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GEOPHYSICS, VOL. 73, NO. 3 �MAY-JUNE 2008�; P. F135–F142, 12 FIGS.10.1190/1.2905835

ncorporating topography into 2D resistivity modelingsing finite-element and finite-difference approaches

rhan Erdogan1, Ismail Demirci1, and Mehmet Emin Candansayar1

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ABSTRACT

We incorporate topography into the 2D resistivity forwardsolution by using the finite-difference �FD� and finite-ele-ment �FE� numerical-solution methods. To achieve this, wedevelop a new algorithm that solves Poisson’s equation usingthe FE and FD approaches. We simulate topographic effectsin the modeling algorithm using three FE approaches and twoalternative FD approaches in which the air portion of themesh is represented by very resistive cells. In both methods,we use rectangular and triangular discretization. Further-more, we account for topographic effects by distorting the FEmesh with respect to the topography. We compare all meth-ods for accuracy and calculation time on models with varyingsurface geometry and resistivity distributions. Comparisonsshow that model responses are similar when high-resistivityvalues are assigned to the top half of the rectangular cells atthe air/earth boundary with the FE and FD methods and whenthe FE mesh is distorted. This result supports the idea that to-pographic effects can be incorporated into the forward solu-tion by using the FD method; in some cases, this method alsoshortens calculation times. Additionally, this study showsthat an FD solution with triangular discretization can be usedsuccessfully to calculate 2D DC-resistivity forward solu-tions.

INTRODUCTION

The development of multichannel and multielectrode systems hasade data acquisition very simple and practical when using the DC-

esistivity method. Generally, data are acquired using sounding-pro-le measurements, and measured data are interpreted using 2D in-ersion algorithms. The DC-resistivity method is widely used foretecting buried archaeological structures and for environmentalnd hydrogeological work, mining exploration, and engineering

Manuscript received by the Editor 20 June 2007; revised manuscript receiv1Ankara University, Faculty of Engineering, Department of Geophysica

ng.ankara.edu.tr; candansa@eng.ankara.edu.tr.2008 Society of Exploration Geophysicists.All rights reserved.

F135

tudies �Candansayar and Başokur, 2001; Friedel et al., 2006; Özür-an et al., 2006�. In these applications, the terrain usually is not flat.opography causes localized dispersion and focusing of the currentistribution near the surface and hence causes artificial, terrain-in-uced conductive and resistive anomalies in field data �Telford et al.,991�. These topographic effects must be incorporated into forwardolutions of the DC-resistivity problem to generate accurate models.

Existing studies of incorporating topography into 2D and 3D DC-esistivity modeling �Fox et al., 1980; Holcombe and Jiracek, 1984;sourlos et al., 1999� and inversion �Tong and Yang, 1990; Sasaki,994� are limited to finite-element �FE� approaches. Coggon �1973�ompares the effects of topography on dipole-dipole and pole-dipolerrays. Fox et al. �1980� offers the first systematic study of topo-raphic effects. They investigate characteristic anomalies associatedith basic topographic features �hill, valley, and slope� for dipole-di-ole arrays and study the correction of topography-related dataoise. Tong and Yang �1990� present an algorithm that incorporatesopography into 2D resistivity inversions. Tsourlos et al. �1999� ex-mine the effect of topography on various resistivity arrays. No pub-ished studies have investigated the possible use of a finite-differ-nce �FD� solution with triangular discretization to calculate 2D re-istivity-model response with terrain topography.

In this study, we incorporate topography by using three FE-basednd two FD-based approaches. Unlike other workers �e.g., Dey andorrison, 1979�, who use an FD forward solution, we use triangular

lements formed by completing the diagonals of the rectangularells, providing model-design flexibility that typically is availablenly when using the FE method �Weaver, 1994�. These triangularells better simulate the air/earth interface on rough terrain for FDolutions.

