Three-dimensional modeling of IP effectsin time-domain electromagnetic data

David Marchant1, Eldad Haber1, and Douglas W. Oldenburg1

ABSTRACT

Understanding the effects of induced-polarization (IP) effectson time-domain electromagnetic data requires the ability to sim-ulate common survey techniques when taking chargeability intoaccount. Most existing techniques preform this modeling in thefrequency domain prior to transforming their results to the timedomain. Even though this technique can allow for chargeablematerial to be easily incorporated, its application for some prob-lems can be computationally limiting. We developed a new tech-nique for forward modeling the time-domain electromagneticresponse of chargeable materials in three dimensions. The fre-quency dependence of Ohms’ law translates to an ordinary dif-ferential equation when considered in the time domain. Thesystem of ordinary-partial differential equations was then dis-cretized using an implicit time-stepping algorithm, that yielded

absolute stability. This approach allowed us to operate directlyin the time domain and avoid frequency to time-domain trans-formations. Although this approach can be applied directly tomaterials exhibiting Debye dispersions, other Cole-Cole disper-sions resulted in fractional derivatives in time. To overcome thisdifficulty, Padé approximations were used to represent the fre-quency dependence as a rational series of integer order terms.The resulting method was then simplified to generate a reducedtime-domain model that can be used to forward model the IPdecay curves in the absence of any electromagnetic coupling.We found numerical examples in which the method producedaccurate results. The potential application of the method wasdemonstrated by modeling the full time-domain electromagneticresponse of a gradient array IP survey, and the occurrence ofnegative transients in airborne time-domain electromagneticdata.

INTRODUCTION

The physical properties of materials often depend on the fre-quency at which they are measured. When the frequency depend-ence, or dispersion, of the electrical conductivity of a material issignificant, the material is considered to be chargeable. Detectingthe presence of chargeable materials has numerous practical appli-cations. In mineral exploration, chargeable materials can be indica-tive of metallic mineralization (Fink et al., 1990). This propertycan be of particular value when exploring for disseminated miner-alization in which chargeability is commonly the most diagnosticphysical property. Recently, studies have explored other potentialapplications, such as hydrocarbon exploration (Davydycheva et al.,2006; Veeken et al., 2009a, 2012), environmental and groundwaterstudies (Slater and Glaser, 2003), and various engineering applica-tions (Kemna et al., 2004; Okay et al., 2013).

The physical mechanism that gives rise to a frequency-dependentconductivity varies depending on the situation (Schön, 2011), andmany parametric models have been developed to describe the natureof the frequency dependence. The model that is most commonlyencountered is the Cole-Cole model (Pelton et al., 1978), whichis given by

ρðωÞ ¼ ρ0

�1 − η

�1 −

1

1þ ðiωτÞc��

. (1)

In this equation, ρ0 is the zero-frequency electrical resistivity, η isthe fractional decrease in the resistivity between the low- and high-frequency asymptotes (this quantity is commonly called the charge-ability), τ is a characteristic time constant, and c is the frequencydependence. The value of c ranges from zero to one and determines

Manuscript received by the Editor 5 February 2014; revised manuscript received 20 May 2014; published online 21 October 2014.1University of British Columbia, Department of Earth, Ocean, and Atmospheric Sciences, Vancouver, British Columbia, Canada. E-mail: dmarchant@eos.ubc

.ca; haber@math.ubc.ca; doug@eos.ubc.ca.© 2014 Society of Exploration Geophysicists. All rights reserved.

E303

GEOPHYSICS, VOL. 79, NO. 6 (NOVEMBER-DECEMBER 2014); P. E303–E314, 14 FIGS.10.1190/GEO2014-0060.1

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how rapidly the transition of the real part of the resistivity takesplace. A low value of c results in a very broad dispersion curve,whereas a high c value results in a rapid transition from thelow- to high-frequency asymptotes (Figure 1). The special casewhen c ¼ 1 is commonly known as the Debye model.The economic importance of chargeable materials has resulted

in a significant body of work on the forward modeling of theelectromagnetic response of chargeable materials. Most early workfocused on modeling the response of geometrically simple polariz-able bodies either by transforming results from the frequency do-main to the time domain (Bhattacharyya, 1964; Morrison et al.,1969; Lee, 1975, 1981; Rathor, 1978; Lewis and Lee, 1984; Waitand Debroux, 1984; Flis et al., 1989; Hohmann and Newman, 1990;Lee and Thomas, 1992) or by treating the time-domain convolutiondirectly (Smith et al., 1988). Zaslavsky and Druskin (2010) and Zas-lavsky et al. (2011) develop a modeling technique based on the ra-tional Krylov subspace projection approach.The effect of dispersive physical properties on the propagation of

electromagnetic waves in the time domain has been studied exten-sively for various engineering applications (Teixeira, 2008). Recentwork has focused on the development of a method for modelingtime-domain wave propagation in media featuring dispersive elec-trical permittivities using an explicit finite-difference time-domainscheme. Rekanos and Papadopoulos (2010) use an approximationof the fractional order differential term to derive a set of auxiliarydifferential equations, which could be used to model the response ofmedia exhibiting arbitrary Cole-Cole dispersion.Frequency-domain techniques can easily incorporate dispersive

conductivities, but they are only efficient when modeling sourcescontaining a small number of frequencies. Accurately modelingsources containing a broadband of frequency content, such asthe square waves commonly used in time-domain electromagneticor induced-polarization (IP) experiments, requires the solution of

