, deeper northern block completelyconcealed beneath Cambrian limestone and recent alluvium.Bounding surfaces to the deposit comprise either postmineral-ization faults or Cambrian and younger erosional surfaces.

Physical characteristics of the ore and host

Comprehensive laboratory petrophysical measurements onCentury core were made on several samples as part of anAustralian Mineral Industry Research Association (AMIRA)project (project P436) to investigate the application of geo-physics in mine planning and mining (Fullagar et al., 1996a;Fullagar and Fallon, 1997). These data have been evaluated inconjunction with extensive geophysical log data collected froma large proportion of the holes drilled. A typical composite geo-physical log through the Century ore sequence and host rocksis presented in Figure 3.

The petrophysical results show that the Century orebodyis typical of many shale-hosted sulfide base-metal deposits

FIG. 2. Century deposit cross-section 46800E.

throughout the world. In particular, the ore is characterizedby distinctive physical properties, including higher density, lownatural radioactivity, and low magnetic susceptibility.

The main difference from many other shale-hosted depositsis that the Century ore, consisting mainly of sphalerite withrelatively low iron sulfide, is not very conductive and in parts ismore resistive than barren host shales. However, petrophysi-cal measurements on core and downhole induced-polarization(IP) surveys show the ore has high electrical chargeability.Figure 4a shows a typical downhole IP log through the deposit,and Figure 4b is a statistical summary of the IP and resistivityresponse recorded in several drill holes. These results clearlyindicate how chargeable the ore units are compared with otherlithologies.

The analysis of physical properties also indicates diagnos-tic contrasts among some of the host lithologies. Althoughthis knowledge is not specifically useful for ore evaluation,it has helped to address other problems arising during theproject development. For example, the high resistivity of theCambrian limestone has been exploited to assist with a hydro-logic problem associated with development of a large pit (seebelow).

Role of geophysics

The discovery of Century was based largely on testing ofa zinc soil geochemical anomaly which was delineated onregional gravity, ground magnetic, and soil sample traverses(Thomas et al., 1992; Broadbent, 1996). No anomalous re-sponse was detected on the gravity or magnetic data or onother geophysical surveys attempted prior to the discovery.


Despite the lack of success of geophysical techniques inthe initial detection of ore, the recognition after the discov-ery of contrasts in the physical properties of the ore andhost sediments provided some confidence that geophysicalmethods could have a role in further exploration and evalu-ation of the deposit. Geophysical surveys subsequently carriedout over and within the Century deposit since its discoveryhave included detailed ground and airborne surveys (gravity,magnetics, electromagnetics, IP/resistivity, reflection seismic),borehole geophysical logging, and interborehole geophysicalimaging

The work resulted in some successes, in particular relatingto defining the overall lateral and vertical extent of mineraliza-tion, mapping lithologic boundaries and structure within thedeposit, providing data for ore-reserve estimates and mineplanning, supporting geotechnical studies for pit design, andaiding ongoing exploration in the area

Detailed surface surveys

Lateral and vertical extent of mineralization.—After the ini-tial discovery of zinc-rich, low-sulfide mineralization and therealisation that EM methods did not detect the mineralization,the IP method was suggested as a possible way of mapping theextent of the mineralization beneath the Cambrian cover. A

FIG. 3. Century deposit composite geophysical log.

limited number of IP/resistivity traverses subsequently werecompleted over the Century deposit and immediate environs,indicating the presence of a good IP anomaly approximately co-incident with the mineralization as known at that time (Thomaset al., 1992). IP/resistivity logging of selected drill holes sub-sequently confirmed the source of the surface IP anomalies.Figure 4a presents an example of downhole IP results in onehole, which clearly indicates the high chargeability response ofthe mineralized zone.

The downhole IP/resistivity surveys thus provided the reas-surance that IP anomalies represented good exploration tar-gets. In fact, initial drilling into the deeper northern ore zone(Figure 2), which is masked completely by hundreds of metersof barren limestone and sediments, essentially was guided bythe presence of the IP anomaly in that area. Although only alimited number of IP lines were completed over the deposit,the IP results, in combination with drilling, helped to confirmthe lateral limits of mineralization far sooner in the evaluationprocess than if IP surveying had not been attempted.

As well as mapping the lateral extent of the orebody, laterwork by CRA Exploration demonstrated that it is possible totransform the IP/resistivity data into an approximate depth“image” of the mineralization (representing the vertical distri-bution of chargeable material). This work used inversion algo-rithms developed by the Geophysical Inversion Facility group

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at the University of British Colombia (Oldenburg et al., 1997).Figure 5 shows the image derived from unconstrained inver-sion of surface IP/resistivity data adjacent to the cross-sectionpresented in Figure 2. The outline of the orebody derived fromsubsequent drilling is superimposed.

This work clearly demonstrates the potential mappingcapabilities—both laterally and vertically—of a surface geo-

a)

FIG. 4. Century downhole IP/resistivity results: (a) log for DDH LH117; (b) statistical summary for all IP/resistivity logs fromCentury deposit.

physical method for which the physical property contrasts havea close association with the mineralization, defining the ap-proximate extent of ore prior to extensive drilling. The imageof the orebody provides the explorer with improved capabilityfor successful drill targeting. However, it also has the poten-tial, by the application of constraints from initial drilling, todefine the ore boundaries and estimate reserves substantially


at a far earlier stage of the evaluation process than hithertopossible.

Geotechnical applications.—Although the predominantlysphaleritic Century ore has little electrical conductivity con-trast with the host sediments, surface electromagnetic methodshave been applied successfully, based on the conductivity con-trast between the Proterozoic sediments and overlying lime-stone (Figure 4), to assist geotechnical work associated withpit design. Two phases of Controlled Source Audio-FrequencyMagnetotelluric (CSAMT) surveys were completed over thedeposit to (1) locate large blocks of detached Proterozoic shalewithin the limestone (the shale presents a geotechnical hazardfor pit-slope stability during initial excavation of the limestoneportion of the proposed pit, and (2) determine the thicknessof the surrounding water-saturated limestone to estimate thelikely water flow into the open pit as it is excavated.

The CSAMT method was chosen in preference to conven-tional TEM methods because of logistical considerations as-sociated with the limestone topography, and the belief thatCSAMT is more sensitive than TEM methods to delineat-ing contrasts in the more resistive lithologies present in theCentury area.

The first work was only partially successful, in that only verylarge blocks of shale—typically greater than 50 × 50 × 50 m—could be delineated. The survey results did not offer sufficientresolution to locate smaller blocks confidently, but improvedsurvey design may have provided better resolution.

FIG. 5. Century deposit Line 46800E IP pseudosection and inversion model compared with orebody location from drilling. IP dataare from 100-m dipole-dipole frequency, domain survey.

The second phase of work proved useful in mapping the baseof the limestone and hence assisting with the hydrologic studyfor the pit design. Figure 6 (after Mayers and Bourne, 1994)shows a plan of the resistivity model obtained from inversionof the CSAMT data and collated at a vertical depth of 100 m.The intense blue represents the presence of resistive limestoneat this depth, and the warmer colors indicate the presence ofless resistive shale and siltstone.

The depth to the base of the limestone is shown in selecteddrill holes. In general, there is a good correlation between theintersected thickness of limestone and depths predicted fromthe CSAMT resistivity model. The results show that postde-positional faulting has led to large variations in the limestonethickness in the vicinity of Century and, importantly, the lime-stone is “necked” to the north and east of the deposit. TheCSAMT result thus gives some confidence that the total vol-ume of water flow into a proposed pit will be significantly lessfrom the thinning of the water-saturated limestone than if thelimestone had been uniformly thick to the north and east ofCentury.

Borehole surveys—geophysical logging

Although geophysics did not feature strongly in the initialdiscovery of ore, downhole geophysical “characterization” log-ging was carried out at Century in several campaigns commenc-ing soon after the initial drill holes were completed. The workwas commissioned for several reasons:

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1) to help geologists and geophysicists gain a better under-standing of the in situ physical properties of the depositand its host rocks as an aid to interpretation of explo-ration and geotechnical data

2) to provide data to assist with the correlation of lithologiesand definition of the structure of the ore zones betweendrill holes

3) to provide quantitative in situ density information to as-sist with estimation of the ore reserves

4) to provide rock-strength information to assist in thegeotechnical assessment of the deposit by miningengineers

The logging programs used standard suites of electrical andnuclear tools (resistivity, resistance, self potential [SP], naturalgamma, density, and neutron), as well as tools such as dipmeter,magnetic susceptibility, and sonic velocity, which had not beenused commonly in base-metal geophysical logging prior to thattime.

Lithologic correlations.—Interpretation of the downholelogs indicates that the major lithologic units and the mineral-ized zones can be distinguished readily by their density, natural

FIG. 6. Century deposit region showing smooth-model inver-sion of CSAMT resistivity at 100-m depth, compared with lime-stone depth from drilling. The more resistive areas (blue) rep-resent limestone greater than 100 m thick.

Table 1. Summary of Qualitative Log Responses.

Prot. Prot. shale and Prot. shaleCambrian sandstone siltstone Ore zone and siltstonelimestone (h/wall) (h/wall) (Units 1–4) (footwall)

Natural gamma (API cps) very low (25) high (160) moderate (125) low-mod (70–150) high (200)Magnetic Susceptibility very low (<10) low (0–50) low-moderate v. low (ore)-mod (waste) low-moderate

(SI × 10−5) (0–100) (0–150) (50–150)Resistivity (ohm-m) high (>1000) low (60) low (75) variable (50–200) low (80)Density (g/cm3) moderate (2.7) low (2.6) moderate (2.7) mod-high (2.8–3.0) moderate (2.6–2.7)Neutron (API cps) high (1500) high (1600) moderate (1200) low (700-variable) moderate (1200)Sonic velocity (m/s) very high moderate moderate (4000) mod-high (4000–5000) moderate (4000)

(5000–6000) (4500)

gamma, neutron, resistivity, sonic, and magnetic susceptibilityproperties. Figure 3 shows a composite log of a typical drillhole through the Century deposit. Table 1 summarizes the re-sponses that characterize the main lithologic units for each ofthe main log types.

The mineralized sequence (units 1 to 4) is differentiated wellby the density, natural gamma, and magnetic susceptibility logs.The density contrast is attributed to the presence of sulfide andan increase in iron carbonate (siderite) content. The internalwaste unit within the ore zone (unit 3.2) is characterized bylower density and generally higher gamma, magnetic suscepti-bility, and resistivity, compared with the mineralized intervals.

Quantitative analysis.—The quantitative analysis of the logdata, which attempts to transform such data into usable physi-cal or chemical quantities such as density, grade, rock strength,etc, is dependent on the equivalence of data from the differentlogging tools used at various times and by various contractors.Differences arise from differences in tool specifications andalso from “environmental” differences, including borehole di-ameter, fluid content and properties, temperature, weathering,smoothness of the hole wall, etc.

Comparisons to a calibration standard, such as the API stan-dard, are used by most logging contractors but do not accountfor most of the environmental factors. For example, an inspec-tion of different API-standard calibrated natural gamma logscan show some marked offsets in absolute values recorded.

The most successful form of calibration achieved at Centuryhas been the use of a “calibration” drill hole, which was resur-veyed at regular intervals during each logging program. Suchaction ensured that the repeatability of logs over a known inter-val could be compared and quantitatively adjusted by meansof a correction factor, if necessary.

Quantitative analyses undertaken on the Century data haveincluded automated lithologic and grade-prediction studies butprimarily have been aimed at determining in situ density androck-strength variations.

Automated lithologic prediction.—Experimental work de-signed to test the possibility of predicting lithology and gradedirectly from a combination of geophysical logs was carried outon selected geophysical logs from Century. Three approacheswere attempted: (1) artificial neural networks (ANN), (2) lin-ear regression and discriminant analysis, and (3) cluster analy-sis (lithology prediction only).

Some encouragement was obtained from the initial predic-tion tests, but it was clear that factors such as measurement


uncertainty (calibration differences, depth discrepancies in thetraining sets, etc.) or the level of data “conditioning” (such asfiltering) can have a large influence on the predictions. Theresults produced from the ANN tests were not sufficiently en-couraging in either lithology or grade prediction for this to beconsidered a viable method at present.

On the other hand, reasonable predictions of lithology wereobtained from the cluster analysis method, using the LogTrans

FIG. 7. Lithology-based LogTrans geologic interpretation of Century drill hole LH643 derived by using statistics from nearby controldrill holes. The interpretation has predicted successfully a normal fault across which part of the orebody is missing.

software developed as part of the AMIRA P436 project(Fullagar et al., 1996a). Figure 7 shows an example of predic-tion of lithology at Century using this method. The predictionfor the hole presented (LH643) is based on a statistical analysisof data from holes nearby, so in effect, the prediction is uncon-strained by the log results and geology from LH643. The resultscompare favorably with the mapped lithologies, even thoughpart of the ore sequence in LH643 is missing because of faulting.

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Improvements to and routine use of such software in thefuture could save considerably on the cost of manual loggingof core, but the greatest saving would result from the use ofpercussion rather than core drilling for a large proportion ofthe drill holes. This approach would be applicable to a depositsuch as Century in which the need for closely spaced drilling isrelated more to structural interpretation than to grade control.

Density.—The most extensive quantitative analysis of theCentury log data has been done for the purpose of densitydetermination for ore-reserve estimation. Several independentmethods were considered and tested to determine a suitableapproach. The methods included:

1) Archimedes’-type measurements on whole core at siteor by a laboratory on core fragments submitted for geo-chemical analyses

2) laboratory measurement of density of pulverized and ho-mogenized drill-core samples by acetone titration

3) physical measurement (dimensions and weight) of wholedrill core at site

4) stoichiometry, based on base metal and sulfur assay data5) geophysical (gamma-gamma) logging of drill holes

Of these methods, geophysical logging is considered to pro-vide the most consistent measure of bulk density, althoughphysical measurement and stoichiometric calculations basedon zinc, lead, iron, manganese, and sulfur assay data also

FIG. 8. Comparison of gamma-gamma log-derived density (showing differences in calibration standards) and density measurementson core for Century drill hole LH483.

proved useful. The titration method proved to be the mostunreliable, yielding many spurious values probably caused bysmall variations in the measurement method.

All methods have limitations. For example, the chemicalassay–based methods yield data restricted to the assay inter-vals only, and the physical-measurement methods are moresubject to measurement error arising from slight variations inthe procedure. Geophysical logging appears to yield very con-sistent results. However, the conversion of the measured countrate to a density value is subject to error because of differencesbetween the material contained within the volume of rock sam-pled by the logging tool and the smaller volume of rock in thecore used as the reference value.

Calibration factors supplied by the logging contractor shouldbe used with caution in hard-rock mining environments, be-cause most of the calibration standards are devised for loggingin lower-density sedimentary environments hosting petroleumor coal deposits. Figure 8 shows the comparison between thecontractor’s derived density and the recalculated density logusing the calibration derived from the physical measurementof core. The result of using the former is a large overestimationof the ore density and therefore the contained tonnage of ore.

The major advantages of using geophysical logging for den-sity determination are that (1) the data are available and con-sistent through the length of the drill hole rather than just overan assayed interval, and (2) data can be obtained for noncoredholes, reducing the requirement for cored holes. However, thenoncored holes must be drilled well so that the hole walls are


reasonably smooth. Caving and irregularities in the wall are themain sources of error with this method. Calliper data collectedin conjunction with the density logs can be used to correct orexclude data affected in this manner.

The main limitation of density logging is the use of a strongradioactive source and the possibility that such a source couldbecome stuck in a drill hole, thus requiring some effort andexpense to recover the source. For the work at Century, a pro-cedure was implemented to use the density tool only after oneor more successful runs with other nonradioactive tools. In thisway, the safety risk was considered manageable, and a high pro-portion of density logs from the available holes was achieved.

Rock strength.—Measurement of the strength and fractur-ing characteristics of both the ore and the host rock (includ-ing the overlying shale, sandstone, and carbonate) is essentialfor mine planning at Century and has implications for pit de-sign, blasting requirements, and milling of the ore. Initially, atraditional approach to prediction and modeling of the rock-strength variations within the deposit was taken, using stan-dard procedures such as measurement of rock quality descrip-tor (RQD) on all cores. These data then were compared withtest measurements of unconfined compressive strength (UCS)and other strength parameters on selected pieces of core.

To assist with this problem, sonic velocity data acquired froma standard slimline sonic tool were investigated to determineif such data could provide more uniform and extensive infor-mation than reliance on RQD and limited core measurements.

FIG. 9. Comparison of geotechnical data derived from laboratory measurements and sonic logs for Century drill hole LH643.

This approach has been used widely in rock-strength analysesin Australian coal-deposit evaluation for many years (Asten,1983; McNally, 1990; Davies and McManus, 1990).

A study (Duplancic, 1995) was initiated as part of theAMIRA P436 research project to address this requirement.The focus of this specific study was to determine the relation-ship between rock strength (measured as UCS) and sonic ve-locity, measured both on core samples in the laboratory and insitu in the borehole.

Figure 9 shows a portion of a sonic velocity log fromCentury, with the laboratory velocity (P-wave measured at1 MHz) and UCS measurements superimposed. The velocitydata show a reasonable correlation, but the correlation withUCS is poor. The analysis found that the relationship betweenvelocity and rock strength is a function of the lithology andthe rock porosity, more than the intrinsic variation in strength.Figure 10 demonstrates this relationship schematically, sug-gesting that it would not be feasible to predict UCS fromthe borehole velocity data without first accounting for theporosity and lithologic variation.

The study also noted that sample quality, laboratory mea-surement problems, and insufficient samples to gauge statisticalreliability could affect these results adversely. However, it mustbe concluded from this work that at best, the prediction of UCSfrom velocity can be regarded as indicative only.

Structural prediction from geophysical logs.—One of theearliest problems recognized at Century was the presence of

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small-scale faulting within the deposit (Waltho et al., 1993).These faults are mostly steeply dipping and trend in several pre-ferred directions. Because surface exposures of these featuresare masked by the irregular structure of the Cambrian lime-stone cover, drill-hole intersections provided the only indica-tion of the presence of these faults initially. However, becausemost of the drill holes are vertical, only limited informationabout the structures is available from drilling.

Some thought was put into methods that may assist withthe prediction of the presence and geometry of such faults be-tween drill holes. Seismic and electromagnetic imaging werepostulated and are discussed below. Because of lack of confi-dence and uncertainty of the availability of these techniques atthe time, an alternate method was sought that used individualboreholes and made predictions from the information in theborehole.

Measurement of the core orientation and the dip of thestrata in the core is the traditional approach to interpretingstructure between drill holes in stratiform deposits. However,the problem at Century was that because most of the holeswere vertical, the reliability of the available core-orientationdata was questionable. To address this issue, tests using a com-bined dipmeter/hole deviation tool developed by BPB Instru-ments were carried out at Century. The results demonstratedthat good-quality oriented dip information could be obtainedon the more laminated units (shales and siltstones), includ-ing the ore sequence. The amount and quality of the data ob-tained enabled the estimation of structural offsets between drillholes.

Blast-hole logging.—The recognition that lithologic contactscould be delineated readily from borehole geophysical logs ledto the possibility that the blast holes planned for the benchmining of the deposit could be logged geophysically to pro-vide more accurate information about ore-waste contacts thanconventional geologic logging from blast cuttings. Such infor-mation could provide a substantial benefit to the mining eco-nomics by optimizing the blast design and the mining of oreand waste. Most of the benefit would come in reduced dilutionof ore for grade-control purposes.

Figure 11 shows the result of logging a test blast hole withina small pit excavated to obtain a bulk sample of the ore (Fig-

FIG. 10. Approximate relationships between sonic velocity,rock strength (UCS), and porosity, Century deposit (afterDuplancic, 1995).

ure 1). Only natural gamma and magnetic susceptibility logswere obtained for this test. The prediction of the ore unit con-tacts was based on a comparison with curve shapes, using atemplate overlay, from nearby reference core holes. The mainore units, as mapped from the pit face and extrapolated to theblast hole, are shown also. The comparison with the interpretedlithologies suggests that the ore contacts can be predicted toa vertical precision of about 10 cm. This result was confirmedfrom the subsequent geophysical logging of about 50 blast holesdrilled in the trial pit.

The possibility of measuring ore grade in blast holes hasbeen considered also and some testing has been conducted.The key to grade estimation is the measurement of density, butthe use of a highly radioactive source in an active mining areawas discouraged strongly. A test program was conducted bythe Commonwealth Scientific and Industrial Research Organ-isation Australia (CSIRO) to try a low-activity spectrometricprobe developed by the CSIRO (Charbucinski et al., 1997)to measure zinc grade directly in blast holes. Such a probewould be an alternative to laboratory chemical analysis forgrade control. Information from such a probe would be avail-able in a much shorter time frame than chemical analyses, andpotentially should provide more accurate information on ver-tical grade variations than would be possible from analysis ofsamples taken from drill cuttings.

Initial results of such tests were promising. The tests suggestthat such a tool could achieve better than ±2.5% Zn deter-mination, with substantial opportunity for improvement (to acalculated limit of about ±1.5% Zn). Ore boundaries can belocated to a vertical resolution of 10 cm. Although the preci-sion of this grade prediction is not as good as with a chemicalassay, it is unlikely that the result obtained from a bulk assayof drill cuttings will reflect the actual grade of the material toany better accuracy than indicated by the probe. Further workon the assessment and development of this technology is rec-ommended.

Borehole surveys—interborehole imaging

Because the Century deposit is faulted locally, the delin-eation of faults within the deposit and the impact these wouldhave on mine design and extraction of the ore have been sub-ject to much investigation. To determine possible options toaddress this problem, some analysis of the potential applica-tion of seismic tomography, radio frequency electromagnetic(RFEM) imaging, and borehole radar was undertaken to de-termine if any of these methods can define the structure of theore zone between drill holes at Century.

