Geophysical Journal InternationalGeophys. J. Int. (2013) 194, 158–169 doi: 10.1093/gji/ggt096Advance Access publication 2013 April 5

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Approximate three-dimensional resistivity modelling using Fourieranalysis of layer resistivity in shallow soil studies

Solene Buvat,1 Cyril Schamper2 and Alain Tabbagh1

1UMR 7619, Sisyphe, UPMC/CNRS, Paris, France. E-mail: solene.buvat@upmc.fr2Department of Geoscience, Aarhus University, Aarhus, Denmark

Accepted 2013 March 7. Received 2013 March 7; in original form 2012 August 8

S U M M A R YThe approximate forward modelling method using Fourier analysis has been used in 2-Dapplications for several decades. It involves decomposition of the terrain parameters, eitherthe resistivity or the layer thickness, into a Fourier series expansion to simplify the problem tothat of a 1-D situation. In this study, the Fourier analysis is applied to 3-D forward modellingfor the purposes of shallow DC resistivity imaging with pole–pole array. Our work is to assessadvantages and drawbacks of the simplified approach by comparing to exact 3-D solutions,method of moments (MoM) and surface integrals and to the Born approximation appliedto MoM. While the Fourier analysis method offers very short calculation times, it shows asignificant, albeit systematic, reduction of the anomaly amplitudes; and its ability to delineateanomaly sources is lower than the other methods. Nevertheless, its rapidity makes it aninteresting first approach in the modelling of DC resistivity results.

Key words: Fourier analysis; Numerical approximations and analysis; Electrical properties.

1 I N T RO D U C T I O N

Electrical resistivity has been shown to be a highly relevant parame-ter for describing underground structure, since it exhibits the widestrange of variations and greatest sensitivity to clay content, watercontent and water mineralization. This fact is particularly advan-tageous in near-surface geophysics where resistivity measurementscan be rapid and inexpensive, consequently both horizontal and ver-tical variations can be measured by a series of electrical multipolearray systems and electromagnetic (EM) multicoil systems (e.g.Samouelian et al. 2004; Ogilvy et al. 2009; Siemon et al. 2009;Dabas et al. 2012). Data interpretation processes, however, mustkeep up with these developments, to ensure that suitable fast 3-Dmodelling techniques can be associated with the acquisition of fielddata. Today the application of 3-D numerical inversion proceduresremains a very time-consuming task (Rucker et al. 2009) even withmassive parallelization (Johnson et al. 2010).

Several strategies can be considered to confront the difficultiesassociated with 3-D inversion. The first one is to reduce the prob-lem to a series of 1-D inversion. Simplifying the problem in sucha way has been applied for the past 20 yr and numerous applica-tions have shown that it is often possible and advisable to makeuse of different types of simple (Meheni et al. 1996; Martinelli& Duplaa 2008; Riss et al. 2011), or sophisticated (Auken &Christiansen 2004; Auken et al. 2005; Viezzoli et al. 2008) 1-Dinterpretations. The reasons for justifying the simplification aretwofold: first, although the density of measured points can be highalong any given profile, it is often too weak between the profiles to

constrain 3-D inversion processes; and, secondly, resistivity changesare far stronger and occur over shorter spatial scales in the verti-cal direction than in the horizontal direction. In other words, mostnaturally occurring media have longer wavelengths in the lateraldirection compared to the vertical direction, and can be sufficientlywell defined by 1-D inversion, whether or not they are laterallyconstrained.

A second strategy (Brinon et al. 2012) consists of apply-ing a two-step process where, after 1-D inversion, a 3-D inver-sion is applied over limited areas where the 1-D model is in-sufficient to properly describe the subsurface. This is adequate,for instance, in archaeological and engineering studies whereone looks for man-made features corresponding to major lateralchanges, and also in soil studies for the detection of cryoturba-tion structures. In such inversions, a limited number of parame-ters are sufficient to describe the geometric pattern and resistivitycontrast.

