locating anomalous seismic attenuation: a mathematical ... · tions of a source seismic wave are...

CANADIAN APPLIEDMATHEMATICS QUARTERLYVolume 12, Number 4, Winter 2004

LOCATING ANOMALOUS SEISMIC

ATTENUATION: A MATHEMATICAL

INVESTIGATION

CATALINA ANGHEL, GARY MARGRAVE, NILIMA NIGAM

Based on work carried out at the Eighth Annual Industrial Problem Solv-

ing Workshop, sponsored by the Pacific Institute for the Mathematical

Sciences, May 17–21, 2004. Original problem submitted by Husky En-

ergy.

1 Introduction Seismic imaging, a technique in which the reflec-tions of a source seismic wave are recorded as it passes through theearth, is a major tool for geophysical exploration. A detonating chargecreates a seismic wave which propagates through the earth, and surfacegeophones record the reflections of this wave from different geologicallayers. Seismic imaging can be used to reconstruct a profile of the ma-terial properties of the earth below the surface, and is thus widely usedfor locating hydrocarbons.

In this paper we are concerned with the detection of anomalous seis-mic attenuation: the loss of energy of a seismic wave as it propagatesthrough the earth. Specifically, attenuation is defined as the loss of en-ergy of a seismic wave as it travels though the earth, which is not causedby geometric spreading, but depends on the characteristics of the trans-mitting media. As an exploration tool, attenuation effects have onlyrecently attracted attention. These effects can prove useful in two ways:as a means of correcting seismic data to enhance resolution of standardimaging techniques, and as a direct hydrocarbon indicator. Many phys-ical processes can lead to the attenuation of a seismic trace. These can

Thanks to: M. al-Khaleel (McGill) L. Dong (University of Calgary), C. Dupuis(UBC), G. Hennenfend (UBC), F. Hermann (UBC), Heejeong Lee (Seoul NationalUniversity), JinwooLee (Seoul National University), Joohee Lee (UNC), NamyongLee (Minnesota State University), C. Montana, M. Peyman (UBC) T. Schaefer(SUNY).Special thanks to: Mr. Kenneth J. Hedlin (Husky Energy) and Prof. Gary Margrave,University of Calgary.

Copyright c©Applied Mathematics Institute, University of Alberta.

439

440 C. ANGHEL, G. MARGRAVE, N. NIGAM

be divided into two main categories: scattering and absorption, depend-ing on the way energy is transformed. Scattering attenuation occursbecause particles in the earth redirect the sound wave in other direc-tions and happens when “the scale of heterogeneities is smaller than thecharacteristic wavelength of the seismic wave” [19]. The energy is notdissipated, merely redirected. On the other hand, intrinsic attenuation iscaused by absorption, or the conversion of acoustic energy into thermalenergy. In other words, intrinsic attenuation is mainly due to frictionlosses. Absorption and scattering effects are difficult to separate; weassume most of the loss in energy is caused by intrinsic attenuation andfocus on these effects exclusively.

Intrinsic attenuation is caused by friction, particularly in porous rocksbetween fluid and solid particles, (see [7, 2]). Though there is no agree-ment on the mechanism involved, it has been observed that porous rocksfilled with fluid considerably attenuate a seismic signal, with the atten-uation being strongest in a partially fluid-saturated rock [17, 18, 22].Attenuation varies with many factors such as water saturation, porosity,pore geometry and pressure.

This work is motivated by a problem originally posed by Husky En-ergy at the 8th Industrial Problem Solving Workshop organized by thePacific Institute of Mathematical Sciences, May 2004. The goal of theworkshop was to find a means of computing seismic attenuation fromrelatively short windows of seismic imaging data, and particularly to beable to identify regions of anomolous attenuation.

The paper is organized as follows. We begin by a detailed descriptionof the attenuation problem in Section 2, collecting important notationand assumptions for easy reference.

In Section 3, we consider the use of frequency-shift techniques to iden-tify anomalous attenuation; several different attributes are tested on realand synthetic data. In Section 4, we present the mathematical ideas be-hind an extension of a Wiener technique. In Section 5, we describe somewavelet-based strategies, and in Section 6 we sketch an optimization-based algorithm. We end the paper in Section 7 with ideas for futurework.

Please note that captions and discussions regarding figures are madeto color values. For a full-color version of this paper, please see theCAMQ website at

http://www.math.ualberta.ca/ami/CAMQ/online.htm

http://www.math.ualberta.ca/ami/CAMQ/online.htm

LOCATING AOMALOUS SEISMIC ATTENUATION 441

2 Description of problem The attenuation of the wave is directlylinked to the different layers that compose the Earth, so that wheneverchanges in the composition of layers occur, the attenutation changes too.This is why we would like to be able to detect changes in attenuation,which would enable us to identify changes in material properties.

In this section, we begin by defining some notation. We then presenta problem statement, and end with a description of the data we will beworking with.

2.1 Notations and definitions A detonation creates the source wave-form we will usually call w(t), which propagates though the earth. Thismay also be called the “signature of the source.” For a finite time, thereflected signal is recorded by a receiver at discrete time intervals. Thissignal, s(t), is called a seismic trace. If we assume the propagation speed,c, to be constant, the time tn will correspond roughly to reflections fromdepth (ctn)/2. Since the waveform w(t) is caused by a detonation, itis not a delta pulse, but stretches over a finite time interval; thereforereflections from layers n−1, and perhaps n−2, etc. will also contributeto the signal at tn. Furthermore, the coefficients of reflection from differ-ent layers in the earth are unknown, as is w(t). Several source-receiverpairs are located along a line and a trace is recorded at each one. Thecollection of traces is called a seismic section.

It might be useful to note that in the literature the models for seismicwave propagation are often illustrated using specific methods of collect-ing seismic data, which are different from surface reflection profiles (e.g.,convolutional model in [23], p. 162). A vertical seismic profile (VSP)refers to measurements made with a source at the surface and many re-ceivers positioned down a vertical well. The receivers record full traces.Sonic logs are vertical seismic profiles where only the arrival time, notthe full trace, is recorded at sparsely-spaced geophones. In both casesthe velocity in the vertical direction is measured, and changes in ve-locity are related to changes in geological layers [12]. These strategiesare expensive, since they require the drilling of wells. As a preliminarystep in exploration, it is cheaper to collect reflection information at thesurface. For this reason, we are interested in surface seismic reflectionprofiles, where we have assumed the propagation speed of the wave tobe constant. Although this will affect resolution, we can still obtaininformation about seismic boundaries.

The ability of a material to attenuate seismic waves is measured by adimensionless quantity Q, called the attenuation factor, which describesthe energy loss of a seismic wave per cycle of oscillation. We follow one


pulse through the medium, and observe how it’s energy decays over awavelenth. Specifically, the attenuation factor is defined by

(1) Q :=energy of seismic wave

energy dissipated per cycle of wave=

2πE

−∆E

where E is the energy of the wave, and ∆E is the change in energy percycle. Thus, lower Q values are related to high energy loss. Typicalvalues of Q range from 5–20 (dirt) through 100 (rock) to 10,000 (steel).The reservoirs of interest, bearing hydrocarbons in fluid or gas form,typically have a Q value of between 20 and 40, and are more attenua-tive than the surrounding rocks. In what follows, we assume that thisattenuation factor is independant of frequency ω in the useful seismicbandwidth, but that it changes with depth..