Maxwell’s equations at a large number of frequencies. For example,Newman et al. (1986) and Flis et al. (1989) report requiring between20 and 50 frequencies to accurately model a step-off response. Ifiterative methods are used, the response at each frequency canbe calculated in parallel. This can make frequency-domain methodsefficient at solving problems with a limited number of sources; how-ever, they can quickly become computationally limiting as the num-ber of sources grows (Streich, 2009; da Silva et al., 2012).Treating the time-domain convolution directly has other compu-

tational shortcomings. The convolution requires that fields through-out the entire space-time domain be stored, which, for largerproblems, can quickly become limiting. Some of these issues havebeen addressed in the context of medical and engineering applica-tions. However, with the resulting explicit techniques developed,stability requirements limit the size of the time step that can be takenmaking them prohibitively computationally expensive for applica-tion to geophysical problems.In this paper, we therefore develop a methodology to forward

model the time-domain response of an electromagnetic experimentin the presence of frequency-dependent conductivities based onimplicit time-stepping techniques. The modeling is carried out di-rectly in the time domain and can accommodate models processingarbitrary dispersion curves through an approximation scheme sim-ilar to that developed in Rekanos and Papadopoulos (2010). Ourapproach can simulate the response of an arbitrary transmitter wave-form and only has to store fields at a small number of previous timesteps. The IP method commonly assumes that the effects of electro-magnetic induction can be neglected. Making the same assump-tions, we develop a reduced model capable of simulating the fulltime-domain response of an IP survey in the absence of induction.Finally, we demonstrate the technique on examples using inductiveand galvanic sources.

THEORY

In this section, we describe the approach taken when modelingthe effects of a chargeable material on time-domain electro-magnetic data. We begin with materials exhibiting Debye dispersion(c ¼ 1) and then extend the approach to include other Cole-Coledispersions.

Debye dispersion

Maxwell’s equations in the frequency domain, assuming an eiωt

time dependence, are

~∇ × ~Eþ iω~B ¼ 0 and

~∇ × ~H − ~J ¼ ~Js; (2)

where ~E and ~H are the electric and the magnetic fields, ~B is themagnetic flux density, ~J is the current density in the ground, and~Js is the applied source current density. The magnetic fields andfluxes are related through the constitutive relationship:

~B ¼ μ ~H. (3)

The current density ~J is related to the electric field ~E through Ohm’slaw:

a)

b)

Figure 1. (a) Real and (b) imaginary Cole-Cole conductivity spec-tra for a range of c values. The smaller the value of c, the broader thedispersion of the conductivity. The frequency of peak dispersioncorresponds to the value of the time constant.

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~J ¼ σðωÞ~E; (4)

where σ is the electrical conductivity and σ ¼ ρ−1. Assuming ma-terials that only exhibit Debye dispersion (Cole-Cole model withc ¼ 1) Ohm’s law becomes

~E ¼ ρ0

�1 − η

�1 −

1

1þ iωτ

��~J; (5)

which can be rewritten as

σ0 ~Eþ τσ0iω~E ¼ ~J þ τð1 − ηÞiω~J. (6)

making use of the Fourier transform pair:

F�∂fðtÞ∂t

�¼ iωFðωÞ; (7)

in which equation 6 is easily transformed back to the time domain.Along with Maxwell’s equations in the time domain, this gives thesystem,

~∇ × ~eþ ∂~b∂t

¼ 0;

~∇ × ~h − ~j ¼ ~js;

and

σ0~eþ τσ0∂~e∂t

¼ ~jþ τð1 − ηÞ ∂~j∂t

. (8)

System 8 is a combination of a partial differential equation (PDE)with a pointwise ordinary differential equation (ODE). Similarequations arise in the context of flow that involves chemical reac-tions (Ascher and Petzold, 1998). Such systems tend to be very stiff,which implies that, to keep stability, a very small time step may beneeded if we are to use explicit techniques.