A third option consists of using approximate forward modellingtechniques where the calculation is fast. While apparently varied,most of them correspond to linear approaches (Li & Oldenburg1992; Møller et al. 2001). Some of them may allow avoiding nu-merous repetitions in the inversion process through direct modelparameter calculations, and all can also be used as a first step toinitiate the interpretation process. However, in order to be able toassess the advantages and drawbacks associated with their use, it isimperative to know the importance of the approximation in termsof magnitude and location of the anomalies, in other words, to de-termine if the error (the risk) is acceptable or not. Consequently, the

158 C© The Authors 2013. Published by Oxford University Press on behalf of The Royal Astronomical Society.

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3-D Fourier electrical approximate model 159

Figure 1. Apparent resistivity maps using a 1 m Pole–Pole array for: (a) a conductive body of resistivity 50 �m, in 200 �m ground, with dimensions 4 × 2 ×1 m3 (white rectangle), centred at a depth of 0.8 m, the A (injection) and M (measurement) poles shown are parallel to the x-axis, (b) the same feature as (a)but with exchange of A and M position, (c) a 200-�m resistive body in 50 �m ground, with the same 4 × 2 × 1 m3 geometry.

Table 1. Dependence of the anomaly extrema and widths on the number of terms in the Fourier series expansionwhen a conductive, 50 �m, 2-m wide body is embedded in a resistive 200 �m medium and when a resistive,200 �m, 2-m wide body is embedded in a conductive 50 �m medium.

Maximum rank of the expansion L = K = 3 L = K = 5 L = K = 8 L = K = 11 L = K = 14

Conductive anomaly extremum (�m) 185.0 178.9 176.5 175.9 176.1Conductive FWHM (m) 5.6 3.7 3.0 3.0 3.0Resistive anomaly extremum (�m) 54.1 55.7 56.3 56.5 56.4Resistive FWHM (m) 5.6 3.6 3.0 3.0 3.0

first step of this option is to compare through a theoretical modellingstudy the results obtained with the considered approximate methodand (at least) one exact method.

The aim of this study is to present one of these approximatemethods and to define its limitations by comparison with two

independent exact forward calculations. The Born approximationis also considered to illustrate the result variability that is intro-duced by different approximations.

The method consists in using Fourier series expansion to describelateral variations of one or several parameters so that, in the spatial

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160 S. Buvat, C. Schamper and A. Tabbagh

Figure 2. Apparent resistivity maps calculated for the conductive elongated 4 × 2 × 1 m3 body (top) and the resistive body (bottom), using the MoM method(left) and the surface integral method (right).

frequency domain, the problem is reduced to a 1-D one and can beanalytically solved.

2 P R I N C I P L E O F F O U R I E R A NA LY S I SF O R A P P ROX I M AT E 3 - D M O D E L L I N G

The concept of modelling with Fourier analysis has been knownin EMs for modelling profiles in magnetotellurics (Mann 1964;Hughes & Wait 1975), and for the study of high-frequency sur-face roughness scattering (Sassi & Tabbagh 1986). The basic con-cepts of these analyses can be described by: (1) lateral variationsin the depth of an interface between two layers, or the electricalproperties of the layers, are expanded in a Fourier series in x, (2)the EM field components are also expanded in a series in x, (3)

each term in this expansion is calculated using a 1-D analyticalinversion in z and (4) the resulting EM field is reconstructed bysumming the successive terms of the series. The past applications,however, have been limited to 2-D structures and to uniform, pri-mary EM fields. Recently, the same method has been applied to 2-Delectrical resistivity tomography inversions (Gyulai et al. 2010).However, for the interpretation of either DC resistivity or electro-magnetic induction conductivity maps (Gebbers et al. 2009), the3-D structure of the ground must be considered, meaning that a 2-D(x and y) Fourier expansion, and (a) moving point source(s) must beused.