2.2 Problem statement The goal of the work conducted at the IPSW,and in this paper, was to locate efficient methods for computing changesin Q values from given seismic trace data. In particular, we are notinterested in the actual Q values, only in being able to identify regionsin space where these values change anomalously. Since we assume thespeed of the wave is constant, this becomes equivalent to locating tem-poral regions where the values change dramatically.

Our strategy was to develop algorithms to detect anomalous atten-uation, and then test these methods on simulated data as well as realseismic data, where the location of the anomaly is known. This helps uscompare the effectiveness of different strategies, at least on these datasets.

2.3 Description of data The data consists of surface seismic reflec-tion profiles. The experimental set-up is assumed to be one dimensional.That is, all geological layers are horizontal, and the signal only travelsalong the vertical direction. We also assume that source and receivergeophones are coincident and positioned directly on the surface of theearth (zero offset). The source emits a pulse at time t = 0, and thegeophones immediately begin collecting data.

Two real data sets were provided during the IPSW workshop. Thefirst comes from Pike Peaks field, discovered in 1970 in east-centralSaskatchewan, where the hydrocarbon production is from the lowerCretaceous Waseca formation that is predominantly quartz, well-sortedchannel sand of 33% porosity. It is about 500 m deep and 10 to 30 mthick. The oil is 12 degree API with a GOR of 15,” [14]. The seismicsection consists of 761 traces. Each trace was sampled for 1 second, at


intervals of time 0.002 s. The reservoir is located at around trace 400,and depth corresponding to a time of about 0.7 seconds.

The second data set is of the Blackfoot reservoir located southeastof Calgary in Alberta. The reservoir is a channel filled with porouscemented sand at 1550 m below the surface and thickness of 10 to 20 m.The seismic section consists of 86 traces. Each trace was sampled for 2.7seconds, at time intervals of 0.002 s. The reservoir is located at traces40–50, and depth corresponding to a time of about 1.2 s.

Figure 1 shows the seismic sections and the traces from each data set.

The data sets we have received have undergone some processing re-lated to the data acquisition. We do not know the precise nature of thisprocessing. For example, a possible step might have been to performthe experiment with different sources at the same location, and then toaverage them into a single seismic profile. The two sets of data mayhave undergone different pre-processing corrections, especially since thedata is also used in other contexts, not just attenuation measurement.In particular, some filtering of higher-frequency modes may have beendone; since attenuation is best measured in this end of the spectrum,the preprocessing rendered the problem quite challenging.

Starting from the data received, our goal is to identify the reservoirlocations, by locating regions of anomalous attenuation.

In order to test ideas, we also generated some synthetic data wherethe attenuation anomaly is known. Two traces are produced. Eachis sampled for t = 2 s, at intervals of time 0.002 s. The reflectioncoefficients are drawn from a random Gaussian noise distribution andthen raised to a integral power, making the sequence more spiky. Thephase velocity is assumed to be normalized so that c = 1 and the sourcesignal w has a dominant frequency of 20 Hz. The two traces differ onlyin the attenuation coefficients of the material of propagation:

• One has a constant attenuation coefficient Q = 100.• The other has a Q = 20 at a depth ct = 1 until ct = 1.1 (where

c = 1), and Q = 100 everywhere else. This simulates a hydrocarbonreservoir of depth 0.1.

The two seismic traces s(t) are calculated as they propagate. Theyappear nearly identical in time, but if we take their difference we canlocate the onset of the anomaly. Figure 2 shows the simulated data.

3 Frequency shift methods for attenuation detection A pop-ular strategy for detecting anomalous attenuation is to study how the


(a) Pikes Peak: anomaly at depth 350, loca-tion x=400.

(b) Blackfoot reservoir: anomaly at deptht=600, location x=40–50.

FIGURE 1: Actual seismic traces: Pikes Peak and Blackfoot.

frequency profile of a pulse changes as it passes through the earth. Wewill see next that higher frequencies are attenuated more than lowerones; the average frequency of the pulse will therefore decrease as it isattenuated. Our strategy in this section is to develop frequency-based“attributes” which are easy to compute, and which highlight abnormal


(a) Profile of “normal” and “anomolous” attentuation.

(b) l-r: normal trace, anomalous trace, traces superim-posed, difference of traces

FIGURE 2: The seismic trace corresponding to the attenuation anomalyis nearly identical to the normal attenuation case.


shifts in the frequency best.Suppose the signal observed is of form s(t) = A(ω, t) cos(v(t)), where

A(ω, t) is the amplitude modulation as a function of frequency ω andtime t. For a medium with linear stress-strain relation, it is known thatthe wave amplitude A is proportional to

√E, where E is the energy of

the wave. Hence, the attenuation factor, which depends on the depth,is

(2) Q(z) =2π(A(z))2

−2A(z) 4 A−→ 1

Q(z)= −∆A

πA

from which we can obtain the amplitude changes due to attenuation.That is, given the initial amplitude A0(ω, t), let λ be the wave lengthgiven in terms of frequency ω and phase velocity c by λ = 2πc/ω, then∆A = λ(dA/dz). Equation (2) becomes

(3)dA

dz= − ω

2cQ(z)A(ω, z)

with the exponential decaying solution

(4) A(ω, z) = A0(ω, t) exp

(− ωz

2cQ(z)

).

Now, from observation of exponentially decaying values of A(ω, z), wecan compute the Q(z) value. That is, from (4), we have

(5) ln

(A(ω, t)

A0(ω, t)

)= −ω

(z

2cQ(z)

)= −ω

(t

2Q(z)

).

Here we assume that the phase velocity c does not depend on frequency,i.e., that there are no dispersion effects. This has the added effect ofcorrelating well the time of travel of the reflected wave with the depthof the layer from which the reflection occurs. Hence, by recording theln(A/A0) versus ω graph, and then estimating the average slope, wecan recover the value of Q(z) at the given depth. This idea is knownas log spectral ratio method. Even though attenuation decreases boththe amplitude as well as the average frequency of a seismic wave asit propagates, amplitude decay is more likely to be affected by noise.Hence the frequency shift is a more reliable method of estimating theattenuation [19]. We will work with the frequency representation of thesignal, and wish to target anomalous high frequency energy loss.


For seismic imaging we can deduce a spectral ratio formula whichincludes the effect of reflections from discrete geological layers. Let thereflection coefficient of the nth layer be rn. Assume a temporally local-ized a seismic pulse w(t) is the source seismic wave, which propagatesthrough the earth, being reflected and attenuated over time. The seis-mic traces s(t) is collected at times t1, t2, . . . , tn. Suppose we hear theecho around times tm, tm+k, tm+2k, etc. These need not be successivetime intervals. In the Fourier domain at frequency ω, we may write

(6) |sk(ω)| = rk |so(ω)| exp(−pωtk/Qk)

where p is a constant, and rk is the coefficient of reflectivity of the kth

layer.Suppose we have similar information about a seismic trace reflected

from layer j, and that the attenuation factor Qk has not changed betweenthe kth and jth layer. Then the log spectral ratio method estimates theattenutation Qk as

log

( |sk||sj |

)= log |rk| − log |rj | +

pω(tj − tk)

Qk.