Seismic tomography.—Although surface seismic has littleapplication at Century because of the rugged limestone to-pography and weathering (Thomas et al., 1992), analysis ofsome test data suggested that reflectors are associated withlithologies within the ore zone. It was postulated therefore thatborehole seismic tomography could map these units and pro-vide a basis for structural interpretation between drill holes.However, seismic tomography was not undertaken at Century,mainly because of the limited support to service such work inAustralia.


Radio imaging.—As an alternative to seismic, RFEM imag-ing was assessed by CRA as having potential application atseveral of its operations. These include delineation of the struc-ture, including offsets caused by faulting, of coal seams or oreunits between drill holes or mine-development access ways,detection and location of geologic or operational hazard zonesbetween drill holes, detection of unknown ore not intersectedby drill holes, assessment of quality of coal or ore between drillholes, and detection and location of large voids (cavities, oldworkings, etc.) between drill holes.

Unlike seismic tomography, several potentially suitable EMtomography systems are available, such as the RIM system(Stolarczyk, 1992), the Russian FARA system, the SouthAfrican RT system, and the Chinese JW-4 system (Fullagaret al., 1996b). The RIM system has advantages in Australianconditions because of its lower frequency capability (down to

FIG. 11. Lithologic prediction from geophysical log in blast hole, based on curve matching from nearby referencehole, Century bulk sample pit.

12.5 kHz). It was also the only commercially available systemin Australia when the work at Century was being considered.

A trial of the RIM system therefore was undertaken atCentury in 1992 to determine its suitability in delineating thestructure of the ore zone between drill holes. A test was carriedout in the vicinity of an exploration shaft which had been sunkinto the orebody to obtain a bulk sample of ore and to provideaccess for geotechnical investigations and mapping of structurefrom drives within the orebody (Figure 12a, b). This mappingprovided direct evidence of a previously unknown fault withthrow about 10–15 m adjacent to holes in which an RIM surveywas carried out.

Initial processed results from this survey did not providemuch encouragement that the method could offer any infor-mation on the structure between holes to the resolution re-quired for mine development. As a result, no further surveys

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FIG. 12. Comparison of mapped geology in underground drive and radio-imaging method (RIM) survey results, Century explorationshaft area: (a) location plan; (b) geologic section through drive; (c) reprocessed RIM tomogram (frequency 520 kHz) from drill-holesection oblique to drive.


have been undertaken. However, alternative data-processingoptions were investigated subsequently, resulting in the im-age shown in Figure 12c. This result was obtained by usingexactly the same data as the initial processing, but in this casethe tomographic inversion used software developed by VIRG-Rudgeofizika for its FARA RFEM system. The main differ-ence with this software is that it uses both the amplitude andphase information. The reprocessed image clearly indicates thepresence of the fault very close to its predicted position. It isinteresting to note that no specific a priori geologic informationavailable from the drill holes or the subsequent undergrounddevelopment was used to constrain the resulting image.

The use of RIM or equivalent radio-imaging systems there-fore could play an important part in evaluation of the structureof an orebody such as Century in advance of infill drilling andmine development. At Century, this has not happened so for,probably because of unfavorable impressions gained by themine-evaluation staff from the initial processing of the data,and the passage of time before better results were obtained.However, it is estimated that if the radio-imaging technologyand processing had been sufficiently developed, proved, andaccepted at the time of the Century discovery, the structuralevaluation of the deposit might have been achieved at a sig-nificantly lower cost ($5 million to $10 million less) and in ashorter time frame. The basis for this substantial cost benefit isin the reduction of the number and type of boreholes required(i.e., noncore), but the main benefit comes from increased con-fidence in the reserves and the subsequent mine design andmining plan.

Borehole radar.—A trial borehole radar survey was carriedout at Century as part of the AMIRA P436 research project.The objectives of this work were to map faults within the Pro-terozoic sediments and determine their geometry; and to de-tect cavities, representing a future mining hazard, within thelimestone.

The use of borehole radar to map faults was a possible al-ternate to radio imaging, on the basis that the higher frequen-cies would give higher resolution of the structures and thatthe radar could be carried out in a single-hole reflection moderequiring less logistical effort. The results of this work were dis-appointing because the penetration distance of the radar signalusing a 60-MHz source frequency through the weakly conduc-tive sediments was very small (<5m), effectively rendering thetechnique of little value for this problem.

The detection of cavities in the limestone with boreholeradar was more promising. A reflection range of about 25 m at60-MHz frequency was obtained, and features that may be re-lated to cavities were noted. No testing or proving of such fea-tures was possible at that time. However, the results were con-sidered sufficiently encouraging to suggest that a radar method(including surface radar) may have application to the locationof potentially hazardous voids during waste-removal activitiesfor open-pit development.

DISCUSSION

The use of geophysics in the evaluation of the Century de-posit was an attempt to influence, change, and improve the pro-cess of resource definition in CRA and eventually integratesuch technology into routine mining practice. The work has

demonstrated some of the potential benefits that could flowfrom the routine use of geophysics during resource evaluationor when integrated with mining. With regard to evaluation of apotentially minable resource, areas in which successful use ofgeophysics could have a significant positive benefit to a projectdevelopment include:

1) Locating the lateral and vertical extent of ore at a muchearlier stage of drill evaluation. The influence this wouldhave on the time frame and accuracy of initial reserveestimations could have a major impact on developmentcriteria.

2) Correlation of mineralized intercepts and prediction ofstructural offsets between drill holes. The informationobtained on the structure and volume of ore should be ofgreat benefit to mine planning and reserve estimation.

3) Substitution of cored with noncored holes. If grade varia-tion is relatively uniform, the use of geophysically loggednoncored holes as a substitute for more expensive andtime-consuming cored holes could represent substantialcost benefits at an early stage of the project.

4) Direct measurement of physical parameters such as den-sity or rock strength. Geophysical logging potentiallyshould provide the most consistent and reliable informa-tion on the in situ rock properties.

5) Geotechnical evaluation of cavities, zones of weak-ness, or incompetent rocks representing potential min-ing hazards. Reliable remote detection of such hazardshas enormous implications for mine design and safetymanagement.

6) Evaluation of hydrologic conditions, and foundations formine infrastructure. High-resolution surface geophysicsintegrated with information available from drilling andwater monitoring can provide potentially more definitivesolutions to some of these problems.

Knowledge obtained during the resource-definition phaseof the project also provides the evidence and confidence forsuccessful use of geophysical technology in the mining process.Areas in which application of geophysics has demonstratedsome potential for integration into mining operations include:

1) Ore-boundary delineation. Detailed information on ore-waste contacts from geophysical logging of productiondrill holes or high-resolution surface surveys can influ-ence the optimization of the blast design or ore extrac-tion, yielding a substantial benefit to the mining eco-nomics, largely because of the reduction in dilution ofore.

2) Grade control. Where a relationship between grade andone or more physical properties is present, direct gradeprediction in production drill holes is possible, yieldingsubstantial benefits because of improved grade control,more rapid availability of data, and reduced laboratorycosts.

3) Hazard detection. The remote detection of cavities, bro-ken ground, or geologic impediments in advance of min-ing can reduce personal and economic risk substantially.

Knowledge and understanding of contrasts in the physicalproperties of the materials being mined are the fundamentalprinciples which determine if there is a role for geophysics in

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the definition, evaluation, and mining of a mineral deposit. Thisknowledge can be gained only from physical measurementson core or geophysical logging of drill holes. Therefore, as aprerequisite to any mine-evaluation program, it is necessary toacquire a suite of information on a range of physical properties,and then use either previous experience or indicative modelingstudies to determine what geophysical methods may achievethe required goals of the resource definition.

CONCLUSIONS

Experience gained by CRA in the early 1990s during theevaluation of the Century deposit and at other Australianmining operations suggests that successful application of geo-physics in mine evaluation or during mining is achievable incertain circumstances. These circumstances are controlled bythe contrasts in the physical properties of the materials beingmined. To use geophysics successfully in mining, it is there-fore necessary to acquire information on a range of physicalproperties, and then use this information to determine whatgeophysical methods may assist the evaluation process or min-ing operation.

The overall economic benefit of successful implementationof geophysical technology into all phases of resource defini-tion and mining at Rio Tinto’s operations is estimated at tensof millions of dollars annually. This projected benefit aloneshould stimulate the need for investigating further technolog-ical improvements and evaluation of available technologies atexisting operations.

Despite this positive assessment, some barriers inhibitingthe routine testing and implementing of such technology ex-ist. These barriers relate largely to the appropriate integrationof the technology with the mining process, and acceptance bymine operators of the economic value of such work, comparedwith traditional approaches to resource definition. Some effortby mine management will be required to address these issuesand support further evaluation of the technology before theuse of geophysics will be accepted as a routine component ofresource definition and mining.

Further anticipated technological developments, cost pres-sures on production, and gradual acceptance by mine operatorseventually will ensure the use of geophysics in mining. The re-ality may be that those mining operations that evaluate andsuccessfully exploit such developments will be the most effi-cient and economically viable mines of the future.

ACKNOWLEDGMENTS

The assistance of the staff of CRA Exploration and CenturyZinc Limited with the work undertaken at Century from 1990

to 1996 is acknowledged gratefully. In particular, I acknowl-edge the efforts of Barry Bourne and Theo Aravanis in ini-tiating some of the innovative work presented here, and theencouragement of John Main, Andrew Waltho, and the CZLstaff to undertake such work. I also thank the AMIRA P436team for its input. This paper is published with the permissionof Rio Tinto Limited and Century Zinc Limited.

This paper was published 1997 in Proceedings of Exploration97: Fourth Decennial International Conference on Mineral Ex-ploration, edited by A. G. Gubins. Some updates have beenmade to this version at the request of the Special Editor forpublication in GEOPHYSICS.

REFERENCES

Asten, M. W., 1983, Borehole log analysis using an interactive com-puter: Bull. Aust. Soc. Expl. Geophys., 14, 3–10.

Broadbent, G. C., 1996, The Century discovery—Is exploration evercomplete?: in Mauk, J. L., and St. George, J. D., Eds., ProceedingsPacrim 95 Congress: Aust. Inst. Min. Metall. Publ. 9/95, 81–86.

Charbucinski, J., Borsaru, M., and Gladwin, M., 1997, Ultra-low ra-diation intensity spectrometric probe for ore body delineation andgrade control of Pb-Zn ore: Gubins, A. G., Ed., Proceedings of Ex-ploration 97: Fourth Decennial International Conference on MineralExploration, 631–638.

Davies, A. L., and McManus, D. A., 1990, Geotechnical applications ofdownhole sonic and neutron logging for surface coal mining: Expl.Geophys., 21, 73–82.

Duplancic, P., 1995, A comparison of mechanical properties of rockwith in situ velocity measurements at the Century deposit: B.E. the-sis, Univ. Queensland.

Fullagar, P. K., and Fallon, G. N., 1997, Geophysics in metallifer-ous mines for orebody delineation and rock mass characterization,Gubins, A. G., Ed., Proceedings of Exploration 97: Fourth DecennialInternational Conference on Mineral Exploration, 573–584.

Fullagar, P. K., Fallon, G. N., Hatherly, P. J., and Emerson, D. W., 1996a,Implementation of geophysics at metalliferous mines—Final reportAMIRA Project P436: Cooperative Research Centre for MiningTech. and Equipment Report MM1-96/11.

Fullagar, P. K., Zhang, P., Wu, Y., and Bertrand, M-J., 1996b, Applica-tion of radio frequency tomography to delineation of nickel sulfidedeposits in the Sudbury Basin: 66th Ann. Internat. Mtg., Soc. Expl.Geophys., Expanded Abstracts, 2065–2068.

Mayers, P. J., and Bourne, B. T., 1994, Geological and geophysical studyof the Lawn Hill limestone annulus: Century Zinc Ltd. Internal Re-port 94/002.

McNally, G. H., 1990, The prediction of geotechnical rock propertiesfrom sonic and neutron logs: Expl. Geophys., 21, 65–72.

Oldenburg, D. W., Li, Y., and Farquharson, C. G., 1997, Geophysical in-version: Fundamentals and applications in mineral exploration prob-lems: in Gubins, A. G., Ed., Proceedings of Exploration 97: FourthDecennial International Conference on Mineral Exploration, 545–548.

Stolarczyk, L. G., 1992, Definition imaging of an orebody with theRadio Imaging Method (RIM): IEEE Trans. on Industry Applica-tions 28, 1141–1147.

Thomas, G., Stolz, E. M., and Mutton, A. J., 1992, Geophysics ofthe Century zinc-lead-silver deposit, northwest Queensland: Expl.Geophys., 23, 361–366.

Waltho, A. E., Allnutt, S. L., and Radojkovic, A. M., 1993, Geologyof the Century zinc deposit, northwest Queensland: Proc. Internat.Symposium—World Zinc ’93, 111–130.


Exploration geophysics at the Pyh¨ asalmi mine and grade controlwork of the Outokumpu Group

Aimo Hattula∗ and Timo Rekola∗

ABSTRACTThe power of geophysics is often realized while survey-

ing barren exploration holes. Integrated interpretationof borehole electromagnetic (EM) and lithogeochemicaldata led to the discovery of a new volcanogenic massivesulfide (VMS) ore deposit at 500 m depth in the Pyha-salmi area, which belongs to the Main Sulfide ore belt inFinland. In the deep exploration program, wide-bandmultifrequency EM ground surveys were successfullyused to detect both new ore lenses and geological struc-tures. Mise-a-la-masse (MAM) borehole and ground sur-veys as well as borehole EM surveys were effectivelyused to correlate intersections between drill holes andto locate new orebodies. The latest modeling of MAMdata resulted in an exploration target at 700 m depth.

The use of geophysics for exploration has been ex-tended to mine production at Outokumpu. Geophysical

logging detects ore-waste boundaries, reduces expen-sive core drilling, and obtains physical property infor-mation quickly on ore intersections. Depending on oretype, geophysical borehole logging can also be appliedto classify mineralization, interpret lithology, and some-times to transform physical responses to metal grades inore. At the Pyhasalmi zinc-copper-sulfur mine, densitylogging in percussion boreholes is used to locate mine-able ore boundaries and to classify drillhole intersectionsas massive or semimassive sulfide ore types. Pyrrhotite-bearing zones are separated from other sulfides by in-ductive conductivity logs. The use of geophysical loggingfor grade estimation and control has been most effectivein the nickel mines at Enonkoski, Finland, and NamewLake, Canada (using conductivity logs), and in the Kemichromium mine, Finland (using gamma-gamma densitylogs).

INTRODUCTION

Economically significant sulfide ore deposits exist on theMain Sulfide ore belt of Finland (Figure 1). The belt is40–150 km wide and at least 400 km long. It occupies an areabetween a gravity low related to a Proterozoic complex and anArchaean basement complex. The Proterozoic portions of thebelt host most of the major sulfide ore deposits and are charac-terized by schists and granitoids. Pyrrhotite-bearing graphiteschists, or black schists, abound in the belt and are responsi-ble for many magnetic and electromagnetic (EM) geophysicaland geochemical anomalies. The Archaean part consists of ex-tensive greenstone belts and granite gneiss. Some 95–97% ofthe bedrock is covered by 6–9-m-thick Quaternary till, sand,and clay. The deposits of the Main Sulfide ore belt containmore than 90% of the known mineral resources in Finland.This confirms that conditions for sulfide ore-forming processesmust have been favorable (Kahma, 1973).

The importance of geophysical techniques in site-specific anddeposit-scale exploration is demonstrated by investigations on

Manuscript received by the Editor May 10, 1999; revised manuscript received June 9, 2000.∗Outokumpu Mining Oy, P.O. Box 15, Kummunkatu 34, FIN-83501 Outokumpu, Finland. E-mail: [email protected]; [email protected]© 2000 Society of Exploration Geophysicists. All rights reserved.

sulfide ore deposits of the volcanogenic massive sulfide (VMS)type in the Pyhasalmi area (Figure 1). Applications are pre-sented to illustrate the successful extension of geophysics intomine production using borehole logging for grade control atmines operated by the Outokumpu Group.

SITE-SPECIFIC EXPLORATION

Geology of Pyhasalmi–Mullikkorame area

Bimodal volcanics, metapelites, tonalite gneisses, pyroxenegranites, gabbros, and granites characterize the geology of themain Pyhasalmi deposit area (Figure 2). Altered volcanics(cordierite, anthophyllite, sericite, chlorite, phlogopite, etc.)are ubiquitous in the area. All of the known VMS-type zinc-copper-sulfur-ore deposits and mineralized zones are locatedin altered rocks (Puustjarvi, 1992).

The massive pyrite-rich zinc-copper-sulfur ore deposit atPyhasalmi is 650 m long and up to 75 m wide, becoming nar-rower to the south and north and dipping almost vertically

1961

1962 Hattula and Rekola

down to a depth of 1450 m. Contacts between ore and wasteare sharp. Acid pyroclastic rocks and quartz porphyries host theore (Maki, 1986). The ore deposit is surrounded by a large alter-ation zone. Sericitized quartzites occur closest to the ore, whilecordierite-mica schists and cordierite-anthophyllite rocks aremore distal. The isoclinally folded alteration zone is over 6 kmlong and 100–500 m wide, and it reaches depths in excess of1450 m. A new deep extension of the ore deposit was discov-ered by down-dip drilling at a depth of 1050 to 1450 m from1996 to 1998. Mine production to date since the mine start-up in1962 has been 33.4 million tons (Mt). Current ore reserves are19.7 (Mt), containing 1.16% copper, 2.25% zinc, 37.7% sulfur,and 0.4 g/ton gold.

The Mullikkorame deposit, a satellite mine of Pyhasalmi, islocated about 7 km to the northeast. The Mullikkorame For-mation is up to 1000 m wide and almost 3000 m long (Fig-ure 3). The stratigraphy of the area from bottom is as follows:mica gneiss (not encountered within the Mullikkorame area)—felsic volcanics—sulfide ore deposits—mafic volcanics. Maficvolcanics with variable amounts of pyrrhotite dominate thewestern part of the formation. To the east, zinc-copper-lead-sulfur ore of the VMS type is commonly hosted by alteredfelsic volcanics within a bimodal sequence (Puustjarvi, 1992).Abundant sericitic and chloritic alteration is typical of the min-eralized horizons. As part of the exploration effort, 148 holestotaling 26.2 km were drilled into the Mullikkorame Forma-tion from 1987 to 1995. Subsequently, 0.27 Mt of ore contain-ing 5.8% zinc, 0.2% copper, 0.3% lead, 22% sulfur, 22g/t silver,and 1.3 g/ton gold were mined from surface in 1991 and 1992.Another orebody, located at a depth of 400 to 600 m, is being

FIG. 1. The Main Sulfide ore belt and ore deposits in Finland.

mined; it contains 0.7 Mt of 8.3% zinc, 0.4% copper, 1.3% lead,17.8% sulfur, and 1.2 g/ton gold. Exploration in the surround-ing areas has been initiated.

The Mullikkorame deposit dips about 60◦ east at surfacebut becomes more flat lying with depth. The plunge of the orelenses is not well established. At greater depths it is almosthorizontal, trending between north and east as inferred frommise-a-la-masse (MAM) data. The volcanics, surrounded bygranites to the east and west, are cut by uralite porphyritedikes, which are controlled by factors not yet fully under-stood. A conductive mylonite layer almost 100 m thick at

FIG. 2. Geological map of the Pyhasalmi deposit area.

Mining Geophysics at Outokumpu Group 1963

the eastern contact between host rocks and granite hampersthe use of electrical methods because of its relatively lowresistivity.

Geophysical surveys in the Pyhasalmiand Mullikkorame deposits

The high-density (4.0 g/cm3) and conductive (10–100 S/m)Pyhasalmi ore deposit was delineated relatively easily usinghorizontal loop electromagnetics (HLEM) and gravity meth-ods (Figure 4). The deposit contains 2–4% pyrrhotite, whichgreatly increases its conductivity compared with the pyrite.Gravity data were interpreted using a prism model to simu-late the orebody. In Figure 5 the cross-section models showhow the orebody is near vertical in upper levels and inclined at

FIG. 3. Geological map of the Mullikkorame area.

deeper levels where a strong D4 shear zone deforms the wholeorebody into a vertical propeller shape.

In the Mullikkorame area, an airborne EM and magneticsurvey (at an altitude of 40 m above terrain) was carriedout in 1978. The first ground follow-up was conducted usingHLEM, magnetic, and gravity surveys. Resulting anomalieswere checked with percussion drill sampling, and a shalloworebody was discovered (Hattula and Rekola, 1997). The de-posit was indicated by a weak −4% HLEM anomaly in bothin-phase and out-of-phase components (Figure 6). StrongerEM responses result largely from the conductive pyrrhotitemineralization (2–10% sulfur) in the west. The poorly conduc-tive zinc-copper-sulfur orebody (0.1–0.5% sulfur) in the east(body A) was weakly indicated only by the higher frequenciesof the out-of-phase component (Figure 7).

IP measurements outline the shallow orebody rather well(Figure 8). The felsic volcanics, with variable amounts of pyrite,are indicated by apparent resistivities in the range of 600–800 ohm-m compared with host rock values of unmineralizedmafic volcanics, which usually exceed 1000 ohm-m.

The Mullikkorame mafic volcanics (2.9 g/cm3) are the sourceof a 1.5-mGal gravity anomaly. These volcanics are interpretedto continue eastward beneath granite (Figure 9). Magnetic andIP surveys suggest the same interpretation. Although ore hasthe highest density (3.6 g/cm3) among rocks in the area, gravitysurveys do not necessarily possess practical value in locatingorebodies because the targets are relatively small compared tothe whole volcanic formation (Rekola, 1992).