If the study is limited to DC electrical resistivity measurements,a point source A, located at the surface of a layered terrain, isconsidered. Within one layer, labelled i, there are lateral vari-ations in resistivity, δρi (x, y), or conductivity, δσ i(x, y). These

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3-D Fourier electrical approximate model 161

variations can be expanded into a Fourier series using the generalformula:

δρi (x, y) = a0 +∞∑

l=1

∞∑k=1

(ak cos

2πkx

XT+ bk sin

2πkx

XT

)

×(

cl cos2πly

Y T+ dl sin

2πly

Y T

), (1)

where XT and YT are the lateral extents, in the x and y directions, ofthe surface under consideration. As an example, the calculations forthe different coefficients of a crenel function in x and y are presentedin Appendix A.

The electrical potential generated on the surface of the layeredterrain, by the injection of a current I into A located at the coordi-nates (xA, yA) in the first layer of ρ1 resistivity, can be separated intotwo components: a primary potential,

Vp(x, y) = ρ1 I

2π

1√(x − xA)2 + (y − yA)2

, (2)

and a secondary potential Vs that verifies

∇2Vs = 0. (3)

If Vs is developed in Fourier series expansion to solve eq. (3)and if the origin of the coordinate system is located at the injectionpoint, the expression for this secondary potential is:

Vs(x, y) = ρ1 I

2π

4

XT .Y T

×∞∑

l=1

∞∑k=1

4π

λ

C(k, l)

e2λe1 − C(k, l)cos

2πkx

XTcos

2πly

Y T. (4)

In this expression: λ = 2π

√k2

X T 2 + l2

Y T 2 , and C(k, l ) is the re-cursively calculated contrast coefficient. The details of this calcu-lation and the definition of the contrast coefficient are given inAppendix B.

For each pair of spatial frequencies (k, l), λ and C are recalcu-lated. In the general formula (1), for each frequency pair (k, l), fourdifferent corresponding values of C(k, l) can be found. However,due to the choice of origin, only the product ak.cl corresponds to thespatial variations of the secondary potential.

The entire calculation process must be repeated for each pointsource, to ensure that the apparent resistivity is determined for anytype of array when it is displaced over the surface of the survey area.Despite its apparent complexity, this forward modelling process isfast and involves analytical calculations only.

3 R E S U LT S

For the purposes of simplicity, we consider a pole–pole array todemonstrate the concept of forward modelling with Fourier analysis.If desired, the models for all other arrays can be reconstructed bytaking the sum of several pole–pole array responses (e.g. Rucker2012). In our set-up, the two moving electrodes (A for the injection,M for the potential) have a fix spacing a = 1 m and the apparentresistivity, calculated by ρa = 2πa VM

I , is affected to their midpoint.All the figures demonstrating the modelling concept are drawn usinga 0.2 × 0.2 m2 mesh, that is with 2601 values of the apparentresistivity.

In Fig. 1 the theoretical apparent resistivity maps are presentedfor two cases: a resistive, 200 �m, elongated body in conductive,50 �m, homogeneous ground; and a conductive, 50 �m, elongatedbody in resistive, 200 �m ground. The bodies have a thickness

of 1 m, a width of 2 m, a length of 4 m and are positioned withtheir centre at a depth of 0.8 m. In both cases the anomalies areeasy to discern from background and follow the shape of the fea-tures with a FWHM (full width half maximum) corresponding to3.0 m. However, it can be seen (Figs 1a and b) that there is a slightdifference in the position of the anomaly, between the case in whichthe injection pole A is placed to the left and the measurement poleM is placed to the right along the x-axis, and the case in whichthe injection pole is placed to the right. This means that this cal-culation does not strictly verify the reciprocity principle, accordingto which the injection and measurement poles can be interchangedand which is always respected by exact methods. Tests were runto understand the source of the reciprocity problem, including thenumber of terms used in the Fourier analysis. The results shown inFig. 1 are calculated until the eighth term of the series expansion in xand y. This limit was adopted after considering the influence of eachsuccessive term on the sum. Beyond the eighth term the apparentresistivity oscillates slightly but does not change significantly. Thedependence of the magnitude of the anomaly and of the FWHM onthe number of terms is given in Table 1. The table shows that evenfor numbers as small as eight, the results are satisfactory, and con-firms the outcome observed in a previous 2-D study (Gyulai et al.2010). In addition, with only eight terms a significant advantagecan be expected in terms of calculation speed. The CPU time forthe calculation, including the output of the 2601 apparent resistivityvalues, was 0.0624 s (using a desk computer with an Intel Core(TM) i5 processor, at 2.5 GHz frequency with 4 GB RAM under64 bits Windows 7).