To use the log spectral ratio method, we divide an observed traceinto several “windows” comprised of a few layers in the temporal di-rection, and assume the attenuation Q is constant over each of these.We then compute the Fourier transform of each of these windows, anduse the log spectral ratio to compute the attenuation factor in each.Variations in this factor across windows should highlight geophysicalvariations. Clearly, we would wish for the windows to be narrow so asto more precisely identify the location of the anomaly. Unfortunately,this windowing procedure contaminates the Fourier transform of the sig-nal. To alleviate this problem, we could also take a Gabor transform ofthe seismic trace at each of these times. Then we get

|sg(ωl, tm)| = rm |w(ωl)| exp(−pωltm/Q),

where p is a constant and sg(ω, tm) is the amplitude of the frequencyω in the Gabor-transformed signal at time tm. The same is true for|s(ωl, tm+k)|. The log spectral ratio estimates Q as:

log

( |sg(ω, tm+k)||sg(ω, tm)|

)= log |rm+k| − log |rm| + pω (tm+k − tm)

Q.


3.1 Centroidal frequency Since the seismic traces are noisy, calcu-lation based on individual frequencies is not robust. However, we expectto observe a downshift in the average of the signal’s frequency spectrum,since the high frequency components are attenuated more rapidly thanthe low frequency components. To find how the centroidal frequencyof the signal s(t) changes as it propagates in the earth, we first choosea (large) number of points in time, t0, t1, . . . , tn and a fixed windowlength. Successive windows should overlap. At each of the points tj , wecalculate the Gabor transform of the windowed signal, then the centroidfrequency for that window:

fc(tj) =

∫ω |sg(ω, tj)| dω∫|sg(ω, tj)| dω

≈∑

k ωk |sg(ωk, tj)|∑k |sg(ωk, tj)|

,

and also the second moment:

fc(tj) =

∫ω2 |sg(ω, tj)| dω∫|sg(ω, tj)| dω

≈∑

k ω2k |sg(ωk, tj)|∑

k |sg(ωk, tj)|.

The centroidal frequency and the second moment are two possible“attributes” of the signal which we can compute with the aim of identi-fying an anomalous attenuation of the signal.

We tested the centroid method on both the synthetic and the realdata. For the synthetic case, we clearly see a down shift in the centroidfrequency in the abnormal trace at 1 s, the onset of the abnormality,in comparison with the normal trace (Figure 3). With the real data wehave no ‘normal’ trace for comparison, yet there are regions of suddendrop in the centroid frequency at the regions where the reservoirs arelocated. In the figures, blue indicates lower frequency and red indicateshigher frequency. For the Pike Peaks data in the centroid figure andeven more clearly in the second moment figure there is a spot of blueat t = 0.7 s, traces 350 to 400, (Figure 4). Similarly, for the Blackfootdata (Figure 5), there is a region of cyan between traces 40–50, at timeof about 1.2 s corresponding to the location of the reservoirs. There isa region of blue at the same location for the second moment, though inthis case the area of low frequency is less sharply resolved.

The denominator (the integral of the Gabor transform in each win-dow) seemed to be a good indicator of the onset of the anomaly for thesynthetic data, though not for the real data. We therefore do not recom-mend it as an attribute to track for the purpose of anomaly detection.

The sg(ω, tj) needed in the calculation of the centroid may be smooth-ed using a convolution with boxcar function b of frequency. If b(z)


is centered around z = 0 and falls away far from zero then, similarto the Gabor transform, the function s(u) near u = t is emphasized,while oscillatory behaviour far away is dampened. However, the resultsdo not seem much different when the smoothing is applied. For the

0 0.5 1 1.5 2 2.50

2

4

6

8x 10−3 Integral of the Gabor transform (denominator)

normalabnormal

0 0.5 1 1.5 2 2.50

20

40

60Centroid frequency

0 0.5 1 1.5 2 2.50

2000

4000

6000Second moment frequency

FIGURE 3: The synthetic signal. The centroid and second momentfor both traces were calculated over many windows. The values for theabnormal trace fall below those of the normal trace.

spectral ratio method, we take the logarithm of the ratio of the Gabortransform of successive windows. Plotted with respect to time the slope,the graph should be linear with slope 1/Q, if the attenuation factor Qremains constant through the earth. However, noise makes the graphvery oscillatory, and often the calculated slope is near to zero, or evennegative. We relaxed this method by first trying to fit the logarithmof the ratio |sg(ωl, tm)| with quadratic or cubic polynomials to capturemore of the changes.

In the next subsection we investigate some other possible attributes.

3.2 Polynomial fit to spectral ratio The frequency spectrum ofeach Gabor window was fitted with a polynomial of the desired degree,using a least-squares fit. We store only the leading order coefficient ofthis polynomial, and track it from window to window. Hence, for a win-dow at time tn we have a leading coefficient an. Either an was stored forwindow n, or an − an−1 was stored. Subtracting successive coefficients


was thought to highlight local changes. We anticipate a steady decreasein high frequency components from the propagation of the wave throughthe rock, but a sharp drop at the location of the reservoir. Comparingsuccessive windows would compensate for the background decay, accen-tuating the sharp drop. However, the leading coefficient an proved tobe a better measure, as opposed to the variation an − an−1.

For the synthetic data, the coefficients change at the onset of theabnormality for the linear and quadratic terms when only the leadingcoefficients are stored, though the change is not large (Figure 6). Forthe Pike Peaks data, the best results were obtained using a quadraticfit. The area of greatest contrast in the leading order coefficient seemsto be around trace 350 and 0.6 seconds (aim: trace 400, time 0.7 s).For the Blackfoot reservoir, the cubic coefficients show a clear low pointbetween trace 45 and 65 and at time 1.2 s, the location of the reservoir.(Figures 7 and 8).

4 Convolutional models and modifications of the Wiener

transform method As a seismic signal propagates through the earth,it is both reflected and attenuated. Mathematically, the observed wave-

(a) Integral of the Gabor transform (denomi-nator).

FIGURE 4: The frequency method applied to the Pike Peaks data.


(b) Centroid frequency.

(c) Second moment frequency.

FIGURE 4: (contd.) The frequency method applied to the Pike Peaksdata. The centroid and second moments method clearly show a blueregion at trace 400, depth corresponding to 0.7 s. This low point corre-sponds to a downshift in the average frequency due to high attenuation.


form is a convolution between the source and the reflectivity coefficients(in the absence of attenuation). The effect of attenuation is mathemat-ically modeled by the action of a pseudo-differential operator on thesource signal. In this section, we study these convolutional models, andseek to use them to identify anomalous attenuation.

4.1 Convolutional model without attenuation We will beginwith the model without attenuation. Our discussion is based on [23],and the lecture notes of Tim Henstock [10], and is included here forcompleteness. Let

s(t) = unattenuated seimic trace received at time t at a receiver,w(τ) = source waveform or signature,r(t) = reflectivity as a function of depth (≡ time).