MAM surveys were also carried out during the explorationprogram. Measurements on near-surface orebodies delineatedthe extent of sulfides fairly well (Figure 10); however, it wasdifficult to distinguish between ore and noneconomic pyriteconductors because of similar resistivities. MAM was one ofthe main methods used to correlate drillhole ore intersectionsand guide drilling during a deep exploration program. First,this led to the intersection of a 45-m-thick pyrite-predominantalteration zone containing 0.6% zinc and 0.12% lead in holeMU-99 (Figure 11, top); later, an ore lens (with a surface pro-jection as shown in Figure 11, top) was intersected by fur-ther drilling, including extending hole 99. When the deep-seated orebody (B) was discovered, a transmitter (current)electrode was placed in an intersection in hole MU-108. Surfaceand downhole MAM results indicated a potential extensionboth north and south (Figure 11). Interpretation by VIRG-Rudgeofizika, a Russian geophysical institute, using analog-digital modeling from the earlier and latest holes showed threeconductive blocks at 700–900 m depth, under the known deeporebody. Correlation with drill results on one profile demon-strates good confirmation of this interpretation (Figure 11,bottom).

In the deep exploration program at Mullikkorame, wide-band multifrequency EM surveys using the Gefinex 400S sys-tem (Aittoniemi et al., 1987) were carried out with a broadsidecoil configuration across the mineralized zone. Coil separationvaried between 200 and 800 m. Resistivity results interpretedfrom the 800-m coil separation survey data are presented sta-tion by station in Figure 12. A distinct indication of a deeporebody (B) at a depth of 500 m was detected. A dipping low-resistivity zone from west to east is related to pyrrhotites in ba-sic volcanics; existence of this sulfide conductor was indicated


by DHEM measurements in hole MU-120 (Figure 12). TheGefinex 400S survey also suggested a low-resistivity structurebelow a depth of 800 m, but this indication has not yet beentested by drilling. Results of the Gefinex 400S measurementscontributed to the decision to drill into the deep extension ofthe surface orebody.

Drillholes MU-98 and MU-99 intersected an alteration zoneat a depth of 200 to 300 m. Thereafter, all holes were loggedwith a Geonics EM-37 borehole EM system. These DHEMcontributed to a decision to deepen an existing hole MU-108from 450 m (Figure 13), resulting in the discovery of the firstlens of the deep orebody B which in turn led to additional sys-tematic drilling. Further DHEM and MAM surveys around thedeep orebody followed; the DHEM measurements indicatedmany additional ore lenses in the deeper parts of the volcanicsformation.

MINE GRADE CONTROL

Excessive dilution often results as waste is mined and pro-cessed with ore (Elbrond, 1994). Geophysical borehole loggingis used in the mines operated by the Outokumpu Group for

FIG. 4. The Pyhasalmi deposit and geophysical results.

FIG. 5. Measured and calculated gravity field over thePyhasalmi deposit.


quickly detecting ore boundaries and quality to reduce rockdilution and subsequent grade reduction.

Outokumpu developed a borehole logging system, OMS-logg, to directly measure differences in physical characteristicsof mineralization from borehole walls (Figure 14). Often whena mineral (or minerals) has a dominating property, the bound-aries of mineralization can be determined. Grade estimationcan be done using an empirical calibration procedure. Probes

FIG. 6. HLEM out-of-phase results in the Mullikkorame area.

FIG. 7. HLEM results with different frequencies in theMullikkorame area.

are customized according to individual orebody characteristics.The objective is to use percussion drilling or production holesfor in-hole geophysics so that slow and expensive diamond coredrilling is no longer needed in detailed stope planning or gradecontrol. Depending on ore type, OMS-logg applications aredivided into the following categories: determination of litho-logical boundaries, definition of ore zones, estimation of drybulk densities, and grade estimation of ore.

For the past 13 years, borehole logging has been in routineuse at mines of the Outokumpu Group worldwide. Density–conductivity–susceptibility logging is currently used at thePyhasalmi zinc-copper-sulfur and Mullikkorame zinc-copper-lead-sulfur mines in Finland and in the Forrestania nickel minein Australia. The Tara zinc-lead mine in Ireland and the Kemichromium mine in Finland utilize gamma-gamma density log-ging. All of these are underground mines except for Kemi,which is an open pit. The Enonkoski and Telkkala nickel minesin Finland, the Namew Lake nickel mine in Canada, and theViscaria copper mine in Sweden used inductive conductivitylogging. In all of these mines, logging was mainly carried out inpercussion and/or production boreholes.

The latest model of the OMS-logg system provides a modein which measurements can be made without the necessity ofa logging cable. Probes are connected to drill rods and movedin the holes by a drill machine. This permits logging long holesin any direction.

FIG. 8. IP data in the Mullikkorame area.


The boundaries of the Pyhasalmi pyrite-rich massive sul-fide ore are defined using gamma-gamma density logging(Figure 15). Parts of the ore contain pyrrhotite that has almostthe same density as typical pyrite ore. For separating barrenpyrrhotite mineralization from economic ore to be stoped, con-ductivity logging combined with gamma-gamma density log-ging is used. The latter method is also used in the Mullikkoramezinc-copper-lead-sulfur mine. High-density blocks accuratelycorrelate with economic ore boundaries (Figure 16).

The OMS-logg conductivity probe was designed to observeresponses from sulfide nickel ores. A 10-cm coil length allowsthe detection of sharp contacts and narrow veins as well as mea-sures close to the bottom of the holes. The Enonkoski nickelmine is an example in which nickel grade was successfully es-timated using conductivity logging calibrated in massive ore(Hattula, 1992). In addition, reliable lithological informationwas obtained on the distribution of disseminated mineraliza-tion and waste. Figure 17 shows profiles of assay and nickel-calibrated logging data in (a) a diamond drill hole and (b) apercussion borehole.

Lithology in the Kemi chromium mine (Figure 18) is classi-fied using the OMS-logg density logging for product control.Density is correlated with Cr2O3 grades to decide which parts

FIG. 9. Gravity data in the Mullikkorame area.

of the mined ore will be separated for upgraded lump ore.Logging is carried out in production holes in the open pit andin holes drilled for a new underground mine development. Astrict calibration procedure between the gamma-gamma log-ging and density for various ore types is fundamental. It isessential to classify ore types but also to detect boundaries forinternal or external waste blocks.

CONCLUSIONS

The Pyhasalmi and Mullikkorame mine examples show howmany and different petrophysical elements are related to VMS-type deposits. The most complicated are EM and electrical pa-rameters because both waste and ore sulfides cause geophysi-cal responses. Wide-band multifrequency EM surveys providemeans of detecting weakly conductive sulfide ore bodies andmineralization as well as highly conductive ore zones. Detailedinterpretation of MAM data assists in correlating drillhole in-tersections but is often useful also in indicating/detecting off-hole lenses. The use of different geophysical methods and their

FIG. 10. MAM data for the Mullikkorame shallow orebody.


geologically focused modeling provides a tool to distinguishwaste and ore sulfide lenses.

Borehole geophysics plays an important role in the final tar-geting and delineation of ore deposits. In many cases, geophys-ical in-hole survey data may result in discoveries before ac-tually intersecting ore. Borehole geophysics can also greatlybenefit mine grade control in various types of ore deposits.Depending on the physical properties of an orebody, boreholelogging provides fast determination of both ore and/or wasteboundaries, although geophysical logging seldom provides re-sults sufficiently accurate for grade estimation. Fallon et al.(2000) present very similar findings in their investigation ofinterpretation of geophysical borehole logs.

FIG. 11. MAM data for the Mullikkorame deep orebody.

FIG. 12. Resistivity cross-section of the wide-band, multifre-quency EM survey using the Gefinex 400S system over theMullikkorame deep orebody structure.

ACKNOWLEDGMENTS

The authors thank Outokumpu Mining Oy for permissionto publish this paper. Kaarlo Makela and Heikki Puustjarvi,project managers for Outokumpu Mining Oy Exploration,and J. P. Matthews, Manager Geophysical Services for PhelpsDodge Exploration Corp., are thanked for critically reviewingthe manuscript.

REFERENCES

Aittoniemi, K., Rajala, J., and Sarvas, J., 1987, Interactive inversionalgorithm and apparent resistivity versus depth (ARD) plot in mul-tifrequency depth soundings: Acta Polytech. Scandinavica, AppliedPhysics Series No. 157.

Elbrond, J., 1994, Economic effects of ore losses and rock dilution:CIM Bull., 87, 131–134.


FIG. 13. DHEM data from boreholes MU-98 and MU-108 inthe Mullikkorame deposit.

Fallon, G. N., Fullagar, P. K., and Zhou, B., 2000, Towards grade esti-mation via automated interpretation of geophysical borehole logs:Expl. Geophys., 31, 236–242.

Hattula, A., 1992, Borehole logging system at Outokumpu EnonkoskiNi mine: Analys i borrhal, Swedish Mineral Industry Research Or-ganisation report MITU 1992:14 T.

Hattula, A., and Rekola T., 1997, The power and role of geophysicsapplied to regional and site-specific mineral exploration and minegrade control in Outokumpu Base Metals Oy: 4th Decennial Inter-nat. Conf. on Min. Expl. Proceedings, 617–630.

Kahma, K., 1973, The main metallogenic features of Finland: Geol.Surv. Finland Bull. 265.

Maki, T., 1986, Lithogeochemistry of the Pyhasalmi zinc-copper-pyrite

FIG. 14. Schematic of the OMS-logg borehole logging system.

FIG. 15. Conductivity logging combined with gamma-gammalogging for separating pyrrhotite zones from ore to be stopedat the Pyhasalmi mine.

deposit, Finland: Prospecting in areas of glaciated terrain 1986: In-stitution of Mining and Metallurgy, 69–82.

Puustjarvi, H., 1992, Exploration in Finland—Search for VMS-depositsin the Pyhasalmi district, central Finland: Irish Assn. Econ. Geol.course notes.

Rekola, T., 1992, Recent geophysical surveys for massive sulfides inthe Pyhasalmi area, central Finland: 54th Meeting and TechnicalExhibition, Eur. Assn. Eng. Geosci. Abstracts, 316–317.


FIG. 16. Correlation of high-density blocks with boundaries ofecomomic ore-grade mineralization in the Mullikkorame mine.

FIG. 17. Comparison of (a) core sample assays from a diamonddrill hole and (b) sludge sample assays from a percussion bore-hole, versus conductivity calibrated to nickel content at theEnonkoski mine.

FIG. 18. Density logging profile to define ore boundaries andclassify ore for product control at the Kemi chromium mine.


Ultralow-radiation-intensity spectrometric probe for orebodydelineation and grade control of zinc-lead ore

Jacek Charbucinski∗, Mihai Borsaru∗, and Michael Gladwin∗

ABSTRACT

A fully spectrometric gamma–gamma probe using anultralow-activity gamma-ray source (i.e., <2 MBq) hasbeen developed. The prototype low-radiation-intensityprobe was tested at a zinc-lead deposit in Queensland,Australia. The probe showed an excellent capability fororebody delineation. Lithological profiles derived fromlogging data showed sharp anomalies both in selectedspectral regions and in spectral ratios during probe tran-sition from orebody to barren rock, or vice versa. In com-parison to standard geochemical analysis, the instrumentdemonstrated good potential for quantitative determi-nation of lead and zinc content in ore with rms error of0.3% and 2.4%, respectively. Delineation of nonlitholog-ical boundaries, through an application of a cutoff-gradealgorithm, has been demonstrated as a practical stand-alone mine control tool.

INTRODUCTION

The mineral exploration and mining industry needs safe,portable, and maneuverable borehole logging equipment fordeposit delineation and grade control, especially during the fi-nal stages of exploration, mine development, and ore produc-tion. Geophysical logging has become an integral part of mod-ern exploration and mine development activities. Among themany different borehole logging techniques, nuclear boreholelogging is practically the only technique that provides on-line,quantitative, in-situ grade control.

Most commercial logging companies use a nonspectrometricgamma–gamma borehole logging-technique (gamma–gammadensity log). This technique is commonly applied for coal-seamevaluation or orebody delineation. When ore-grade determi-nations are required, it is assumed that ore grade and ore bulkdensity are correlated. While ore grade and density may becorrelated quite well in some deposits, that correlation is oftentoo weak or the polymetallic nature prevents a reliable deter-

Manuscript received by the Editor February 3, 1999; revised manuscript received May 12, 2000.∗CSIRO Exploration and Mining, P.O. Box 883, Kenmore, Queensland 4069, Australia. E-mail: [email protected]; [email protected]; [email protected]© 2000 Society of Exploration Geophysicists. All rights reserved.

mination of ore grade or chemical concentrations of key im-purities. Spectrometric techniques have proven useful for bothidentifying elemental constituents and quantitatively estimat-ing ore. Spectrometric methods can differentially detect andrecord different probe-response events, e.g., those from dif-ferent gamma-ray energies or events having different arrivaltimes. This capability to use gamma-ray energies is importantwhen identifying and estimating some elemental constituentsof the ore, e.g., the prompt gamma neutron activation analysis(PGNAA) technique.

The commercial gamma–gamma density logging tools use ra-dioactive sources with high activities (around 2000–6000 MBq).These levels of radioactivity present difficulties for man-aging the safety and security of mine personnel and theenvironmental impact if a probe is irretrievably stuck in aborehole. CSIRO-developed Sirolog spectrometric boreholelogging probes have achieved significant use and recognitionin the mining industry because of their superior quantita-tive performance (Eisler et al., 1990; Charbucinski, 1993a;Borsaru et al., 1994). However, many mining companies stillconsider that the risk associated with the irretrievable lossof even a standard Sirolog probe is too high (even thoughits source activity of ∼40 MBq is about 1/100th of a con-ventional commercial probe). These companies would pre-fer to reduce the source intensity by another order of mag-nitude. The states of Victoria, New South Wales, and Queens-land in Australia define 3.7 MBq as the minimum activity ofa gamma-ray source requiring a license for possession, use,and transport of radioactive substances. Therefore, require-ments for operations using logging equipment with a radioac-tive source of activity lower than 3.7 MBq would be muchsimpler and would minim the risk associated with radioactivesources.

The Sirolog instrumentation goes one step further towardintroducing to the mining industry a spectrometric boreholelogging system based on applying an ultralow-activity gamma-ray source (<2 MBq). Tests have been reported for coal-seam delineation and on-line quantitative determination ofash in coal (Charbucinski, 1993b; Borsaru and Ceravolo 1994;

1970

Low-Radiation-Intensity Probe 1971

Charbucinski et al., 1996) and for iron orebody delineation(Borsaru et al., 1995).

The objective of our work was to optimize probe design andto demonstrate the delineation and quantitative capabilitiesof the low-radiation-intensity spectrometric probe for in-situanalysis of lead and zinc.

TECHNIQUE OVERVIEW

Two different low-activity probe configurations have beendeveloped for the coal mining industry. In the Low Activitytool (Borsaru et al., 1994), a single 137Cs gamma-ray sourceof 1.8 MBq activity was positioned axially from the detectorbehind a very short (30 mm) shield of tungsten and iron. In thesecond configuration, the Zero Probe (Charbucinski, 1993b),three 137Cs microsources (each 0.37 MBq) were placed radiallyaround the detector at virtually zero distance from the crystal(measured along the borehole axis). This ultimately shortdistance provided the probe with the best possible verticalresolution.

For mineral applications these ultralow activity probes re-quired redesign of probe configurations and development ofnew calibration and inversion algorithms. This was necessarybecause the probes developed for the black coal industry weredesigned to perform optimally in low-density, low-equivalent-atomic-number environments. The zinc and lead ores have sig-nificantly higher equivalent atomic numbers and much higherdensity than the host lithology. The higher densities requirednew optimization of source–receiver distances. Planned largerdiameter holes (>150 mm) allowed further optimization ofscintillator crystal volumes. However, the optimized geome-try still featured a very short source–receiver distance, so theprobe operated in the preinversion zone of the universal cali-bration curve (Artsybashev, 1972). In that zone, the count rate(recorded in the density window) of gamma quanta scatteredin a high-density medium (ore) is higher than the count raterecorded in a lower density scatterer (barren rock).

We need to emphasize that the low-radiation-intensity probeis the low-activity version of the Sirolog spectrometric gamma–gamma probe. As such, the probe is unable to measure theconcentration of zinc directly. The gamma–gamma probe re-sponse is related to the overall contributions given by the majorcomponents with a high atomic number present in the ore, e.g.,lead, zinc, iron, manganese. Figure 1 shows spectra recorded

FIG. 1. Spectra recorded with a 76-mm diameter × 76-mm de-tector in large (200-liter) ore samples of various zinc, lead, andiron content.

in five large geophysical models. Four models contained zinc-lead ore of various concentrations, and the fifth model (W2)contained waste. The prominent features of the spectra are the180◦ backscatter peak around 210 keV and lead’s characteris-tic X-ray peak around 87 keV. Not shown in the figure is thephotopeak of primary radiation from the probe’s 137Cs source(667 keV), which was utilized to stabilize the logging systemshardware gain. The probe was designed to use single scatteredgamma rays (180◦) to provide information about the bulk den-sity of the logged formation, while multiscattered quanta oflower energies convey information on the average chemicalcomposition of the ore. The lead X-ray peak helps determinepercent lead content (%Pb).

The main way the major ore components contribute to thevalue of a spectral ratio used to correlate with the averagechemical composition can be expressed in percent zinc-metalequivalent (%ZME) units. ZME units provide a useful mea-sure of the response of the probe to the total composition ofthe material. They are defined from equivalence factors (EF),which are numerical values that translate contents of otherheavy elements (e.g., lead, iron, manganese) into equivalentzinc content. The EF is a multiplier such that gamma radiationscattered and/or absorbed by a compound of a multiheavy-element medium (and measured through a given spectral pa-rameter) results in the same value as that of the spectral param-eter being measured in a monoheavy-metal ore (e.g., zinc only).

For percent zinc content (%Zn), the %ZME value is

%ZME = %Zn + %Pb ∗ EFPb/Zn

+ %(Fe + Mn) ∗ EF(Fe+Mn)/Zn.

In other words, gamma-ray interaction with matter (abovethe energy of the lead X-ray peak) in a multiheavy-elementmedium should be the same as the interaction in a hypotheticalmonoheavy-element medium described by the same %ZME.EFs are usually established empirically and have differing val-ues for different spectral ratios or windows.

The quantitative determination of zinc content in ore, af-ter correcting for lead content’s spectral contribution (X-raypeak), is possible only if the iron and manganese content areeither relatively constant, at least within a given ore unit,or can be measured by a complementary physical or chem-ical technique. The iron component requires an alternativemeasurement technique—perhaps a magnetic susceptibilityprobe—that can be incorporated into the measurement sys-tem to provide a single-pass logging operation. Pyrite, whichis fairly common in lead-zinc sulfides, does not give a responseto susceptibility; hence, iron from pyrite cannot be estimatedthis way.

INSTRUMENTATION

The large hole size (142 mm) permitted application ofa higher volume crystal than the standard detectors of the60-mm Sirolog probes. A probe with an external diameterof 100 mm was designed and manufactured. This probe usesa 76 × 76-mm NaI crystal. The auxiliary equipment consistsof a portable single conductor cable and winch with a winchcontroller, the surface electronics unit (Siromca), and a ded-icated PC computer. Because of the very weak gamma-raysource applied (<2 MBq), there was no need for a source

1972 Charbucinski et al.

container/transporter. The Siromca unit combines the func-tions of pulse shaper, nuclear amplifier, and digitally gain-stabilized multichannel analyzer.

In the new generation of Sirolog instrumentation, theSiromca surface electronics unit has been eliminated. AllSiromca functions and additional features have been incor-porated directly into the logging probe, and all Sirolog spec-trometric systems use essentially the same hardware. Recentmicroprocessor developments have allowed the design of ageneric multichannel analyzer (MCA) system that can fulfillthe needs of these low-count-rate and low-power-consumptionsystems while coping with the higher count rates of logging(∼100,000 cps). The system was designed to fit inside a 60-mmprobe housing and uses serial communications to transmit com-plete spectra up the cable in digital format to a PC.

FIELD TRIALS

To establish the bed resolution capability of the loggingprobe, static measurements were performed in a two-layermodel simulating barren rock and zinc-lead ore. The measure-ment points were established by moving the probe uphole in1-cm increments. Figure 2 presents the change of count raterecorded in the spectral window (112–135 keV) for the ore-to-rock transition. A value of 120 mm is a distance for which afull transition from ore count rate to barren rock count ratetakes place. This value is the distance from the barren rock/oreboundary, where the probe will record information reflectingthe intrinsic value of the spectral parameter for the lithologicalunit.

Two cored holes, reamed later to a diameter of 142 mm(which is close to the anticipated diameter of future productionholes) were available for the tests. Both holes were water filled.Cors obtained from those holes were subdivided according tothe observed lithological classification. Over 80 sections of corewere assayed, and the results of laboratory analysis were usedas reference points for assessing the probe’s potential to deter-mine lead and zinc content.

Delineation of the orebody

The borehole tests were performed at logging speeds be-tween 1.5 and 2.0 m/min, implying integration times of 1.5 s for asampling interval of 5 cm. The spectral parameters, which wereapplied as delineation indicators, were the same as the best in-

FIG. 2. Ore-to-rock transition response profile (static measure-ments, 10-mm increments).

dicators found during laboratory tests—namely, the spectralwindow (112–135 keV) and the spectral window ratio (190–235/112–135 keV). Figure 3 shows a geophysical profile basedon this spectral ratio plotted alongside the zinc chemical assaylog. The spectral ratio changes significantly with the transitionfrom barren rock to zinc ore, from around 1.4–1.5 to 2.3–2.5,providing a good means for ore delineation. A comparison withpreviously acquired commercial logs indicate the Sirolog low-radiation-intensity probe offers better boundary definition.