This approach is first applied to a magnetotelluric �MT� forwardolution �Weaver, 1994�; in our study, we use it for DC-resistivityodeling. We compare FE and FD solution results for various 2D re-

istivity models with undulating surface topographies and resistivityistributions. We show that an FD solution of the Poisson equationhich uses triangular discretization allows us to simulate surface to-ography as well as an FE solution. Our study also shows that an FD

ovember 2007; published online 2 May 2008.eering, Ankara, Turkey. E-mail: eerdogan@eng.ankara.edu.tr; idemirci@

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olution can be an alternative method for obtaining solutions moreuickly than and as accurately as FE methods.

CALCULATING THE 2D FORWARD SOLUTION

Poisson’s equation describes the partial-differential equation thatoverns the behavior of the electric potential for 2D resistivity struc-ure:

�� · � 1

��x,z�� ��x,y,z�� � I� �x � xs�� �y�� �z� , �1�

here ��x,z� is the resistivity in 2D space, ��x,y,z� is the electric po-ential, I is the point current source in the surface, and � denotes thenit Dirac �impulse� function and shows the source location.

Equation 1 is transformed from �x,y,z� space to �x,ky,z� spaceith a Fourier-cosine transform and generally is solved by using FEr FD numerical methods for each wavenumber ky. Then, by apply-ng an inverse Fourier cosinus transform, unknown potentials arealculated in �x,y,z� space. In our new algorithm, the system of lin-ar equations is solved by using LU decomposition. We use the ap-roach developed by Xu et al. �2000� to obtain the ��x,y,z� values.he following eight ky and g values were calculated using the Xu etl. �2000� algorithm and are used in our FD and FE solutions:

ky � �0.003694, 0.031109, 0.105288, 0.291071,

0.766626, 1.994081, 5.172940, 13.672040� ,

a)

b)

c)

d)

igure 1. Schematic of typical 2D modeling meshes that incorporateopography: �a� FE full-air, �b� FE half-air and FD half-air, �c� FEistorted-mesh, and �d� FD full-air. Here we used only one cell be-ween adjacent electrodes. However, in practice, for more accurateesults, a mesh �FE or FD� would have two or more cells insertedorizontally between adjacent electrodes. The near-surface layersre also subdivided vertically by several mesh lines �Loke, 2000�.

g � �0.007699, 0.028744, 0.071942, 0.181920,

0.468466,1.210086, 3.147197, 8.763861� .

The FE and FD methods are widely used numerical-solution tech-iques for solving partial differential equations. In many cases, theD algorithm is faster than typical FE algorithms. The most impor-

ant differences between these methods are �1� when using the FEethod, the solution region can be divided into desired shapes and

2� the FE method is more powerful than the FD method when con-idering complex earth models �Rao, 2005�.

Using the FE method, a 2D solution region is divided into a finiteumber of linear triangular elements that are connected to each otherith nodes. The �unknown� potential is approximated in each ele-ent using a linear polynomial that is defined by using the nodal val-

es and linear systems of equations set for each element. These lin-ar systems are joined to create a global matrix equation. The Neu-ann boundary condition is applied to the top part �air/earth bound-

ry� of the modeling mesh, and Dirichlet boundary conditions arepplied to all other boundaries in our FE solution. When applying theirichlet boundary conditions, at the bottom and side boundaries,

he potential ��x,ky,z� is set equal to zero �Zienkiewicz, 1971; Rijo,977� to protect against inhomogeneity. Finally, the global matrixquation is solved and unknown potentials are calculated �Coggon,971�.