Let us recall two of the assumptions mentioned briefly in the datasection. We assume that:

1. The earth consists of stacks of horizontal layers, with the same com-position in each layer.

0 10 20 30 40 50 60 70 80 90

0

0.5

1

1.5

2

2.5

3

trace

Regular integral

time

(a) Integral of the Gabor transform (denomi-nator).

FIGURE 5: The frequency method applied to the Blackfoot data. Thecentroid graph shows a low region at 1.2 s, traces 45–65 which is nearlocation of the reservoir at traces 40–50.


0 10 20 30 40 50 60 70 80 90

0

0.5

1

1.5

2

2.5

3

trace

Centroid

time

(b) Centroid frequency.

0 10 20 30 40 50 60 70 80 90

0

0.5

1

1.5

2

2.5

3

trace

Second moment

time

(c) Second moment frequency.

FIGURE 5: (contd.) The frequency method applied to the Blackfootdata. The centroid graph shows a low region at 1.2 s, traces 45–65 whichis near location of the reservoir at traces 40–50. For the second moment,the low area is less sharply resolved.


0 0.5 1 1.5 2 2.5

−1

−0.5

0

x 10−4

time

coef

ficie

nts

Linear coefficients graph

normalabnormal

0 0.5 1 1.5 2 2.5

0

1

2

3

4

x 10−7

time

coef

ficie

nts

Quadratic coefficients graph

0 0.5 1 1.5 2 2.5

−4

−2

0

2

x 10−9

time

time,

equ

ival

ent t

o de

pth

Cubic coefficients graph

(a) Leading coefficients

0 0.5 1 1.5 2 2.5

−5

0

5

x 10−6

time

coef

ficie

nts


normalabnormal

0 0.5 1 1.5 2 2.5

−5

0

5

x 10−8

time

coef

ficie

nts


0 0.5 1 1.5 2 2.5

−4

−2

0

2

4

x 10−10

time

time,

equ

ival

ent t

o de

pth


(b) Subtract leading coefficients of consecu-tive windows

FIGURE 6: Coefficients of the linear, quadratic, and cubic polynomialsfit to the absolute Gabor transform over many windows for the synthetictrace.


0 100 200 300 400 500 600 700 800

0

0.2

0.4

0.6

0.8

1

1.2

trace number


time,

equ

ival

ent t

o de

pth

(a) Linear coefficients

0 100 200 300 400 500 600 700 800

0

0.2

0.4

0.6

0.8

1

1.2

trace number


time,

equ

ival

ent t

o de

pth

(b) Quadratic coefficients

FIGURE 7: The polynomial fit of the Pike Peaks data. The Gabortransform was taken for successive windows, and fitted with a polyno-mial for each window. This graph represents the leading coefficients.The abnormal areas seem to be at trace 350 and 0.6 seconds, insteadof at trace 400 and 0.7 seconds. However, the quadratic coefficientsplot shows a high (dark red region) at the required location. From theprevious graph, we expect the quadratic coefficients to give the bestresults.


0 100 200 300 400 500 600 700 800

0

0.2

0.4

0.6

0.8

1

1.2

trace number


time,

equ

ival

ent t

o de

pth

(c) Cubic coefficients

FIGURE 7: (contd.) The polynomial fit of the Pike Peaks data. TheGabor transform was taken for successive windows, and fitted with apolynomial for each window. This graph represents the leading coeffi-cients. The abnormal areas seem to be at trace 350 and 0.6 seconds,instead of at trace 400 and 0.7 seconds. However, the quadratic coeffi-cients plot shows a high (dark red region) at the required location. Fromthe previous graph, we expect the quadratic coefficients to give the bestresults.

2. The source generates a compressional wave (no shear wave) whichtravels only in the vertical direction. That is, all the incident andreflected waves are normal to the horizontal interfaces.

The combination of these two assumptions makes the seismic traceone dimensional. Attenuation and dispersion cause an amplitude decayin the source wave as it travels though the earth. The time-dependentchange in the waveform is called nonstationarity. For the simple convo-lutional model we make a third assumption.

3. The source waveform does not change as it propagates though theearth.

From assumption (1), the source wave will only be reflected at theboundary between two layers of different materials and the amplitude


of the reflection is dependent on the properties of the materials. So inthis simple model, the sequence of reflectivity coefficients correspondsto the earth’s impulse response. Reflectivity is measured as a functionof two-way time travel to that interface, where the time corresponds todepth.

Suppose we have a reflectivity sequence r(t) and a source wave w(τ),and a trace recorded at t0. If w(τ) is a simple spike (a delta function),then we simply have

s(t0) = r(t0).

However, a detonation, which is the typical initial signal, has a short,but finite width. For a finite width pulse, only the beginning part ofthe wavelet, w(0), will reach the reflection boundary and return to thesurface at t0. But the parts of the pulse at τ > 0 have not had time toreach the reflection layer r(t0), so that the contribution from τ withinthe wavelet is w(τ)r(t0 − τ) [10]. The total contribution is the integral

0 10 20 30 40 50 60 70 80 90

0

0.5

1

1.5

2

2.5

3

trace number


time,

equ

ival

ent t

o de

pth

(a) Linear coefficients

FIGURE 8: The polynomial fit for the Blackfoot data. In this case thequadratic coefficients are not a good indicator of the reservoir location.However, for the cubic coefficients graph, a dark blue region appears attrace 45–65 and time 1.2 s, at the location of the reservoir. This regionis well resolved, similar to the centroid method result.


0 10 20 30 40 50 60 70 80 90

0

0.5

1

1.5

2

2.5

3

trace number


time,

equ

ival

ent t

o de

pth

(b) Quadratic coefficients

0 10 20 30 40 50 60 70 80 90

0

0.5

1

1.5

2

2.5

3

trace number


time,

equ

ival

ent t

o de

pth

(c) Cubic coefficients

FIGURE 8: (contd.) The polynomial fit for the Blackfoot data. In thiscase the quadratic coefficients are not a good indicator of the reservoirlocation. However, for the cubic coefficients graph, a dark blue regionappears at trace 45–65 and time 1.2 s, at the location of the reservoir.This region is well resolved, similar to the centroid method result.


of these:

s(t0) =

∫∞

−∞

w(τ)r(t0 − τ)dτ ,

or

(7) s(t) = w(t) ∗ r(t),

a convolution. Our (restated) goal is to estimate the earth’s response,or r(t), but even with this simple model it is difficult to identify closelyspaced reflecting boundaries given only s(t).

To summarize, for the model without attenuation we have

s(t) = w(t) ∗ r(t) + n(t),

where n(t) is noise, which we assume is uncorrelated. External sourcesof noise include wind motion, environmental noise and bad coupling ofgeophone to the ground, and internal noise from the recording instru-ments. We now have one equation, three unknowns w(t), r(t) and n(t).In order to proceed, we make a fourth and fifth assumption:

4. The noise component n(t) is negligible.5. We know the source wave w(t).

The trouble is, we don’t know w(t). For marine seismic data theseismic wavelet produced from an air-gun can be estimated from a mea-surement near the source, but for detonations w(t) is unknown. Someargue that w(t) should be estimated from source signature measurements(e.g., [24]), but the common procedure is to obtain w(t) by working inthe frequency domain and replacing assumption five by

5’.The earth’s reflectivity response has a white spectrum. The spec-trum is constant amplitude where all frequencies are present. This issatisfied by a random sampling from a Gaussian distribution.