Cole-Cole dispersions using Padè approximations

For materials that exhibit a Cole-Cole-like dispersion, Ohm’s lawin the frequency domain can be written as

~E ¼ ρ0

�1 − η

�1 −

1

1þ ðiωτÞc��

~J; (9)

which can be rearranged as

σ0 ~Eþ σ0τcðiωÞc ~E ¼ ~J þ ð1 − ηÞτcðiωÞc~J. (10)

Because c < 1, the transformation of this expression into the timedomain will result in fractional derivatives, that is, a combination ofintegrals and derivatives in time. Rather than treat the fractionalderivatives directly, we approximate the frequency-dependent por-tion of equation 9 with a Padé approximation (Baker and Graves-Morris, 1996). This is a similar approach to that taken by Weedonand Rappaport (1997) to model the effects of dispersive media onwave propagation in biological materials. Using the approximation

ðiωÞc ≈ PðiωÞQðiωÞ ¼

P0 þP

m¼1M PmðiωÞm

1þPNn¼1 QnðiωÞn ; (11)

Ohm’s law becomes

σ0 ~EðωÞþσ0XNn¼1

QnðiωÞn ~Eþτcσ0P0~Eþτcσ0

XMm¼1

PmðiωÞm ~E.

¼ ~JþXNn¼1

QnðiωÞn~Jþτcð1−ηÞP0~J

þτcð1−ηÞXMm¼1

PmðiωÞm~J. (12)

By defining

Pk ¼ 0; if k > M; Qk ¼ 0; if k > N (13)

and K ¼ max ðM;NÞ, this can be rewritten as

σ0ð1þ τcP0Þ~Eþ σ0XKk¼1

ðQk þ τcPkÞðiωÞk ~E

¼ ð1þ τcð1 − ηÞP0Þ~J þXKk¼1

ðQk þ τcð1 − ηÞPkÞðiωÞk~J;

(14)

and then simplified to

A0~Eþ

XKk¼1

AkðiωÞk ~E ¼ B0~J þ

XKk¼1

BkðiωÞk~J; (15)

where

A0 ¼ σ0ð1þ τcP0Þ;Ak ¼ σ0ðQk þ τcPkÞ;B0 ¼ 1þ τcð1 − ηÞP0;

and

Bk ¼ Qk þ τcð1 − ηÞPk. (16)

Because the powers of iω are now exclusively integer order,equation 15 can now be easily transformed to the time domainyielding the k-order, ODE as

A0~eþXKk¼1

Ak∂k~e∂tk

¼ B0~jþ

XKk¼1

Bk∂k~j∂tk

. (17)

Ohm’s law for a Debye media (equation 8) is a special case ofequation 17, with K ¼ 1, P0 ¼ 0, P1 ¼ 1, and Q1 ¼ 0.We chose to follow the approach outlined in Rekanos and Papa-

dopoulos (2010) for calculating the Padé coefficients. The coeffi-cients Am and Bm are found by setting

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TNþMðsÞ −P

m¼0M Pmsm

1þPNn¼1 Qnsn

¼ 0; (18)

where TNþMðsÞ is the N þMth order Taylor series expansion of sc

computed about a point s0. Rearranging ( equation 18), we obtain

TNþMðsÞ�1þ

XNn¼1

Qnsn�−XMm¼0

Pmsm ¼ 0; (19)

which is a linear system for Pm and Qn given TNþMðsÞ.The accuracy of the resulting Padé approximation is determined

by the order of the approximation used, and the center point se-lected. Examples of various approximations of different ordersare shown in Figure 2 and approximations with different centerfrequencies are shown in Figure 3. It is clear that changing thecenter frequency of the approximation changes the frequency atwhich the approximation is most valid. Also, it is evident that in-creasing the order of the approximation increases the width of therange of frequencies over which the approximation adequately rep-resents the original conductivity spectrum.The frequency dependence plays the most important role in

determining how difficult it will be to represent the frequency spec-trum with a Padé approximation. Debye models (c ¼ 1) can be per-fectly represented by a Padé approximation of order N ¼ M ¼ 1.The lower the value of c, the less accurate an approximation ofa given order becomes. The time constant τ shifts the frequencyof peak dispersion; so, changes in it have no effect on the accuracyof the approximation as long as the center frequency used tocalculate the coefficients is also changed accordingly. The fre-quency-dependent part of the conductivity is linearly dependenton the chargeability η, so changes in ηwill change the absolute errorin the approximation proportionately.Other techniques for generating the Padé approximations are

available. The use of multipoint Padé approximations should be

able to slightly improve the quality of the resulting approximation.However, in this case in which the required order of the approxi-mation is quite low, this simple single-point approximation schemewas found to produce excellent results.