Traditionally, an FD mesh consists of rectangular cells that do noteadily fit actual topographies found in field conditions. To over-ome this, we divide the rectangular cells into subtriangular ele-ents �Weaver, 1994�. These cells also are connected to each otherith nodes. The unknown potentials at each node are calculated by

eferring to the neighboring nodes’ values. The coefficient of eachotential is defined with respect to the geometry and conductivityistribution of the model, and the coefficient matrix is set. Finally,nknown potentials are calculated by solving the system of linearquations. For the FD algorithm, we use discretization by area, orig-nally presented by Dey and Morrison �1979�. Their FD solutionses the mixed-boundary condition, but here we modified the Deynd Morrison �1979� coefficient formulas for triangular discretiza-ion as shown inAppendix A.

ncorporating topography into the FE approach

Following techniques established in previous studies on variousopographic models, we incorporated topography into the forwardolution by using three approaches based on the FE method. First,e assigned high-resistivity values �representing the air portion� to

ll triangular elements that constituted the cells at the air/earthoundary. This FE full-air approach �Figure 1a� represents the topo-raphic boundary in a staircase fashion.

Second, using the advantage gained by applying triangular ele-ents, we simulate the topographic boundary wherever it crossed

ectangular cells. We assign a high-resistivity value to the triangularlements at the air/earth boundary cells. This FE half-air approachepresents the topographic edge more smoothly and accurately �Fig-re 1b�. The resistivity contrast used for the air portion is 105 timeshe resistivity for the earth portion �Fox et al., 1980�. The topograph-c surface can be accommodated by allowing the z-coordinates of the

esh nodes to vary according to elevation �Sasaki, 1994�.

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Third, we uniformly distort the FE mesh with respect to the sur-ace geometry to simulate the topographic effect �Holcombe andiracek, 1984; Loke, 2000�. This approach is called FE distortedesh �Figure 1c�.

ncorporating topography into the FDpproach with triangular discretization

Traditionally, an FD mesh consists of rectangular cells and cannote reshaped; topography can be simulated only by using resistiveurface cells that represent the air effect in the FD method. The firstpproach we used to simulate topography was to assign high-resis-ivity values to all rectangle cells at the air/earth boundary, as withhe FE full-air approach. We named this approach FD full-air �Figured�. Then we used Weaver’s �1994� formulation and divided theectangular cells into the subtriangular elements �Figure 2�, usinghe same mesh as for the FE method in Figure 1b. We named this ap-roach FD half-air. Appendix A gives the formulation of the FDethod with triangular elements.Figure 1 shows the computational mesh with four cells added to

he boundaries for 20 electrodes. The top panels for all of the model-ng meshes in Figure 1 also show the first and last electrode posi-ions. We inserted two cells between each consecutive electrode. Welso added six more cells to represent the left, right, and bottom edg-s of the modeling meshes so we could apply boundary conditionsver the entire modeling mesh. The first added cells were the sameidth and height as the previous cell. The additional cells were two

imes wider than the previous cell in the x-direction and two timesigher than the previous cell in the z-direction

MODELING STUDY

We compared accuracy and calculation times for the FE full-air,E half-air, FE distorted-mesh, FD full-air, and FD half-air ap-

a)

b)

igure 2. �a� FD half-air mesh. Each cell is subdivided into four tri-ngular elements. �b� FD triangular discretization of a cell betweenhe air/earth interface.

roaches on various models of geometry and resistivity distribution.his article presents forward-solution results for three models withifferent surface topographies �ridge, valley and ridge, and slope�nd some buried resistivity structures. We used 20 electrodes in therst and second models and 30 electrodes in the third model.For each topographic feature, we computed the anomaly for aodel with 30° slope. We assumed a 1-m electrode-separation dis-

ance for models 1 and 2 and a 4-m separation for model 3. The cellsetween two electrodes were divided into two cells, and extra cellsere added to the bottom, right, and left edges to remove boundary

ffects. We calculated apparent resistivities using a dipole-dipole ar-ay, which provides better lateral resolution, and a Wenner-Schlum-erger array, which is more suitable for resolving resistivity changesith depth �Dahlin and Zhou, 2004�. For the homogeneous modelith a 30° sloped-hill surface topography, we found that the FE half-

ir and FD half-air approaches performed poorly because they didot fit the topography readily. Therefore, we did not include those re-ults for the last two models.

