Hindawi Publishing CorporationInternational Journal of GeophysicsVolume 2013 Article ID 931876 9 pageshttpdxdoiorg1011552013931876

Research Article3D DC Resistivity Inversion with Topography Based onRegularized Conjugate Gradient Method

Jian-ke Qiang Xue Han and Shi-kun Dai

School of Geosciences and Info-Physics Central South University Changsha 410083 China

Correspondence should be addressed to Xue Han 313915246qqcom

Received 25 March 2013 Revised 30 July 2013 Accepted 21 August 2013

Academic Editor Salvatore Piro

Copyright copy 2013 Jian-ke Qiang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

During the past decades we observed a strong interest in 3D DC resistivity inversion and imaging with complex topography Inthis paper we implemented 3D DC resistivity inversion based on regularized conjugate gradient method with FEM The Frechetderivative is assembledwith the electric potential in order to speed up the inversion process based on the reciprocity theorem In thisstudy we also analyzed the sensitivity of the electric potential on the earthrsquos surface to the conductivity in each cell underground andintroduced an optimizedweighting function to produce new sensitivitymatrixThe syntheticmodel study shows that this optimizedweighting function is helpful to improve the resolution of deep anomaly By incorporating topography into inversion the artificialanomaly which is actually caused by topography can be eliminated As a result this algorithm potentially can be applied to processthe DC resistivity data collected in mountain area Our synthetic model study also shows that the convergence and computationspeed are very stable and fast

1 Introduction

In DC resistivity exploration method complex topographycan generate artificial anomalies which will cause difficultyfor the data interpretation Based onQiang and Luorsquos work [1]on 3D DC finite element resistivity modeling with complextopography we conducted 3D regularized inversion andimaging for this complex model

The efficiency for 3D inversion problem depends primar-ily on 3 factors efficient inversion algorithm method forcomputing sensitivity matrix and the solver for a large linersystem Tripp et al [2] introduced a method for calculatingthe sensitivitymatrix based on the relationship between elec-tric potential and model parameter This developed methodfor computing sensitivity matrix was successfully applied to a2D DC resistivity inversion problem

Park and Van [3] introduced 3D inversion based on finitedifferencemethod Sasaki [4] also described similar 3D inver-sion algorithm but based on finite element method Theseintroduced 3D inversion methods work well to recover shal-low resistivity anomalies but fail to produce high-resolutionimage for the deep anomalous bodies The synthetic model

studies show that the recovered resistivity imaging is quitedifferent from the true model if the anomaly is located deepunderground Zhang et al [5] conducted the research on 3DDC resistivity inversion using conjugate gradient methodLoke and Barker [6] introduced the E-SCAN (pole-polearray) 3D inversion technique where the Frechet derivativematrix can be obtained by the halfspace analytical solutionThe computation cost can be reduced significantly and itmakes the 3D inversion become feasible Wu and Xu [7ndash9]and Liu et al [10] tested the application of E-SCANmethod ofmeasurement to 3D inversion problem with complex topog-raphy and synthetic study shows that this method works wellHowever it is difficult to obtain the raw 3D data in real explo-ration since this method of measurement is time and moneyconsumingMoreover E-SCANmethod is based onAM (twoelectrodes) array and the resolution is lower than gradientarray Papadopoulos et al [11] improved the efficiency ofcomputing Jacobi matrix which can eliminate some unusedparameters for inversion save memory space and speed upinversion without losing the accuracy Tsourlos and Ogilvy[12] studied the 3D inversion of borehole-to-surface resis-tivity and IP data Based on the smoothness constrained

2 International Journal of Geophysics

FEM algorithm they get reasonable result Huang et al[13] increased the accuracy of inversion by introducing thevolume factor and switching from global inversion to localinversion However this modified inversion process can onlybe applied to flat surface Gunther et al [14] achieved 3Dresistivity inversion for arbitrary topography based on finiteelement with unstructured mesh Oldenborger and Routh[15] also introduced 3D resistivity inversion with the assis-tance of point spread function