A reduced model for low frequencies

IP phenomena are rarely modeled by the full Maxwell’s equa-tions, and most modeling is done in the frequency domain, usingthe static Maxwell’s equation assuming a frequency-dependent con-ductivity; that is, the common frequency-dependent model is

~J ¼ σðωÞ ~∇ϕ ∇ · ~J ¼ ~Js. (20)

A similar simplified model can be used in time to generate IP decaycurves and discuss the conditions of its validity.The system (equation 8) contains time dependence in two differ-

ent equations. First, Maxwell-Faraday’s law contains the derivative∂~b∕∂t, and second, Ohm’s law contains time-dependent derivatives.If we assume that we are given a finite time source and that after

some short time: ∂~b∕∂t ≈ 0, whereas ∂~e∕∂t and ∂~j∕∂t are not, thenthe condition ~∇ × ~e ≈ 0 implies that ~e ≈ ~∇ϕ. Taking the divergenceof Ampere’s law, we eliminate the ~∇ × ~h term and obtain a simpli-fied time-dependent system that can be solved for the electric po-tential ϕ and the flux density ~j as

∇ · ~j ¼ ∇ · ~js σ0 ~∇ϕþ τσ0 ~∇∂ϕ∂t

¼ ~jþ τð1 − ηÞ ∂~j∂t

. (21)

The system (equation 21) is the time-dependent equivalent of theusual simplified frequency modeling of IP phenomena when EMeffects are neglected. It assumes that the IP effects are still presentat times when ∂~b∕∂t fields are small and can be neglected. Clearly,this assumption breaks down in the presence of large conductors,and the contamination by significant induction is often referred to as

a)

b)

Figure 2. (a) Real and (b) imaginary Cole-Cole conductivity spec-tra (black line) compared with Padé approximations of varying or-der. The Padé coefficients used in each case were computed using aTaylor series calculated at 100 Hz.

a)

b)

Figure 3. (a) Real and (b) imaginary Cole-Cole conductivity spec-tra (black line) compared with Padé approximations whose coeffi-cients were calculated using Taylor series calculated about differentcentral frequencies f0. All of the approximations have N ¼ M ¼ 4.

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electromagnetic coupling. Therefore, using the reduced system as amodeling and inversion tool should be done judicially. We discussthis point in our numerical experiments.

The time-domain electromagnetic response of a Padémodel

It is possible to test the accuracy of time-domain response of aPadé model through the use of a synthetic model. The vertical com-ponent of the magnetic field measured at the surface of a uniformhalf-space, a distance p away from a vertical magnetic dipole, isgiven by (Ward and Hohmann, 1988)

Hz ¼ −m

2πk2p5½9 − ð9þ 9ikp − 4k2p2 − ik3p3Þe−ikp�.

(22)

In this expression, k is the wavenumber given by k ¼ ð−iσμ0ωÞ1∕2.The frequency-domain response of a true Cole-Cole dispersion andits Padé approximation can be calculated, transformed into the timedomain, and compared. The frequency-domain to time-domaintransform is accomplished through the use of digital filters (Gup-tasarma, 1982; Anderson, 1983).Results of this test are shown in Figures 4 and 5. The calculated

response of a uniform half-space exhibiting a Cole-Cole dispersionis compared with the response of Padé approximations of differentorders. The coefficients of all of the approximations were calculatedusing a Taylor series calculated at 10 Hz. All of the approximationsadequately reproduce the early time results, but the lower order ap-proximations do not provide accurate results at late times. For thelarger value of c (Figure 4), a third-order approximation canadequately reproduce the response over this time range. For thesmaller value of c (Figure 5), more terms are required to achievean adequate result. It is important to note that the accuracy ofthe response of a particular approximation strongly depends onthe frequency content of the transmitter and the time range of in-terest. An approximation that accurately predicts the response of agiven transmitter waveform and a particular time range may not beaccurate for other waveforms and times.

DISCRETIZATION

In this section, we discuss the discretization that is used to solveour system of differential equations. We first consider the Debyesystem (system 8) before extending the method to the generalizedPadé system.In space, a staggered grid is used to discretize the system (Yee,

1966). Physical properties are placed at the cell center, ~b is placedon cell faces, and ~e and ~j are placed on cell edges (Figure 6). Let b,e, and j be grid functions that are the staggered discretization of ~b, ~e,and ~j. Using a standard staggered discretization of the differentialoperators, we obtain the following semidiscretized differentialequation:

Ceþ ∂b∂t

¼ 0

C⊤Mfμ−1b −Mej ¼ Mejs

Meσ0eþMe

τσ0

∂e∂t

¼ MejþMeτð1−ηÞ

∂j∂t. (23)

In these equations, C and C⊤ are the discrete curl operators goingfrom edges to faces or faces to edges, respectively, and Me and Mf

are mass matrices containing physical properties (denoted by thesubscript) averaged onto either edges or faces (denoted by thesuperscript). For further details on the formation of these matrices,see Madden and Mackie (1989), Wang and Hohmann (1993), Drus-kin and Knizhnerman (1994), Newman and Alumbaugh (1995),Tavlove (1995), Smith (1996), Haber et al. (2000), and Haberand Ascher (2001).This system is discretized in time using a backward Euler discre-

tization with a step size of δt. The resulting discrete system is

Ceðtþ1Þ þbðtþ1Þ−bðtÞ

δt¼0;

C⊤Mfμ−1b

ðtþ1Þ−Mejðtþ1Þ ¼Mejðtþ1Þs ;

Meσ0e

ðtþ1Þ þMeτσ0

eðtþ1Þ−eðtÞ

δt¼Mejðtþ1Þ þMe

τð1−ηÞjðtþ1Þ− jðtÞ

δt;

(24)

a)

b)

c)

Figure 4. Step-off response 100 m from a vertical magnetic dipolesource placed at the surface of a uniform chargeable half-space withc ¼ 0.75. The response of the true Cole-Cole system is shown inblack, and the response of a Padé approximation is shown in gray.The Padé coefficients were calculated using a Taylor series calcu-lated at 10 Hz. The thick gray line shows the response of a non-chargeable half-space with the equivalent σ∞ (1.67 × 10−2 S∕m).Dashed lines indicate a negative response.