Probe quantitative performance

The analysis of the spectrometric data was aimed at selectinga set of spectral ratios and/or spectral windows to be used toquantitatively analyze acquired data. Initially, the windows andratios used for to analyze the laboratory data were applied.Then, a number of new windows and ratios were evaluated andsubjected to an analysis where step-by-step changes of Pb/Znand (Fe + Mn)/Zn EFs were investigated. The final stage ofselecting the optimal spectral parameter for quantitative zincpredictions (after fitting the right EFs) was a fine-tuning ofthe spectral boundaries of the best spectral ratio found in thepreceding stage. Figure 4 shows correlation between values of%ZME derived from logging, applying the optimal spectralparameter (based on the special ratio presented above) for alinear calibration equation and the values of %ZME based onchemical assays.

The rms deviation between laboratory assays and the tool’spredictions was 2.4%ZME. The standard deviation for the pop-ulation, s, was 10.4%ZME, and the number of assays in the re-gression equation was 72. The correlation coefficient was 0.97.Only 72 samples were considered in the regression equation.The reason for not including all 80+ samples was to have a moreuniform distribution of samples in the regression. Therefore, anumber of low-grade samples (ZME < 9) were (randomly) notincluded in the analysis. If all of the samples were included inthe equation, the data points would have been clustered in thearea of low ZME.

Determination of lead content

The 87-keV characteristic X-ray peak showing prominentlyin the backscattered gamma-ray spectrum was used to deter-mine lead content. The results of the laboratory investigationsof the potential of the Sirolog low-radiation-intensity probefor lead determination has been reported by Almasoumi et al.(1997). The spectral parameter based on a ratio of net lead peakarea to gamma-ray intensity of energy just above the lead peakprovided a good measure of lead content in the ore. Figure 5 isa crossplot of the predicted percent of lead content (obtainedfrom a two-variable calibration equation) against the labora-tory assays. The rms deviation between the field estimates andthe laboratory assays was 0.3%Pb, and the standard deviationof the population was 1.7%Pb. The correlation coefficient forthe regressed data was 0.98.

Determination of zinc content

To assess the feasibility of zinc-grade predictions fromlog-derived values of %ZME and lead, and also iron andmanganese contributions to %ZME, predictions need to be


estimated. Spectrometric logging data permit the applicationof an intrinsic correction of the lead contribution by measur-ing the net area of the characteristic lead X-ray peak. Becausethe iron and manganese contributions to the total value of%ZME cannot be measured from the spectrometric gamma–gamma log, the corrections for iron and manganese were basedon statistical information from the geological data base. Fig-ure 6 shows a crossplot between zinc content derived from%ZME and the chemical assays. The correlation coefficientfor the data set was equal to 0.90, and the rms deviation wasequal to 2.4% zinc. Only when the lead contribution was takeninto account were the quoted values 0.88 and 2.6%Zn, cor-respondingly. However, a noticeable improvement may bepossible when percent iron (and percent manganese) con-tent is measured directly by another physical or chemicaltechnique.

Reproducibility tests

Each hole was logged at least twice to assess how repro-ducible the obtained profiles and quantitative predictions were.Figure 7 shows logs of duplicate runs in one of the holes.The logs show spectral intensity measured in energy window112–135 keV. For better clarity of comparison, run 2 has beenshifted vertically along the y-axis. Runs 1 and 2 are almostidentical, proving a high quality of delineation reproducibility.

Reproducibility of quantitative predictions was tested bycomparing values of %ZME obtained from run 1 with the cor-responding values obtained from run 2. The 83 logging splits,corresponding to the core samples taken for chemical analysis,

FIG. 3. Chemical assay for the test hole and the logging profile based on the spectral ratio: 140–235/110–135 keV (T. Petrov, 1996,unpublished ATD Rio Tinto–C.R.A. report).

were used to test reproducibility. Values of the spectral ratioused for analyses were derived from both logging runs; subse-quently, pairs of %ZME were calculated using the calibrationequation. Finally, the values from run 2 were regressed againstthe values from run 1. The correlation coefficient for the dataset was equal to 0.99, and the standard deviation (rms) for theregression was equal to 0.85% ZME, indicating high repro-ducibility of the logging data.

Determination of cut-off boundary

The application of the Sirolog low-radiation-intensity probefor predicting the cut-off boundaries of zinc ore was tested on

FIG. 4. Crossplot of log-derived %ZME versus %ZMEobtained from chemical assays.

1974 Charbucinski et al.

FIG. 5. Comparison between log-derived %Pb and %Pb fromchemical assays.

FIG. 6. Comparison between %Zn, log-derived (%Zn-log),and %Zn from chemical assays (%Zn-chem). The log-derivedvalues of %Zn are Pb and Fe + Mn corrected.

one of the ore units. By definition, the cutoff boundary is thedepth where the integrated zinc value from the top of the oreunit reaches the lowest economically acceptable value for themined unit. The integrated values of %Zn were calculated fromthe top of the ore unit in 5-cm-depth increments. We assumedthat the concentrations of zinc, lead, iron, and manganese wereconstant for each of the 5-cm splits that belonged to a chem-ically assayed sample interval. This assumption might not al-ways be accurate. Figure 8 shows profiles of integrated (fromthe top of the ore unit) values of % Zn derived from chemicalassays (%Zn-chem) and two profiles of %Zn derived from logs(%Zn-log), one corrected on only %Pb and the second addi-tionally corrected by the average (for a particular unit) value of%Fe + %Mn. The profiles of the log-derived %Zn-log matchthe profiles of %Zn obtained from chemical assays. The %Znlog profiles show more structure than %Zn-chem for the topsection of the ore unit.

CONCLUSIONS

The spectrometric low radiation intensity probe is effec-tive in the zinc-lead ore application. Nonspectrometric probeswould be incapable of equivalent performance standards. Theprobe utilizes a gamma-ray source of activity <2 MBq and pro-vides minimal risk to safety and environmental integrity.

The probe showed excellent delineation characteristics andindicated considerable variation in the log. The logging datashowed sharp definition, both in selected spectral regions and

FIG. 7. Profiles of duplicate logs in one of the test holes (profileof run 2 has been shifted vertically on the count-rate axis).

FIG. 8. Profiles of integrated values of %Zn from the top ofthe ore unit for both chemical assays and log-derived values.

in spectral ratios during probe transition from ore body to bar-ren rock, or vice versa. Placing the source a very short distancefrom the detector provided the probe with the best possiblevertical resolution.

The quantitative performance of the probe was satisfacto-rily demonstrated, providing determination of % Zn to within2.4% rms and %Pb within 0.3% rms. Delineation of nonlitho-logical boundaries by applying a cutoff grade boundary wasalso demonstrated as a practical stand-alone mine control tool.

ACKNOWLEDGMENTS

The authors are grateful for the collaboration and supportgiven by RTZ CRA. The authors thank A. Hugill and T. Petrovfor their active collaboration—in particular, for providing for-ward modeling of the ore body. Contributions from A. Mutton,A. Waltho, and L. Partridge are also acknowledged. The au-thors are also grateful for the technical support provided byR. Dixon and Z. Jecny of CSIRO.

REFERENCES

Almasoumi, A., Borsaru, M., and Charbucinski, J., 1997, The applica-tion of the gamma–gamma technique for the determination of leadconcentration in boreholes in a Pb-Zn deposit using very low activ-ity gamma-ray sources: Internat J. Appl. Rad. Isotop., 49, No. 1–2,125–131.

Artsybashev, V. A., 1972, Yaderno-geophizitshaya razvedka: Atomiz-dat.

Borsaru, M., and Ceravolo, C., 1994, A low activity spectromet-ric gamma–gamma borehole logging for the coal industry: Nucl.Geophys., 9, 343–350.


Borsaru, M, Charbucinski, J., and Eisler, P. L., 1994, Nuclear in-situanalysis techniques for the mineral and energy resources miningindustries: Proc. 9th Pacific Basin Nuclear Conference, 1, 391–398.

Borsaru, M., Ceravolo, T., and Tchen, T., 1995, The application of thelow activity borehole logging tool to the iron ore mining industry:Nucl. Geophys., 9, 55–62.

Charbucinski, J., 1993a, Comparison of spectrometric neutron-gammaand gamma–gamma techniques for in-situ assaying for iron ore gradein large diameter production holes: Nucl. Geophys., 7, 133–141.

——— 1993b, The ZERO PROBE low radioactivity borehole loggingtool: Nucl. Sci. Symp., 2, 855–859.

Charbucinski, J., Ceravolo, C., and Tchen, T., 1996, Ultra-low activ-ity spectrometric probe for the coal mining industry: J. Radiochem.Nucl. Chem., 206, No. 2, 311–319.

Eisler, P. L., Charbucinski, J., and Borsaru M., 1990, CSIRO boreholelogging research and development for the solid resource miningindustries: Proc. Internat Symp. Internat. Atomic Energy Agency,IAEA-SM-308/59, 299–318.


Case History

The Spectrem airborne electromagnetic system—Further developments

Peter B. Leggatt∗, Philip S. Klinkert∗, and Teo B. Hage∗

ABSTRACT

The Spectrem airborne electromagnetic (AEM) sys-tem has been in full production in Canada andAfrica since 1989. The prototype, built by A-cubed ofMississauga, Canada, was subject to two major upgradesby Anglo American Corp. staff to produce the Spec-trem II instrument that located the Photo Lake andKonuto Lake orebodies in Manitoba, Canada. Subse-quent work in tropical Africa has made it desirable tofurther increase the transmitter power and use greateravailable computer power in the aircraft to substantiallyimprove the ability of the system to reject sferic noise,although there appears to be a limiting sferic level abovewhich current filtering methods are ineffective.

Although designed to detect massive sulfide bodiesdeep below conductive overburden, techniques havebeen developed to map the regolith—in particular, tocompute resistivity–depth sections using an approximatebut fast method. A comparison is presented of such a sec-tion with the same section flown and processed using afrequency-domain helicopter AEM system.

The latest upgrade of the system, Spectrem 2000, witha 50% increase in transmitter power, was completedin October 1999 and is currently at work in northernCanada.

INTRODUCTION

The Spectrem airborne electromagnetic (AEM) system hasbeen flying surveys in Canada and Africa since 1989. In thatyear, the prototype, described by Annan (1986), was sevenyears old and the original computer hardware then consider-ably out of date. A review of operational performance in 1990identified major inefficiencies attributable to the poor avail-ability of spare parts for the gasoline-powered engines of theDC3 (built in 1940). As a result, Basler Turbine Conversions

Manuscript received by the Editor June 14, 1999; revised manuscript received June 9, 2000.∗Spectrem Air Ltd., P.O. Box 457, Hanger 204, Lanseria Airport, Lanseria 1748, South Africa. E-mail: [email protected]; [email protected]© 2000 Society of Exploration Geophysicists. All rights reserved.

Inc. of Oshkosh, Wisconsin was hired to modify the airframeand fit modern PC16 turbine engines. While this work was inprogress, almost all of the electronic hardware was redesignedand replaced by Anglo American Corp. staff, as was all of thecomputer hardware.

The new AEM system was named Spectrem II (described byKlinkert et al., 1997), and operations commenced in Manitoba,Canada, in 1993. The rms transmitter dipole moment had beenincreased by a factor of two to about 300 000 Am2. The princi-pal result was the discovery of the Photo Lake orebody, whoseprofile is shown in Figure 1. Subsequently, at least two newconductors located by this survey have been followed up andevaluated as massive sulfide orebodies. These are now produc-ing mines or are about to be developed.

The success of this effort is largely because of vigorousground follow-up of the conductive targets by the Flin Flonmine exploration staff.

It is worth examining the results of this early success to iden-tify the best method likely to improve the system.

SIGNAL PROCESSING

The AEM system described here differs from the pulse sys-tems which evolved from the original Input time-domain AEMsystem in that the transmitter waveform approximates a mag-netic step rather than the magnetic impulse of the pulse sys-tems. Having a step transmitter waveform means effectivelythat the pulse width of the inducing signal can be almost 100%of the time for a half-waveform. Liu (1998) gives a good ex-position of the merits of different waveforms. He states, “Theratio of the target voltage response due to two pulses of differ-ent shape is approximately equal to the ratio of the areas thatthese pulses enclose with the time axis for a target of large timeconstant.”

The drawback of having the transmitter always active is thatone must effect the separation of the transmitter primary signalfrom the ground secondary signal at the receiver by signal pro-cessing. This is unlike pulse systems, which effectively measurethe secondary signal when the transmitter is off. The primary

1976

Spectrem AEM Developments 1977

and secondary signals are rather similar in shape. Unlike thesituation with ground EM systems with a static geometry, theamplitude of the transmitter primary signal at the receiver can-not be exactly computed from the static geometry. This is be-cause the exact geometry at any moment is unknown becauseof the unpredictable relative motion between the towed birdand the aircraft. These primary signal variations are generallyorders of magnitude larger than the secondary-field amplitudesand, if not removed, will entirely swamp the latter.

In Figure 1 the X primary field (bottom trace) shows a varia-tion of about 10% just to the right of the Photo Lake body overa width much the same as the anomaly. In the more turbulentconditions to be found in the tropics, these variations are evenlarger. Nearly all this variation in primary signal is from rota-tion of the transmitter and receiver coils with respect to eachother.

To deconvolve the received signals and compute the impulseresponse, one must construct an operator which, when con-volved with the received primary signal, will reduce the latter

FIG. 1. The Photo Lake orebody detected by the AEM system in 1993. Two groups of profiles are shown. The upper group is the Zchannel (coil oriented vertically), while the lower is the X channel (coil oriented horizontally along the line of flight). Within eachgroup the early time channels are at the top. The bottom channel in each group is the primary field. Vertical scales decrease withlater time. The scales are chosen so that residual low-frequency effects show the same amplitude on all channels to aid recognition.

to an impulse. It has long been known in the seismic indus-try (Robinson, 1978) that this is normally possible only whenthe primary signal is minimum phase. The transmitter primarywaveform has accordingly been designed so the half-waveformis minimum phase.

Once the impulse response has been determined, the step re-sponse of both primary and secondary signals is calculated. Forthe step response, the secondary signal resulting from confinedconductors can be closely approximated by a sum of exponen-tial decays (Kaufman, 1978), while the primary signal takes theform of a dc level as designed—this level being the amplitude ofthe step which that signal has become. In this form the primarysignal is easier to separate from the desired secondary signal.Effectively, to within a scaling constant, one computes the mag-netic field that results from a transmitter whose current wave-form is a perfect step. This has advantages in interpretationand, of course, in the computation of conductivity–depth sec-tions using a technique pioneered by Macnae and Lamontagne(1987).

1978 Leggatt et al.

An example of the deconvolution is shown in Figures 2and 3. Figure 2 shows the signal, stacked down to a singlehalf-waveform after removal of the low-frequency and sfericnoise. This plot of the first 200 samples of the 256-sample half-waveform of the combined primary and secondary signals isalmost indistinguishable from a stack devoid of any secondarysignal because the latter is so small. To make this point, thelower of the two traces shown in Figure 2 is the combinedprimary and secondary signal over Photo Lake superimposedon the primary signal alone. (The latter is recorded at a greatheight away from ground signals.)

A least-squares adjustment of the scale of the prerecordedprimary reference signal has been applied to equalize the am-plitudes of the two signals superimposed in the lower trace. Theupper trace is the same scaled reference signal, shifted upward20 mV to facilitate comparison. One can see, with suitable mag-nification, the added secondary signal as a slight thickening ofthe lower trace—particularly at early time.

In Figure 3 one sees the same data converted to step re-sponse with the primary signal removed. The final 56 pointshave been omitted to simplify the plots. The last 128 samplesof the step response are used as an initial estimate of the levelof the primary step, which has been subtracted in the plot. Thefirst 128 points of the waveform are averaged down to sevenwindows, shown as boxes in Figure 3. Usually only these sevenvalues and the value of the primary signal are saved for furtherprocessing and interpretation from profile plots. A final valueof the eighth secondary signal window (occurring in samples129 to 256), the late time-decay constant, and the corrected

FIG. 2. Sample of the 200-ms X channel data filtered andstacked to a single half-waveform at the peak of the secondarysignal over the Photo Lake orebody. The lower trace showsthe primary and secondary superimposed; the upper trace isthe primary alone. To aid comparison the upper trace has beendisplaced upward by 20 mV. Only the first 200 of the 256 pointsof the waveform are plotted here. The secondary signal is sosmall that it can only be identified by the slight thickening ofthe lower trace.

primary signal can be computed from this data further downthe processing chain.

In the AEM system signals shown here, there is no specificsferic removal applied in the stream data apart from removingall frequencies that are not odd multiples of the fundamentaltransmitter frequency. (The transmitter signal has half-wavesymmetry, so its spectrum contains only frequencies that areodd multiples of the primary frequency.) Figures 2 and 3 illus-trate how careful one must be not to damage the tiny secondarysignal by data processing.

NEW AREAS, NEW PROBLEMS

In 1997 the aircraft returned from Canada to its home inAfrica and surveys were flown in subtropical and later tropicalareas with large areas of conductive cover that presented avery different prospecting environment than northern Canada.More recently, work has been completed in West Africa.

All the African surveys, flown within about 15◦ latitude of theEquator, encountered an unprecedented level of sferic noisecompared to that experienced in higher latitudes.

The experience confirms the analysis of Buselli and Cameron(1996), who report sferic levels in northern Australia some 100times worse than levels in the south. While we have tried manyof the robust methods of sferic reduction mentioned in theirarticle, we have had most success using the methods of Strack(1990). This is a simple selective stack where only a proportionof the data close to the median of the stack is used. However, wehad not anticipated that the recently encountered high sfericlevels could persist for days at a time.

The practice of simply waiting for unusually high levels ofsferics to pass, like a magnetic storm, leads to unacceptable loss

FIG. 3. The data of Figure 2 converted to a display of the stepresponse. The last 128 samples of the waveform (not all ofwhich are shown) have been used to estimate the dc valuerepresenting the primary field. The boxes are a display of theseven values of secondary signal saved, the height of each boxbeing the average of all the samples between its ends.


of production in equatorial latitudes. James Macnae (personalcommunication, 2000) advises that heavy sferics are inevitablein all seasons in equatorial Africa (see thunder.msfc.nasa.gov).

A 200-ms sample of data with moderate sferic noise is shownin Figure 4, and an example of data from tropical Africa isshown in Figure 5, both using 25-Hz data. Turbulent motion

FIG. 4. A 200-ms buffer of moderately noisy data recorded at25 Hz with 7680 digitized values from the Y channel of the re-ceiver. After removal of low frequencies, the result is displayed1 V below the time axis. After removing sferics, the result isdisplayed 1 V above the time axis.

FIG. 5. Data similar to Figure 4 with very large amounts ofsferic and turbulence noise. Note the huge low-frequency signalwhich comes from turbulent motion of the bird in the earth’smagnetic field.

of the towed receiver coils causes them to rotate with respectto the transmitter. This has two undesirable effects: (1) theamplitude of the primary signal changes because of the changein coupling with the transmitter coil and (2) the rotation itselfinduces low-frequency signals from coupling changes with theearth’s magnetic field. In Figure 5 the Y channel is displayed(horizontal component perpendicular to the flight direction).This component is nominally null coupled to the transmitter;not only can the amplitude of the primary field change, but alsoits sign.

A two-stage filtering process is demonstrated. First, the low-frequency (<25 Hz) signal is removed with a linear filter. Theresult is displayed along with a −1.0-V displacement. Thenthe five negative half-waveforms of the ten in the stack areinverted. For each of the 768 samples in the half-waveform,we have ten specimens. The median of these ten specimens iscomputed, and the six specimens with values furthest from themedian are discarded. The other four are averaged to produce asingle point of the 768-point waveform. The ten half-waveformsshown displaced +1.0 V above zero are a replication of thisresultant single half-waveform.

The implied inference in this procedure is that one can ex-pect no sferics for about half the time. Figure 5 shows this is afallacy in some parts of the world. Considering the small sec-ondary signal of an orebody shown in Figure 2, the accurateremoval of sferics and other noise is very important. Whilelower fundamental transmitter frequencies improve the depthpenetration, lowering the frequency from 75 to 25 Hz reducesthe number of half-waveforms to stack from 30 to 10, with aconsequent reduction in noise rejection capability.

NEW TRANSMITTER WAVEFORM

Early in October 1999 a new system, identified as Spectrem2000, was completed. A new transmitter using quasi-resonanttechniques has been developed to attempt an rms dipole mo-ment of 1 million Am2. For the time being the rms dipole mo-ment has been increased to 450 000 Am2, since realizing fullpower will involve rather costly modifications to the loop wires,their suspension, and mounting on the airframe.

The transmitter waveform is approximately a square wavebecause of the desirable properties of that waveform as previ-ously discussed (Liu, 1998). The new waveform is an improvedapproximation to a square wave, with a faster rise time and aflatter top. The higher rms dipole moment increases power inthe lower frequency components, which facilitates deep pene-tration, while the faster rise time permits better resolution ofshallow features.

The better resolution assumes greater importance when sur-veying prospects having a complex conductivity structure, asdistinct from discrete conductors in resistive host rock. A mas-sive sulfide orebody in the latter category gives an anomalywith a sharp and localized peak. But the complex conductiv-ity structure of Africa, with deep and conductive weathering,requires more emphasis on mapping the regional conductivestructure and interpreting it in combination with other datasuch as magnetics and radiometrics integrated with mappedgeology.