In this paper we will focus on the study of fast 3D resis-tivity inversion problem for complex topography based onweighted regularized conjugate gradient method The mod-ified sensitivity matrix can be used to recover relatively deepanomalous body

2 3D DC Resistivity Forward Modeling andInversion Theory

21 3119863 Resistivity Forward Modeling with Complex Topogra-phy The forwardmodelingmethod is based onfinitemethodwith triangular prism discretization [1] which can simulatevariable topography We approximated the unknown electricpotential within each element by dual-linear function andmixed boundary condition [16] is implemented By assem-bling each element equation we can formulate a global linearsystem of equations as follows

119870(119909119910119911120590)119880(119909119910119911120590) = 119875(119909119904119910119904119911119904120591Ω) (1)

where 119909 119910 and 119911 are the coordinate for the node 120590 is theconductivity for the element Ω and 120591 indicate the boundaryat infinity and the earthrsquos surface 119870 is the element equationmatrix which is related to the location of nodes and elementconductivity 119880 is the electric potential which is a functionof node location and conductivity in the node 119875 containsthe information of source location and boundary conditionsEquation (1) can be solved using incomplete Cholesey con-jugate gradient method (ICCG) By decomposing the matrix119870 and substituting back to (1) we can compute the electricpotential 119880 (these values need to be stored to formulate theJacobi matrix in inversion) at every node corresponding to allpoint electric sources Apparent resistivity can be computedfor different source-receiver array configuration

The algorithm is implemented by Fortran language on PCmachine We adopted triangular prism discretization of thesubsurface in order to simulate the topography and complexanomalous bodies underground

22 AdaptiveWeighed Regularization Inversion Algorithm Ina compact form the forward modeling process describedpreviously can be written as follows

119889119894 = 119860 (119898119895) 119894 = 1 2 119898 119895 = 1 2 119899 (2)

where 119889119894 is the observed data 119898119895 is the model parameter(conductivity in the element) and 119860 is a nonlinear operatorwhich is implemented using finite element method Theinverse problem in (2) to recover model parameter from

observed data is an ill-posed problem with nonunique solu-tion

To obtain stable solution for this ill-posed problem oneneeds to consider regularization theory The regularized the-ory has been applied to 3D DC resistivity inversion for a longtime [11 17ndash22] In our study we consider a minimization ofTikhonov parametric functional as follows

119875120572(119889119898) = 119860 (119898) minus 119889

2+ 120572119904 (119898119898apr) = min (3)

where 120572 is regularization parameter (Lagrange parameter)119904(119898) is a nonnegative stabilizing functional of model param-eter with the property of monotonic decreasing and 119898apr issome a priori information of model parameters

It was shown by Zhdanov [23] that the regularizationparameter can be selected using an adaptive scheme as fol-lows

120572119896 = 1205720119902119896 119896 = 1 2 3 119899 0 lt 119902 lt 1 (4)

where 119902 is a real number smaller than 1 and 1205720 is the initialvalue for regularization parameter which is selected in sucha way to balance the misfit functional and the stabilizer asfollows

1205720 =119860(119898) minus 119889

2

10038171003817100381710038171003817119898 minus 119898apr

10038171003817100381710038171003817

2 (5)

The quality and amplitude of different data vary in geo-physical applicationDataweightingmatrix is preferred in theinversion process in order to increase the contribution of datawith good quality without suppressing geophysical informa-tion from data with low amplitude After taking into accountof data weighting the Tikhonov parametric functional in (3)can be modified as following

119875120572(119889119898) = (119882119889119860 (119898) minus119882119889119889)

119879(119882119889119860 (119898) minus119882119889119889)

+ 120572(119882119898119898 minus119882119898119898apr)119879(119882119898119898 minus119882119898119898apr)

= min(6)

where119882119889 is the data weighting matrix which can be selectedaccording to the amplitude and quality of different data thesuperscript 119879 indicates the complex transpose 119882119898 is themodel weighting matrix which results in practically equalresolution of the inversion with respect to different param-eters of the model It was shown that the model weightingmatrix can be computed as follows in such equal resolutioncriteria