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or

Ceðtþ1Þ þ 1

δtbðtþ1Þ ¼ 1

δtbðtÞ;

C⊤Mfμ−1b

ðtþ1Þ −Mejðtþ1Þ ¼ Mejðtþ1Þs ;

Meeeðtþ1Þ −Me

τσ0eðtÞ ¼ Me

j jðtþ1Þ −Me

τð1−ηÞjðtÞ; (25)

where Mee ¼ Me

τσ0 þ δtMeσ0 and Me

j ¼ Meτð1−ηÞ þ δtMe, and ð·ÞðtÞ

denotes the tth time step.Next, jðtþ1Þ and eðtþ1Þ can be eliminated from system 25 to give

an equation for only bðtþ1Þ in terms of the fields and current densityat the previous time step:

�Mf

μ−1CMee−1Me

jMe−1C⊤Mf

μ−1 þ1

δtMf

μ−1

�bðtþ1Þ

¼ 1

δtMf

μ−1bðtÞ −Mf

μ−1CMee−1Me

τσ0eðtÞ

þMfμ−1CM

ee−1Me

τð1−ηÞjðtÞ þMf

μ−1CMee−1Me

j jðtþ1Þs . (26)

Once equation 26 has been solved for bðtþ1Þ, the electric field andthe current density at the next time step can be calculated using:

eðtþ1Þ ¼ Mee−1Me

jMe−1C⊤Mf

μ−1bðtþ1Þ þMe

e−1Me

τσ0eðtÞ

−Mee−1Me

τð1−ηÞjðtÞ −Me

e−1Me

j jðtþ1Þs ;

jðtþ1Þ ¼ Mej−1Me

eeðtþ1Þ −Mej−1Me

τσ0eðtÞ þMe

j−1Me

τð1−ηÞjðtÞ.

(27)

This completes the computation for the fields and fluxes attime ðtþ 1Þ.

The above approach for the Debye model naturally extends in themore general Cole-Cole case when considering conductivities rep-resented by a Padé approximation. Discretizing equation 17 for thediscrete spacial variables in time gives

A0eðtþ1Þ þXKk¼1

Ak

�∂ke∂tk

�ðtþ1Þ¼ B0jðtþ1Þ þ

XKk¼1

Bk

�∂kj∂tk

�ðtþ1Þ;

(28)

where ð∂mf∕∂tmÞðtÞ is the backward Euler approximation of themthorder derivative at time step ðtÞ given by

�∂mf∂tm

�ðtÞ≈P

mk¼0 ð−1Þk m

k fðt−kÞ

δtm. (29)

Figure 6. Magnetic fields are placed on cell faces. Electric fieldsand current densities are placed on cell edges. Physical propertiesare placed at cell centers.

a) c)

b) d)

e)

Nonchargeable

Nonchargeable Nonchargeable

Nonchargeable

Nonchargeable

Figure 5. Step-off response 100 m from a verticalmagnetic dipole source placed at the surface of auniform chargeable half-space with c ¼ 0.25. Theresponse of the true Cole-Cole system is shown inblack, and the response of a Padé approximationis shown in gray. The Padé coefficients were cal-culated using a Taylor series calculated at 10 Hz.The thick gray line shows the response of a non-chargeable half-space with the equivalent σ∞(1.67 × 10−2 S∕m). Dashed lines indicate a neg-ative response.

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Expanding the summations in equation 28 and collecting like termsallows equation 28 to be rewritten as

Aeðtþ1Þ þ eðtþ1Þp ¼ Bjðtþ1Þ þ jðtþ1Þ

p ; (30)

where

A ¼XKk¼0

Ak

δtk; B ¼

XKk¼0

Bk

δtk; (31)

and eðtþ1Þp and jðtþ1Þ

p contain the influence of the electric field andcurrent density at previous time steps and are given by

eðtþ1Þp ¼

XKn¼1

ð−1Þn�XK

k¼n

Bk

δtkkn

�eðtþ1−nÞ

and

jðtþ1Þp ¼

XKn¼1

ð−1Þn�XK

k¼n

Ak

δtkkn

�jðtþ1−nÞ. (32)

Equation 30 can then be discretized in space to arrive at a linearsystem for bðtþ1Þ as�Mf