It is important to note that we are presenting the apparent-resistiv-ty pseudosections with respect to the surface topographies. In thepparent-resistivity pseudosections in all figures, the left verticalxis shows the n-spacing �where receiver-transmitter dipole sep-ration is n times the potential electrode separation and where n

1,2, . . . ,6� and the right vertical axis shows the real height for theeference zero point.

odel 1

We compared the five methods with model 1, which had a 100-hm-m homogeneous half-space. The surface topography of modelwas a 30° sloped hill �Figure 3�. In all solutions, the dipole-dipole

rray produced a centrally located high-apparent-resistivity anoma-y below the area of the hill �Figure 4b� because the existence of theill between the transmitting and receiving dipoles caused a high ap-arent resistivity attributable to current-focusing effects �Fox et al.,980�. The Wenner-Schlumberger array �Figure 4a� produced an ef-ect opposite that of the dipole-dipole array. Here, we see the dipole-ipole array is more distorted than the Wenner-Schlumberger arrayecause of the undulating terrain topography.

The pseudosections �Figure 4� and relative-difference sectionsFigure 5� show that the modeling results from FE full-air and FDull-air methods are similar, as are the results from FE distorted-

igure 3. Model 1:A30° sloped, homogeneous hill model.

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esh, FE half-air, and FD half-air methods. However, the FE half-ir and FD half-air results and the FE full-air and FD full-air resultsre not similar.

As a validity test for each solution, we determined that the averageelative differences for the dipole-dipole array are 1.2% and 1.7%etween the FE distorted-mesh solution and the FE half-air and FDalf-air solutions, respectively, using the FE distorted-mesh solutions the reference �Figure 5�. This shows that FE solutions �FE distort-d-mesh and FE half-air� and the FD half-air solution yield very sim-lar results. These three numerical solutions also are very similar re-arding the resistivity models given below. The comparisons showhat our new algorithms that uses FE distorted-mesh, FE half-air, andD half-air methods generate correct solutions.For the dipole-dipole array, the average relative differences are

3.9% and 12.6% between the FE distorted-mesh solution and theE full-air and FD full-air solutions, respectively. These differencesre slightly smaller for Wenner-Schlumberger array data for theame model because that array has fewer data points than does theipole-dipole array. We conclude this is because the air/earth inter-ace is better distinguished in FD half-air and FE half-air methodshan in the FD full-air and FE full-air methods. Therefore, we do notresent the modeling results of the FE full-air and FD full-air meth-ds for the other topographic models.

a) b)

igure 4. Apparent-resistivity pseudosections of �a� Wenner-chlumberger and �b� dipole-dipole array data sets for model 1.

odel 2

We used a more complex model �model 2� to compare the FE half-ir, FD half-air, and FE distorted-mesh approaches. The surface to-ography of model 2 was a 30° sloped joint valley and hill �Figure 6�.wo 500-ohm-m resistive cells were present in the 100-ohm-m ho-

a) b)

igure 5. Relative difference between the FE distorted-mesh solu-ion and the FE half-air, FD half-air, FE full-air, and FD full-air solu-ions for �a� Wenner-Schlumberger and �b� dipole-dipole arrays.

Electrode Location

igure 6. Model 2: 30° sloped, joint valley/hill model.

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Incorporating topography into two dimensions F139

ogeneous half-space. The first cell was between 5 and 6 m, and theecond one was between 13 and 14 m along the profile direction.oth cells were 1m below the surface. The FE half-air, FD half-air,nd FE distorted-mesh methods yielded approximately the sameodel responses as shown in the apparent-resistivity pseudosections

Figure 7�.Again, if we use the FE distorted-mesh results as a reference, the

verage relative differences between the FE distorted-mesh and theE half-air and FD half-air solutions were 2.5% and 2.2%, respec-

ively. The relative-difference sections in Figure 8 also show noarge differences between results for n�3. Overall, the relative dif-erences that the three methods produced almost the same response.hese results also show the FD approach can be used as accurately as

he FE method for calculating topographic effects, at least for theodels investigated in this study.

odel 3

Model 3 had more complex surface topographies and resistivityistributions than the other two models. We used two-step, 30° slopeill models �Figure 9�. A 100-ohm-m surface layer, which was 1 mhick for the first 56 m, expanded to 1.5 m thick after 60 m in therofile direction. In the first layer in the profile direction, we placed a

a) b)

igure 7. Apparent-resistivity pseudosections of �a� Wenner-chlumberger and �b� dipole-dipole array data sets for model 2.