119882119898 = diag (119865lowast119865)18 (7)

where 119865 is the Frechet derivative matrix corresponding tothe nonlinear forward modeling operator 119860 in (2) 119865lowast is thetranspose of 119865

International Journal of Geophysics 3

We used regularized conjugated gradient method [23] forthe minimization of (6)

r119899 = 119860 (119898119899) minus d

l120572119899 = 119865lowast1198822

119889 r119899 + 1205721198822

119898 (119898119899 minus 119898apr)

120573120572

119899 =

1003817100381710038171003817l120572119899

1003817100381710038171003817

2

1003817100381710038171003817l120572119899minus11003817100381710038171003817

2

l120572119899 = l120572119899 + 120573120572

119899 l120572

119899minus1

119896120572

119899 =

(l120572119899 l120572119899)

(10038171003817100381710038171003817119882119889119865119898

119899

l11989910038171003817100381710038171003817

2+ 120572

10038171003817100381710038171003817119882119898 l119899

10038171003817100381710038171003817

2)

119898119899+1 = 119898119899 minus 119896120572

119899 l120572

119899

(8)

where r119899 is the residual vector between the observed data andpredicted data

Based on the algorithm in (8) we implemented 3D regu-larized inversion for DC resistivity with complex topography

23The Formulation of Frechet DerivativeMatrix Weusuallyfix the size of cells for model parameter in the DC resistivityinversion and assume that the apparent resistivity on theearthrsquos surface is only function of the conductivity distribu-tion in the cells As such the Frechet derivative matrix iscomputed as follows

119865 =120597120588si120597120590119895

(9)

where 120588si is the measured apparent resistivity data and 120590119895 isthe conductivity value for the 119895th cell

As we know the measured apparent resistivity on theearthrsquos surface is related to the electric potential in the obser-vation position Take the dipole-dipole array as an example

120597120588119860119861119872119873119878

120597120590=

minus119866119860119861119872119873

119868119860119861

(120597119880119860119872

120597120590minus120597119880119860119873

120597120590minus120597119880119861119872

120597120590+120597119880119861119873

120597120590)

(10)

where 119866119860119861119872119873 is array coefficient which is a function of

electrode spacing 119868119860119861 is the injected current and 119880119860119872 is the

electric potential at 119872 when current is injected from elec-trode 119860 Equation (10) indicates that the derivative of appar-ent resistivity to the model conductivity can be transformedto the derivative of electric potential to the conductivity ineach cell The electric potential on the earthrsquos surface can becomputed from (1) By considering the first order derivativeof both sides of (1) to the model parameter we can obtain thefollowing formula

K120597U120597120590

+120597K120597120590

U = 0 (11)

which can be written as follows

K120597U120597120590

= minus120597K120597120590

U (12)

Equation (12) is in a similar to that of as (1)The right sideof (12) can be treated as an auxiliary currentWe can solve thelinear system of equations in (1) to obtain the derivative of theelectric potential on the earthrsquos surface to the conductivity ineach cell This derivative can also be computed by using thefollowing equations

120597119880119860119872

120597120590119890

= minus 1198801198601 1198801198602 1198801198603 1198801198604 1198801198605 1198801198606

sdot

[[[[[[[[[[[[[[[[[[[[[[[[[[

[

12059711989611

120597120590

12059711989612

120597120590

12059711989613

120597120590

12059711989614

120597120590

12059711989615

120597120590

12059711989616

120597120590

12059711989621

120597120590

12059711989622

120597120590

12059711989623

120597120590

12059711989624

120597120590

12059711989625

120597120590

12059711989626

120597120590

12059711989631

120597120590

12059711989632

120597120590

12059711989633

120597120590

12059711989634

120597120590

12059711989635

120597120590

12059711989636

120597120590

12059711989641

120597120590

12059711989642

120597120590

12059711989643

120597120590

12059711989644

120597120590

12059711989645

120597120590

12059711989646

120597120590

12059711989651

120597120590

12059711989652

120597120590

12059711989653

120597120590

12059711989654

120597120590

12059711989655

120597120590

12059711989656

120597120590

12059711989661

120597120590

12059711989662

120597120590

12059711989663

120597120590

12059711989664

120597120590

12059711989665

120597120590

12059711989666

120597120590

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

sdot

1198801198601

1198801198602

1198801198603

1198801198604

1198801198605

1198801198606

(13)