μ−1CMeA−1Me

BMe−1C⊤Mf

μ−1 þ1

δtMf

μ−1

�bðtþ1Þ

¼ 1

δtMf

μ−1bðtÞ þMf

μ−1CMeA−1ðeðtþ1Þ

p − jðtþ1Þp þMe

Bjðtþ1Þs Þ;(33)

where MeA and Me

B are mass matrices containing A and B, respec-tively, averaged onto cell edges. Current densities and electric fieldscan be updated using

eðtþ1Þ ¼ MeA−1Me

BMe−1C⊤Mf

μ−1bðtþ1Þ

−MeA−1eðtþ1Þ

p þMeA−1jðtþ1Þ

p −MeA−1Me

Bjðtþ1Þs ;

jðtþ1Þ ¼ MeB−1Me

Aeðtþ1Þ þMe

B−1eðtþ1Þ

p −MeB−1jðtþ1Þ

p . (34)

Reduced model

Using the same methodology, we can discretize the reducedmodel (equation 21) for the low-frequency approximation. Stag-gered grids imply mimetic properties of the operators (Hymanand Shashkov, 1999; Haber and Ascher, 2001), and this allowsus to derive a discretization for this equation in discrete space. Set-ting e ¼ Gϕ where G is a discretization of the nodal gradient (seeHaber and Ascher [2001] for details of such discretization) and fol-lowing the same procedure as before, taking the (discrete) diver-gence of the final result yields a linear system describing thebehavior of ϕ with time:

G⊤Mej−1Me

eGϕðtþ1Þ ¼G⊤Mej−1Me

τσ0GϕðtÞ

−G⊤Mej−1Me

τð1−ηÞjðtÞ−G⊤jðtþ1Þ

s . (35)

Once this has been solved for ϕðtþ1Þ, the current densities at the nexttime step can be calculated using

jðtþ1Þ ¼ Mej−1Me

eGϕðtþ1Þ −Mej−1Me

τσ0GϕðtÞ

þMej−1Me

τð1−ηÞjðtÞ. (36)

Implementation and validation

The forward modeling was implemented in the Python program-ming language. The linear systems in equations 26, 33, or 35 aresolved using the multifrontal massively parallel solver (MUMPS)developed by the CERFACS group (Amestoy et al., 2001).The application of direct methods has distinct strengths and

weaknesses when compared with iterative methods. Although theforward modeling matrix is sparse and requires relatively littlememory, factoring this matrix comes with significant memoryrequirements. The strength of direct methods comes when we con-sider problems with multiple transmitters and a uniform, or semi-uniform, discretization in time. The forward modeling matrix doesnot change as long as the time step δt remains constant. This allowsfor a single factorization to be used for all transmitters and multipletime steps. The use of direct methods for time-domain electromag-netic problems, and its comparison with iterative methods, is con-sidered in detail in Oldenburg et al. (2013).To test the implementation, the step-off responses of a vertical

magnetic dipole source at the surface of various uniform half-spacemodels were simulated and compared with the analytic results. Theanalytic responses were obtained by transforming the output ofequation 22 into the time domain using digital filters. The goodagreement for three different half-spaces is shown in Figure 7.

EXAMPLE 1: EFFECTS OF ELECTROMAGNETIC-COUPLING ON INDUCED-POLARIZATION DATA

Traditionally, chargeable material is mapped using the IP tech-nique (Bleil, 1953; Seigel, 1959). The IP method can be appliedeither in the time or frequency domain. Current is injected intothe ground through a pair of transmitter electrodes, and the voltageresponse of the earth is measured across a separate pair of receiverelectrodes. In the time domain, a steady current is injected, abruptlyinterrupted, and the decay of the voltages is measured at the receiv-ers. At sufficiently late times (to allow for inductive responses todissipate), the presence of a slowly decaying response is typicallyassumed to be indicative of chargeable materials. The ratio of thesecondary voltage (measured after transmitter shut off) to the pri-mary voltage (measured in the on time) is proportional to charge-ability (Seigel, 1959). This observation forms the basis of most IPinterpretation and inversion methodologies (see, for example,Oldenburg and Li, 1994; Li and Oldenburg, 2000). The neglectof induction also forms the basis for the associated MIP techniquein which magnetic fields, rather than electric fields, are measured(Seigel, 1974; Chen and Oldenburg, 2006).Neglecting the effects of induction is often reasonable, but this

assumption can break down in more conductive environments (Deyand Morrison, 1973; Hallof, 1974). Attempts have been made tomitigate these issues by estimating and removing the inductive ef-fects from collected data (Fullagar et al., 2000; Routh and Olden-burg, 2001; Veeken et al., 2009b) or designing arrays to minimizetheir effects (White et al., 2003). Nevertheless, the effects of EMcoupling still poses a problem for many surveys.