00-ohm-m resistive cell between electrodes 11 and 12 and a-ohm-m conductive cell between electrodes 23 and 25. Figure 9hows the dimensions of the cells. There also were two main lateral

a) b)

igure 8. Relative difference between the FE distorted-mesh solu-ion and the FE half-air and FD half-air solutions for �a� Wenner-chlumberger and �b� dipole-dipole arrays.

igure 9. Model 3:A30° sloped, two-step hillside model.

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iscontinuities: 50-ohm-m conductive and 500-ohm-m resistiveasements separated by a 4-m-wide fault with 5-ohm-m resistivitynder the first layer.

The FE distorted-mesh approach and the FE half-air and FD half-ir approaches give similar results for both the dipole-dipole and theenner-Schlumberger arrays �Figure 10�. For dipole-dipole arrays,

he average relative differences between the FE distorted-mesh solu-ion and the FE half-air and FD half-air solutions are 1.7% and 2.3%,espectively �Figure 11�. These differences are slightly smaller for

enner-Schlumberger array data for the same case. These differenc-s also show that the FD half-air approach �with triangular discreti-ation� can be used as an alternative to FE solutions �FE distorted-esh and FE half-air approaches� to calculate resistivity-model re-

ponse when undulating topographies are present.

omparison of CPU times in calculatingE distorted-mesh and FE half-airnd FD half-air approaches

We compared calculation times for increasing numbers of cells inhe z-direction for FE distorted-mesh, FE half-air, and FD half-airolutions �Figure 12�. In the comparison, the number of cells in the-direction was fixed at 48, and the number of cells in the z-directionnitially was 11. �The total number of cells was 528.� The FE distort-d-mesh approach does not require that extra cells be added to simu-ate the topographic effect, but the slope of the surface topography

a) b)

igure 10. Apparent-resistivity pseudosections of �a� Wenner-chlumberger and �b� dipole-dipole array data sets for model 3.

ometimes required us to add extra cells in the z-direction to repre-ent air in the models for the FE half-air and FD half-air approaches.

Hence, the CPU time for the FE distorted-mesh solution was useds a reference level. Using 11 cells in the z-direction for the first mod-l, the FE distorted-mesh and FE half-air solutions were found in.65 s of CPU time, whereas the FD half-air solution was found in

a) b)

igure 11. Relative difference between the FE distorted-mesh solu-ion and the FE half-air and FD half-air solutions for �a� Wenner-chlumberger and �b� dipole-dipole arrays.

igure 12. CPU time per number of cells in the z-direction. Here, theumber of cells in the x-direction is fixed.

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Incorporating topography into two dimensions F141

.3 s. We increased the hill slope in 5° intervals while we increasedhe number of cells in the z-direction for the FE half-air and FD half-ir methods.

The FE distorted-mesh and FD half-air solutions were found inhe same amount of CPU time �0.65 s� as when the FD half-air meth-d had 27 cells in the z-direction. �The total number of cells is 1296.�eanwhile, the FE half-air solution took much longer ��3 s� for the

ame number of cells in the z-direction. This comparison showedhat the FD half-air method provides results similar to the FE distort-d-mesh method but in less CPU time. The ratio between the totalumber of cells used in the FD half-air and the FE distorted-meshethods was 1:2.45 for this example. We can assert that if this ratio

s �2.45, we can use the FD half-air method to find a solution in lessPU time. Figure 12 shows total CPU time with respect to the num-er of cells in the z-direction.

