Aria Abubakar, Peter M. van den Berg-on-linear three-dimensional inversion of cross-well electrical...

Non-linear three-dimensional inversion ofcross-well electrical measurements1

Aria Abubakar2 and Peter M. van den Berg2

Abstract

Cross-well electrical measurement as known in the oil industry is a method fordetermining the electrical conductivity distribution between boreholes from theelectrostatic field measurements in the boreholes. We discuss the reconstruction of theconductivity distribution of a three-dimensional domain. The measured secondaryelectric potential field is represented in terms of an integral equation for the vectorelectric field. This integral equation is taken as the starting point to develop a non-linearinversion method, the so-called contrast source inversion (CSI) method. The CSImethod considers the inverse scattering problem as an inverse source problem in whichthe unknown contrast source (the product of the total electric field and the conductivitycontrast) in the object domain is reconstructed by minimizing the object and data errorusing a conjugate-gradient step, after which the conductivity contrast is updated byminimizing only the error in the object. This method has been tested on a number ofnumerical examples using the synthetic ‘measured’ data with and without noise.Numerical tests indicate that the inversion method yields a reasonably goodreconstruction result, and is fairly insensitive to added random noise.

Introduction

Interest in cross-well tomography (imaging) of the earth’s electrical conductivity hasincreased because of improvements in field instrumentation, computing power andmethods of interpretation. Cross-well electromagnetic logging is a technique toinvestigate the geological properties of the region between boreholes using theelectromagnetic measurements made in these boreholes, either at low frequencies (lessthan 100 kHz), so-called induction logging, or at zero frequency, so-called electricallogging. One of the important parameters to be determined from the measurements isthe conductivity because of its sensitivity to porosity, pore-fluid type and saturation.

During the last decade, important progress has been made in solving the inverseproblem for cross-well configurations. Some notable contributions in this area include

q 2000 European Association of Geoscientists & Engineers 109

Geophysical Prospecting, 2000, 48, 109–134

1 Paper presented at the 60th EAGE Conference – Geophysical Division, Leipzig, June 1998. ReceivedOctober 1998, revision accepted June 1999.

2 Laboratory of Electromagnetic Research, Center for Technical Geoscience, Delft University ofTechnology, PO Box 5031, 2600 GA Delft, The Netherlands.

those of Torres-Verdin and Habashy (1994) and Alumbaugh and Morrison (1995).Torres-Verdin and Habashy (1994) developed a fast inverse solution based on a non-linear approximation of the integral equation. This approximation is used in theirforward calculations to produce the predicted fields from the latest model update. Theapproximation compares favourably with full solutions for conductivity contrast aslarge as 50 to 1, for frequencies below 100 kHz. Alumbaugh and Morrison (1995)developed a multifrequency imaging procedure for the reconstruction of two-dimensional variations of conductivity excited by electric line sources. In theirapproach, the Green’s function was fixed for a certain background while the unknownelectric field was updated after each iteration. A slightly different class of iterativemethods undertakes repeated modifications of the Green’s function after eachiteration. This method is known as the distorted Born iterative method (Chew andWang 1990) and it was also used in a three-dimensional cross-well problem byNewman (1995). Note that in this method a full forward problem must be solved ineach iteration. The necessity of solving a forward problem in each iteration was avoidedby Torres-Verdin and Habashy (1995) by using a non-linear inversion technique,known as the iterative extended Born approximation, to investigate a two-dimensionalobject with conductivity contrast. They showed that, with the same computationalefficiency as the first-order Born approximation, the extended Born approximationenables a much wider class of two-dimensional inverse scattering problem to be solved.

All the schemes mentioned above are used to invert low-frequency electromagneticmeasurements. Unlike induction logging, electrical logging is not so widely used. In theelectrical logging technique, the conductivity distributions are determined from thestatic field measurements (the electric current or the voltage) made in the borehole.Such a measurement system enables us to increase the investigation range whilesacrificing the resolution. This technique was used by Daily and Owen (1991) to invertresistivity data in two-dimensional cross-well configurations with the help of aniterative modified least-squares inversion based on a finite-element forward solution ofLaplace’s equation. In contrast to the work of Daily and Owen (1991), Abubakar(1996) used the modified gradient method to reconstruct three-dimensional layeredsingle-well and cross-well configurations. In the modified gradient method developedby Abubakar (1996), the vector electric current density or the scalar electric potentialfield and the conductivity contrast are updated simultaneously by a non-linearconjugate-gradient algorithm that minimizes the error in both the object equation andthe data equation. Furthermore, it was also shown that inversion based on the vectorintegral equation for the electric current density leads to substantially better resultsthan inversion based on the scalar integral equation for the electric potential.

Recently, we have developed a method to reconstruct a three-dimensionalconductivity distribution from the cross-well electrical measurements (Abubakar andVan den Berg 1998). This method is called the contrast source inversion (CSI) methodand was originally introduced by Van den Berg and Kleinman (1997) to handle thetwo-dimensional wave problem. Unlike most non-linear inversion methods, the CSImethod does not require any artificial regularization techniques to deal with the

110 A. Abubakar and P.M. van den Berg

q 2000 European Association of Geoscientists & Engineers, Geophysical Prospecting, 48, 109–134

problems of non-uniqueness in the inversion of data. It attempts to overcome thisproblem by recasting the problem as an optimization problem, in which it seeks notonly the contrast source (the product of the total field and the conductivity contrast)but also the conductivity contrast itself to minimize a cost functional consisting of twoterms, the L2 errors in the data equation and in the object equation. An alternativeiterative method of solving this optimization problem is proposed, in which first thecontrast source is updated in the conjugate-gradient direction, weighted so as tominimize the cost functional, and then the conductivity contrast is updated to minimizethe error in the object equation using the updated contrast source. This latterminimization can be performed analytically which allows easy implementation of thepositivity constraint for the conductivity. An ambiguity in the work of Abubakar andVan den Berg (1998) as to whether the object error was actually reduced is removed inthe present version by performing an extra line minimization to update theconductivity contrast in the optimization process. Furthermore, numerical tests alsoindicate that, by using sources and receivers located in four boreholes around a three-dimensional domain, we can improve the reconstruction results of Abubakar and Vanden Berg (1998). In view of the efficiency of the CSI method, three-dimensionalinverse problems can now be handled with moderate computer power.