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In this paper, we develop two procedures for computing time-do-main EM fields when the conductivity is complex. In the first, wehave the full EM response, and in the second, we obtain the re-sponses when the effects of induction are neglected. This allowsus to evaluate the effects of EM coupling on IP data and vice versa.As an example, we simulate the responses of a typical gradient arrayIP survey. A single transmitter oriented in the east–west directionwith an A-B offset of 1225 m is simulated. The transmitter wire isrun just south of the area of interest. This survey geometry is shownin Figure 8.In the first example, a single chargeable block (dark-gray square

in Figure 8) is buried in a uniform background. The block exhibitsDebye dispersion with σ0 ¼ 0.1 S∕m, τ ¼ 1 s, and η ¼ 0.1, and thebackground has a conductivity of 0.005 S∕m. The top of the blockis located 100 m below the surface, and it extends to a depth of200 m. Three simulations were run: the electromagnetic responseof the model with no chargeability present, the reduced model of theIP response in the absence of electromagnetic induction, and thetotal response, including both the IP effects and electromagnetic

induction. Figure 9a shows the x-component of the electric fieldsfrom the three simulations observed at the point (0, 0). Due to thevery resistive background model, induction effects decay rapidly,and the reduced model accurately predicts the response from ap-proximately 2 × 10−2 s onward.In a second example, 50 m of conductive overburden is placed at

the surface of the model shown in Figure 8. Three additional con-ductive blocks are also placed directly under the overburden,extending to a depth of 150 m below the surface. The over-burden and conductive blocks have a conductivity of 1 S∕m.

a)

b)

c)

Figure 7. The calculated (gray) and analytic (black) step-off re-sponses of the vertical component of the magnetic field, a dipolesource at the surface of a uniform half-space. Fields shown are100 m from the transmitter. The models were (a) a nonchargeablehalf-space, (b) a chargeable half-space exhibiting Debye dispersion,and (c) a chargeable half-space exhibiting a Cole-Cole dispersion. Afifth-order Padé approximation calculated using a Taylor series cen-tered at 250 Hz was used to represent the dispersion.

Figure 8. Layout of synthetic gradient array experiments. The thickline shows the path of the transmitter wire, and dots show receiverlocations. The dark-gray box shows the extents of the chargeabletarget, and the light-gray boxes show the location of additional con-ductive blocks present in the second example.

a)

b)

Figure 9. The x-component of the electric fields recorded at (0, 0)for (a) a chargeable block in a resistive half-space and (b) a charge-able block in a resistive half-space with a conductive overburden.

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The chargeable block and the background havethe same properties as in the first example.The x-component of the electric field at (0,0)

is shown in Figure 9b. This time, the inductiveeffects take much longer to decay due to the pres-ence of the conductive units. The reduced modelstill accurately predicts the late time response ofthe system, but, in this case, inductive effectsdominate until approximately 2 × 10−1 s in thislocation. Figure 10 shows the time evolution ofthe x-component of the electric fields during thesimulation. Immediately following the switch offof the transmitter current (t ¼ 1.0 × 10−3 s),electric fields are induced directly below the pathof the transmitter wire. These fields completelymask the presence of the chargeable target. Thesefields diffuse through the area, interacting withthe other conductive units. It is not until latetimes (t ¼ 2.5 × 10−1 s) that the response ofthe chargeable block is easily identified. At theend of the simulation (t ¼ 0.1 s), the electricfields resemble those of a dipole, centered atthe location of the chargeable block. These fieldsagree with the response of the reduced model.In each of the examples, the conductivity

was discretized onto rectilinear mesh with55 × 55 × 54 cells in the x-, y-, and z-directions,respectively. This included a uniform core regionof 25 × 25 × 24 cells that are 25 m on a side aswell as 10 padding cells in each direction. Thepadding cells expand away from the core regionwith an expansion rate of 1.3. This results in acore region that is 625 × 625 × 600 m withapproximately 5.4 km of padding in every direc-tion. Time was discretized into four intervals ofconstant δt: 1 × 10−4 s, 5 × 10−4 s, 2.5 × 10−3 s,and 6.25 × 10−3 s, with 65 steps taken foreach δt.

Example 2: Negative transientsin airborne time-domain data

Electromagnetic surveys using inductive sources are affected bychargeable materials. Unfortunately, for most survey geometries,the effects are often impossible to recognize in the data. In thesesituations, the resulting IP effects are essentially noise, which hinderthe interpretation of the collected data. For the particular case ofcoincident-loop time-domain EM data, negative transients, sound-ings with a reversal in sign of the observed data, are diagnostic ofchargeable material. Although early papers speculated that this ef-fect could be caused by particular conductivity distributions or mag-netic effects, Weidelt (1982) shows that, for a coincident loopsystem with a step-off primary field, the measured secondary fieldmust be nonnegative over nonpolarizable ground, regardless of thesubsurface distribution of conductivity. Observed negative transi-ents in coincident-loop EM systems can therefore only be attributedto polarization effects. In many cases, this property can be extendedto center-loop systems, including many airborne platforms (Smithand Klein, 1996). An example of negative transients, taken from a

Figure 10. Time evolution of the x-component of the electric fields recorded above themodel shown inFigure 8.The extent of the chargeableblock is shownas a dashedwhite line.

a)

b)

Figure 11. Example of a negative transient observed in a VTEMsurvey at the Mt. Milligan deposit in British Columbia: (a) Lineof data above the deposit and (b) single sounding at 850 m alongthe line. Dashed lines indicate a negative value.