In the Spectrem I system, data were stacked for a periodof 200 ms down to a single half-waveform of 128 points. Af-ter deconvolution, the data volume was further reduced by

1980 Leggatt et al.

averaging (or binning) the 128 points into seven binary spacedwindows (sometimes called gates), each twice as wide as itspredecessor. Apart from a few test flights, the 128-point stackwas not recorded. In the current AEM receiver, the stackedhalf-waveform (now 256 samples because of faster digitizers)is routinely saved to tape for postflight analysis. This has givenopportunities for new processing insights, the most importantbeing that there are categories of noise that, once blurred into acommon stack, cannot be removed effectively. In consequencethe latest receiver records all data from all digitizers in an unin-terrupted stream (prestack) during each active traverse. Datastorage requirements have increased from 800 bytes/s (binneddata) to 21 280 bytes/s (stacked half-waveform data) and mostrecently 738 880 bytes/s (prestack data).

RESISTIVITY MAPPING

The mapping of regional conductivity variations has provedrather useful in Africa. The volume of data collected during asurvey is too great to permit a full 3-D inversion of an entiresurvey, even if this were desirable. An approximate method,described by Macnae et al. (1991) and termed conductivity-depth imaging, is used to compute a conductivity or resistivity–depth parasection.

The method strictly applies only to a horizontally layeredearth. The changing amplitude of the secondary EM signal as afunction of time is used to compute a slowness (reciprocal ve-locity) function at the position of an assumed receding imagesource. In turn, conductivity as a function of depth is com-puted from the slowness of the downward receding image. Themethod, akin to resistivity mapping, explicitly assumes that theearth model is a 1-D layered earth.

To honor the layered-earth model, one must assume thatthe data reflect a model that is at least approximately a layeredearth to within, say, twice the horizontal coil spacing, or about250 m. The AEM system has a vertical transmitter–receiverseparation of 35 m and a horizontal spacing of 125 m, geome-try achieved by use of a very high-drag receiver bird. This smallvertical separation results in the Z -component anomaly beingeffectively symmetrical for profiles across horizontal conduct-ing slabs. The EM response at the edges of such slabs consistsof negative side lobes about 100 m wide when flying at 100 msurvey height. As these negative-amplitude features, togetherwith other narrow responses from culture, power lines, andshallow discrete conductors, degrade the conductivity calcula-tions, they are removed using a low-pass filter prior to com-puting the conductivity parasection. While there is no doubtthat this reduces the horizontal resolution, it is a requirementforced by limitations of the layered-earth model.

Figure 6 shows a resistivity–depth parasection computedfrom synthetic data. The model is a 500 × 500 × 50-m-thick hor-izontal conductive slab of conductivity 1 S/m embedded in ahalf-space of 0.004 S/m. The top of the slab is on the ground sur-face. The upper part of the figure shows the filtered Z -channelsecondary amplitudes, while the lower part shows the conduc-tivity parasection on which is superimposed the traces of theimage depths computed from each channel. Lateral filteringhas removed the side lobes at the edges of the body, and theminor asymmetry that remains is less noticeable in the imagethan in the EM amplitude data. The resistivity–depth parasec-tion represents the body as being more narrow than it really is

and has rounded the upper surface quite severely. The thicknessis represented as about twice the actual thickness. The repre-sentation of the body is reasonably symmetrical so that onewould have difficulty identifying whether the line was flownleft to right or right to left. The most serious asymmetry is thesmall displacement of the entire body to the right. This is easilyrectified in plan presentations by applying a lag to the entireprofile.

The numbers displayed along the ground surface are sample(fiducial) numbers; in this case, we have 200 samples across thefull 1 km of the section. The horizontal scale is about 10 timesfiner than one would normally use to display a real section.

Accepting the limitations of the unrealistic earth model used,conductivity–depth data are independent of aircraft terrainclearance and flying direction. So the depth parasections canbe used to produce maps of several derived parameters, in-cluding contours of depth to a given interface, the conductivityor thickness of the overburden, or the conductivity below agiven depth. In practice the limitations of the model are lessrestrictive than the synthetic example might suggest. In fact, forfeatures of lateral dimension larger than the aforementioned250-m restriction, considerable vertical variations can bedisplayed.

Figure 7 shows a resistivity–depth parasection flown by bothfixed-wing AEM and a frequency-domain helicopter electro-magnetic (HEM) system (Dighem) for comparison. The heli-copter data have been processed with two different algorithmswhich give slightly different results. Details of fixed-wing AEMprocessing are given in Leggatt and Pendock (1993). HEM de-tails are given in Huang and Fraser (1996) and Sengpiel (1988).While both systems give parasections that are broadly similar,

FIG. 6. A resistivity–depth parasection computed from syn-thetic data from the Emigma modeling package. The modelwas a 500 × 500 × 50 m horizontal slab of 1 S/m conductivityembedded in the top of a half-space of 0.004 S/m conductivity.The numbers along the surface of the ground are sample num-bers, and each represents 5 m of traverse. The entire traverse is1 km long. The true position of the body is shown by the rect-angular outline. The resistivity–depth parasection has roundedthe shoulders of the body significantly and overestimated itsthickness. The major asymmetry in the resistivity–depth para-section is the small displacement of the model to the right.


three boreholes were selected to examine the section at pointswhere the EM results differed most from each other.

At borehole 6750, fixed-wing AEM shows a thin surface con-ductor over a resistive layer about 90 m thick with another con-ductive layer below it. The HEM data image the resistive layerpoorly and fail to detect the deeper conductive layer below it.Unfortunately, the hole was not drilled deep enough to testthe thickness of the lower conductor defined by the fixed-wingsystem.

Borehole 6650 confirmed the presence of the deep conduc-tor and showed the interface at its top and bottom at depthspredicted by fixed-wing AEM. The HEM survey did not detectthe base of this conductor. Borehole 6500 confirmed both theAEM conductivity and the Sengpiel inversion of HEM for theupper part of the section, while both methods of processingHEM failed to resolve the resistive layer lower down. One ex-pects a helicopter EM system to be slightly better at resolvingshallow horizons than a fixed-wing AEM system, but this is notevident in this example. The greater low-frequency componentof the AEM square-wave signal, designed to maximize pene-tration, is shown to advantage in the resolution of structuresbelow 90 m.

FIG. 7. Comparison of resistivity–depth parasections computed for HEM (Dighem) and fixed-wing AEM (Spectrem) systems. Thedrillholes were located at points where the two systems differ. The fixed-wing system running at a transmitter frequency of 75 Hzseems to have shallow-layer vertical resolution as good as HEM and better depth penetration. Despite inherent limitations in lateralresolution of the fixed-wing system discussed in the text, the horizontal resolution seems no worse than for the HEM parasection.

Parasections of AEM data usually display conductivity, butin this example, resistivity has been plotted using colors approx-imately the same as those used for HEM, for ease of compari-son. Even in view of the lateral filter applied to the fixed-wingAEM data (15 fids), the lateral resolution of the fixed-wingsystem appears at least as good as the HEM system in thisparticular example.

CONCLUSIONS

AEM survey work in Africa encounters more problems ofnoise from geological sources than is the case in the CanadianShield. The tropical environment creates more turbulent airconditions when flying over hot areas. The heat itself (occasion-ally >50◦C in the aircraft) makes operating conditions moredifficult and unpleasant and is harder on the equipment. In theIntertropical Convergence Zone, the level of sferics is oftenso high that good data acquisition can be very difficult. Thereis a critical level of sferic noise which, when exceeded, makesrecovery impossible within the short (200 ms) time availablefor an AEM reading. This results in data dropouts.

Conductivity–depth imaging is a useful tool for approximat-ing the structure of an essentially horizontally layered earth,

1982 Leggatt et al.

which has useful applications in mapping the subsurface ge-ology. Despite an inherent limitation in lateral resolution,considerable detail can be deduced from a parasection thatis confirmed by drilling.

ACKNOWLEDGMENTS

The authors thank the Anglo American Corp. for permis-sion to publish this work and Steve Lynch for calculating theEmigma model. Emigma is an EM forward-modeling softwarepackage from PetRos Eikon Inc. Thanks are also extended toJames Macnae and an anonymous reviewer for constructiveand appreciated criticism.

REFERENCES

Annan, A. P., 1986, Development of the prospect 1 airborne electro-magnetic system, in Palacky, G. J., Ed., Airborne resistivity mapping:Geol. Surv. Canada, 63–70.

Buselli, G., and Cameron, M., 1996, Robust statistical methods for re-ducing sferics noise contaminating transient electromagnetic mea-surements: Geophysics, 61, 1633–1646.

Huang, H., and Fraser, D. C., 1996, The differential parameter method

for multifrequency airborne resistivity mapping: Geophysics, 61,100–109.

Kaufman, A., 1978, Frequency and transient response of electromag-netic fields created by currents in confined conductors: Geophysics,43, 1002–1010.

Klinkert, P. S., Leggatt, P. B., and Hage, T. B., 1997, The Spectremairborne electromagnetic system—latest developments and field ex-amples: 4th Internat. Conf. on Min. Expl., GEO/FX, Proceedings,557–563.

Leggatt, P. B., and Pendock, N., 1993, Conductivity-depth imaging ofairborne electromagnetic step-response data using maximum en-tropy, in Mohammad-Djafari, A., and Demoment, R. G., Eds., Max-imum entropy and Bayesian methods, Kluver Acad. Publ.

Liu, G., 1998, Effect of transmitter current waveform on airborne temresponse: Expl. Geophys., 29, 35–41.

Macnae, J. C., and Lamontagne, Y., 1987, Imaging quasi-layered con-ductive structures by simple processing of transient electromagneticdata: Geophysics, 52, 545–554.

Macnae, J. C., Smith, R. S., Polzer, B, D., Lamontagne, Y., and Klinkert,P. S., 1991, Conductivity-depth imaging of airborne electromagneticstep-response data: Geophysics, 56, 102–114,

Robinson, E. A., 1978, Multichannel time series analysis with digitalcomputer programs: Hobden Day.

Sengpiel, K. P., 1988, Approximate inversion of airborne EM data froma multi-layered earth: Geophys. Prosp., 36, 446–459.

Strack, K. M., 1990, A practical review of deep transient time electro-magnetic exploration: Harbour Dome Consulting.


Advanced inversion methods for airborne electromagnetic exploration

Klaus-Peter Sengpiel∗ and Bernhard Siemon‡

ABSTRACT

Airborne electromagnetic (AEM) surveys can con-tribute substantially to geologic mapping and targetidentification if good-quality multifrequency data areproduced, properly evaluated, and displayed. A set ofmultifrequency EM data is transformed into a set of ap-parent resistivity (ρa) and centroid depth (z∗

p) values,which then are plotted as a sounding curve. These ρa(z∗

p)curves commonly provide a smoothed picture of the ver-tical resistivity distribution at the sounding site. We havedeveloped and checked methods to enhance the sensitiv-ity of sounding curves to vertical resistivity changes byusing new definitions for apparent resistivity and cen-troid depth. One of these new sounding curves with en-hanced sensitivity to vertical resistivity contrasts is plot-ted from ρNB , z∗

s values derived from differentiation ofthe ρa( f ) curve with respect to the frequency f . This ap-

proach is similar to the Niblett-Bostick transform used inmagnetotellurics. It not only enhances vertical changesin resistivity but also increases the depth of investigation.

Sounding curves can be calculated directly from EMsurvey data and can be used to generate a resistivity-depth parasection. Based on such a section, it can be de-cided whether a Marquardt-type inversion of the AEMdata into a 1-D layered half-space model is adequate.Each sounding curve can be transformed into an initialstep model of resistivity as required for the Marquardtinversion. We have inverted data from sedimentary se-quences with good results. For data from a dipping con-ducting layer and a dipping plate, we have found that theresults depend on the right choice of the starting model,in which the number of layers should be large ratherthan too small. Complex resistivity structures, however,often are represented better by using the sounding-curveresults than with the parameters of a layered half-space.

INTRODUCTIONThe applicability of the airborne electromagnetic (AEM) in-

duction method for exploration depends on the tools availablefor a detailed study of the ground resistivity distribution. Suchstudies include the determination of the boundaries of con-ducting bodies and zones in three dimensions, determinationof the thickness and type of sediment above the bedrock, trac-ing of paleochannels that are filled with sediments of differentconductivity than the surroundings, and identification of faultsystems and the structure of the bedrock.

A detailed investigation of the resistivity pattern requires ahigh-resolution measuring device. Maximum horizontal reso-lution is achieved for “coincident” transmitter and receivercoils (Duckworth et al., 1993), but in practice, it is suffi-cient if the coil spacing is smaller than the height of the sen-sor above the ground. For frequency-domain electromagnetic(FEM) systems, the vertical resolution depends on the num-ber of frequencies used. Because FEM systems are housed in

Published on Geophysics Online July 19, 2000. Manuscript received by the Editor March 15, 1999; revised manuscript received May 12, 2000.∗Federal Institute for Geosciences and Natural Resources (BGR), Stilleweg 2, D-30655 Hannover, Germany. E-mail: [email protected].‡Presently at Geophysik GGD, Hamburger Allee 12-16, D-30161 Hannover. E-mail: [email protected]. Permanent address: FederalInstitute for Geosciences and Natural Resources (BGR), Stilleweg 2, D-30655 Hannover, Germany.c© 2000 Society of Exploration Geophysicists. All rights reserved.

a “bird,” space and thus the number of the coil systems arelimited.

For helicopter electromagnetic (HEM) data, methods forderiving apparent resistivity ρa from the measured secondaryfield data were developed by Fraser (1978) and by Mundry(1984). The first relation that assigns a depth to the apparentresistivity using the centroid depth (z∗) concept for dipole in-duction was published by Sengpiel (1988). Since then, it hasbeen possible to transform HEM data into ρa(z∗) soundingcurves. A color-coded representation of all ρa(z∗) curves alonga flight line provides a resistivity-depth parasection (Sengpiel,1990). Because apparent resistivity values are used in thesesections, only a smoothed image of the true resistivity patternis obtained.

Although 1-D inversion procedures based on a layered half-space model have been used to interpret frequency-domainHEM data for more than a decade (e.g., Holladay et al.,1986; Paterson and Reford, 1986; Bergeron et al., 1987, 1989;

1983

1984 Sengpiel and Siemon

Best, 1990; Fraser, 1990; Hogg and Boustead, 1990; Kovacsand Holladay, 1990; Won and Smith, 1991; Palacky et al.,1992; Huang and Fraser, 1996; Qian et al., 1997; Fittermanand Deszcz-Pan, 1998), examples of detailed resistivity cross-sections rarely are presented in literature. BGR scientistshave developed a new 1-D inversion procedure (Fluche andSengpiel, 1997; Sengpiel and Siemon, 1998) that has been ap-plied successfully to HEM survey data since 1993. Comparedwith a ρa(z∗) sounding curve, the 1-D inversion emphasizesvertical resistivity contrasts. Particularly in groundwater explo-ration surveys, this inversion procedure has become a very use-ful tool for the interpretation of HEM data (compare Siemonand Sengpiel, 1997).

In this paper, we present our recent inversion proceduresand show some inversion results based on synthetic and fielddata.

NEW TYPES OF MULTIFREQUENCY SOUNDING CURVES

Mundry (1984) has shown that the well-known integral de-scribing the (normalized) secondary magnetic field Z (Wait,1982) measured with a horizontal coplanar coil system abovea homogeneous or layered half-space ρ(z),

Z = r3∫ ∞

0Ro( f, λ, ρ(z))λ2e−2λh Jo(λr) dλ, (1)

can be simplified if the coil spacing r is smaller than the sen-sor height h (r ≤ 0.3 h). In the “Mundry integral,” the Besselfunction Jo is replaced by 1 and the coil spacing r is no longerpresent in the integral (Ro is the reflection factor, f the fre-quency, and λ the wavenumber). Physically, this is equivalentto the “superposed dipole condition,” as postulated by Fraser(1978).

For a uniform half-space, the Mundry integral depends onlyon the ratio δ = h/p (Mundry, 1984), where p is the skin depth,

p =√

2ρ/ωµo (2)

in a half-space, where ω = 2π f and µo = 4π10−7 Vs/Am. Thisleads to a straightforward inversion of measured data into thehalf-space parameters apparent resistivity ρa and apparent dis-tance Da of the coil system from the top of the conductinghalf-space. The above-mentioned third half-space parameter,namely the centroid depth z∗, originally was defined as

z∗ = da + Da Re(C), (3)

using the real part of a complex transfer function C (Sengpiel,1988) and the apparent depth da = Da − h (Fraser, 1978).Siemon (1996) found that by inserting ρ = ρa into equation (2),the centroid depth can be calculated simply, as follows:

z∗p = da + pa/2, (4)

where pa is the apparent skin depth. (ρa, z∗p) defines a point

on a standard sounding curve ρa(z∗p) for each measured fre-

quency. The sounding curves ρa(z∗) and ρa(z∗p) are compared

in Figure 1. Although the upper part of the model is approxi-mated sufficiently by both curves, only ρa(z∗

p) shows the lowerconducting layer at the correct depth.

A number of sounding curves which are more sensitive tovertical resistivity contrasts (called enhanced sounding curves

here) have been published by Siemon (1996). The basic ideaswere adopted from magnetotellurics and modified for dipoleinduction. Two enhanced sounding curves, ρε

a (z∗p) and ρNB (z∗

s ),are shown in Figure 1. The first is derived from the ratio of thequadrature and in-phase components of the HEM data and thetrue flight altitude h above the ground, leading to

ρεa = ρa(h/Da)2. (5)

The corresponding centroid depth is z∗p (equation 4). The sec-

ond curve is defined as

ρN B = ρa1 + m ′

1 − m ′ , m ′ = m

(m + c

1 + c

),

m = − f

ρa· dρa

d f(6)

where c = 3 log(5) was obtained empirically from model calcu-lations, and the corresponding centroid depth is

z∗s = da + pa/

√2. (7)

Here, z∗s differs slightly from z∗

s = √2z∗

p , previously publishedby Siemon (1996) and by Sengpiel and Siemon (1998). Huangand Fraser (1996) presented a similar formulation, but theydifferentiated a conductance curve with respect to an effectivedepth ze f f (ze f f is a function of da and pa), yielding a ρ�(z�)sounding curve (Figure 1). While their differentiation is basedon discrete ρa and ze f f values of two neighboring frequencies,the differentiation in our case is conducted after a spline inter-polation through the ρa( f ) values, leading to stable results.

The enhanced sounding curves are more sensitive to verticalresistivity contrasts than the standard sounding curve ρa(z∗

p).The most sensitive one is ρε

a (z∗p), but this sounding curve has

two disadvantages. First, it overemphasizes the resistivity of

FIG. 1. Sounding curves for a layered half-space model and forfrequencies between 50 Hz and 300 000 Hz showing apparentresistivities ρa versus centroid depths z∗ and z∗

p , as well as three“enhanced” sounding curves ρε

a (z∗p) and ρNB (z∗

s ), and, for com-parison, ρ�(z�), after Huang and Fraser (1996). The enhancedsounding curves show a better fit to the model resistivities thanthe ρa sounding curves.

Inversion Methods for AEM Exploration 1985

very thick, homogeneous layers; i.e., the half-space below acover layer is not approximated correctly even at extremely lowfrequencies. Second, it requires exact altitude measurements.

Sounding curves for a large number of frequencies are shownin Figure 1. In practice, however, the number of availablefrequencies is limited. Modern HEM systems operate at fivefrequencies, for example, the BGR system manufactured byGeoterrex-Dighem. Are a few frequencies enough to outlinea conductor embedded in a resistive host rock?

FIG. 2. Apparent resistivity parasections derived from five-frequency HEM data of 1-D models: (a) 51 modelswith three layers (the thickness of the first layer, d1, varies from 0 to 200 m), (b) ρa(z∗

p) parasection, (c) ρεa (z∗

p)parasection, and (d) ρNB (z∗

s ) parasection. The coil spacing for all systems is 6.7 m, and the sensor height is 30 mthroughout. The centroid depth values z∗ (z∗

p or z∗s ) for the frequencies f1– f5 are indicated by black dots.

To answer this question, we calculated the HEM responses of1-D models with three layers at five frequencies ( f1 = 375 Hz,f2 = 1792 Hz, f3 = 8600 Hz, f4 = 41 000 Hz, f5 = 195 000 Hz).In a second step, these data were inverted by using the ρa(z∗

p),ρε

a (z∗p), and ρNB (z∗

s ) algorithms. In Figure 2a, 51 1-D three-layermodels are shown as color-coded resistivity columns. The mod-els consist of a conducting layer (ρ2 = 10 ohm-m, d2 = 20 m) in aless-conducting host (ρ1 = ρ3 = 100 ohm-m). The only parame-ter varied in these 51 models is the thickness of the upper layer


(d1 = 0–200 m), gradually shifting the conductor to greaterdepth.