The results obtained when the surrounding medium is a layeredground are in agreement with those described above. For example,if instead of a single homogeneous layer with ρ = 50 �m, theground is assumed to have three layers with ρ1 = 70 �m, e1 =0.25 m, ρ2 = 50 �m, e2 = 3 m and ρ3 = 100 �m, the anomalygenerated by a resistive (200 �m) body is found to have a very sim-ilar 2.8 m FWHM with only a slight increase in apparent resistivitymaximum resulting from the general increase in apparent resistivityinduced by ρ1. Considering three layers increases the CPU time up to0.11 s.

Although all of these results are satisfying, since they are achievedwith short computation times and provide plausible apparent resis-tivity variations, it is difficult to evaluate the errors resulting fromthe underlying approximations involved in this method: the spatialfrequencies are assumed to be decoupled, and the total secondarypotential is obtained by summing the components calculated at eachfrequency. To make an initial assessment of the use of the Fouriermethod, before comparing to the values given by the exact cal-culations, a simple 1-D interpretation was made of the precedingsynthetic results. At the extrema (i.e. above the centre of the 3-Dbody), this led to a value of 156 �m instead of 50 �m for theconductive body in resistive (200 �m) ground, and 63 �m insteadof 200 �m for the resistive body placed in conductive (50 �m)ground. This suggests that in the case of the Fourier approxima-tions, the computed magnitudes of the anomaly are significantlysmaller than the correct values.

4 A P P ROX I M AT I O N A S S E S S M E N T

4.1 Comparison with two independent exact methods

Two independent 3-D numerical modelling methods were used toassess the results of the Fourier series expansion one. The first

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162 S. Buvat, C. Schamper and A. Tabbagh

Figure 3. Apparent resistivity profile in y direction at the centre of the 2-m wide bodies, (a) for conductive bodies, (b) for resistive bodies. The curves calculatedusing the MoM (red line) and Surface integral (blue line) are nearly identical, whereas those calculated with the Fourier series expansion (black line) indicatesmaller magnitudes in apparent resistivity changes.

of these is the method of moments (MoM), in which the anoma-lous body is replaced by a set of secondary sources distributedthroughout its volume. Originally proposed in EM (Raiche 1974),this technique has since been extended to DC prospection (Dabaset al. 1994). With the second method, called surface integral method(Alfano 1959; Spahos 1979; Li & Oldenburg 1991; Boulanger &Chouteau 2005), electrical charges are located on the surface of thebody(ies) where exist resistivity changes. As can be seen in Figs 2and 3, both methods are in perfect agreement with respect to theapparent dimensions of the anomaly and the reciprocity principleis strictly verified. The MoM method took 21.7 s for calculatingthe apparent resistivity map and the surface integral method took94.4 s when the surrounding medium is homogeneous. When athree-layer ground is considered, the corresponding CPU times sig-nificantly increases (due to the fact that the Green’s functions mustthen be calculated using Hankel transforms) to 63 s and 161 s,respectively.

However, the computed resistivity variations are approximatelytwo to three times greater in magnitude than those obtained withthe Fourier series expansions (Fig. 1). This discrepancy in anomalymagnitude disqualifies the use of the Fourier method for a quan-titative inversion process. Nevertheless, one must observe that ifthe magnitudes are clearly lower the FWHM are not very different,which suggests that, using this calculation, the location of lateralchange would be quite similar to that of the exact methods. Be-fore developing this point, it is wise to compare the results of theFourier series expansion method to another approximate method,the Born approximation, where the linearization approach isdifferent.

4.2 Comparison with the Born approximation applied tothe MoM modelling scheme

The Born approximation (see Appendix C for more details) is a sim-plification that can be applied to most of the modelling schemes.