Integral representations

A theoretical model of the cross-well configuration is shown in Fig. 1. We define aninhomogeneous domain D with conductivity j(x) in an unbounded homogeneousbackground medium with conductivity j0. The source is a small spherical electrodeemitting a DC current I, located in domain S. We measure the secondary electric

Non-linear 3D inversion of cross-well electrical measurements 111


Figure 1. Object domain D with conductivity j(x) in the unbounded homogeneous backgroundof conductivity j0.

potential fields V s at the various electrode locations in domain S. In the present cross-well electrical problem we want to determine the conductivity distribution j(x) insidethe domain D from the secondary electric potential field V s measurements made indomain S.

The governing equations originating from Maxwell’s equations at zero frequency areof the form:

=V þ E ¼ 0; ð1Þ

=· ðjEÞ ¼ qext; ð2Þ

where V is the electric potential field, E is the electric field, j is the electricalconductivity and qext is the external source. Here, = ¼ (∂1, ∂2, ∂3) denotes spatialdifferentiation with respect to the Cartesian position vector x ¼ (x1, x2, x3).

To obtain (1) and (2) in an integral form, we assume that the actual configuration inwhich the field is to be computed (the object domain D) is embedded in a medium forwhich the point source solution can be determined analytically. This point sourcesolution {V G, EG} is also known as the Green’s state. The simplest medium in thiscategory is the unbounded homogeneous medium with conductivity j0. Substitutingqext ¼ d(x) into (1) and (2), we arrive at

V GðxÞ ¼ j¹10 GðxÞ and EGðxÞ ¼ ¹j¹1

0 =GðxÞ; ð3Þ

where G is the scalar Green’s function for the static field, given by

GðxÞ ¼1

4pjxj: ð4Þ

We define Vp and Ep as the primary electric potential field and the primary electricfield, respectively, measured in the background configuration and excited by a smallspherical electrode emitting a DC current I. The primary fields are the fields that wouldbe present in the entire configuration if domain D showed no contrast with theembedding background medium (j(x) ¼ j0). The small spherical electrode can bemodelled as a point source (Lovell 1993). Using qext ¼ Id(x ¹ xs) in (1) and (2), theprimary fields are obtained as

V pðxÞ ¼ j¹10 IGðx ¹ xsÞ and EpðxÞ ¼ ¹j¹1

0 I=Gðx ¹ xsÞ: ð5Þ

Starting from (1) and (2), and using the spatial Fourier transform as the mathematicaltool (De Hoop 1995), we arrive at the integral equation for the scalar electric potentialfield

V ðxÞ ¼ V pðxÞ þ =·�

X0[DGðx ¹ x0Þxðx0Þ=0V ðx0Þdv; x [ D; ð6Þ

where V is the total electric potential field in D, and

xðxÞ ¼jðxÞ ¹ j0

j0: ð7Þ



In (6), =0 denotes spatial differentiation with respect to x0. This is an integral equationfor V and =V. Although these quantities are coupled by a spatial differentiation, weprefer an integral equation in which the same unknown occurs both inside and outsidethe integral. Abubakar (1996) showed that inversion based on the vector integralequation for the electric current density (electric field) leads to substantially betterresults than inversion based on the scalar integral equation for the electric potentialfield. The integral equation for the vector electric field is then given by

EðxÞ ¼ EpðxÞ þ ==·�

X0[DGðx ¹ x0Þxðx0ÞEðx0Þdv; x [ D; ð8Þ

where E is the total electric field in D. Note that in (6) and (8) the target is assumed tobe localized such that the integration is over the object domain D. Outside the objectdomain D the conductivity contrast x is zero, and then the total electric field is exactlythe primary electric field.

In the electrical logging problem we are interested in the secondary electric potentialfield V s in data domain S at xR. The secondary electric potential field is the differencebetween the total electric potential field and the primary electric potential field due tothe presence of the object,

V s ¼ V ¹ V p: ð9Þ

Using (9) in (6), and (1), the secondary electric potential field can be given in anintegral representation form,

V sðxRÞ ¼ ¹=·�

X0[DGðxR ¹ x0Þxðx0ÞEðx0Þdv: ð10Þ

Equation (10) is an integral equation of the second kind, and it relates the conductivitycontrast x to the secondary electric potential field V s (measurement data). In forwardmodelling, the conductivity contrast x is known and the secondary electric potentialfield V s can be calculated after the total electric field E in the object domain D has beenobtained by solving (8). We observe that the integral equation (8) is a singular integralequation, in which the grad-div operator acts on a normalized vector potential, definedas the spatial convolution of the Green’s function and the product of the conductivitycontrast and the total electric field. Numerical implementation of such an integralequation must be carried out carefully. This discretization procedure is discussed inAppendix A.

In the inverse problem to solve for x, the field values V s are known only over a limitedset of space points which lie outside the object domain D, and one must solve for boththe conductivity contrast x and the total electric field E inside D. This inverse problemis not linear.

Forward problem

In Appendix A we present the discretized version of the integral equation for theelectric field given in (8). We discretize the object domain D using a mesh uniformly



subdivided in the x1-, x2- and x3-directions with spacings Dx1, Dx2 and Dx3. From theresults of Appendix A, by substituting (A16) into (A11)–(A13), and using the resultsin (A10), we obtain a linear system of equations for E1, E2 and E3 when theconductivity contrast x is known. This linear system of equations can be writtencompactly in an operator notation. The operator expression L E is directly obtainedfrom the left-hand side of (A10) of Appendix A.