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VTEM survey performed over the Mt. Milligan deposit in BritishColumbia is shown in Figure 11.Similar results can be produced by a simple conductive charge-

able block in a nonchargeable background. A block, measuring 140×140 × 80 m is buried 40 m below the surface of a 3 × 10−4 S∕mhalf-space in Figure 12. The block possesses the Cole-Cole param-eters of σ0 ¼ 1 × 10−2 S∕m, τ ¼ 0.2 s, η ¼ 0.5, and c ¼ 0.5. Thefirst time derivative response of vertical component of the magneticflux density was simulated in a single line of colocated transmittersand receivers running in the x-direction directly over the center ofthe block, 30 m above the surface. The transmitters were modeled aspoint dipole sources with a step-off waveform.

The conductivity model was discretized onto a rectilinear meshwith 61 × 41 × 60 cells in the x-, y-, and z-directions, respectively.This included a uniform core region of 31 × 11 × 30 cells that are20 m on a side as well as 15 padding cells in each direction. Thepadding cells expand away from the core region with an expansionrate of 1.3. This results in a core region that is 620 × 220 × 600 m

with approximately 3.3 km of padding in every direction. Timewas discretized into three intervals of constant δt: 1 × 10−5 s,5 × 10−5 s, and 2.5 × 10−4 s, with 35 steps taken for each δt.The conductivity spectra of a block was approximated by a Padé

approximation with M ¼ N ¼ 5 and calculated using a Taylorseries expansion calculated at 160 Hz.To choose these parameters, the true response of a uniform half-

space exhibiting this particular Cole-Cole dispersion was calculatedby transforming the frequency-domain solutions (calculated with a1D code) to the time domain. The predicted response of the Padémodel was computed using the same approach. The order of theapproximation and the central point used in the Taylor seriesapproximation were adjusted until the approximation produced re-sults of a sufficient accuracy. These coefficients were then used inthe 3D modeling.

Figure 12. Geometry of synthetic airborne time-domain example.A block measuring 140 × 140 × 80 m is buried 40 m below thesurface of a 3 × 10−4 S∕m half-space. The block possesses theCole-Cole parameters of σ0 ¼ 1 × 10−2 S∕m, τ ¼ 0.2 s, η ¼ 0.5,and c ¼ 0.5. Transmitter/receiver pairs are located in a single linepassing over the center of the block, 30 m above the surface of themodel.

a)

b)

Figure 13. Simulated dB∕dt response of the model shown in Fig-ure 12. (a) Line of data above the center of the chargeable block.(b) Single sounding located at x ¼ 0 m. Dashed lines indicate anegative response.

a)

b)

c)

Figure 14. The y-component of current densities produced from thetransmitter located at x ¼ 0 m and z ¼ 30 m. Initially, (a) a strongpositive response is observed in the block and the background. Atlater times, (b) direction of current flow has reversed in the block.When the direction of current flow in the block has reversed and isdecaying back to zero, the decay of the remaining positive back-ground response prevents the observation of a negative responseuntil later times (c).

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The simulated data are shown in Figure 13. The nature of theobserved negative transient is very similar to that observed inthe Mt. Milligan VTEM data. The y-component of the computedcurrent densities at a few times is shown in Figure 14.

CONCLUSION

In this paper, we have developed a new technique for modelingthe response of time-domain electromagnetic experiments in thepresence of chargeable materials exhibiting Cole-Cole-like disper-sions. We have also developed a reduced model for the modeling oftime-domain IP decay curves in the absence of electromagneticcoupling. These techniques work directly in the time domain,eliminating the need of Fourier transforms and solving only real,symmetric, positive, definite systems rather than complex, non-Hermitian systems that are needed for frequency-domain modeling.The techniques were demonstrated to produce accurate results andthen used for a pair of synthetic examples. As demonstrated, usingboth models can be an effective method to study the effect of EMcoupling when designing geophysical surveys. Our method uses thePadé approximation and works extremely well for c’s that are largerthan zero and approaching one. For small c’s, our approach requiresmore terms in the Padé series; however, we have observed that byusing even a moderate number of terms, we are able to fit mostrelevant decay curves. Our methodology opens the door for thequestion of the identifiability and recoverability of Cole-Coleparameters from an IP-effected time-decay curve.

ACKNOWLEDGMENTS

This research was supported by the Natural Sciences and Engi-neering Research Council of Canada and by the University of Brit-ish Columbia.

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