For the MoM one, it corresponds to the situation in which thecross-coupling between secondary sources corresponding to het-erogeneous body(ies) is neglected. With this simplification, theCPU time is 0.3 s, which is significantly shorter than that ofthe MoM model but five times longer than with the Fourier se-ries expansion for the case of a homogeneous medium. How-ever, when a three-layer embedding medium is considered, thetime for modelling with the Born approximation significantly in-creases to 41.4 s, about 400 times longer than with Fourier seriesexpansion.

Comparisons between the MoM calculation, the Born approx-imation and the Fourier analysis are presented for a number ofresistive targets embedded in a homogeneous conductive medium(Fig. 4), and for conductive targets embedded in a homogeneousresistive medium (Fig. 5). The targets were generated by increasingcontrasting ratios with the host medium. The results show that forboth resistive and conductive targets, the FWHM are similar but themagnitudes are clearly different, the differences between the threemodelling methods depending on the target’s contrast with the host.Specifically for the resistive targets, when the contrast correspondsto a factor of two (a 100 �m body in a 50-�m host medium),the magnitude of the anomaly determined by the Fourier method isnear a quarter (0.26) of that given by the Born approximation anda sixth (0.16) of that given by the exact calculation. For a contrastof four (a 200 �m body in a 50-�m host medium), the ratios in-crease to one-half and one-fifth, and it reaches unity and one-thirdfor a contrast of eight (a 400 �m body in a 50-�m host medium).The result of the modelling establishes that for anomaly magnitudesboth approximate methods would be equivalent in the case of veryhigh contrasts, whereas for lower contrasts the Born approximationwould be preferable.

In the case of conductive targets (Fig. 5), the Born approxima-tion delivers unlikely results for highly contrasted ratios, showinga negative apparent resistivity extreme for the case of a 50 �mtarget in a 200 �m medium (Fig. 5a). A decrease in the contrastratio between target and host reduces this abnormal magnitude, but

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3-D Fourier electrical approximate model 163

Figure 4. Comparison between the apparent resistivity variations in the centre of the 2-m wide resistive body, computed using the Born approximation (redlines), the Fourier series expansion (black lines) and an exact calculation (blue lines) for different resistivity contrasts: (a) 100 �m body in a 50-�m hostmedium (ratio = 2), (b) 200 �m body in a 50-�m host medium (ratio = 4) and (c) 400 �m body in a 50-�m host medium (ratio = 8).

still with a higher magnitude difference than that of the exact cal-culation. Regardless of contrast intensity, the models representingFourier expansion show minima clearly smaller than those of theexact method: from one-sixth (0.16) for the 150 �m body (Fig. 5d)to a bit more than one-third (0.39) for the 50 �m body (Fig. 5a).

4.3 Comparisons for resolution of lateral changes

As can be observed in Figs 1 and 2, the lateral extensions of theanomaly are quite similar for all the methods either exact or ap-

proximate. This suggests that both approximate methods could beof value in the determination of the location of lateral resistivitychanges. To go deeply into this point we studied the coalescenceof anomalies generated by several neighbouring bodies. For thestudy, four bodies are centred at each corner of a square of vary-ing size in order to observe the limit at which separate anomaliesappear. Each body corresponds to a 2 × 2 × 1 m3 slab of 200�m resistivity embedded in a 50 �m medium. The results usingMoM modelling method with AM = 1 m and AM being parallel tothe x-axis are shown in Fig. 6: centres of the bodies separated by2.2 m and adjacent neighbouring sides by 0.2 m (Fig. 6a), the

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164 S. Buvat, C. Schamper and A. Tabbagh

Figure 5. Comparison between the apparent resistivity variations in the centre of the 2-m wide conductive body, computed using the Born approximation (redlines), the Fourier series expansion (black lines) and an exact calculation (blue lines), for different resistivity contrasts: (a) 50 �m body in a 200-�m hostmedium (ratio = 0.25), (b) 70 �m body in a 200-�m host medium (ratio = 0.35), (c) 110 �m body in a 200-�m host medium (ratio = 0.55) and (d) 150 �mbody in a 200-�m host medium (ratio = 0.75).

centres separated by 2.4 m and the sides by 0.4 m (Fig 6b), the cen-tres separated by 3 m and the sides by 1 m (Fig. 6c) and the centresseparated by 4 m and the sides by 2 m (Fig. 6d). One can observethat, as expected with the pole–pole electrodes configuration, thecoalescence is more marked in the AM direction but remains limitedwhen the centre distance reaches 2.4 m (Fig. 6b) where it is clear thatfour different bodies are present. For each case the CPU time was102 s.