Since the matrix operator consists of spatial convolutions we can use advantageouslyfast Fourier transform (FFT) routines (Zwamborn and Van den Berg 1994). However,we then need an iterative solution, and the conjugate-gradient method seems to be oneof the most efficient methods. With this so-called conjugate-gradient FFT techniquewe are able to solve complex three-dimensional problems efficiently. Furthermore, italso gives the basis of our solution of the inverse problem. We observe that the matrixdescribing this linear system of equations is non-symmetric. Therefore we also needthe adjoint operator in order to set up the conjugate-gradient scheme (Van den Berg1981). The matrix form of this adjoint operator is given in Appendix B. In sucha scheme we also need the definition of the norm and inner product. In view ofthe definition of the discretized field quantities in Appendix A, the norm on D isdefined as

kE k2D ¼ Dx1Dx2Dx3

X3

k¼1

XMm¼1

XNn¼1

XP

p¼1

Ek;m;n;pEk;m;n;p: ð11Þ

With these definitions we are now able to apply a conjugate-gradient iterative schemeto solve the operator equations

ðL EÞk;m;n;p ¼ Epk;m;n;p; k [ f1; 2; 3g; ð12Þ

for m ¼ 1, . . ., M, n ¼ 1, . . ., N, and p ¼ 1, . . ., P. Once the normalized error,

F 12 ¼

kr kDkEpkD

; ð13Þ

where

r ¼ Ep ¹ L E; ð14Þ

is small enough, the approximate solution of E is substituted in (10), to arrive at thesecondary potential field at the receiver points:

V sðxRÞ ¼ Dx1Dx2Dx3

X3

k¼1

XMm¼1

XNn¼1

XP

p¼1

GRk ðxR

1 ¹ x1;m; xR2 ¹ x2;n; x

R3 ¹ x3;pÞxm;n;pEk;m;n;p;

ð15Þ

where

GRk ðxR

1 ¹ x1;m; xR2 ¹ x2;n; x

R3 ¹ x3;pÞ ¼ ¹∂R

k GðxR1 ¹ x1;m; x

R2 ¹ x2;n; x

R3 ¹ x3;pÞ; ð16Þ

in which ∂Rk denotes the spatial differentiations with respect to xR

k .



Inverse problem

We now assume that the inhomogeneous domain D is irradiated successively bya number (i ¼ 1, . . ., I ) of known primary electric fields. For each excitation, theforward scattering problem may be reformulated as the domain integral equation,see (12),

LEði Þ ¼ Ep;ði Þ; on D; ð17Þ

where the operator L depends on the conductivity contrast x. To show this explicitly,we rewrite this equation as

Eði Þ ¹ KDxEði Þ ¼ Ep;ði Þ; on D: ð18Þ

In (18), KDxE(i ) follows from (A11)–(A13) by replacing Ak;m, n, p with A(i)k;m, n, p.

Equation (18) is referred to as the object equation that holds in the object domain D.Furthermore, in the inverse problem the secondary potential field is known or

measured at the measurement points xR. We assume that all the measurement pointsare located in the data domain S, outside D. We can also write (15) in the shorthandnotation,

KS·xEði Þ ¼ V s;ði Þ; on S; ð19Þ

where V s, (i ) follows from (15) after replacing Ek;m, n, p with E(i )k;m, n, p. Equation (19) is

referred to as the data equation that holds in the data domain S.The data equation contains both the unknown total electric field and the unknown

conductivity contrast, but they occur as a product which can be considered as acontrast source that produces the secondary electric potential field at the measurementpoints; hence there is no unique solution to the problem of inverting the data equationby itself. The CSI method attempts to overcome this difficulty by recasting the problemas an optimization, in which we seek not only the contrast sources but also theconductivity contrast itself to minimize a cost functional consisting of two terms, the L2

errors in the data equation and in the object equation, rewritten in terms of theconductivity contrast and the contrast sources rather than the electric fields. Analternative method of solving this optimization problem iteratively is proposed in whichfirst the contrast sources are updated in the conjugate-gradient step, weighted so as tominimize the cost functional, and then the conductivity contrast is updated to minimizethe error in the object equation using the updated sources. This latter minimization canbe carried out analytically which allows an easy implementation of the positivityconstraint for the conductivity.

To this end we introduce the contrast source W(i) as follows:

Wði Þ ¼ xEði Þ: ð20Þ

The data equation becomes

KS·Wði Þ ¼ V s;ði Þ; on S: ð21Þ



Substituting (18) into (20), we obtain an object equation for the contrast source ratherthan for the field, namely,

Wði Þ ¹ xKDWði Þ ¼ xEp;ði Þ; on D: ð22Þ

These latter two equations are the basic equations for the CSI method.The CSI method consists of an algorithm to construct sequences of contrast sources

{W(i )k } and of conductivity contrasts {xk} which iteratively reduce the value of the cost

functional

F ¼SikV s;ði Þ ¹ KS·Wði Þk2S

SikV s;ði Þk2Sþ

SikxEp;ði Þ ¹ Wði Þ þ xKDWði Þk2DSikxEp;ði Þk2D

; ð23Þ

where the norm on S is given by

kV ði Þk2S ¼X∀ XR

V ði ÞðxRÞV ði ÞðxRÞ; ð24Þ

and the norm on D is given by (11). The normalization is chosen so that both terms areequal to one if the contrast source W(i) is zero. The first term measures the error in thedata equation, and the second term measures the error in the object equation. This is aquadratic functional in W(i), but it is highly non-linear in x. Note that the objectequation acts as a regularization for the data equation, and we have not employed anyother regularization techniques.

The algorithm involves the construction of sequences {W(i)k and {xk}, k ¼ 1, 2, . . ., in

the following manner. Define the data error and the object error at the kth step to be

rði Þk ¼ V s;ði Þ ¹ KS·Wði Þ

k and r ði Þk ¼ xkEði Þ

k ¹ Wði Þk ; ð25Þ

where

Eði Þk ¼ Ep;ði Þ þ KDWði Þ

k : ð26Þ

Now suppose W(i)k¹1 and xk¹1 are known; we update W(i) as follows:

Wði Þk ¼ Wði Þ

k¹1 þ aði Þk wði Þ

k ; ð27Þ

where a(i)k is a constant, and w(i)

k is the Polak–Ribiere gradient direction,

wði Þ0 ¼ 0; wði Þ

k ¼ ∂wði Þk þ

h∂wði Þk ; ∂wði Þ

k ¹ ∂wði Þk¹1iD

k∂wði Þk¹1k

2D

wði Þk¹1; k $ 1; ð28Þ

where ∂w(i)k is the gradient (Frechet derivative) of the cost functional F with respect to

W(i) evaluated at W(i)k¹1 and xk¹1. Explicitly, the gradient is found to be

∂wði Þk ¼ ¹nSK¬

Srði Þk¹1 ¹ nD;k¹1ðr

ði Þk¹1 ¹ K¬

Dxk¹1r ði Þk¹1Þ; ð29Þ

where

nS ¼X

i

kV s;ði Þk2S

!¹1

and nD;k¹1 ¼X

i

kxk¹1Ep;ði Þk2D

!¹1

: ð30Þ



The various operators are defined as (omitting the iteration index k¹1)