The fifth assumption is justified because s(ω), the spectrum of thesignal in the frequency domain has a uniform background, but the basicshape resembles the spectrum of the source wave. The small oscillationsare due to the reflectivity.

The white reflectivity assumption means that

r ∗ r =

∫∞

−∞

r(s)r(t − s)ds = δ(t)


where δ(t) is the Dirac measure and r(s) = r(−s). In other words, theautocorrelation function of r is a delta function. The reflection at somedepth is not related to what is reflected at another depth. Given thisassumption, we are able to recover the amplitude of the source signalfrom the trace

(8) s ∗ s = (w ∗ r) ∗ (w ∗ r) = (w ∗ w) ∗ δ.

For general f , in the Fourier domain,

f(t) = f(−t),

f(ω) = F [f(t)],

˜f(ω) = F

[f(t)

]= f∗(ω),

F [s(t) ∗ s(t)] = F [s(t)]F [s(t)] = f(ω)f∗(ω) = |f(ω)|2.

The superscript ∗ denotes conjugation, while regular ∗ denotes convolu-tion. Taking the Fourier transform of (8), we have that

(9) |s(ω)|2 = |w(ω)|2.

This is an experimental observation as well concerning the source signaland the observed trace. We see that we can arrive at it theoretically ifwe assume that the reflectivity is a random sequence which has a whitespectrum in the frequency domain.

We have the amplitude spectrum (or gain), |w(ω)|, but to get back tow(ω) and the time representation w(t), we also require the phase. Manyprofiles can have the same amplitude spectrum as w(ω). Our choicesreduce to one if we make the following assumption about the phase:

6. The source wavelet is minimum phase.

The definition of minimum phase is as follows: a stable causal se-quence is minimum phase if its Laplace Z-transform has no zeros withinthe unit circle in the Z-plane. If w(t) is a causal function, then w(t) = 0for t < 0 and w(t) 6= 0 for some t ≥ 0. Causality makes intuitive sensebecause physical systems respond to an excitation after that excitation.All the signals we collect are discrete time series. So if we let Z = eiω∆t,


then we can rewrite the signal in the time and frequency domain as

s(t) =

n∑

k=0

skδ(t − k∆t),

s(ω) =

n∑

k=0

skeiwk∆t =

n∑

k=0

skZk,

respectively. Then from (9), we have that:

|s(ω)|2 = |w(ω)|2

= (s0 + s1Z + · · · + snZn)(s∗0 + s∗1Z

−1 + · · · + s∗nZ−n)

=

n∏

k=1

(Z − Zk)(Z−1 − Z∗

k

).

The product was factored into 2n roots which occur in pairs of Zk and1/Z∗

k . Half of them are in the unit circle and half of them are outsidethe unit circle. To get back w(ω) ≈ s(ω), we have to choose n of theseroots. We do not know which ones to choose. This is what is meantwhen we say we have lost information about the phase. The minimumphase wavelet has all of its roots outside the unit circle. Since n out ofthe 2n are outside the unit circle, there is only one wavelet defined bythese roots. Another definition of minimum phase is that the wave hasall of the energy concentrated at its onset, or has the least energy delay.For detonation sources, this assumption is pretty good, because most ofthe energy will be concentrated at the front of the wavelet produced.This means that the front end of the seismic wavelet gives the traveltime from the source to the reflector and back to the receiver.

The assumptions above mean that the source signal is causal, invert-ible, and possesses minimum phase in the sense that if we write thesignal in the frequency domain

(10) w(ω) = A0(ω) exp(iφ(ω)),

we can find the phase φ(ω) by using a Hilbert transform. With theseassumptions, and in the absence of attenuation, we could recover thesource signal w from a given trace s using the Wiener process: Wealready know the amplitude of the source signal w from equation (8)and equation (10), with the minimum phase assumption allows us toretrieve the phase of w.


Unfortunately, the Wiener process is not so simple in the presenceof attenuation. Our attempt in the next subsection is to extend thismethod to the case with attenuation via a perturbation technique.

4.2 A convolutional model with attenuation As mentioned be-fore, Wiener process does not simplify so easily in the case when thesignal is attenuated. The following Wiener model with attenuation wasoriginally proposed by Tobias Schaefer [15].

Since Assumption 3 from the previous subsection no longer applies,the attenuated trace is now the result of a modified convolution

(11) s(t) = w(t) ∗ a • r(t) :=

∫wα(τ, t − τ)r(τ)dτ

where

wα(u, v) =

∫w(η) α(u, η) eiηv dη,

α(u, η) = exp

(− ηu

2Q

)exp

(iu

2Q

∫∞

−∞

e

η − ede

)

= exp

(−u sgn(η)η

2Q(u)+ i

(u sgn(η)H(η)

2Q(u)

)).

The H denotes the Hilbert transform. The action of attenuation is thusmodeled as a pseudodifferential operator. The source wavelet w is firstmodified by attenuation which depends on the time, and then convolvedwith reflectivity. The wa is similar to the Fourier transform of w, exceptfor the factor α(u, η) in front which describes the attenuation. In α(u, η)the negative exponential, exp (−ηu/(2Q)), causes the decay in amplitudeand the remaining part represents the phase.

To simplify this to something manageable, perform the followingsteps. First expand α(u, η) into a linear Taylor polynomial approxi-mation. This is valid for small attenuation, or large Q. Then integrateonly with respect to η in the expression for wa. Finally, find s, thens ∗ s, and then the Fourier transform F [s(t) ∗ s(t)]. (This idea is due toTobias Schaefer).


The Taylor expansion for α(u, η) is:

α(u, η) = exp

((u

2Q(u)

)(− sgn(η)η + i sgn(η)H(η))

)

=

∞∑

k=0

1

k!

(u

2Q(u)

)k

(− sgn(η)η + i sgn(η)H(η))k

=

∞∑

k=0

1

k!

(u

2Q(u)

)k

κ(η)k

where

κ(η) = − sgn(η)η + i sgn(η)H(η).

In expression for wa, u and η are uncoupled so that the integration canbe performed only with respect to η:

wα(u, v) =

∞∑

k=0

1

k!

(u

2Q(u)

)k ∫w(η)κ(η)keiηv dη

=

∫w(η) eiηvdη +

(u

2Q(u)

) ∫w(η)κ(η)eiηv dη + · · ·

= w(v) +u

2Q(u)J1(v) +

1

2

(u

2Q(u)

)2

J2(v) + · · ·

where

Jn(v) =

∫w(η)kn(η)eiηv dη.

Now express s(t) in terms of w, r, J1 and t. Using

s(t) =

∫wα(τ, t − τ)r(τ) dτ

and

wα(τ, t − τ) = w(t − τ) +τ

2Q(τ)J1(t − τ),


we have

s(t) =

∫w(t − τ)r(τ) dτ +

∫τ

2Q(τ)J1(t − τ)r(τ) dτ

= w ∗ r +

∫ (τr(τ)

2Q(τ)

)J1(t − τ) dτ

= w(t) ∗ r(t) +

(tr(t)

2Q(t)

)∗ J1(t).