The results of the Born approximation for the same position ofresistive bodies as those in Fig. 6, are shown in Fig. 7. When thedistances between the centres are 2.2 m or 2.4 m (Figs 7a and b),the image of individual bodies are not readily apparent. The fourbodies appear when the centre distance is 3 m or greater (Figs 7cand d). For each case the CPU time is 0.36 s.

The Fourier series expansion results are presented in Fig. 8, whichshows six different distances between the centres of the bodies: 2.2,2.4, 3, 4, 5 and 6 m. The apparent resistivity shows a single bodyfor centres separated up to 3 m (Fig. 8c). The centre distance mustreach 5 m in the AM direction (Fig. 8e) for the presence of fourindividual bodies to be observed. For each case the CPU time was0.07 s.

5 C O N C LU S I O N

In this work we investigated 3-D forward resistivity modelling usingFourier expansion. The results of the Fourier expansion were com-pared to exact solutions using surface integral method and MoM,

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3-D Fourier electrical approximate model 165

Figure 6. Apparent resistivity maps, obtained using the exact MoM method, in presence of four resistive bodies centred at the corners of a square, each bodycorresponds to a 2 × 2 × 1 m3 slab of 200 �m resistivity embedded in a 50 �m medium, with AM = 1 m, AM being parallel to the x-axis: (a) the centres ofthe bodies are separated by 2.2 m so their neighbouring sides being by 0.2 m, (b) the centres of the bodies are separated by 2.4 m so their neighbouring sidesby 0.4 m, (c) the centres of the bodies are separated by 3 m so the neighbouring sides by 1 m and (d) the centres of the bodies are separated by 4 m so theneighbouring sides by 2 m.

as well as to another approximate modelling method, the Born ap-proximation. As expected, the exact modelling methods producedanomalies that appeared to represent target anomaly magnitudesand extents with greater fidelity. However, the results from theFourier expansion could be used for a first-order rough interpre-tation trial or in an initial step of a multistep inversion process, ifthe balance between their advantages and drawbacks can be con-sidered as favourable. We, thus, summarize the characteristics ofthe Fourier series expansion modelling method in DC electricalresistivity prospecting:

(1) First, Fourier series expansion is a very rapid modellingtechnique for a homogeneous host medium and even more if alayered ground is considered. To be exact, the Fourier expansionis 600 times quicker than the exact calculation for a three-layerground and this advantage remains when comparing to the timenecessary to complete the Born approximation model (400 timesfaster).

(2) The second characteristic is the significant reduction ofanomaly amplitudes. This is the main weakness of the method.However, this underestimation is consistent for both conductive

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166 S. Buvat, C. Schamper and A. Tabbagh

Figure 7. Apparent resistivity maps, obtained using the Born approximate method, in presence of four resistive bodies centred at the corners of a square, eachbody corresponds to a 2 × 2 × 1 m3 slab of 200 �m resistivity embedded in a 50 �m medium, with AM = 1 m, AM being parallel to the x-axis: (a) the centresof the bodies are separated by 2.2 m so their neighbouring sides being by 0.2 m, (b) the centres of the bodies are separated by 2.4 m so their neighbouringsides by 0.4 m, (c) the centres of the bodies are separated by 3 m so the neighbouring sides by 1 m and (d) the centres of the bodies are separated by 4 m so theneighbouring sides by 2 m.

and resistive targets, whereas the Born approximation shows verydifferent behaviours for the different targets. For the Fourier seriesthe underestimation of the anomaly amplitudes is between one-sixth and one-third depending on the target’s resistivity contrastcompared to the host medium.