ðK¬SrðiÞÞk;m;n;p ¼

X∀ XR

GRk ðxR

1 ¹ x1;m; xR2 ¹ x2;n; x

R3 ¹ x3;pÞr

ðiÞðxRÞ; ð31Þ

ðK¬Dxr ðiÞÞk;m;n;p ¼ CðiÞ

k;m;n;p; ð32Þ

where C(i)k;m,n,p is obtained from (B3) by replacing rk;m,n,p in (B4)–(B6) with

xm, n, pr(i)k;m,n,p. The motivation for choosing the Polak–Ribiere gradient direction

rather than the Fletcher–Reeves gradient direction is the presence of the conductivitycontrast x itself in the cost functional in (23) which is also updated during theoptimization process and hence disturbs the orthogonality properties of the gradients.Moreover, the Polak–Ribiere gradient direction is more robust when the differencebetween the successive gradients becomes small.

With the update directions completely specified, the constant a(i)k is determined to

minimize the cost functional

F k ¼ nS

Xi

krðiÞk¹1 ¹ aðiÞ

k KS·wðiÞk k2S þ nD;k¹1

Xi

kr ðiÞk¹1 ¹ aðiÞ

k ðwðiÞk ¹ xk¹1KDwðiÞ

k Þk2D;

ð33Þ

and is found to be

aðiÞk ¼

nShrðiÞk¹1;KS·wðiÞ

k iS þ nD;k¹1hrðiÞk¹1;w

ðiÞk ¹ xk¹1KDwðiÞ

k iD

nSkKS·wðiÞk k2S þ nD;k¹1kwðiÞ

k ¹ xk¹1KDwðiÞk k2D

: ð34Þ

Once the contrast source W(i)k is determined, the field E(i)

k is obtained using (26). Toupdate x we proceed in two steps. First we determine xk to minimize the numerator ofthe second term in (23), which we write as

F D ¼X

i

kr ðiÞk k2D ¼

Xi

kxEðiÞk ¹ WðiÞ

k k2D: ð35Þ

Minimizing (35) with respect to x, we arrive at

xk ¼SiðE

ðiÞk ·WðiÞ

k Þ

Si jEðiÞk j2

: ð36Þ

As noted by Van den Berg and Kleinman (1997), it is not certain whether this choicewill necessarily reduce the error in the second term of (23). We ensure that this choiceis reduced by a line minimization. To this end we write

xk ¼ xk¹1 þ vðxk ¹ xk¹1Þ ¼ xk¹1 þ vdk; ð37Þ

and choose the real parameter v to minimize

F D ¼ nD;k

Xi

kr ðiÞk k2D ¼

SikxkEðiÞk ¹ WðiÞ

k k2DSikxkEp;ðiÞk2D

¼av2 þ bv þ c

Av2 þ Bv þ C; ð38Þ



where

a ¼P

i kdkEðiÞk k2

D; A ¼P

i kdkEp;ðiÞk2D;

b ¼P

ihxk¹1EðiÞk ¹ WðiÞ

k ; dkEðiÞk i2

D; B ¼P

ihxk¹1Ep;ðiÞ; dkEp;ðiÞi2D;

c ¼P

i kxk¹1EðiÞk ¹ WðiÞ

k k2D; C ¼

Pi kxk¹1Ep;ðiÞk2

D:

This is the quotient of two quadratics which, using elementary analysis, may be shownto attain its minimum when

v ¼¹ðaC ¹ AcÞ þ

��ðaC ¹ AcÞ2 ¹ 4ðaB ¹ AbÞðbC ¹ BcÞ

p2ðaB ¹ AbÞ

: ð39Þ

This completes the definitions of the conductivity contrast x if there is no a prioriinformation. If it is known that j is a positive quantity, this information may easily beincorporated by choosing x to be ¹1 if (37) yields a value less than ¹1.

The last point in the description of the algorithm is the choice of the starting valuesW(i)

0 . Observe that we cannot start with W(i)0 ¼ 0 since then x0 ¼ 0 and the cost

functional F in (23) is undefined for k ¼ 1. Therefore we choose as starting values thevalues that minimize only the data error (the first term in (23)),

WðiÞ0 ¼

kK¬SV s;ðiÞk2

D

kKSK¬SV s;ðiÞk2

S

K¬SV s;ðiÞ: ð40Þ

Note that K*SVs,(i) is the back-propagation of the data from the data domain S into the

object domain D, and is often called a back-propagation of the field data. With thisinitial estimate for W(i)

0 , the conductivity contrast estimate x0 is obtained using (26) and(36). This completes the description of the inversion algorithm.

Numerical examples

We demonstrate three examples of inversion of synthetic data. Although we realizethat the ultimate goal is to invert field data, we show the results of our inversionscheme using synthetic data. The synthetic data were simulated by solving theforward scattering problem with the conjugate-gradient method in which all discreteconvolutions are calculated using FFT routines. In order to avoid the possibility of an‘inverse crime’, we also carry out some experiments with noise added to thesesynthetic data. In addition, in our last example, we generate the data by a differenttype of integral equation with a discretization different from that used in the inverseproblem, namely, the scalar electric potential integral equation in (6) using a finerspatial discretization grid, where the mesh size is half the size employed in theinversion scheme. The reconstruction was nearly identical to the reconstruction byinversion of the synthetic data generated by solution of the vector electric fieldintegral equation.



Reconstruction of a homogeneous anomaly

Figure 2 shows the test (object) domain and the set-up of the sources and receivers.We used 42 stations; 21 stations are located in the borehole on the left-hand side ofthe test domain and the other 21 stations are located on the other side. The distancebetween the two boreholes is 100 m. The vertical distance between two stations is 5 mwith each station serving successively as a point source and all other stations acting aspoint receivers. Thus for the reconstruction of the profile we have used 1764 datapoints.