This last modeling test shows that each approach has a unique setf advantages over other approaches. For example, we need to addxtra cells according to the slope of the surface topography in FEalf-air and FD half-air methods but not in the FE distorted-mesh ap-roach. Therefore, computing time is fixed when using the FE dis-orted-mesh method, but it is variable for the FE half-air and FD half-ir methods, depending on how many cells are added to the modelesh to represent changing surface topography.

CONCLUSIONS

In this paper, we examined new approaches for incorporating to-ography into 2D DC-resistivity modeling. We showed that the FEull-air and FD full-air approaches might not represent topographicffects well. However, our modeling studies demonstrated the effi-acy of the FE half-air approach, the FD half-air approach using tri-ngular discretization, and the FE distorted-mesh approach. Theseethods produced almost the same model responses and represent-

d topographic effects more accurately than either of the full-air ap-roaches. This study shows that the FD half-air method with triangu-ar discretization can be used effectively to simulate topography; inome cases, it computes solutions faster than and as accurately as thether approaches. It is important to note that with models that havexcessively undulating topography, the solutions from the FD half-ir and FE half-air methods are less accurate than an FE distorted-esh solution.Furthermore, when using 2D inversion algorithms, an optimum

orward-solution method can be selected by controlling the numberf cells. Very steep surface topography requires a large number ofells in the modeling mesh to simulate the surface topography for FDalf-air and FE half-air solutions. In this case, these solutions requireonger computation times than with an FE distorted-mesh solution.n the other hand, for gently sloped surface topography, we can use

n FD half-air solution, which requires less computation time but iss accurate as an FE distorted-mesh solution.

We recommend two topics for continuing research regarding theD half-air solution: �1� the FD half-air forward solution, which isast, accurate, and well suited to integration into an inversion schemehen considering topography, and �2� application of the general

deas used in finding the FD half-air solution to develop 3D model-ng and inversion algorithms that consider surface topographies.

ACKNOWLEDGMENTS

This work represents sections of MS theses undertaken by the firstnd second authors and supported by the Scientific and Technical

esearch Council of Turkey �TÜBÍTAK� under grant 104Y073. Theorward-solution algorithm was developed in the Geosciences Datarocessing Laboratory �YEBVIL� of Ankara University, Depart-ent of Geophysical Engineering. The first and second authors ac-

nowledge TÜBÍTAK for providing scholarships under project05G145. The authors would like to thank Bülent Tezkan for hisuggestions. They also owe many thanks to the Associate Editor foris constructive comments about improving the manuscript.

APPENDIX A

In this study, we used area discretization in an FE solution, fol-owing the method of Dey and Morrison �1979�. We change Dey and

orrison’s �1979� formulas 17, 18.1, 18.2, 18.3, 18.4, and 18.5 aselow for triangular discretization:

CTij � ���xi�1� i�1,j�1

�r� � �xi� i,j�1�l�

2�zj�1� , �A-1�

CBij � ���xi�1� i�1,j

�r� � �xi� i,j�l�

2�zj� , �A-2�

CRij � ���zj�1� i,j�1

�b� � �zj� i,j�t�

2�xi� , �A-3�

CLij � ���zj�1� i�1,j�1

�b� � �zj� i�1,j�t�

2�xi�1� , �A-4�

CPij � ��CB

ij � CTij � CR

ij � CLij � A�� i,j,Aij�� , �A-5�

nd

A�� i,j,Aij� � Ky2�

�� i�1,j�1�b� � � i�1,j�1

�r� ��xi�1�zj�1

8

��� i,j�1

�l� � � i,j�1�b� ��xi�zj�1

8

��� i,j

�t� � � i,j�l���xi�zj

8

��� i�1,j

�r� � � i�1,j�t� ��xi�1�zj

8

� ,

�A-6�

here superscripts l, r, t, and b represent conductivities for four tri-ngular cells �left, right, top, and bottom, respectively� in a rectangu-ar grid �Figure 2b�.

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