Similarly,

s(t) = w ∗ r +

((tr)tilde

2Q

)∗ J1(t).

We assume that Q is symmetric about t = 0 s.We have already seen in the discussion of the white-reflectivity as-

sumption that r has the nice property that r(t) ∗ r(t) = δ(t) . Togeneralize this, use

tnr ∗ r = βnδ(t).

Also note that 1

(a ∗ b)tilde = a ∗ b

and

F[δ(t)

]= F∗ [δ(t)] = 1 =⇒ δ(t) = δ(t).

These properties allow us to simplify s(t) ∗ s(t) as follows:

s(t) ∗ s(t) = w ∗ r ∗ w ∗ r +

(tr

2Q

)∗ J1 ∗ w ∗ r +

((tr)

tilde

2Q

)∗ J1 ∗ w ∗ r

= w ∗ w +

(tr ∗ r

2Q

)∗ J1 ∗ w +

((tr)

tilde ∗ r

2Q

)∗ J1 ∗ w

= w ∗ w +

(β1

2Q

)∗ J1 ∗ w +

(β1

2Q

)∗ J1 ∗ w.

The last step is to simplify the Fourier transform of the last two termsin the expression above.

1Observation by Vinicius Anghel: (a∗ b)tilde = a∗ b, although (ab)tilde 6= ab. Thiscorrected one sign in the expression for s ∗ s.


Given that

κ(η) = − sgn(η)η + i sgn(η)H(η),

Jn(υ) =

∫w(η)κn(η)eiηυdη.

Calculate

F

[β1

2QJ1 ∗ w +

β1

2Qw ∗ J

].

First term:

F [J1 ∗ w] = J1w = w(η)κ(η)(w∗(η)) = |w|2 κ.

Second Term:

F [w ∗ J1] = wJ1 = J∗

1 w = w∗κ∗w = |w|2 κ∗.

Add them together to obtain

F

[β1

2QJ1 ∗ w +

β1

2Qw ∗ J1

]=

β1

2Q|w|2 (κ + κ∗)

=β1

2Q|w(ω)|2 (−sgn(ω)ω).

We conclude that

F [s(t) ∗ s(t)] = |w|2 +β1

Q|w|2 (Re(κ)) ,

|s|2 = |w|2 − β1

Q|w|2 (sgn(ω)ω) .(12)

This makes sense because then |s|2 is real. Also, the amplitude decreasesas Q decreases. If Q → ∞, then the equation reduces to the classicalequation (9).

4.3 A discrete model It is useful to think about the extent of theill-posedness in the problem by viewing a slight restatement. In prac-tice, seismic trace data is sampled at discrete time intervals, for a finiteduration of time. We therefore describe a discrete version of the convolu-tional model above: suppose we know the initial source signal, as well asthe attenuation and reflectivity properties of the medium being sampled.


Let the data be sampled at times t1, t2, . . . , tn. From this, we can con-struct a matrix Wα, and a vector of reflectivities ~r = (r1, r2, . . . , rn)T ,where ri is the reflectivity of the layer at depth cti. Then, the discreteversion of equation (11) is

Wα~r = (~w1 | ~w2 | . . . | ~wn)~r = ~s := (s1, s2, . . . , sn)T .

The entries wij of matrix Wα have the following properties:

• If ti > tj , wji = 0 (causality assumption).• If ti < tj , wji = wα(ti, tj − ti) where wα was defined in equation (11).

Therefore, Wα is lower triangular, and the amplitude spectra of columnvectors ~wi attenuate by an exponential factor from left to right.

The forward seismic problem is: given Wα, ~r, find the seismic tracevector ~s.

The inverse seismic problem is: given ~s, find Wα, ~r. In our specificcase, we have to find Wα, specifically the amount of attenuation betweenthe amplitude spectra of the columns of Wα. As is easy to see, the inverseproblem is quite ill-posed.

4.4 Experiments with the convolutional model of attenuation

A few coding experiments were done using the Wiener model with at-tenuation.

From equation (9), in the case without attenuation we see that in thefrequency domain, the amplitude of the collected signal is the same asthe amplitude of the source signal. The first step was to see how theamplitude of the collected signal compares with |w(ω)| if attenuation ispresent. For the synthetic trace, the w is known, so we can calculateF [s ∗ s] to see how it compares with |w|2 for different Q’s. The F [s ∗ s]and |w|2 values were normalized by dividing by their respective meanvalues. The relative error is calculated as the absolute difference dividedby the maximum of |w|2. The difference between the two is very big anddecreases only slightly as Q increases (Figure 9).Note that as expectedfrom equation (9), the spectrum of the signal in the frequency domainhas the same basic shape as the spectrum of the source wave, withsuperimposed rapid, small amplitude oscillations.

There are two ways to calculate F [s ∗ s]: either by taking a Fouriertransform (in our case, Gabor transform) of a convolution, or by cal-culating only the the Fourier (Gabor) transform of s and multiplyingit by the conjugate. From now on, G will be used to denote a Gabortransform. To test that G[s∗ s] was calculated correctly, we calculated it


using the two methods. For the method using the convolution, however,the result oscillates from negative to positive at every point, so that mul-tiplying it by a vector of [1,−1, 1,−1, . . .] gives a positive result. Thismeans that there is a mistake in the phase. After this correction, thetwo methods give the same result. So from now on G[s ∗ s] is calculatedas G∗[s]G[s], where the superscript ∗ denotes conjugation and * denotesconvolution.

The next step was to see how G∗[s]G[s] changes over time for thenormal and abnormal trace. First, the time is divided into overlappingwindows, at which G∗[s]G[s] is calculated. For each time window, storethe maximum value of G∗[s]G[s]. Taking the maximum might not bethe best method because |G[s]| has its maximum at low frequency, andwe are interested in how the high frequencies decay. But we do see thatfor the abnormal trace, the value of G∗[s]G[s] is lower than that for thenormal trace after t = 1 s where the abnormality occurs (Figure 10).This is consistent with equation (12), because since Q is smaller at theabnormality, 1/Q is larger, and the value of G∗[s]G[s] decreases.

When this process is applied to the real trace, the resulting graphsseem similar to those for

∫|sg(ω, tj)| dω, recorded in the process of us-

ing the centroid method. (Figure 11, compared to Figure 4(a) and Fig-ure 5(a). They are also similar to the wavelets and polynomial fit graphs.

To see how G∗[s]G[s] varies as Q varies, its maximum in one windowwas calculated for different values of Q and plotted with respect to 1/Q.We would like the relationship to be linear, though we expect it not tobe since higher order terms are ignored (Figure 12).