(3) A third and important aspect of the comparison study isthe ability to precisely locate the resistivity variation. Forwardmodelling of neighbouring bodies shows the Fourier series ex-

pansion method is less efficient than exact methods or Bornapproximation.

Globally, the computation efficiency of the approximate Fourierseries expansion modelling method is the main advantage, whichleads to use it as a first interpretation tool. In most cases it will needto be followed by the application of more time-consuming exactmethods that remain unavoidable when an accurate interpretation isrequired.

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Figure 8. Apparent resistivity maps, obtained using the Fourier series expansion approximate method, in presence of four resistive bodies centred at the cornersof a square, each body corresponds to a 2 × 2 × 1 m3 slab of 200 �m resistivity embedded in a 50 �m medium, with AM = 1 m, AM being parallel to thex-axis: (a) the centres of the bodies are separated by 2.2 m so their neighbouring sides being by 0.2 m, (b) the centres of the bodies are separated by 2.4 mso their neighbouring sides by 0.4 m, (c) the centres of the bodies are separated by 3 m so the neighbouring sides by 1 m, (d) the centres of the bodies areseparated by 4 m so the neighbouring sides by 2 m, (e) the centres of the bodies are separated by 5 m so the neighbouring sides by 3 m and (f) the centres ofthe bodies are separated by 6 m so the neighbouring sides by 4 m.

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A C K N OW L E D G E M E N T S

The text of this paper has been significantly improved by the workachieved by two reviewers one anonymous and Dale Rucker. ThePhD work of Solene Buvat was funded by the FIRE (FederationIle-de-France de Recherche sur l’Environnement, CNRS, France).

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A P P E N D I X A : C A L C U L AT I O N O F T H EF O U R I E R S E R I E S E X PA N S I O NC O E F F I C I E N T S C O R R E S P O N D I N G T O AL AT E R A L C H A N G E I N R E S I S T I V I T Y

If the resistivity change in the ith layer corresponds to a crenelfunction of width XL in the x direction, width YL in the y direc-tion and amplitude δρcr centred in xc, with the injection locatedat (xA = 0, yA = 0), the expressions for the different coefficientsare:

ak = 2

XT

∫ X T/2

−X T/2δρi (x, y) cos

2πkx

XTdx

= 2

XTδρcr

∫ Xc+X L/2

Xc−X L/2cos

2πkx

XTdx

= δρcr

kπ

(sin

2πk(Xc + X L/2)

XT− sin

2πk(Xc − X L/2

XT

),

bk = δρcr

kπ

(cos

2πk(Xc − X L/2)

XT− cos

2πk(Xc + X L/2)

XT

),

cl = δρcr

lπ

(sin

2πl(Y c + Y L/2)

Y T− sin

2πl(Y c + Y L/2)

Y T

),

dl = δρcr

lπ

(cos

2πl(Y c − Y L/2)

Y T− cos

2πl(Y c + Y L/2)

Y T

).

A P P E N D I X B : C A L C U L AT I O N O F T H EC O E F F I C I E N T S F O R T H E S E C O N DA RYP O T E N T I A L E X PA N S I O N

The following general expression is first adopted for Vs:

Vs(x, y) =∞∑

l=1

∞∑k=1

(v1k cos

2πkx

XT+ v2k sin

2πkx

XT

)

×(

w1l cos2πly

Y T+ w2l sin

2πly

Y T

).

Following application of the Laplace equation, Vs and any coeffi-cient from its series expansion thus verifies:

∂2Vs

∂z2− λ2Vs = 0,

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3-D Fourier electrical approximate model 169

with:

λ = 2π

√k2

XT 2+ l2

Y T 2.