The resistivity columns of the ρa(z∗p), ρε


s ) in-versions are shown in Figure 2b through 2d. These columns arederived from the apparent resistivity values ρa, ρ

εa , and ρNB at

the corresponding centroid depths z∗p and z∗

s (black dots) by in-terpolation or extrapolation. It is evident from Figure 2b thatthe three-layer model is approximated sufficiently by ρa(z∗

p)only if d1 is less than 60 m. For greater thicknesses of thecover layer, a decrease in ρa with greater depth still is indi-cated. This is valid even for extremely great thicknesses of thecover layer, but there is no indication of an increase in resistiv-ity below the conductor; i.e., the conductor seems to have no

FIG. 3. Apparent resistivity–depth parasections derived from five-frequency HEM data of a dipping plate [500 m(strike) × 400 m (dip), conductance = 2 S, dip angle = 30◦, depth of the upper edge = 1 m] in 100-ohm-m host:(a) anomalous HEM data, (b) ρa(z∗

p) parasection, (c) ρεa (z∗

p) parasection, and (d) ρNB (z∗s ) parasection. The coil

spacing for all systems is 6.7 m, and the sensor height is 30 m throughout. The centroid depth values z∗ (z∗p or z∗

s )for the frequencies f1 through f5 are indicated by black dots.

lower boundary. The ρεa (z∗

p) inversion (Figure 2c) outlines theconductor better than the standard ρa(z∗

p) inversion. There is astronger decrease in resistivity, and the more resistive substra-tum is indicated for d1 < 70 m. The best results are obtainedwith the ρNB (z∗

s ) inversion. The conducting layer is outlinedclearly down to 80-m depth, and it is still visible at d1 = 200 mbecause of the increase of z∗

s with respect to z∗p [equations (4)

and (7)].In Figure 3, the model is a dipping plate (500 m [strike] ×

400 m [dip], conductance = 2 S, dip angle = 30◦, depth of theupper edge = 1 m) in 100-ohm-m host rock. The correspond-ing HEM responses (Figure 3a) are calculated by using a 3-Dthin-sheet program (Weidelt, 1981). In contrast to the 1-D case,


edge effects appear in the 3-D case. Using a horizontal coplanarcoil system, a vertical plate or a horizontal cable causes sym-metric double-peak HEM responses if the flight direction isperpendicular to the strike of the body. For a dipping plate, theHEM responses are asymmetric, with the greater peak abovethe plate.

The ρa(z∗p), ρε


s ) parasections are shown inFigures 3b through 3d. The dip and depth of the conduct-ing plate (black line) are approximated sufficiently by allinversions, except for great depths and in the vicinity of theplate edges, where the 3-D effect dominates. Again, the bestresults are achieved by using the ρNB (z∗

s ) algorithm, which out-lines the limited thickness of the 3-D body down to 150-m platedepth.

As shown above, buried conductors can be delineated in theenhanced sounding curves better than in the ρa(z∗

p) soundingcurves. These resistivity-depth parasections are calculated eas-ily and rapidly. On the other hand, even the enhanced soundingcurves do smooth sharp resistivity contrasts, and they are notvery sensitive to resistive layers which are covered by a con-ductor. It will be discussed in the following section whethermultilayer inversion can provide better results for the modelsof Figures 2 and 3.

IMPROVING MULTILAYER INVERSION

A significant improvement of HEM multifrequency sound-ing was achieved by the development of our Marquardt-typedata inversion for a layered half-space model (Fluche andSengpiel, 1997; Sengpiel and Siemon, 1998). This procedurerequires an initial model with a given number of layers (N).The layer parameters are changed gradually during the inver-sion until an adequate fit of the calculated data to the measuredHEM data is obtained. Each starting model is derived individu-ally from a sounding curve which is approximated by an N -layermodel in the depth range z∗

5 − z∗1 (centroid depth of the high-

est and lowest frequency). The necessary forward calculation isvery fast, because the Mundry integral (and its derivatives) canbe solved numerically by using a Laplace transform (Fluche,1990), which is calculated more than 10 times faster than thefast Hankel transform.

In our recent paper (Sengpiel and Siemon, 1998), we demon-strated that the increase in the number of frequencies used (fiveinstead of three) for the 1-D inversion provides a better verti-cal resolution of resistivity. In this paper, we compare standardand enhanced sounding curves with 1-D inversion results, andwe discuss the influence of the starting model on the inversionresult.

Our first example is based on the 51 1-D models shown inFigure 2a. The corresponding HEM data are inverted by usingdifferent starting models derived from ρa(z∗

p) sounding curves.As shown in Figure 2b, the conducting layer is indicated downto 60-m depth in the ρa(z∗

p) parasection and thus in the start-ing models for the 1-D inversion. Subdivision of the soundingcurve into a layered starting model is a very critical step inthe modeling process. An optimum starting model should havelayer thicknesses on the order of those of the original model. Inpractice, however, this information is not available, and the set-ting of the layer boundaries is arbitrary. In our examples, wesubdivided the range between z∗

5 and z∗1 into logarithmically

equidistant layers.

The results of a three-layer inversion are shown in Figure 4a.The inversion is exact for d1 < 40 m. At depth d1 ≥ 40 m, theconductive layer appears as the substratum with gradually in-creasing resistivity toward depth. But the upper boundary ofthe conducting layer is reproduced reasonably, even for depthsas great as 200 m.

If we increase the number of layers and thus decrease thethicknesses of these layers in the starting model, the inversionresults are clearly better. In Figure 4b, in which a five-layerstarting model was used, the conducting layer is reproducedcorrectly down to 104 m.

Five-layer inversion results are shown in Figure 4c whichare based on starting models using ρNB (z∗

s ) sounding curves.In Figure 4b through 4d, the indicated number of model layers(five or nine) is often not discernible in the parasections wherea three-layer model apparently prevails. The missing bound-aries exist but do not appear because of very small resistivitycontrasts between adjacent model layers. Because the centroiddepth values z∗

s are larger than z∗p [equations (4) and (7)], the

starting model is extended to greater depth and now yieldscorrect inversion results down to 172-m depth.

A nine-layer starting model, again based on a ρNB (z∗s )

sounding curve, provides good inversion results even downto 184-m depth (Figure 4d). The 1-D inversion models nowappear smoother than the original 1-D models, but the con-ducting layer is indicated clearly. In this case, the number ofmodel parameters (nine resistivities and eight thicknesses) isgreater than the number of measured values (five frequencies× two EM components); i.e., underdetermined equation sys-tems have to be solved. It is evident from Figure 4d that ourinversion procedure can handle this problem.

When synthetic HEM data are used, the relative rms errorof the inversion is negligible, except where the actual resistivitydistribution does not correspond to the 1-D model. Therefore,the rms error is a useful measure of the applicability of the 1-Dinversion model.

In the following discussion, we examine the application of1-D inversion to synthetic HEM data for the dipping platemodel in Figure 3. The results are shown in Figure 5. Thethree-layer inversion using starting models derived from ρa(z∗

p)sounding curves (Figure 5b) is only a crude representation ofthe original model. The rms error is on the order of 4%.

Again, better results are achievable if the number of layersis increased and if the starting models are derived from ρNB (z∗

s )sounding curves (Figure 5c). The plate model, in this case, isreproduced more adequately than before. The error q is re-duced to less than 2%. A small error for a particular location,however, is not an indicator that the true resistivity distributionthere is in fact 1-D (Ellis, 1998).

For such a 3-D model, the advantage of the multilayer in-version results (Figure 5) over those of the resistivity-depthparasections (Figure 3) is not as clear as it is for the 1-D mod-els (Figures 2 and 4). For investigation of an unknown area,it is best to calculate both. In spite of the remaining inaccura-cies, the result can serve as a basis for an exploration drillingprogram.

INVERSION OF SURVEY DATA

As mentioned above, several examples for 1-D inversionresults of survey data already have been presented by Sengpiel


and Siemon (1998). These inversion results are based on three-frequency HEM responses.

Since 1998, BGR has used a five-frequency HEM systemdeveloped in cooperation with Geoterrex-Dighem. This systemuses only horizontal, coplanar coils. Internal calibration coilsprovide absolute calibration and phase adjustment during flightwithout interference from the ground. This modern equipment,

FIG. 4. 1-D inversion results derived from five-frequency HEM data of 51 1-D models as indicated in Fig-ure 2a: (a) three-layer inversions, starting models derived from ρa(z∗

p) sounding curves, (b) five-layer in-versions, starting models derived from ρa(z∗

p) sounding curves, (c) five-layer inversions, starting modelsderived from ρNB (z∗

s ) sounding curves, and (d) nine-layer inversions, starting models derived from ρNB

(z∗s ) sounding curves. The coil spacing for all systems is 6.7 m, and the sensor height is 30 m throughout.

combined with the described inversion procedures, provides awide range of new applications of airborne EM.

One of the first inversion results obtained with the new HEMsystem is shown in Figure 6. The HEM data (Figure 6a) for this3-km-long north-south profile was collected in a landscape cha-racterized by terminal moraines in northeastern Germany. Theρa(z∗

p) and ρNB (z∗s ) parasections are presented in Figures 6b


and 6c and the 1-D inversion results in Figure 6d. In all sec-tions, the dotted line (bird in Figure 6) is the sensor elevation inmeters above mean sea level (m.s.l.) as derived from the baro-metric altimeter measurements. The (apparent) ground eleva-tion (solid line marked topo) is obtained from the differencebetween the sensor elevation and the radar (or laser) altitudeof the bird. The ground elevation, however, may be erroneousbecause of the presence of trees or buildings, for example.

In the apparent resistivity–depth parasections (Figures 6band 6c), the upper boundary of the colored area is defined by

FIG. 5. 1-D inversion results derived from five-frequency HEM data of the dipping plate (compare Figure 3):(a) anomalous HEM data, (b) three-layer inversions, starting models derived from ρa(z∗

p) sounding curves, and(c) six-layer inversions, starting models derived from ρNB (z∗

s ) sounding curves. The relative rms error q of theinversion is plotted below the inversion results. The coil spacing for all systems is 6.7 m, and the sensor height is30 m throughout.

the calculated apparent distance Da for the highest frequency( f5 = 192, 424 Hz) below sensor elevation (line marked bird).The lower boundary of the colored area is determined by ex-trapolation below the centroid depth z∗

1 of the lowest frequency( f1 = 383 Hz). All centroid depth values are marked by blackdots within the colored areas in Figures 6b and 6c.

The starting models for parasection 6d were derived from theρNB (z∗

s ) sounding curves at each of the 150 measuring pointsof the profile, providing independently calculated inversion re-sults. The layer thicknesses and color-coded resistivities of the


1-D inversion models (Figure 6d) are plotted downward fromthe topo line. The first (resistive) model layer includes treesand resistive ground.

Because of the lack of boreholes along the profile, a definitegeologic interpretation of the resistivity-depth parasections isnot possible. However, boreholes a few kilometers north and

FIG. 6. Resistivity parasections and associated curves along a flight line in northeastern Germany: (a) measuredfive-frequency HEM data, (b) ρa(z∗

p) parasection, (c) ρNB (z∗s ) parasection, and (d) 1-D inversion results and

relative rms error q of the inversion. The coil spacing for all systems is about 6.7 m. The sensor height (bird) isplotted (black dots) above each parasection. The color-coded inversion results are referred to the topography(topo), including vegetation (forest). The centroid depth values z∗ (z∗

p or z∗s ) for the frequencies f1 through f5 are

indicated by black dots in the apparent resistivity–depth parasections b and c.

south of the profile show that the groundwater table lies at theboundary between the first and second layers (in Figure 6d).Thus, the highly resistive cover layer (light blue) represents thevegetation (forest) and dry sand and gravel. The groundwatertable does not appear in the apparent resistivity–depth para-sections, but the fact that the apparent depth da of the highest


frequency is positive indicates that the cover layer is more re-sistive than the underlying one. The green and blue areas ofthe resistivity-depth parasections can be interpreted as water-saturated sand and gravel. It is supposed that thin silt layersexist in that region, but location and depth are unknown. Al-though there is no indication of thin conducting layers in theρa(z∗

p) parasection, the ρNB (z∗s ) parasection and the 1-D inver-

sion results indicate such layers at 50–60 m above m.s.l. Thereare prominent layer boundaries at about 0 m and 60 m belowm.s.l. While the yellow layer is assumed to contain sand with arelatively high silt content, the bottom layer (red) has an ele-vated clay content. This strong decrease in resistivity is alsovisible in the ρNB (z∗

s ) parasection, but it is not so clear in thestandard ρa(z∗

p) parasection. In the central part of the profile,the conductive layer appears to be more than 180 m belowground level. The scattering of the parameter values for thebottom layer probably is caused by its great depth.

CONCLUSIONS

The results of airborne EM surveys still are presented com-monly in the form of apparent resistivity maps (for differentfrequencies) and anomaly maps. The apparent resistivity mapis always useful, but it lacks lateral resolution (because of thegridding), and it generally does not contain a depth reference.Therefore, we overlay our apparent resistivity maps with a con-tour plan of the centroid depth of the pertinent frequency.

Our main product, however, has become the resistivity-depth parasection along a flight line containing the measuredEM data, the bird altitude, the topography, and the color-codedresistivity distribution. The bird altitude and the topographyshould be related to sea level (measured by a barometric andlaser altimeter in the bird and a reference barometer at a basestation). The topography along the flight line is very usefulinformation for evaluation of the HEM results, especially forgroundwater investigation.

The standard sounding curve ρa(z∗p) and the enhanced

sounding curve ρNB (z∗s ) do not need the (often distorted) sen-

sor altitude as an input. This is an advantage over the ρεa (z∗

p) andρ�(z�) sounding curves, which require the true sensor heightabove ground.

In general, our resistivity-depth parasections result from 1-Dmultilayer inversion. This type of inversion uses starting modelswhich are derived from standard or enhanced sounding curves.It yields the best fit to the true resistivity pattern if the ground isnot too complex or if a certain layer prevails. We have demon-strated that a thin, dipping conductor can be traced better inthe ground if the number of layers (N) in the starting modelis increased, whereby N may be greater than the number ofinduction frequencies.

A complex (2-D or 3-D) resistivity pattern often is repre-sented better by the simpler inversion procedures based ona uniform half-space model. In particular, vertical resistivityboundaries are resolved much better in the approximate in-version than in the 1-D inversion, which forces boundaries tobe horizontal.

The development of more sophisticated inversion proce-dures (2-D, 3-D) should be accompanied by steady improve-ment of survey data quality. This is because there are stillsome error sources such as calibration errors, zero-level drift,and mechanical instabilities of the coil arrangement which can

degrade the inversion results. With the new five-frequencyEM system, which incorporates only coplanar coils and higherdipole moments of the transmitters, as well as in-flight ampli-tude calibration and phase adjustment, airborne EM explo-ration has been brought forward a significant step.

ACKNOWLEDGMENTS

The authors thank the BGR helicopter geophysics staff forcooperation, and in particular, J. Pielawa for his assistance inproducing the figures. Furthermore, we thank the two review-ers for their constructive comments, as well as the BGR trans-lators R. C. Newcomb and H. Toms for their thorough checkingof the English manuscript.

REFERENCES

Bergeron Jr., C. J., Ioup, J. W., and Michel II, G. A., 1987, Application ofMIM to inversion of synthetic AEM bathymetric data: Geophysics,52, 794–801.

——— 1989, Interpretation of airborne electromagnetic data using themodified image method: Geophysics, 54, 1023–1030.

Best, M. E., 1990, Synthetic modeling and airborne electromagneticinterpretation: in Fitterman, D. V., Ed., Proceedings of the USGSworkshop on developments and applications on modern airborneelectromagnetic surveys, 1987, U.S. Geol. Surv. Bull., 1925, 21–32.

Duckworth, K., Krebes, E. S., Juigalli, J., Rogozinski, A., and Calvert,H. T., 1993: A coincident-coil frequency-domain electromagneticprospecting system: Can. J. Expl. Geophys., 29, 411–418.

Ellis, R. G., 1998, Inversion of airborne electromagnetic data: Expl.Geophys., 29, 121–127.

Fitterman, D. V., and Deszcz-Pan, M., 1998, Helicopter EM mappingof saltwater intrusion in Everglades National Park, Florida: Expl.Geophys., 29, 240–243.

Fluche, B., 1990, Verbesserte Verfahren zur Losung des direkten unddes inversen Problems in der Hubschrauber-Elektromagnetik: inHaak, V., and Homilius, J., Eds., Protokoll Kolloquium Elektromag-netische Tiefenforschung, 249–266.

Fluche, B., and Sengpiel, K.-P., 1997, Grundlagen und Anwendungender Hubschrauber-Geophysik: in Beblo, M., Ed., Umweltgeophysik:Ernst und Sohn, 363–393.

Fraser, D. C., 1978, Resistivity mapping with an airborne multicoilelectromagnetic system: Geophysics, 43, 144–172.

——— 1990, Layered-earth resistivity mapping, in Fitterman, D. V.,Ed., Developments and applications of modern airborne electro-magnetic surveys: U. S. Geol. Surv. Bull., 1925, 33–41.

Hogg, R. L. S., and Boustead, G. A., 1990, Estimation of overbur-den thickness using helicopter electromagnetic data: in Fitterman,D. V., Ed., Proceedings of the USGS workshop on developmentsand applications on modern airborne electromagnetic surveys, 1987,U. S. Geol. Surv. Bull., 1925, 103–115.

Holladay, J. S., Valleau, N., and Morrison, E., 1986, Application ofmultifrequency helicopter electromagnetic surveys to sea-ice thick-ness and shallow-water bathymetry: in Airborne resistivity mapping,Palacky, G. J., Ed., Geol. Surv. Can., Paper 86-22, 91–98.

Huang, H., and Fraser, D. C., 1996, The differential parameter methodfor multifrequency airborne resistivity mapping: Geophysics, 61,100–109.

Kovacs, A., and Holladay, J. S., 1990, Sea-ice thickness measurement us-ing a small airborne electromagnetic sounding system: Geophysics,55, 1327–1337.

Mundry, E., 1984, On the interpretation of airborne electromagneticdata for the two-layer case: Geophys. Prosp., 32, 336–346.

Palacky, G. J., Holladay, J. S., and Walker, P., 1992, Inversion ofhelicopter electromagnetic data along the Kapuskasing transect,Ontario: in Current research, Part E, Geol. Surv. Can., Paper 92-1E, 177–184.

Paterson, N. R., and Reford, S. W., 1986, Inversion of airborne elec-tromagnetic data for overburden mapping and groundwater explo-ration: in Airborne resistivity mapping, Palacky, G. J., Ed., Geol.Surv. Can., Paper 86-22, 39–48.

Qian, W., Gamey, T. J., Holladay, J. S., Lewis, R., and Abernathy,D., 1997, Inversion of airborne electromagnetic data using an Oc-cam technique to resolve a variable number of multiple layers: inProceedings of high-resolution geophysics workshop, SAGEEP ’97,735–744.

Sengpiel, K.-P., 1988, Approximate inversion of airborne EM data froma multi-layered ground: Geophys. Prosp., 36, 446–459.


——— 1990, Theoretical and practical aspects of ground-water ex-ploration using airborne electromagnetic techniques: in Fitterman,D. V., Ed., Proceedings of the USGS workshop on developments andapplications on modern airborne electromagnetic surveys, 1987, U.S.Geol. Surv. Bull., 1925, 149–154.

Sengpiel, K.-P., and Siemon, B., 1998, Examples of 1-D inversion ofmultifrequency HEM data from 3-D resistivity distribution: Expl.Geophys., 29, 133–141.

Siemon, B., 1996, Neue Verfahren zur Berechnung von schein-baren spezifischen Widerstanden und Schwerpunktstiefen inder Hubschrauberelektromagnetik: in Bahr, K., and Junge, A.,

Eds., Protokoll Kolloquium Elektromagnetische Tiefenforschung,89–100.

Siemon, B., and Sengpiel, K.-P., 1997, Helicopter-borne electromag-netic groundwater exploration survey in the Kuiseb Dune area, Cen-tral Namib Desert: in Proceedings of fifth technical meeting of theSouth African Geophysical Association, 137–142.

Wait, J. R., 1982, Geo-Electromagnetism: Academic Press Inc.Weidelt, P., 1981, Dipolinduktion in einer dunnen Platte mit leitfahiger

Umgebung und Deckschicht: BGR report 89727.Won, I. J., and Smith, K., 1991, Airborne electromagnetic bathymetry:

Geoexploration, 27, 297–319.


The use of airborne gamma-ray spectrometry—A case studyfrom the Mount Isa inlier, northwest Queensland, Australia

Prasantha Michael Jayawardhana∗ and S. N. Sheard∗

ABSTRACT

An airborne survey was undertaken on the MountIsa inlier in 1990–1992. During this survey, both air-borne magnetic and gamma-ray spectrometric data wererecorded over 639 170 line-km. Because of perceivedvalue of the radiometric data, stringent calibration pro-cedures, including the creation of a test range, wereadopted. In addition to the data from the newly-flownareas, 76 760 line-km of existing data were acquired fromother companies, and were reprocessed and merged withthe Mount Isa survey. The total area covered by theMount Isa airborne survey was 151 300 km2.

Over the last five years, several studies have beenundertaken that seek to exploit the Mount Isa regiongamma-ray database and maximise the use of radio-metrics for mineral exploration. This paper highlightsthe results of these studies by focussing on radiomet-ric signatures of major mines in the Mount Isa Inlier,radioelement contour maps, geomagnetic/radiometricinterpretation maps, lithological mapping, regolith map-ping, geochemical sampling, and spatial modeling usinggeographical information systems (GIS).

Due to the recent introduction of GIS technology andbetter techniques for handling high quality digital data,there has been a revived interest in making more useof image data sets. The integration of raster and vectordata sets for both spectral and spatial modeling has max-imized the potential of this approach.

INTRODUCTION

M.I.M Exploration Pty Ltd (MIMEX) has been using air-borne radiometric techniques to assist in mineral explorationsince the late 1960s. The initial work was largely for uraniumexploration, but a gradual evolution to mapping using multi-channel systems occurred. Unfortunately, the data quality of-ten varied between surveys and companies, and thus the datawas compromised and too often not interpreted properly.

Manuscript received by the Editor March 12, 1999; revised manuscript received June 20, 2000.∗M.I.M. Exploration Pty Ltd, Level 2, Boundary Court, 55 Little Edward Street, Spring Hill, Brisbane, Queensland 4000, Australia. E-mail:100253.132@ compuserve.com; [email protected]© 2000 Society of Exploration Geophysicists. All rights reserved.