Figure 3a illustrates one of the three-dimensional test model examples we have usedto investigate the performance of the inversion scheme. The model consists of ahomogeneous anomaly of conductivity 0.9 S/m embedded in a 0.3 S/m background.The dimensions of the anomaly are 30 ×30 ×30 m3. In this example we take a testdomain of 28 × 28 ×28 subdomains with a side length of 2.5 m; hence the total numberof conductivity contrast unknowns is equal to 21 952, and the dimensions of the testdomain are 70 × 70 ×70 m3. Note that the test domain is much larger than the actualanomaly. Figure 3 shows, from top to bottom, the conductivity distribution (j) in the(x1, x2)-plane at different x3-levels. The x3-levels, from top to bottom, are ¹27.5,¹12.5, ¹2.5, 2.5, 12.5 and 27.5 m.

Figure 3b is a plot of the conductivity distribution inverted after 512 iterations withthe present inversion method using the synthetic data. Although the total number ofiterations is large, note that we do not solve a full forward problem in each iteration ofthe inversion procedure. The resolution of the reconstructed object is very low, andfurther iterations do not improve the results. After 256 iterations the error in theconductivity has decreased to ERRk ¼ 0.30708. The error in conductivity at the kth



Figure 2. Source–receiver set-up with 42 sources and receivers located in two boreholes.

iteration is given by

ERRk ¼kjk ¹ jexactk2D

kjexactk2D: ð41Þ

The square root of the cost functional F k in (33) has now decreased to 0.94%. Weobserve clearly that although the error with respect to the data is very small, theconductivity contrast is not well reconstructed; this is due to the insensitivity of the datato the changes in the conductivity contrast. Figure 4 shows the conductivity of theoriginal profile and reconstruction after 512 iterations in the (x2, x3)-plane at x1 ¼ 0.

Abubakar and Van den Berg (1998) showed that if we include some a priori



Figure 3. Conductivity distribution (j) in the (x1,x2)-plane for different x3-levels of a three-dimensional homogeneous anomaly. (a) The original profile; (b) the reconstructions after 512iterations.

information about the location of the anomaly, we can improve the reconstructionresults significantly. On the other hand, this a priori information is very often notpresent. If we have a three-dimensional well coverage about the target, it might beexpected that the results would be greatly improved and the need for prior informationgreatly reduced. In order to improve the reconstructed profile, we now use 16 sourcesand 28 receivers located in four boreholes. The boreholes are located at (0, ¹50, x3),(0, 50, x3), (¹50, 0, x3) and (50, 0, x3) as shown in Fig. 5. The vertical positions of thesources in a borehole are x3 ¼ ¹15, ¹5, 5 and 15 m. The vertical positions of thereceivers in a borehole are x3 ¼ ¹30, ¹20, ¹10, 0, 10, 20 and 30 m. Thus for thereconstruction of the profile we have used only 448 data points.

The original profile is shown again in Figs 6a and 7a. The reconstruction after 512iterations is shown in Figs 6b and 7b. We observe that by using fewer sources andreceivers located in four boreholes, better reconstruction results are obtained than



Figure 4. Conductivity distribution (j) in the (x2,x3)-plane at x1 ¼ 0 of a three-dimensionalhomogeneous anomaly. (a) The original profile; (b) the reconstructions after 512 iterations.

those shown in Figs 3b and 4b. The position of the anomaly is now accuratelydetermined, but the conductivity value of the anomaly is still lower than the originalone. The error in the conductivity has decreased to ERRk ¼ 0.28976 and the squareroot of the cost functional F k has now decreased to 1.07%. The computation time forone iteration in the inversion scheme of this example was 210 seconds on a personalcomputer with a 400 MHz Pentium processor.

Reconstruction of a vertically layered configuration

As the second example we consider a more complicated example. The measurementset-up used is shown in Fig. 5. Thus for the reconstruction of the profile we use 448data points. The configuration which is to be reconstructed is embedded in ahomogeneous background medium of conductivity 0.3 S/m. A 35 × 35 ×35 m3 testdomain is used and discretized into 14 ×14 × 14 subdomains. Thus each subdomainagain has the dimensions 2.5 ×2.5 × 2.5 m3, and the total number of conductivitycontrast unknowns is 2744.

The original profile which is used to produce the synthetic data is shown in Fig. 8a.This figure shows, from top to bottom, the original profile in the (x1, x2)-plane atdifferent x3-levels. The x3-levels from top to bottom are ¹13.75, ¹11.25, ¹1.25, 1.25,



Figure 5. Source–receiver set-up with 16 sources and 28 receivers located at four boreholesaround the three-dimensional test domain.

11.25 and 13.75 m. The model consists of three vertical layers. Each layer has thesame dimensions of 30 ×10 × 30 m3. The conductivity of the ‘dark blue’ vertical layer is0.1 S/m, that of the ‘blue’ vertical layer is 0.3 S/m, and that of the ‘red’ vertical layer is0.9 S/m. Thus, the conductivity contrasts x from left to right in Fig. 8a are ¹0.667, 0and 2.

Figure 8b shows the conductivity distribution that was obtained after 256 iterations.We observe that the reconstruction of the middle (blue) vertical layer is not as good asthe other two vertical layers. In Fig. 9 we show the conductivity profile of the originalconfiguration and reconstructions after 256 iterations in the (x2, x3)-plane at x1 ¼ 0. Wenote that after 256 iterations, the error in the conductivity has decreased toERRk ¼ 0.22450, the square root of the cost functional F k has decreased to 0.78%,and further iterations do not improve the reconstruction results. The computation time



Figure 6. As Fig. 3, but using the measurement set-up shown in Fig. 5.

for one iteration in the inversion scheme of this example was 10 seconds on a personalcomputer with a 400 MHz Pentium processor.

Reconstruction of a horizontally layered configuration

We have considered two examples that are quite far removed from actual cases ofpractical cross-well electrical logging problems. Thus, as the next example, we considera more realistic configuration. The configuration consists of three different layers in thevertical direction (x3-direction) with dimensions of 30 ×30 × 10 m3. The conductivityof each layer varies from 0.9 S/m (red) to 0.3 S/m (blue) and 0.1 S/m (dark blue). Theinversion is performed by discretizing a test domain of 35 ×35 × 35 m3 into a grid of14 ×14 × 14 subdomains, and hence the total number of conductivity contrastunknowns amounts to 2744. The source and receiver set-up is the same as before.



Figure 7. As Fig. 4, but using the measurement set-up shown in Fig. 5.