After these preliminary investigations, we would like to get back toequation (12) to find a way to use it for estimating |w|2 and then Q. Ifwe use the Gabor transforms, for our discrete case we have that

G[s ∗ s] = |w(ω, ti)|2 −β1

Q(ti)|w(ω, ti)|2 (sgn(ω)ω)

where ω is the frequency. We can use G∗[s]G[s] from a window near thebeginning of the trace to estimate |w|2, since the signal has not traveleda far enough distance to be attenuated greatly. Dividing the G∗[s]G[s]value from every other window by the estimate of |w|2 obtained from the

first window should give an estimate of[1 − β1

Q(ti)(sgn(ω)ω)

]. We are

assuming that |w|2 changes little in time, and that the value of G∗[s]G[s]is affected most by changes in Q. The values obtained for the abnormaltrace are smaller than those for the normal trace (Figure 13). There arelarge spikes due to dividing by small numbers.


0 50 100 150 200 2500

2

4

6

8

10

12

14Wiener method: |G[s]| for middle window (1 sec), Q =100

0 50 100 150 200 2500

1

2

3

4

5

6Checking the Wiener amplitude: |F(w*w)|1/2

0 50 100 150 200 250−10

−5

0

5Difference between the two amplitudes

0 50 100 150 200 2500

0.5

1

1.5

2Relative error between the two amplitudes, max =1.6094

(a) Q = 100

0 50 100 150 200 2500

2

4

6

8Wiener method: |G[s]| for middle window (1 sec), Q =10000

0 50 100 150 200 2500

1

2

3

4

5

6Checking the Wiener amplitude: |F(w*w)|1/2

0 50 100 150 200 250−6

−4

−2

0

2

4Difference between the two amplitudes

0 50 100 150 200 2500

0.2

0.4

0.6

0.8

1Relative error between the two amplitudes, max =0.79305

(b) Q = 10000

FIGURE 9: The top left graph is a graph of |G[s]| over a large window.The bottom left is a graph of | bw|. The top and bottom left are thedifference and the relative error. The maximum of the relative erroris 1.61 for Q = 100 and 0.79 for Q = 10000.

An idea for future work would be to find an approximation to |w|2 bytaking a Fourier transform of the entire trace. Then this value can becompared to the G∗[s]G[s] of each window along the trace. Clearly the


0 0.5 1 1.5 2 2.50

1

2

3

4

5x 10−4 F*[s]F[s], Q =100

time

normalabnormal

FIGURE 10: The maximum value of G∗[s]G[s] over many different win-dows for the synthetic trace. The values for the normal trace are greaterthan those for anomalous trace.

method needs to be refined before tests on the real data are meaningful.

5 Wavelets applied to the seismic trace Multi-resolution anal-ysis can be used to remove noise from a seismic trace. Random noiseand the reflectivity response of the earth are both high frequency com-ponents of the signal. The low-frequency approximation to the signalfilters out the noise and reflectivity, leaving only the information on theattenuation and the source function. De-noising using wavelets ampli-fies the difference between signals that have suffered different amountsof attenuation [15]. Unfortunately, the effect of attenuation is more pro-nounced in the high-frequency end of the spectrum, which is filtered outduring denoising.

The high frequency components of a signal can also be treated ina similar manner as low frequency components. Since the attenuationdampens high frequencies, abrupt changes in the detail components mayindicate changes in attenuation. For the synthetic data, the secondand third detail component reconstructions of the anomalous trace havesmaller amplitude after time t = 1 s (Figure 14). The discrete Meyerwavelet was chosen by trial and error, because the amplitude differencebetween the anomalous and normal traces seemed largest.


(a) Pike Peaks reservoir

0 10 20 30 40 50 60 70 80 90

0

0.5

1

1.5

2

2.5

3

Blackfoot: Maximum amplitude F*[s]F[s] = |S(omega)|2 for different windows

trace

time

(b) Blackfoot reservoir

FIGURE 11: The maximum value of G∗[s]G[s] over many different win-dows for the real data.


0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020

0.5

1

1.5

2

2.5x 10−4 Max of [F(s)conj(F(s))] around t =0.98288

1/Q

FIGURE 12: The maximum value of G∗[s]G[s] for one window (t = 0.98)for the normal trace plotted with respect to 1/Q.

The fine detail reconstruction for the Pike Peaks data shows greatestvariation at t = 0.65 s, traces 300 to 500 (Figure 15). This is similar tothe location found from the graphs of frequency shift and polynomial fit.This location is slightly above the location of the reservoir. The secondand third detail reconstructions (not shown) are similar to the first detailreconstruction, though the irregularity is more clearly featured. Thefourth detail reconstruction shows another area of variation closer tothe position of the reservoir. A region of large oscillation appears belowthe t = 0.6 s region, at t = 0.7 s and around trace 400.

For the Blackfoot data there are faint irregularities in the oscillations.The fourth detail reconstruction contains horizontal regions of greatervariation at t = 1.1 s, traces 55–75 and at t = 1.7, traces 10–30. Thearea at t = 1.1 s is close to the anomaly in the Blackfoot cubic coefficientgraph.

6 Denoising using a minimization technique Recall that weare seeking to locate −A(ω, t) from noisy data d, where

d = log |s| = −A(ω, t) + log(|r|) = m + n.

We have already discussed a wavelet-based denoising strategy above.Another possible means of removing the noise from the data is to solve


0 100 200 300 400 500 6000

10

20

30

40

50

60

70

80

90snorm: t =1.0832

frequency

(a) Normal trace

0 100 200 300 400 500 6000

10

20

30

40

50

60

70

80

90sanom: t =1.0832

frequency

(b) Abnormal trace

FIGURE 13: An estimate of | bw|2 was obtained from the Gabor trans-form from the first window. The Gabor transform of every otherwindow was divided by this value, in order to give an estimate of1− β1

Q(ti)(sgn(ω)ω) for that time. These windows are at t = 1.08, at the

time corresponding to the low Q region for the abnormal trace.


0 0.5 1 1.5 2−0.02

0

0.02Snorm, Orig. signal and approx. 1 to 5.

0 0.5 1 1.5 2−0.02

0

0.02Snorm, Orig. signal and details 1 to 5.

0 0.5 1 1.5 2−2

0

2x 10−3 A(5)

0 0.5 1 1.5 2−5

0

5x 10−3 D(5)

0 0.5 1 1.5 2−5

0

5x 10−3 A(4)

0 0.5 1 1.5 2−0.01

0

0.01D(4)

0 0.5 1 1.5 2−0.02

0

0.02A(3)

0 0.5 1 1.5 2−5

0

5x 10−3 D(3)

0 0.5 1 1.5 2−0.02

0

0.02A(2)

0 0.5 1 1.5 2−5

0

5x 10−3 D(2)

0 0.5 1 1.5 2−0.02

0

0.02A(1)

0 0.5 1 1.5 2−2

0

2x 10−3 D(1)

(a) Normal Trace

0 0.5 1 1.5 2−0.02

0

0.02Sanom, Orig. signal and approx. 1 to 5.

0 0.5 1 1.5 2−0.02

0

0.02Sanom, Orig. signal and details. 1 to 5.