In the ith layer, the expression for Vs is thus: Vs = αi e−λz + βi eλz .In a soil represented by N layers, due to the continuity of the po-

tential and the vertical current density, the impedance: Zi = λ Vs

− ∂Vsρi ∂z

can be defined. At the deepest interface, ZN(k, l) = ρN(k, l), andthe recursive formula: Zi (k, l) = ρi (k, l) Zi+1(k,l)+ρi (k,l)thλei

ρi (k,l)+Zi+1(k,l)thλeiis found

(ei being the thickness and ρ i the resistivity of the ith layer). Fromthis, Z2 corresponds to the interface at z = e1. In the first layer, α1

and β1 are calculated after expressing the primary potential in theFourier domain, since the boundary conditions apply to the totalpotential:

Vp(x, y, z) = ρ1 I

2π

1√x2 + y2 + z2

= ρ1 I

2π

4

XT .Y T

∞∑l=1

∞∑k=1

2π

λe−λz cos

2πkx

XTcos

2πly

Y T.

Then, by posing C(k, l) = Z2(k,l)−ρ1Z2(k,l)+ρ1

, α1 = β1 = 2π

λ

C(k,l)

e2λe1 −C(k,l), thus

leading to the final expression:

Vs(x, y) = ρ1 I

2π

4

XT .Y T

×∞∑

l=1

∞∑k=1

4π

λ

C(k, l)

e2λe1 − C(k, l)cos

2πkx

XTcos

2πly

Y T.

This expression can be verified by comparing it with the Hankeltransform normally used in 1-D resistivity calculations. From thecorresponding Fourier transform expression,

Vs(x, y) = ρ1 I

2π

∫ ∞

−∞

∫ ∞

−∞

1√u2 + v2

2C

e2λe1 − Ce2π i(ux+vy)dudv,

where u is the spatial frequency in x, and v is the spatial frequencyin y. Since there is no lateral variation, C(k, l) no longer depends onthe spatial frequency and becomes C.

By changing to a polar coordinate system: x = r.cosθ , y = r.sinθ ,u = s.cosϕ, v = s.sinϕ, r2 = x2 + y2, s2 = u2 + v2, λ = 2πs, thefollowing result is found:

Vs(r ) = ρ1 I

2π

∫ ∞

0

∫ 2π

0

1

s

2C

e2λe1 − Ce2π isr cos(ϕ−θ)sdsdϕ.

By applying the identity:

1

2π

∫ 2π

0eiλr cos αdα = J0(λr ),

where J0 is the Bessel function of the first kind, the well knownformula: Vs(r ) = ρ1 I

2π

∫ ∞0

2Ce2λe1 −C

J0(λr ) dλ is obtained.

A P P E N D I X C

The Born approximation named after Max Born is a general methodused in quantum mechanics and in EM domains. In this approxi-mation, a total field (respectively a total potential) is replaced bythe corresponding incident, or primary field, and the scattered, orsecondary, field is neglected.

In the present work the Born approximation is applied to theDC MoM (Dabas et al. 1994), this means that the fictitious currentdensity sources equivalent to the heterogeneous body are calculatedby �js = (σs − σi ) �Ep and not by �js = (σs − σi ) �E , σs being the con-ductivity of the body and σi the conductivity of the host. In otherwords, the calculation step where the total field E is calculated (amatrix inversion) is bypassed and the total field E is replaced by theprimary field Ep generated by the injection pole in a homogeneousor layered terrain.

When using the surface integral approach where the elec-trical charges equivalent to the presence of the heterogeneousbody are located on the surface of the body, the equation defin-ing the charge strength is: ∇2V (r ) = −∇σ (r ).∇V (r )

σ (r ) − Iσ (r ) δ(r − rA),

where the first term of the right-hand side corresponds to thesecondary charges source term and the second term to the pri-mary injection located in rA. If one considers that σ (r ) = σiμ(r )(Li & Oldenburg 1992), this secondary sources term reduces to−∇ ln μ(r ).∇V (r ) and the application of the Born approximationreplaces ∇V (r ) by ∇Vp(r ), which is independent of the secondarycharges.

While the complete calculations of the MoM and the surfaceintegral methods deliver the same results, see Figs 2 and 3, there isa priori no reason for having the same results after applying the Bornapproximation to both methods because none of them is linear andthe results here obtained indeed differ from those presented in theLi & Oldenburg (1992) paper. A full discussion of the advantagesand limits of each of the 3-D methods and of the application to bothof the Born approximation would merit to be the subject of a futurepaper.

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