We use 16 sources and 28 receivers located in four boreholes. The boreholes arelocated at (0, ¹50, x3), (0, 50, x3), (¹50, 0, x3) and (50, 0, x3) as shown in Fig. 5. Thevertical positions of the sources in a borehole are x3 ¼ ¹15, ¹5, 5 and 15 m. Thevertical positions of the receivers in a borehole are x3 ¼ ¹30, ¹20, ¹10, 0, 10, 20 and30 m. Thus for the reconstruction of the profile we use 448 data points. Theconductivity of the background medium is 0.3 S/m.

The original profile which is used to produce the synthetic data is shown inFig. 10a. This figure shows, from top to bottom, the conductivity distribution inthe (x1, x2)-plane at different x3-levels. The x3-levels from top to bottom are ¹13.75,¹11.25, ¹1.25, 1.25, 11.25 and 13.75 m. Figure 10b shows the reconstructionafter 256 iterations. We observe that after 256 iterations we have already obtained a



Figure 8. Conductivity distribution (j) in the (x1,x2)-plane for different x3-levels of a verticallylayered configuration. (a) The original profile; (b) the reconstructions after 256 iterations.

good reconstruction result. We note that after 256 iterations, the error in theconductivity has decreased to ERRk ¼ 0.20412, the square root of the cost functionalF k has decreased to 0.53%, and further iterations do not yield substantialimprovement of the reconstruction results. In order to investigate the verticalresolution, we show in Fig. 11 the conductivity of the original profile and thereconstructed profile after 256 iterations in the (x2, x3)-plane at x1 ¼ 0 m. Weobserve that the resolution of the reconstruction of the middle layer is lower than theothers.

So far, the results presented are for noiseless synthetic data. In order to simulate amore realistic field experiment we have added 5%, 10% and 20% random noise to thesynthetic data. In our analysis we have taken the secondary electric potential fieldV s,(i)(xR) as the data quantities (see (15)); then the noisy secondary electric potential



–10 0 10

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(b)

0.9

0.9

Conductivity (S/m)

Conductivity (S/m)

x3

(m)

x3

(m)

x2 (m)

Figure 9. Conductivity distribution (j) in the (x2,x3)-plane at x1 ¼ 0 of a vertically layeredconfiguration. (a) The original profile; (b) the reconstructions after 256 iterations.

field is given by

V s;ðiÞnoiseðx

RÞ ¼ ½1 þ bfnoiseÿVs;ðiÞðxRÞ; ð42Þ

where fnoise is a random number between ¹1 and 1 and b¼ 0.05, 0.1 or 0.2.The 5% and 10% noise do not alter the reconstruction results significantly. The

results of the reconstruction after 256 iterations with noisy data (20% noise) are shownin Figs 12b and 13b. By comparing Figs 10 and 11 with Figs 12 and 13, we observe thatthe 20% noise only altered the region with positive contrast (the red layers). Wenote that after 256 iterations, the error in the conductivity of the reconstruction



Figure 10. Conductivity distribution (j) in the (x1,x2)-plane for different x3-levels of ahorizontally layered configuration. (a) The original profile; (b) the reconstructions after 256iterations.

with noisy data has decreased to ERRk ¼ 0.20950, the square root of the cost functionalF k has decreased to 10.08%, and further iterations do not improve the reconstructionresults.

Conclusions

We have demonstrated that the present inversion algorithm can reconstruct a three-dimensional electrode logging problem over a wide range of conductivity contrastsusing moderate computer power. The main advantage of the present non-linearinversion method is that we do not have to solve a full forward problem in eachiteration, and we do not need an artificial regularization technique to deal with the non-uniqueness problem of the inversion of data. Without using any a priori information



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0.9

(a)

(b)

x3

(m)

x3

(m)

x2 (m)

Conductivity (S/m)

Conductivity (S/m)

Figure 11. Conductivity distribution (j) in the (x2,x3)-plane at x1 ¼ 0 of a horizontally layeredconfiguration. (a) The original profile; (b) the reconstructions after 256 iterations.

(except for the positivity of the conductivity), the present algorithm can reconstruct theunknown conductivity contrast up to an acceptable level of accuracy. Because of theinsensitivity of the data with respect to changes in the conductivity contrast, the use of athree-dimensional well coverage about the target is crucial. Furthermore, the numericaltests indicate that this inversion algorithm using synthetic data with 20% noise still givesreasonably good reconstruction results. We expect that the present inversion scheme iscapable of handling a realistic three-dimensional inverse problem.

Future research will be directed at using the present inversion method to reconstructthe conductivity distribution from the measurements in a single-borehole configura-tion. In addition, we also plan to test our inversion method for the non-static case,namely, the induction logging problem.



Figure 12. As Fig. 10, but with 20% noise.

Appendix A

Discretization procedure

Before discussing the discretization procedure of the integral equation for the electricfield, we first write (8) as

Ek ¹ Bk ¼ Epk ; xk [ D; k [ f1; 2; 3g; ðA1Þ

where the vector Bk is given by

Bk ¼ ∂k½∂1A1 þ ∂2A2 þ ∂3A3ÿ; ðA2Þ

in which the normalized vector potential Ak is given by

Akðx1; x2; x3Þ ¼

�X0[D

Gðx1 ¹ x01; x2 ¹ x0

2; x3 ¹ x03Þxðx0

1; x02; x

03ÞEkðx

01; x

02; x

03Þdv; ðA3Þ



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0.9 Conductivity (S/m)

Conductivity (S/m)

x3

(m)

x3

(m)

(a)

(b)

x2 (m)

Figure 13. As Fig. 11, but with 20% noise.

and the Green’s function is given by

Gðx1; x2; x3Þ ¼1

4pðx21 þ x2

2 þ x23Þ

12: ðA4Þ

We assume that the domain D as shown in Fig. 1 is a rectangular domain withboundaries along the x1-, x2- and x3-directions. We discretize the domain into arectangular mesh. The mesh is uniformly spaced in the x1-, x2- and x3-directions. Therectangular subdomains with a width of Dx1 in the x1-direction, Dx2 in the x2-direction,and Dx3 in the x3-direction1