0 0.5 1 1.5 2−2

0

2x 10−3 A(5)

0 0.5 1 1.5 2−5

0

5x 10−3 D(5)

0 0.5 1 1.5 2−5

0

5x 10−3 A(4)

0 0.5 1 1.5 2−0.01

0

0.01D(4)

0 0.5 1 1.5 2−0.02

0

0.02A(3)

0 0.5 1 1.5 2−5

0

5x 10−3 D(3)

0 0.5 1 1.5 2−0.02

0

0.02A(2)

0 0.5 1 1.5 2−5

0

5x 10−3 D(2)

0 0.5 1 1.5 2−0.02

0

0.02A(1)

0 0.5 1 1.5 2−2

0

2x 10−3 D(1)

(b) Abnormal Trace

FIGURE 14: The multi-resolution decomposition of the synthetic traces.The abnormal trace has smaller amplitudes in the detail graphs aftert = 1 s.


(a) Pike Peak, 1st level of detail

(b) Pike Peak, 4th level of detail

FIGURE 15: Some of the detail reconstruction using wavelets for thePike Peak and Blackfoot data.


(c) Blackfoot, 4th level of detail

(d) Blackfoot, 5th level of detail

FIGURE 15: (contd.) Some of the detail reconstruction using waveletsfor the Pike Peak and Blackfoot data.


a minimization problem

minm

1

2‖C−1/2

n (d − m)‖22

where Cn is the covariance matrix of the noise Cn = E[nnT ]. We canuse a wavelet transform W to convert the minimization problem into

minem

1

2‖Γ−1(d − m)‖2

2 + λ‖m‖p

where

d := Wd, m := Wm, n := W (n), E(nnT ) = Γ2.

The noise thresholding used is called hard or soft, depending on whetherp = 1 or 2, respectively. Solving the minimization problem and subse-quently inverting would allow us to reconstruct an approximation to m.This strategy has not yet proved computationally efficient, and remainsonly a possible future direction.

7 Conclusions and future work This paper explores differentsignal analysis techniques for reservoir location: the centroid method,polynomial fit, wavelet analysis techniques, and the modified Wienermethod.

Many of the results from different techniques were consistent. Forthe Pike Peaks data, we observe two regions of irregularity. The oneabove the location of the reservoir at t = 0.6 s appears in the centroidmethod, the polynomial fit, and the wavelet method. The lower area,at the location of the reservoir, is sometimes more difficult to find. Thecontrast scale for the centroid method, the order of the polynomial, orthe level of detail for the wavelet method have to be adjusted correctly.The Blackfoot data is even more sensitive to these adjustments.

The Wiener technique is promising because, like for the centroidmethod, the theory of the method is directly used in its application;it is less heuristic than the wavelet method or the polynomial fit. Wedeveloped a theoretical model for the Wiener method in the presence ofattenuation. We arrived at the equation

|s|2 = |w|2 − β1

Q|w|2 (sgn(ω)ω)


which can be used to determine how Q changes over time. However, mostof the work done so far was directed at understanding the backgroundfor the method (e.g., the general behaviour in time of G∗[s]G[s]). Thenext step would be to implement the estimation of |w|2 from the Fouriertransform for the entire trace and using this to find how β1

Q (sgn(ω)ω)changes. Also, the method should be tested on the real data.

Another promising direction is the use of change-point methods fromstatistics to analyze these problems, which is the subject of ongoingwork.

REFERENCES

1. K. Hedlin, L. Mewhort and G. Margrave, Delineation of steam flood usingseismic attenuation, 71st Ann. Inter. Mtg. Soc. of Expl. Geophys. (2001), 1572–1575.

2. G. Kumar, M. Batzle and R. Hofmann, Effects of fluids on the attenuation ofelastic waves, 73rd Ann. Inter. Mtg. Soc. of Expl. Geophys. (2003), 1592–1595.

3. S. R. Pride and J. G. Berryman, Linear dynamics of double-porosity dualpermeability materials, I. Governing equations and acoustic attenuation, Phys.Rev. E 68.

4. S. R. Pride and J. G. Berryman, Linear dynamics of double-porosity dualpermeability materials, II. Fluid transport equations, Phys. Rev. E 68.

5. S. R. Pride, J. Harris, D. L. Johnson, A. Mateeva, K. Nehei, R. L.Nowack, J.Rector, H. Spetzler, R. Wu, T. Yamamoto, J. Berryman and M. Fehler, Per-meability dependance of seismic amplitudes: The leading edge, 22(6) (2003),518–525.

6. Y. Quan and J. M. Harris, Seismic attenuation tomography using the frequencyshift method: Geophysics, Soc. of Expl. Geophys. 62 (1997), 895–905.

7. K. Winkler and A. Nur, 1982, Seismic attenuation: effects of pore fluids andfrictional sliding: Geophysics 47(1) (1982), 1–15.

8. Leon Cohen, Time Frequency Analysis, Prentice Hall, 1995.9. Patrick Flandrin, Time-Frequency/Time-Scale Analysis Academic Press, 1999.

10. Tim Henstock, Soes6004 applied and marine geophysics: seismic section, lec-ture notes. http://www.soes.soton.ac.uk/teaching/courses/soes6004/seismic/,September, 2004.

11. Einar Kjartansson, Constant q-wave propagation and attenuation, Journal ofGeophysical Research 84(B9) (1979), 4737–4747.

12. Christopher L. Liner, Elements of 3-D Seismology, PennWell Publishing, 1999.13. Stephane Mallat, A Wavelet Tour of Signal Processing, Second Edition, Aca-

demic Press, 1998.14. G. Margrave, et al., Delineation of steam flood using seismic attenuation CSEG

Recorder (2002), 27–30.15. Nilima Nigam, et al., Seismic attenuation problem, In Proceedings of the

Eighth PIMS-MITACS Industrial Problem Solving Workshop (2004), 53–66.16. Robi Polikar, Wavelet tutorial, http://users.rowan.edu/˜ polikar/WAVELETS/

WTtutorial.html.


17. Steven R. Pride and James G. Berryman, Linear dynamics of double-porositydual-permeability materials: I. governing equations and acoustic attenuation,Physical Review E 68(3) (2003).

18. Steven R. Pride and James G. Berryman, Linear dynamics of double-porositydual-permeability materials: I.i. fluid transport equations, Physical Review E68(3) (2003).

19. Youli Quan and Jerry Harris, Seismic attenuation tomography using the fre-quency shift method, Geophysics 62(3) (1997), 895–905.

20. M. M. Reda Taha, Introduction to the use of wavelet multiresolution analysisfor intelligent structural health monitoring, Canadian Journal of Civil Engi-neering 31 (2004), 719–731.

21. Enders Robinson, Electrical engineer, an oral history conducted in 1997, An-drew Goldstein, IEEE History Center, Rutgers University, New Brunswick,NJ, USA, 1997.

22. Kenneth W. Wrinkler and Amos Nur, Seismic attenuation: effects of porefluids and frictional sliding, Geophysics 47(1) (1982), 1–15.

23. Oz Yilmaz, Seismic Data Analysis: Processing, Inversion and Interpretationof Seismic Data, Vol I and II, Society of Exploration Geophysics, 2001.

24. Anton Ziolkowski, Seismic wavelet estimation without the invalid whitenessassumption, CSEG Recorder (2001), 18–28.

Corresponding author

Nilima Nigam

Department of Mathematics and Statistics, McGill University, Montreal,

Quebec, Canada H3A 2K6

E-mail address: [email protected]

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