In 1990, MIMEX embarked upon a major airborne geo-physical program in northwest Queensland, Australia (Fig-ure 1). A significant component of this program was theneed to collect high-quality radiometric data which would as-sist mapping and target generation in this highly prospectiveregion.

To maintain radiometric data quality throughout the survey(which occurred over a 12-month span and used four differentaircraft), stringent survey specification and calibration proce-dures were adopted (Figure 2). These specifications were es-tablished in conjunction with World Geoscience CorporationLtd (WGC), the contractors. The adherence to these standardsallowed the collection of 639 170 line-km of data converted toelemental concentrations. This allowed accurate comparisonacross areas using different acquisition aircraft and detectorsystems. The paper gives a brief description of the collectionprocedures and shows different processing and interpretationtechniques used to assist exploration in the region.

SURVEY PROCEDURES

The survey was largely flown east-west using 200-m line spac-ing, a terrain clearance of 80 m, and a sampling interval of 60 m.Areas where the interpreted depth to magnetic basement wasgreater than 300–400 m were flown at 400-m line spacing. Thiswider spacing was chosen due to the reduced prospectivity oflocating economic mineralisation at these depths.

The four spectrometers used were Geometrics ExplorationGR800B units with detector volumes of 33.56 liters. Two hun-dred and fifty-five (255) channels of data were measured from0.006 to 3.0 MeV, and one channel, the cosmic channel, from3.0 to 6.0 MeV. All channels were recorded, but only the countsin the energy ranges shown in Table 1 were used for data pro-cessing and interpretation.

To maintain data quality the following routine quality con-trol procedures were adopted.

1) Daily spectral plots were made using a standard cesiumsource to monitor crystal tuning and drift.

2) Preflight and postflight ground calibration was done ina preselected location. A thorium source, a uranium

1993

1994 Jayawardhana and Sheard

source, and background recordings were made. The tho-rium and uranium source data were corrected for back-ground and used to monitor spectral stability.

3) Weekly spectral drift and resolution checks of eachindividual detector were made using cesium and thoriumsources.

4) At the commencement of the survey at two-monthintervals thereafter, at times of crystal changes, andat completion the following additional checks were

FIG. 1. Mount Isa inlier radiometrics survey location diagram.

FIG. 2. Sample calibration range profile (field copy) showing ground/airborne and geochemical assaying alongthe same ground profile.

Table 1. Spectrometer energy ranges used for interpretation.

Energy (MeV)

Name From To

Total count 0.402 3.000Potassium-40 1.373 1.562Uranium 1.668 1.858Thorium 2.414 2.804Cosmic 3.000 6.000

MIM Airborne Gamma Ray Spectrometry 1995

undertaken: pad calibration to determine stripping ratios(Compton scattering), high-altitude stack to determinebackground coefficients, and test range to determineheight attenuation coefficients and to determine sensi-tivity coefficients. The pads used for calibration weremanufactured by Dr. R. L. Grasty of the GeologicalSurvey of Canada (Grasty et al., 1991). The test rangeestablished for this survey allowed accurate conversionof data to percent potassium, equivalent parts per millionuranium, and equivalent parts per million thorium. Anexample of the results from an in-field calibration run isshown in Figure 2, where the concentrations computedfrom surface geochemical sampling are plotted againstground and airborne counts.

The rigorous implementation of the above procedures atthe time of survey 1990 has resulted in a data base uniquein Australia, which allows quantitative interpretation of theworld’s largest continuous commercial airborne survey at thetime. The collection of 256 channel data will allow reprocessingof data if and when required.

DATA ENHANCEMENT AND INTERPRETATION

Radioelement contour maps

Radioelement contour maps over the survey area were gen-erated by the contractors at 1:100 000 and 1:25 000 scales. Theradiometric data was corrected for system dead time, back-ground radiation, channel interaction (i.e., stripping), heightattenuation, and system sensitivity, and then gridded using a70-m cell size. These would have been different for each of thefour aircraft. System parallax was removed and microlevellingwas applied. Figure 3 shows part of the Quamby 1:100 000-scaleradioelement contour maps over the Dugald River deposit.Contour maps were generated for total count (50 counts persecond contour interval), potassium (0.1% contour interval),equivalent thorium (1.0 ppm contour interval) and equivalenturanium (0.5 ppm contour interval). These maps are used toaid in the visual interpretation of potential exploration areas.Hard copies of these maps were produced at 1:100 000 and1:25 000 scales for subsequent interpretation of the airbornegamma-ray data.

Geomagnetic/radiometric interpretation maps

Fifty-three 1:100 000-scale geomagnetic/radiometric inter-pretation maps were compiled from radiometric and magneticcontour maps. An example is shown in Figure 4. Geologicalcontrol was from Australian Geological Survey Organisation(AGSO) maps published at 1:100 000 scale and localised maps.The area shown in Figure 4 is approximately 20 km north-west of Mount Isa and on the southern part of the KennedyGap 1:100 000 scale map sheet. The large dome-shaped bodyin the center is the Sybella batholith and can be distinguishedclearly in the radiometrics. The eastern and marginal parts ofthe batholith are nonmagnetic and highly enriched in potas-sium (K). The west and central part of the batholith showsconsistent weak, noisy magnetic patterns and variable radioele-ment response. Enrichment in uranium (U) and thorium (Th)is also common. These geomagnetic/radiometric interpretationmaps have been useful in first-pass targeting.

Radiometric signatures for major depositsin the Mount Isa inlier

Detailed visual analysis of the airborne radiometrics was car-ried out over 80 mineral deposits. An example from this compi-lation is the Dugald River deposit situated 300 km northwest ofMount Isa on the Quamby 1:100 000-scale map sheet. DugaldRiver is a zinc/lead/silver deposit with an estimated resource of60 million tons at 10% zinc, 1% lead, and 30 g/ton silver. Thedeposit type is a sheeted mineralisation of Proterozoic age.The host rock is the Dugald River Shale Member containingblack, commonly carbonless shales. Mineralisation occurs infine-grained black slate with abundant sulfides. The primarymineralogy is pyrrhotite, sphalerite, pyrite, and galena. Theairborne radiometrics for this deposit (Figure 5) highlightthe presence of a 2-km north-south lineament centered aboutthe deposit and enriched in U and K. The large K high to thewest of the deposit is due to a Knapdale Quartzite ridge, whichis dominated by potassium feldspar. The magnetic data overthis deposit (Figure 6) shows a prominent north-south mag-netic lineament, 6 km long and with a peak amplitude of 660 nT,associated with a pyrrhotite-enriched stratigraphy surroundingand including the mineralisation. This has been mapped as asteep synform in the Corella Formation.

Lithological mapping

Radiometrics has been a useful tool in helping geologistswith the delineation of lithological boundaries and geologi-cal mapping mismatches (Graham and Bonham-Carter, 1993;Jaques et al., 1997). This interpretation procedure involvesoverlaying of geological maps or traced geological boundariesonto radiometrics and Landsat Thematic Mapper images tohighlight areas of variation and possible mismatch. While thiscan be achieved using hard copies of maps and images, recentimprovements in computer hardware and software capabili-ties and the introduction of geographical information system(GIS) technology, has enabled the overlaying of data sets onthe computer screen. Figure 7 shows the published AGSO ge-ological boundaries (white) overlain on a ternary image of thegamma-ray data. The area shown in Figure 7 is approximately70 km north of Mount Isa and on the northeastern corner of theKennedy Gap 1:100 000 sheet. The dark areas (low K, Th, andU) are part of the orthoquartzite component of the LeanderQuartzite. The bright area (high K, Th, and U) in the lowerleft of Figure 7 differentiates the Gunpowder Creek Forma-tion and, in particular, the micaceous siltstone and ferruginoussiltstone components.

In areas of good outcrop, the radiometric data is very goodat delineating the geology and is readily interpreted (Darnleyand Grasty, 1971). Interpretation of this data for lithologicalmapping becomes more difficult in deep, transported soils. Inthese areas, radiometrics have proven more useful for regolithmapping and planning geochemical surveys.

Regolith mapping

The awareness of the value of regolith mapping in mineralexploration is rapidly increasing (Wilford, 1995; Cook et al.,1996; Wilford et al., 1997). By understanding the regolith, grea-ter insight into the underlying geology can usually be attainedand, importantly, exploration techniques to explore below the


regolith can be devised. When the radiometric data are com-bined with Landsat Thematic Mapper data, aerial photogra-phy, and digital elevation models, a better understanding ofthe regolith landforms can be attained. An example of the useof the Mount Isa airborne radiometrics data set to identifydepositional areas is shown in Figure 8. The various regolithboundaries are in gray and have been overlaid onto the ternaryimage (K:Th:U = R:G:B). The lighter colored arcuate channelin the top center of the figure highlights an area of alluvial claysand silts within a depositional zone. By using the radiometrics,these depositional zones are easily identified in similar regolithsettings elsewhere (Wilford et al., 1997).

FIG. 3. Radioelement contour maps over the Dugald River deposit (black circle). (a) Total count with a contour interval of 50counts per second. (b) Potassium with a contour interval of 0.1%. (c) Thorium with a contour interval of 1.0 ppm. (d) Uraniumwith a contour interval of 0.5 ppm.

Geochemical sampling

Radiometric data has been recognised (but too often ig-nored) as useful in planning exploration geochemical surveysand in interpreting previous ill-constrained surveys. By recog-nising the context of regolith landforms—depositional, ero-sional, or residual—the sample site significance can be betterused in the interpretation (Dickson and Scott, 1997).

Spatial modeling using GIS

The introduction of GIS technology has enabled the devel-opment of semiautomated spatial analysis models to aid in the


interpretation and analysis of large quantities of the airborneradiometric data within a reasonable time frame. The tech-niques described below were developed as part of the datasynthesis, regional interpretation, and targeting for the MountIsa block and involved MIMEX staff and B. Dickson of CSIRO(Exploration and Mining).

The aim was to identify anomalous radiometric (enrichmentor depletion) responses within lithological units (B. Dickson,personal communication, 1994). Figure 9 shows the radiometricnormalisation methodology adopted. The technique requiredthe development of a GIS-based model to analyse and nor-malise the radiometric data within each lithological unit. Litho-logical boundaries were obtained from the AGSO 1:100 000-scale published maps.

The GIS model generated normalised images for a given1:100 000-scale map sheet. The radioelement statistics for eachlithological unit were also automatically generated. These in-cluded gcode (unique number for each lithological unit), min-imum, maximum, mean and standard deviation (std). Oncenormalised images were created, standard processing was car-ried out to stretch the images to highlight areas of enrichment(≥3 std) and depletion (≤−2 std). Different cutoffs were cho-sen for enrichment and depletion as the normalised distribu-tion showed a negative skew. The results were saved as a newnine-band image: bands 1–3 contain K, Th, and U from theoriginal image; bands 4–6 show areas of enriched K, Th, andU; and bands 7–9 show areas of depleted K, Th, and U. The re-sulting nine-band image was taken back to the GIS for secondpass processing. In the second pass processing, the enrichmentand depletion areas from pass one were masked out of the ra-diometrics and the normalisation rerun to highlight the subtlehighs and lows. The results from the second pass were com-bined with the first pass to create radiometric enrichment anddepletion maps (Figures 10 and 11). Figure 10 shows an ex-ample of part of the Mount Isa 1:100 000-scale radioelementenrichment map around the Mount Isa deposit.

FIG. 4. An extract from the southern part of the Kennedy Gap 1:100 000-scale geomag-netic/radiometric interpretation map.

The Mount Isa deposit is a lead/zinc/silver/copper deposit inthe Mount Isa inlier mineral province. The estimated resourceis 150 million tons at 7% zinc, 5–6% lead, and 140 g/ton sil-ver, and 255 million tons at 3.3% copper. The deposit types are

FIG. 5. The ternary gamma-ray spectrometric (K = red, Th =green, U = blue) image (K:Th:U = R:G:B) over the DugaldRiver deposit (white circle).


stratabound for lead/zinc and discrete tabular bodies for cop-per. The host rock for the lead/zinc and copper orebodies isthe Urquhart Shale, which contains dolomitic and variable car-bonaceous siltstone rich in fine-grained pyrite and numeroustuff beds. The copper orebody has a silica dolomite alterationhalo. In Figure 10, the lithological boundaries are in gray andthe ternary image (K:Th:U = R:G:B) highlights areas of K, Th,and U enrichment.

FIG. 6. Pseudocolor image of total magnetic intensity for theDugald River deposit.

FIG. 7. Radiometric colour image (K:Th:U = R:G:B) overlaidwith geological boundaries (white) from the northeastern cor-ner of the Kennedy Gap 1:100 000-scale published mapping.

FIG. 8. Radiometrics ternary image (K:Th:U = R:G:B) with re-golith boundaries overlaid in gray. The lighter colored arcuatechannel running across the top center of the figure is an areaof alluvial clays and silts within a depositional profile.

FIG. 9. Radiometric normalisation methodology.


An interesting finding from this technique is the large areaof K enrichment to the south of Mount Isa. This occurs on theNative Bee Siltstone unit, which is stratigraphically above theUrquhart Shales and composed of bedded dolomitic siltstone,laminated siltstone, minor tuff, and chert. Field assessment us-ing a handheld spectrometer confirmed this area as an area ofhigh K alteration. This could be interpreted as a possible fluidchannel for the ore-bearing fluids of the Mount Isa deposit.

The technique has also highlighted lithological boundaries.For example, the area of Th enrichment on the western edge of

FIG.10. An extract from the Mount Isa 1:100 000-scale radioelement enrichment map. The center of the Mount Isadeposit is shown by a black circle. The lithological boundaries are in gray and the ternary image (K:Th:U = R:G:B)highlights areas of K, Th, and U enrichment.

FIG. 11. An extract from the Mount Isa 1:100 000-scale radioelement depletion map. The lithological boundariesare in gray, roads in gold, and the ternary image (K:Th:U = R:G:B) highlights areas of K, Th, and U depletion.

Figure 10 was due to a mapped geological boundary mismatchbetween the Sybella Granite and Cainozoic cover in the AGSOdatabase.

Figure 11 shows an example of part of the Mount Isa1:100 000-scale radiometric depletion map around the MountIsa deposit. The lithological boundaries are in gray, roads are ingold, and the ternary image (K:Th:U = R:G:B) highlights areasof K, Th, and U depletion. The depletion results identify a largearea of Th depletion coincident with the surface projection ofthe lead/zinc orebody. The Th depletion is mainly within the


Urquhart Shale and could not be due to geological boundaryeffects. This area is covered by mining infrastructure, and thesource which could be geological or manmade has thus far notbeen identified.

Another example of geological mismatch is the sliver ofK depletion to the west of Mount Isa. This resulted from alithological boundary mismatch between the Eastern CreekVolcanics and the Mount Guide Quartzite.

This semiautomated spatial modeling technique has resultedin defining areas of enrichment/depletion and thus prospectivefor mineralisation. It has also assisted in more precise geologi-cal mapping of areas considered previously to be well mapped.

CONCLUSION

Each of the products and techniques described have pro-vided valuable information to assist mineral exploration in thecomplex geological environment of the Mount Isa block. Theoverall compilation of the radiometric signatures for the 80 ma-jor mines in the Mount Isa inlier was bound into a three-volumeset for internal company use. Hard copies of the radioelementcontour maps at 1:100 000 and 1:25 000 scale form a convenientsupplement when interpreting the airborne radiometric survey.The geomagnetic/radiometric interpretation maps have beenuseful in first-pass targeting, as have the Mount Isa airbornesurvey data set in helping geologists with the delineation oflithological boundaries and identification of geological map-ping mismatches. When the radiometric data are combinedwith Landsat Thematic Mapper, aerial photography, and dig-ital elevation models, a better understanding of the regolithlandforms can be attained. In terms of geochemical sampling,the radiometric data have been recognised (but too often ig-

nored) as useful in planning exploration geochemical surveysand in interpreting previous ill-constrained surveys. GIS anal-ysis has enabled the development of a semiautomated spatialmodeling technique for defining areas of enrichment/depletionand hence for generation of new prospects for mineralisation.It has also assisted in more precise geological mapping of areasconsidered previously to be well mapped.

By adopting internationally established airborne radiomet-ric survey standards, the large airborne radiometric survey is ofa consistent high quality and is timeless in its value for mineralexploration.

REFERENCES

Cook, S. E., Corner, R. J., Groves, R. J. and Grealish, G., 1996, Applica-tion of airborne gamma radiometric data for soil mapping: Austral.J. Soil Research, 43, 183–194.

Darnley, A. G., and Grasty, R. L., 1971, Mapping from the air bygamma-ray spectrometry: Proc. 3rd Internat. Geochemical Symp.:Canadian Inst. Mining and Metallurgy Special Volume 11, 485–500.

Dickson, B. L., and Scott, K. M., 1997, Interpretation of aerial gamma-ray surveys—Adding the geochemical factors: AGSO J. Austral.Geol. Geophys., 17, 187–200.

Graham, D. F., and Bonham-Carter, G. F., 1993, Airborne radio-metric data: a tool for reconnaissance geological mapping using aGIS: Photogrammetric Engineering and Remote Sensing, 59, 1243–1249.

Grasty, R. L., Holman, P. B., and Blanchard, Y. B., 1991, Transportablecalibration pads for ground and airborne gamma-ray spectrometers:Geological Survey of Canada Paper 90-23.

Jaques, A. L., Wellman, P., Whitaker, A., and Wyborn, D., 1997,High-resolution geophysics in modern geological mapping: AGSOJ. Austral. Geol. Geophys., 17, 159–173.

Wilford, J. R., 1995, Airborne gamma-ray spectrometry as a tool forassessing relative landscape activity and weathering development ofregolith, including soils: AGSO Research Newsletter, 22, 12–14.

Wilford, J. R., Bierwirth, P. N., and Craig, M. A. 1997, Application of air-borne gamma-ray spectrometry in soil/regolith mapping and appliedgeomorphology: AGSO J. Austral. Geol. Geophys., 17, 201–216.


The detection of potassic alteration by gamma-rayspectrometry—Recognition of alteration relatedto mineralization

Robert B. K. Shives∗, B. W. Charbonneau∗, and K. L. Ford∗

ABSTRACT

Canadian case histories document the use of airborneand ground gamma-ray spectrometry to detect and mappotassium alteration associated with different stylesof mineralization. These include: volcanic-hosted mas-sive sulfides (Cu-Pb-Zn), Pilley’s Island, Newfoundland;polymetallic, magmatic-hydrothermal deposits (Au-Co-Cu-Bi-W-As), Lou Lake, Northwest Territories; and por-phyry Cu-Au-(Mo) deposits at Mt. Milligan, BritishColumbia and Casino, Yukon Territory. Mineralizationin two of these areas was discovered using airbornegamma-ray spectrometry.

In each case history, alteration produces potas-sium anomalies that can be distinguished from nor-mal lithologic potassium variations by characteristiclows in eTh/K ratios. Interpretations incorporating air-borne and ground spectrometry, surficial and bedrockgeochemistry and petrology show that gamma-ray spec-trometric patterns provide powerful guides to mineral-ization. This information complements magnetic, elec-tromagnetic, geological, and conventional geochemicaldata commonly gathered during mineral explorationprograms.

INTRODUCTION

In Canada, many studies (see Shives et al., 1995, and refer-ences therein) document application of gamma-ray spectrom-etry to surficial (glacio-fluvial deposits, tills, soils) and bedrockmapping at regional to deposit scales, exploration for a widevariety of commodities (rare, base, and precious metals, gra-nophile elements, and industrial minerals) and environmentalstudies. In Australia and elsewhere, where the effects of trop-ical weathering and landform development may significantly

Manuscript received by the Editor April 12, 1999; revised manuscript received June 19, 2000.∗Geological Survey of Canada, Mineral Resources Division/Airborne Geophysics Section, 601 Booth Street, Ottawa, Ontario K1A 0E8, Canada.E-mail: [email protected]; [email protected]; [email protected]© 2000 Society of Exploration Geophysicists. All rights reserved.

modify bedrock radioactive element distribution (Dickson andScott, 1997), the patterns from airborne gamma-ray surveysprovide important information for soil, regolith, and geomor-phology studies used for land management and mineral explo-ration strategies (Wilford et al., 1997).

The use of gamma-ray spectrometry to determine concen-trations of elemental potassium, regardless of the associatedpotassium mineral species, enables alteration mapping in awide range of geological settings. For example, potassic al-teration in the form of sericite is commonly associated withmany types of volcanic-associated massive-sulfide base-metaland gold deposits (Franklin, 1996; Poulsen and Hannington,1996). Potassium feldspar alteration has been documented as aregional alteration product at volcanic-associated base-metaldeposits in the Bergelagen district, Sweden (Lagerblad andGorbatschev, 1985), in the Mount Read volcanics, Tasmania,Australia (Crawford et al., 1992), and at the Que River massive-sulfide deposit, Tasmania, Australia (Offler and Whitford,1992). Potassium alteration is common in shear-hosted golddeposits, such as those at Hemlo, Ontario, Canada (Kuhns,1986) and Red Lake, Ontario, Canada, (Durocher, 1983), andof many other deposit types (Hoover and Pierce, 1990).

Many alkaline and calc-alkaline porphyry Au-Cu (±Mo)deposits have extensive potassic hydrothermal alteration ha-los (Davis and Guilbert, 1973; Schroeter, 1995) which varymineralogically, both laterally and vertically, with changes inpressure, temperature, eH, and pH during magmatic, hypo-gene, and subsequent supergene processes. A well-establishedalteration-mineralization potassic zoning sequence (Lowelland Guilbert, 1970) may be evident within a single de-posit, ranging from a central, orthoclase and/or biotite core(± sericite as fracture controlled and pervasive replacements)outwards through successive phyllic (sericitic), argillic, andpropylitic zones. Although phyllic zones may contain less bulkpotassium gain than potassic cores, their peripheral distribu-tion commonly offers much larger targets for detection bygamma-ray spectrometry.