2 are given by

Dm;n;p ¼ ðx1; x2; x3Þ [ R3kx1;m ¹ 12 Dx1 < x1 < x1;m þ 1

2 Dx1;�

x2;n ¹ 12 Dx2 < x2 < x2;n þ 1

2 Dx2; x3;p ¹ 12 Dx3 < x3 < x3;p þ 1

2 Dx3; ðA5Þ

for m ¼ 1, . . ., M, n ¼ 1, . . ., N, p ¼ 1, . . ., P, where

x1;m ¼ x1;12

þ ðm ¹ 12ÞDx1; m ¼ 1; . . . ;M;

x2;n ¼ x2;12

þ ðn ¹ 12ÞDx2; n ¼ 1; . . . ;N ;

x3;p ¼ x3;12

þ ðp ¹ 12ÞDx3; p ¼ 1; . . . ;P;

ðA6Þ

in which x1;1/2 is the lower x1-bound of the contrasting domain D, x2;1/2 is its lower x2-bound, and x3;1/2 is its lower x3-bound. In each subdomain Dm,n,p with centre points(x1;m, x2;n, x3;p), we assume the conductivity contrast x to be constant, with the samevalue as that at the centre point, xm, n, p ¼ x(x1;m, x2;n, x3;p). We assume that theboundary of the object domain D lies completely outside the contrasting object(target). Using this spatial discretization grid, the pertaining field quantities are definedas follows:

Ek;m;n;p ¼ Ekðx1;m; x2;n; x3;pÞ; ðA7Þ

Epk;m;n;p ¼ Ep

k ðx1;m; x2;n; x3;pÞ; ðA8Þ

Bk;m;n;p ¼ Bkðx1;m; x2;n; x3;pÞ; ðA9Þ

for k [ {1, 2, 3}. Then (A1) is discretized as

Ek;m;n;p ¹ Bk;m;n;p ¼ Epk;m;n;p; ðA10Þ

for m ¼ 1, . . ., M, n ¼ 1, . . ., N, and p ¼ 1, . . ., P, where Bk, given in (A2), is computed withthe finite-difference rule (Abramowitz and Stegun 1970), and the results are given as

B1;m;n;p ¼ ðDx1Þ¹2ðA1;m¹1;n;p ¹ 2A1;m;n;p þ A1;mþ1;n;pÞ

þ 14 ðDx1Dx2Þ

¹1ðA2;m¹1;n¹1;p ¹ A2;m¹1;nþ1;p ¹ A2;mþ1;n¹1;p þ A2;mþ1;nþ1;pÞ

þ 14 ðDx1Dx3Þ

¹1ðA3;m¹1;n;p¹1 ¹ A3;m¹1;n;pþ1 ¹ A3;mþ1;n;p¹1 þ A3;mþ1;n;pþ1Þ;

ðA11Þ



B2;m;n;p ¼ ðDx2Þ¹2ðA2;m;n¹1;p ¹ 2A2;m;n;p þ A2;m;nþ1;pÞ

þ 14 ðDx1Dx2Þ

¹1ðA1;m¹1;n¹1;p ¹ A1;m¹1;nþ1;p ¹ A1;mþ1;n¹1;p þ A1;mþ1;nþ1;pÞ

þ 14 ðDx2Dx3Þ

¹1ðA3;m;n¹1;p¹1 ¹ A3;m;n¹1;pþ1 ¹ A3;m;nþ1;p¹1 þ A3;m;nþ1;pþ1Þ;

ðA12Þ

B3;m;n;p ¼ðDx3Þ¹2ðA3;m;n;p¹1 ¹ 2A3;m;n;p þ A3;m;n;pþ1Þ

þ 14 ðDx1Dx3Þ

¹1ðA1;m¹1;n;p¹1 ¹ A1;m¹1;n;pþ1 ¹ A1;mþ1;n;p¹1 þ A1;mþ1;n;pþ1Þ

þ 14 ðDx2Dx3Þ

¹1ðA2;m;n¹1;p¹1 ¹ A2;m;n¹1;pþ1 ¹ A2;m;nþ1;p¹1 þ A2;m;nþ1;pþ1Þ:

ðA13Þ

Next we have to replace the continuous representation of the normalized vectorpotential Ak by a discrete one. In order to handle the singularity of the Green’s function,we first take the spherical mean of the normalized vector potential. We integrate Ak overa spherical domain with centre at the point (x1;m, x2;n, x3;p). The radius of these patchesis taken to be 1

2Dx ¼ 12 min(Dx1, Dx2, Dx3). The results are divided by the volume of the

spherical domain of radius 12Dx. We may then write

Ak;m;n;p ¼ Akðx1;m; x2;n; x3;pÞ

¼

�X0[D

Gðx1;m ¹ x01; x2;n ¹ x0

2; x3;p ¹ x03Þxðx0

1; x02; x

03ÞEkðx

01; x

02; x

03Þdv; ðA14Þ

where we have interchanged the order of integrations, such that

Gðx1; x2; x3Þ ¼6

pðDxÞ3

�jX0j<1

2DxGðx1 þ x0

1; x2 þ x02; x3 þ x0

3Þdv;

¼

34pDx

jxj ¼ 0;

1

4pðx21 þ x2

2 þ x23Þ

12

jxj > 12 Dx:

8>>><>>>: ðA15Þ

In fact, G(x1, x2, x3) is the mean value of the Green’s function over a spherical domainwith centre at (x1, x2, x3). Note that this weakening of the singularity is different fromthe technique used by Van Bladel (1991), where the spatial differentiations act on theGreen’s function directly, while we first compute the normalized vector potential Ak,in which the Green’s function has been weakened by taking its spherical mean, andsubsequently the differentiations are carried out numerically on the normalizedvector potential Ak. This has proved (Zwamborn and Van den Berg 1994) to yield anefficient, stable and accurate algorithm. After this weakening procedure, we are nowable to compute the integral of (A14) numerically. In view of the functionalproperties of Ek, we approximate the integral of (A14) using a midpoint rule. We then



arrive at

Ak;m;n;p ¼ Dx1Dx2Dx3

XMm0¼1

XNn0¼1

XP

p0¼1

Gm¹m 0 ;n¹n 0 ;p¹p 0xm 0;n 0 ;p 0Ek;m 0 ;n 0 ;p 0 ; ðA16Þ

for m ¼ 0, . . ., Mþ1, n ¼ 0, . . ., Nþ1, and p ¼ 0, . . ., Pþ1, where

Gm¹m 0 ;n¹n 0 ;p¹p 0 ¼ Gðx1;m ¹ x1;m 0 ; x2;n ¹ x2;n 0 ; x3;p ¹ x3;p 0 Þ: ðA17Þ

Note that Ak;m, n, p are discrete convolutions in m0, n0, and p0 and can be computedefficiently by FFT routines (Press et al. 1992).

Appendix B

Adjoint operator

The adjoint operator is defined through the relationship

hr ;L EiD ¼ hL ¬r ;EiD; ðB1Þ

where r and E are both in the same vector space (see (14)). Substituting the expressionof the operator L E in the left-hand side of (B1) and interchanging the varioussummations, the adjoint operator is recognized as

ðL ¬r Þk;m;n;p ¼ rk;m;n;p ¹ xm;n;pCk;m;n;p; ðB2Þ

for m ¼ 1, . . ., M, n ¼ 1, . . ., N, and p ¼ 1, . . ., P, in which

Ck;m;n;p ¼ Dx1Dx2Dx3

XMþ1

m 0¼0

XNþ1

n 0¼0

XPþ1

p 0¼0

Gm 0¹m;n 0¹n;p 0¹pFk;m 0;n 0;p 0 ; ðB3Þ

where

F1;m;n;p ¼ ðDx1Þ¹2ðr1;mþ1;n;p ¹ 2r1;m;n;p þ r1;m¹1;n;pÞ

þ 14 ðDx1Dx2Þ

¹1ðr2;mþ1;nþ1;p ¹ r2;mþ1;n¹1;p ¹ r2;m¹1;nþ1;p þ r2;m¹1;n¹1;pÞ

þ 14 ðDx1Dx3Þ

¹1ðr3;mþ1;n;pþ1 ¹ r3;mþ1;n;p¹1 ¹ r3;m¹1;n;pþ1 þ r3;m¹1;n;p¹1Þ; ðB4Þ

F2;m;n;p ¼ ðDx2Þ¹2ðr2;m;nþ1;p ¹ 2r2;m;n;p þ r2;m;n¹1;pÞ

þ 14 ðDx1Dx2Þ

¹1ðr1;mþ1;nþ1;p ¹ r1;mþ1;n¹1;p ¹ r1;m¹1;nþ1;p þ r1;m¹1;n¹1;pÞ

þ 14 ðDx2Dx3Þ

¹1ðr3;m;nþ1;pþ1 ¹ r3;m;nþ1;p¹1 ¹ r3;m;n¹1;pþ1 þ r3;m;n¹1;p¹1Þ; ðB5Þ

F3;m;n;p ¼ ðDx3Þ¹2ðr3;m;n;pþ1 ¹ 2r3;m;n;p þ r3;m;n;p¹1Þ

þ 14 ðDx1Dx3Þ

¹1ðr1;mþ1;n;pþ1 ¹ r1;mþ1;n;p¹1 ¹ r1;m¹1;n;pþ1 þ r1;m¹1;n¹1;p¹1Þ

þ 14 ðDx2Dx3Þ

¹1ðr2;m;nþ1;pþ1 ¹ r2;m;nþ1;p¹1 ¹ r2;m;n¹1;pþ1 þ r2;m;n¹1;p¹1Þ: ðB6Þ



Since according to (B3), m0 runs from 0 to Mþ1, n0 runs from 0 to Nþ1 and p0 runsfrom 0 to Pþ1, we set in (B4)–(B6)

rk;m;n;p ¼ 0; m ¼ ¹1; 0;M þ 1;M þ 2; ∀ n∀ p;

rk;m;n;p ¼ 0; n ¼ ¹1; 0;N þ 1;N þ 2; ∀ m∀ p;

rk;m;n;p ¼ 0; p ¼ ¹1; 0;P þ 1;P þ 2; ∀ m∀ n:

ðB7Þ

Note that Ck;m, n, p is discrete convolution in m0, n0, and p0 and these convolutions canalso be computed efficiently by FFT routines (Press et al. 1992).

References

Abramowitz M. and Stegun I.A. 1970. Handbook of Mathematical Functions, p. 884. DoverPublications Inc.

Abubakar A. 1996. Non-linear inversion in electrode logging using the modified gradient method.MSc thesis, Delft University of Technology, The Netherlands.

Abubakar A. and Van den Berg P.M. 1998. Three-dimensional non-linear inversion in cross-wellelectrode logging. Radio Science 33, 989–1004.

Alumbaugh D.L. and Morrison H.F. 1995. Theoretical and practical considerations for crosswellelectromagnetic tomography assuming a cylindrical geometry. Geophysics 60, 846–870.

Chew W.C. and Wang Y.M. 1990. Reconstruction of two-dimensional permittivity distributionusing the distorted Born iterative method. IEEE Transactions on Medical Imaging 9, 218–225.

Daily W. and Owen E. 1991. Cross-borehole resistivity tomography. Geophysics 56, 1228–1235.De Hoop A.T. 1995. Handbook of Radiation and Scattering of Waves, pp. 719–729. Academic

Press Inc.Lovell J.R. 1993. Finite element methods in resistivity logging, pp. 47–48. PhD thesis, Delft

University of Technology, The Netherlands.Newman G. 1995. Crosswell electromagnetic inversion using integral and differential equations.

Geophysics 60, 899–911.Press W.H., Teukolsky S.A, Vetterling W.T. and Flannery B.P. 1992. Numerical Recipes in Fortran:

The Art of Scientific Computing, pp. 490–525. Cambridge University Press.Torres-Verdin C. and Habashy T.M. 1994. Rapid 2.5-dimensional forward modeling and

inversion via a new non-linear scattering approximation. Radio Science 29, 1051–1079.Torres-Verdin C. and Habashy T.M. 1995. A two-step linear inversion of two-dimensional

electrical conductivity. IEEE Transactions on Antennas and Propagation 43, 405–415.Van Bladel J. 1991. Singular Electromagnetic Fields and Sources. Clarendon Press, Oxford.Van den Berg P.M. 1981. Iterative computational techniques in scattering based upon the

integrated square error criterion. IEEE Transactions on Antennas and Propagation 32, 1063–1071.

Van den Berg P.M. and Kleinman R.E. 1997. Contrast source inversion method. Inverse Problems13, 1607–1620.

Zwamborn A.P.M. and Van den Berg P.M. 1994. Computation of electromagnetic fields insidestrongly inhomogeneous objects by the weak-conjugate gradient FFT method. Journal of theOptical Society 11, 1414–1421.



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