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As thorium enrichment generally does not accompany potas-sium during hydrothermal alteration processes, eTh/K ratiosprovide excellent distinction between potassium associatedwith alteration and anomalies related to normal lithologicalvariations (Galbraith and Saunders, 1983). This important cor-relation of low eTh/K ratio with alteration is evident in count-less studies worldwide, including the four Canadian examplesdescribed below (Figure 1).

VOLCANIC-HOSTED MASSIVE SULFIDES—PILLEY’SISLAND, NEWFOUNDLAND

On the Island of Newfoundland, the Geological Surveyof Canada has collected more than 65 000 km of combinedgamma-ray spectrometric, total-field magnetic, and very lowfrequency electromagnetic (VLF-EM) airborne data at 1000-mline spacing covering approximately 50% of the island.

Within the major volcanic belts, a number of potassiumanomalies are directly associated with felsic volcanic unitsand past-producing or prospective volcanic-hosted massive sul-fides (VHMS) alteration systems. Figure 2b shows the potas-sium distribution map for the western part of Notre DameBay. Corresponding geology for the area is shown in Fig-ure 3a (Colman-Sadd et al., 1990). High potassium concen-trations west of Green Bay correspond to felsic intrusive unitswithin the Siluro-Devonian, post-tectonic King’s Point Com-plex, which comprises units ranging from gabbro to syenite andperalkaline granite. The prominent area of low K values northof Halls Bay accurately reflects the distribution of primitive-arcophiolitic mafic volcanic rocks. Southwest of Halls Bay, a broadregion of moderate-to-high potassium values corresponds tofelsic volcanic units of the Silurian Springdale Group (Coyleand Strong, 1987).

The northern part of the Robert’s Arm Group (RAG,Figure 3a) contains lower tholeiitic and upper calc-alkalinesuites of mafic volcanics with localised felsic volcanic centers(Bostock, 1988). Figure 2b shows that small potassium anoma-lies overlie these felsic centers. Ground follow-up studies onsouthern Pilley’s Island (Ford, 1993), including ground gamma-ray spectrometry, geological mapping, and litho-geochemicaland petrological investigations, have shown that two of thestrongest potassium anomalies (Figure 2c) are associated with

FIG. 1. Location of examples discussed in this paper.

hydrothermally altered felsic pyroclastics and massive daciticflows (Figure 3b; Tuach et al., 1991). Potassium anomalies atBumble Bee Bight, Mansfield Showing, and the 3B deposit areassociated with narrow zones of hydrothermally altered pil-low basalts, dacitic flows, and pyroclastics in the hanging walls.The large range in K values (Figure 4d) reflects sporadic dis-tribution of potassium enrichment. Maximum potassium con-centrations at Bumble Bee Bight represent a two-to three-foldincrease over the average potassium values (3.2% K ±1.0%;Figure 4a) for equivalent unaltered to weakly-altered felsic vol-canic sequences within the Robert’s Arm Group. Potassiumenrichment at Spencer’s Dock (Figure 4b) and the Old Minearea (Figure 4c) shows similar variability with maximum con-centrations of approximately 12% K.

Lithogeochemical alteration indices (Figure 4f) definedby [(K2O + MgO) × 100]/(K2O + MgO + CaO + Na2O) gen-erally range between 70 and 98 for samples containing greaterthan 7% K, and between 30 and 60 for less altered samples con-taining 2–6% K. Maximum potassium concentrations are notassociated with widespread sericitic alteration, but are asso-ciated with an unusual fine-grained K-feldspar alteration thatThurlow (1996) described as volumetrically equivalent to thesericitic alteration. In the Spencer’s Dock area, Thurlow de-scribed felsic volcanic units affected by this alteration as “un-remarkable, light grey-green, pyrite-free, hard, dacitic lavashaving a weakly silicified aspect.” K-feldspar alteration in thehanging wall of the Old Mine, Spencer’s Dock, and Bumble BeeBight areas is overprinted by sericitic microfractures, suggest-ing that the K-feldspar alteration may precede the pervasivesericitization and massive sulfide mineralization (Santaguidaet al., 1992). Notably, no potassium enrichment is associatedwith barren pyritic gossans hosted by pillow basalts on SundayCove Island (Figure 4e).

Although the areas of intense K-feldspar enrichment are lessdirectly associated with VHMS mineralization than sericiticand chloritic alteration, the proximity of potassium anomalies(Figure 2c) to mineralization provides evidence of significantfluid/rock interaction associated with the mineralizing event.The ability to map and quantify this potassium alteration fromthe air and on the ground has important implications for VHMSexploration and mapping.

POLYMETALLIC MAGMATIC-HYDROTHERMAL DEPOSITS(Au-Co-Cu-Bi-W-As), LOU LAKE, NORTHWEST

TERRITORIES

In 1974, the magnetite breccia/rhyodacite ignimbrite–hostedSue-Dianne Cu-U-Au deposit (to date, 8 million tons grading0.8% Cu, open to depth and along strike) was discovered as aresult of eU and eU/eTh anomalies on a gamma-ray spectro-metric survey with 5-km line spacing (Richardson et al., 1974;Charbonneau, 1988). The deposit is located 20 km north of theLou Lake area (Figure 1) in the southern part of the 1870–1840Ma, Proterozoic volcano-plutonic Great Bear magmatic zone(GBmz) in northern Canada. Subsequent metallogenic studies(Gandhi, 1994) documented similarities between Sue-Dianneand the giant Olympic Dam polymetallic deposit in Australia,and promoted the exploration potential of the entire magmaticzone.

In 1993, the Geological Survey of Canada (GSC), encour-aged by the success of the Sue-Diane discovery, surveyed aselected part of the southern GBmz using 500-m line spacing,

Detecting Ore Using GRS and K Alteration 2003

combining gamma-ray spectrometric, magnetic total field, andVLF-EM sensors in a Skyvan fixed-wing aircraft. Results werepublished in 1994 in digital format and as a bound bookletcontaining twelve 1:100 000–scale color interval maps (K, eU,

FIG. 2. Potassium maps compiled from airborne gamma-ray spectrometer surveys flown using 1-km line spacing: (a) Island ofNewfoundland (LMIS = Lake Michael Intrusive Suite, DBG = Deadman’s Bay Granite, AG = Ackley Granite, FG = FrancoisGranite, NBGS = North Bay Granite Suite), (b) Notre Dame Bay area, (c) Pilley’s Island.

eTh, eU/eTh, eU/K, eTh/K, total count, ternary radiometric,magnetic total field, calculated magnetic vertical gradient,VLF-EM total field, and quadrature), stacked profiles,and a color geology map with mineral occurrences (Hetu

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et al., 1994). This more detailed survey provided a regionalframework for ongoing geological mapping and metallogenicstudies, and delineated several new and existing explorationtargets. The survey clearly showed the previously unrealized

FIG. 3. Simplified geology, corresponding to airborne potassium maps in Figures 2b, c: (a) Notre Dame Bay area, showing distributionof major Cambrian to Ordovician volcanic rocks, derived from Colman-Sadd et al. (1990); (b) general geology, southern Pilley’sIsland, derived from Tuach et al. (1991).

significance of several known, small, scattered mineral oc-currences at Lou Lake (Figure 1), placing them within thecontext of a large, potassium and iron-enriched, polymetallic(Au-Co-Cu-Bi-W-As) hydrothermal system. Publication of the


FIG. 4. Variations in equivalent thorium and potassium concentrations as measured by in-situ gamma ray spectrometry: (A–E)unaltered and variably K-altered mafic and felsic volcanics of the Robert’s Arm Group, (F) variation of lithogeochemical alterationindex with increasing potassium concentrations as measured by in-situ gamma ray spectrometry.

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survey prompted extensive new exploration activity through-out the GBmz.

Airborne radioactive element and magnetic signatures ofthe Lou Lake area are shown in Figure 5 and in profile formatin Figure 6. The broad potassium anomaly (Figure 5a) covers

FIG. 5. Airborne geophysical patterns (A–D) and geology (E–F) for the Lou Lake area, Northwest Territories.

a 3 × 4 km area and contains contoured values which locallyexceed 7% K southeast of Lou Lake. Corresponding profiledata (Figure 6) across the same area are not subject to averaginginherent in the contouring process, resulting in values whichexceed 8% K.


The potassium anomaly is coincident with a high mag-netic total-field anomaly, which has a peak intensity exceed-ing 2000 nT (Figure 5d). Polymetallic mineralization occurswhere coincident potassium and magnetic intensities are great-est. The southeast trending axis of the magnetic anomaly par-allels a belt of lower Proterozoic metasedimentary rocks con-taining synsedimentary, stratiform magnetite beds and lensesand later, hydrothermal polymetallic magnetite veins and dis-seminations.

The strong potassium anomaly is characterized by eTh/K ra-tios of less than 2.5 × 10−4 (Figure 5b). In general, low eTh/Kratios are excellent indicators of potassium alteration (Shiveset al., 1995). Unaltered lithologies typically reflect the normalratio of crustal abundances of K and Th, of approximately5 × 10−4 (Galbraith and Saunders, 1983). During the processof potassium alteration, however, thorium does not usually ac-company potassium. The resulting low eTh/K ratio, as observedat Lou Lake, thus enables distinction of potassium anomaliesthat have exploration significance from those related solely tolithological variations.

Uranium enrichment, evident on the eU/eTh ratio map (Fig-ure 5c), is peripheral to the potassium anomaly and relatesto numerous small pitchblende veins. This mineralization mayrepresent lateral movement of uranium away from the hy-drothermal center of the system.

Ground spectrometry across the mineralised zones at LouLake (N. Prasad, Geological Survey of Canada, personal com-munication, 1995; Gandhi et al., 1996) measured potassiumconcentrations as high as 15% K in K-feldspar altered rhy-olitic units and up to 6% K in biotite-altered metasediments(Figure 7). Unaltered equivalents of these rocks contain lessthan 4% K.

In addition, company reports indicate a 3-mgal gravityanomaly is coincident with the potassium and magnetic air-borne anomalies, and with resistivity lows that outline themineralized zones. Currently, in-pit mineable reserves for the“Bowl” zone include 42.145 million tons grading 0.1% Co,0.5 grams/ton Au, and 0.12% Bi. The inferred geologicalresource is more than 100 million tons (R. Goad, FortuneMinerals Ltd., personal communication, 2000).

FIG. 6. Airborne geophysical stacked profile across Lou Lakearea, Northwest Territories (line 16, position indicated inFigure 5).

Based on geological mapping and metallogenic studies, thenumerous polymetallic mineral occurrences at Lou Lake areinterpreted as hydrothermal, related to a deep-seated gran-ite pluton (Figure 8). It was suggested (Gandhi et al., 1996)that the mineralizing solutions moved upwards through metal-rich argillaceous metasedimentary beds, scavenged the metalsand redepositing them at the unconformity with the overlyingLou Lake volcanics. The discovery of the large potassium andiron enrichment zone by the airborne spectrometric and mag-netic survey further emphasizes the hydrothermal characterand large size of the mineralizing system. This example under-scores the importance of combining geophysical/geochemicalinformation provided by regional airborne surveys with groundstudies, good geological control, and metallogenic models toidentify and delimit mineralization.

PORPHYRY Au-Cu-(Mo) DEPOSITS—MT. MILLIGAN,BRITISH COLUMBIA, AND CASINO, YUKON TERRITORY

In the Canadian Cordillera, exploration for porphyry cop-per and molybdenum boomed in the 1960s and 1970s, respec-tively, when several successful mines were discovered. Dur-ing those periods, some researchers used ground gamma-rayspectrometry to measure potassium enrichment in porphyrydeposits (Moxham et al., 1965; Davis and Guilbert, 1973; Port-nov, 1987), and a few enlightened explorationists attempted touse the technique to detect mineralizing intrusions or associ-ated alteration. Unfortunately, despite the research successes,some of these early exploration tests were seen as failures, per-haps due to some combination of inferior instrumentation, in-correct techniques, or inappropriate application to alkali-poor(low-potassium) targets. Other geophysical techniques, such asmagnetic and induced polarization surveys, proved far moresuccessful and became well established, while spectrometrywas virtually forgotten.

During the late 1980s, when exploration turned towardsgold-bearing alkalic porphyry systems, several deposits were

FIG. 7. Potassium versus thorium concentrations determinedby in-situ gamma-ray spectrometry, Lou Lake area, NorthwestTerritories.

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reevaluated for their gold potential, and new discoveries weremade, including those at Mt. Milligan, British Columbia, andCasino, Yukon Territory (Figure 1). Despite the obvious potas-sium enrichment associated with these alkalic systems, groundgamma-ray spectrometers generally remained dust covered incompany storage rooms. Field crews relied on qualitative stain-ing of rock slabs and drill core, or sparse whole-rock geochem-ical analyses to map potassic alteration.

In 1990, authors Shives and Ford conducted brief groundspectrometric surveys over the Mt. Milligan and Mt. Polley(250 km south of Mt. Milligan) alkalic porphyry copper-golddeposits in British Columbia to reestablish the applicabilityof the technique to porphyry deposit exploration. The re-sults, which indicated strong potassium variations (Figure 9),

FIG. 8. Geological cross-section along line A-B (see Figure 5 for section location and geological legend). NoteK and Fe enrichment halo associated with hydrothermal Au-Co-Cu-Bi-W-As mineralization. Modified fromGandhi et al. (1996).

FIG. 9. Potassium versus thorium concentrations determined by in-situ gamma-ray spectrometry over deposits at Mt. Milligan (a)and Mount Polley (b) (approximately 250 km to southwest) in British Columbia. Note progressive K enrichment, with maximumvalues associated with mineralization.

demonstrated the ability of the ground technique to detectpotassium alteration associated with the deposits and to dis-tinguish various related intrusive and extrusive lithologies.This work encouraged subsequent GSC fixed-wing and con-tracted helicopter- borne surveys over several porphyry sys-tems throughout the Cordillera. Two examples are summarizedbelow.

Mt. Milligan deposit area

The large low-grade Mt. Milligan porphyry copper-gold de-posits occur within the lower Mesozoic Quesnel terrane incentral British Columbia. The deposits are associated withlower to middle Jurassic, alkalic, porphyritic monzonite stocks


which intrude latitic, andesitic to high-potassium basaltic andtrachytic volcanic rocks of the Witch Lake Formation (WL inFigures 10a, b; Nelson et al., 1991).

Mineralization occurs in several zones (combined resource299 million tons grading 0.45 grams/ton Au, 0.22% Cu) aspyrite, chalcopyrite, magnetite, bornite, molybdenite, and gold.A detailed description of the exploration history, geology, al-

FIG. 10. Airborne geophysical data compiled from surveys using 500-m line spacing over the Mt. Milligan deposit area, BritishColumbia (A–C) and Casino deposit area, Yukon Territory (D–F). Mt. Milligan area bedrock geology after Nelson et al. (1991);surficial geology after Kerr (1991).

teration, and mineral zoning of the deposits is provided bySketchley et al. (1995).

In 1991, the GSC conducted the first public domain, high-sensitivity gamma-ray spectrometric survey flown in BritishColumbia, combined with magnetic total-field measurements,over the Mt. Milligan deposit area (Geological Survey ofCanada, 1992). Magnetic total-field and potassium maps for

2010 Shives et al.

part of this survey, flown with 500-m line spacing, are shownin Figure 10. A broad, regional magnetic high is associatedwith exposed and buried portions of the Mt. Milligan IntrusiveComplex (Figure 10a). High potassium concentrations are as-sociated with bedrock exposures of the complex at Mt. Milligan(Figure 10b). To the south, discrete K anomalies associatedwith the Mt. Milligan deposits provide smaller, better-focusedexploration targets relative to the regional magnetic signature.Elsewhere, despite generally few outcrop exposures and ex-tensive, often thick overburden (till, glaciofluvial, and collu-vial deposits locally exceeding 100 m), potassium anomaliesoverlie several previously known and new mineral showings orprospects, providing significant exploration vectoring. Alongthe eastern margin of the survey, coincident uranium, thorium,and potassium concentrations characterize rocks within theWolverine Metamorphic Complex (WMC in Figure 10b).

Based on detailed ground spectrometry and correlation ofthe airborne survey with regional surficial geological mapping(Kerr, 1991), a threshold value of 1.2% potassium typically dis-tinguishes higher concentrations associated with glaciofluvialdeposits from lower values over glacial tills (Figure 10c). Thisoffers substantial aid to ongoing surficial mapping. Althoughall potassium anomalies throughout the survey warrant care-ful field investigation, above-threshold values in tills overlyingbedrock mapped as andesitic may be considered first-orderanomalies.

Two of these anomalous-K till sites occur west of the Mt. Mil-ligan deposits, in the Phillips Lakes area (labelled K5 andK6 in Figure 10). Ground follow-up at these sites includedtill, bedrock, and biogeochemical sampling, ground spectrom-etry, and magnetic susceptibility measurements. Bleached,K-altered andesitic volcanics outcropping in the K5 area con-tain pyrite and chalcopyrite in quartz carbonate veins. Al-though no outcrop occurs in the K6 area, many large, angular,sulfide-bearing, quartz-veined, K-feldspar-altered porphyriticintrusive boulders define a granite till, containing numerousgold grains and high Cu and Au concentrations. These resultsprompted exploration activity, and drilling has confirmed thepresence of blind, low-grade Au and Cu mineralization in aK-altered porphyritic intrusion, extending into the enclosingvolcanic host rocks.

Casino deposit area

The Casino Au-Cu-Mo porphyry deposit is located in west-central Yukon Territory (Figure 1) in deeply weathered,unglaciated terrain (extremely rare in Canada). The depositis associated with the Upper Cretaceous Casino IntrusiveComplex, which intrudes middle-Cretaceous granodiorites ofthe Dawson Range Batholith. A preserved, 70-m thick, gold-bearing, leached cap overlies a Cu-rich supergene zone, withthe base of weathering extending to 300 m below the sur-face. Geological reserves include 28 million tons grading 0.68g/ton Au, 0.11% Cu, and 0.024% Mo in the leached cap andsupergene-oxide zone; 86 million tons grading 0.41 g/ton Au,0.43% Cu, and 0.031% Mo in the supergene-sulfide zone; and445 million tons grading 0.27 g/ton Au, 0.23% Cu, and 0.024%Mo in the hypogene zone. A central, mineralized, potassic alter-ation zone consisting of K-feldspar and biotite is surroundedby barren phyllic (sericite) and propylitic alteration. Boweret al. (1995) provide detailed description of the deposit history,geology, alteration, and mineralization.

FIG. 11. Potassium versus thorium concentrations determinedby in-situ gamma-ray spectrometry over the Casino depositarea, Yukon Territory.

Ground spectrometry over the deposit and surrounding area(Figure 11) demonstrated that despite deep weathering, ra-dioactive element concentrations differentiate host litholo-gies and alteration. Maximum bedrock K concentrations (6–8% K) were measured in mineralized felsic microbrecciaswithin the leached cap. Maps of airborne potassium concentra-tions and eTh/K ratios for part of a 1993 helicopter-borne sur-vey (Geological Survey of Canada, 1994) using 500-m line spac-ing are shown in Figure 10. A unique, low eTh/K ratio bullseyeanomaly (Figure 10e) clearly distinguishes high-potassium val-ues associated with altered, mineralised bedrock in the Casinodeposit from similar high-potassium patterns (Figure 10d) as-sociated with felsic volcanic rocks in the older Yukon Meta-morphic Terrane, which has normal eTh/K values. These ra-dioactive element and magnetic features are best viewed, indetail, on stacked profiles (Figure 10f).

CONCLUSIONS

The ability of gamma-ray spectrometry to map potassium,uranium, and thorium enrichment or depletion provides pow-erful exploration guidance in a wide variety of geological set-tings. The case histories presented highlight the use of gamma-ray spectrometry to measure and map potassium enrichmentrelated to volcanic-hosted massive sulfide, polymetallic, andporphyry mineralization in Canada. Potassium enrichment inthese and many other geological settings is characterised byanomalously low eTh/K ratios relative to normal lithologicalsignatures, thus providing significant exploration vectors.

ACKNOWLEDGMENTS

For the examples presented, the authors appreciate thecooperation of Fortune Minerals Limited, Imperial Met-als Corporation, Continental Gold Corporation, Pacific Sen-tinel Gold Corporation, and Archer, Cathro and Associates.N. Prasad is thanked for field data and digital images for theLou Lake area. F. Santaguida supplied valuable field guidanceand petrological information regarding Pilley’s Island.

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There are several corrections to the equations in Appen-dix B. Please note that, once corrected, they still supportthe claim that the scaled Poisson’s ratio change is equiva-lent to the fluid factor, as stated in the paper.

Equation (B-5)

∆ρ/⟨ρ⟩ ≈ g ∆ VP / ⟨V

P⟩

Equation (B-9)

B =54— A (1 – γ ) – 4γ 22 ∆VS———

⟨V ⟩S

,

Equation (B-9a)

γ5γ

= ———— – —A (1 – γ ) B2∆VS———⟨V ⟩S 4γ

.

Equation (B-11)

∆ F = [1.6 – 0.232 (1 – γ ) / γ] A + B.0.29γ——2

ErrataTo: “Effective AVO crossplot modeling: A tutorial” (C. P. Ross, GEOPHYSICS, 65, 700-711)

Equation (B-12)

∆F = αA + βB.

Equation (B-12a)

a = 1.6 – 0.232 ———2

[ [(1 – γ )γ

Equation (B-13)

∆F = 1.252A + 0.58B,

Equation (B-14)

∆F = k(0.50A + 0.23B),

Equation (B-15)

k-1∆F ≈ 0.50A + 0.23B.

anatomy inverse problems

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