As we can see the computation cost in formulating the Jacobimatrix is trivial since the electric potential in all the nodescorresponding to different source location already precom-puted at the stage of forward modeling and the stiffnessmatrix is also computed in forward modeling process Thecomputation of Jacobi matrix is only a transformation whichutilizes the forward modeling result In real application weonly need the forward modeling result on the nodes whichlocate on the earthrsquos surface instead of the electric potentialon all nodes Also one needs to notice that only one forwardmodeling is required to obtain Jacobi matrix As such thecomputation cost for inversion is reduced dramatically andit makes the inversion be implemented on a PC machine

4 International Journal of Geophysics

3 Discussions for Inversion

31 Stability Issue for Inversion In 3D resistivity inversionthe misfit for inversion will stay in a relative large level ifthe initial model is not properly chosen which will cause thebackground conductivity to be either too small or too largeIn this study we implemented some computation to calculatethemaximum andminimumvalue of apparent resistivity andselect reasonable background conductivity from this infor-mation to be our initial model for inversion

Another main issue is that the inversion will be unstableif the variation of inversion parameter is either too large ortoo small In order to solve this problem we will take thelogarithmof themodel parameter and do the inversion in thisnew space As a result the variation of model parameter isin control Moreover the upper and lower limits for modelparameter can be set in the inversion by applying thismethod

32 Speed for Inversion The modern PC machine is alreadyvery powerful now with the development of CPU andincrease of memory However it can still take hours and evendays to run an inversion if the number of nodes increases dra-matically especially in 3D problem As a result it is veryimportant to choose proper method of data storage and fastsolver for linear system of equations Incomplete Choleseyconjugate gradient method (ICCG) works well for DC resis-tivitymethodwithmanymoving electric source By using thismethod we only need to decompose the stiffnessmatrix oncewhich takes approximately several minutes and the substitu-tion is also very fast The computation of Jacobi matrix is atime-consuming process In this paper we used the methodof electric potential combination to form the Jacobi matrixwhich can reduce the computation time dramaticallyWe runthe inversion in a PC machine and the model discretizationis 44 times 22 times 17 times 2 which generates 32912 cells The numberof observation is 1505 with 5 profiles and 155 points to injectcurrents The forward modeling takes around 5 minutes andeach iteration of inversion takes approximately 6 minutesThe formulation of Jacobimatrix takes around 30 seconds andit takes other 30 seconds to update the model The total timefor inversion is around 1 hour if the number of iteration is setto be 10

33 Sensitivity Analysis The sensitivity we mentioned heredescribes how sensitive the measured apparent resistivity tothe change of conductivity in each element is Aswe know theelectric potential decreases as the distance between the obser-vation point and the source increasesThe apparent resistivityalso decreases as the observation point moves away from thesource since the apparent resistivity is proportional to theelectric potential Figure 1 shows a sensitivity distribution in119883119885 vertical section As we can see from Figure 1 the datais more sensitive to shallow structure than deep structureFigure 2 shows the sensitivity changes as a function of depthat 119883 = 0 The curve in Figure 2 concaves upward and thesensitivity decreases dramatically as depth increases As aresult the inverted anomaly will move upward if we usethe sensitivity shown in Figures 1 and 2 We can modifythe sensitivity distribution to make it decay slower than

00 40 80 120

02 04 06 08 10 12Sensitivity (J)

minus20

minus60

minus100

minus120 minus80 minus40

X (m)

Z(m

)

Figure 1 Original sensitivity distribution in119883119885 section

0000

16

08

04

12

minus20 minus60 minus80minus40 minus100

Sens

itivi

ty (J

)

Z (m)

Figure 2 The original sensitivity changes as a function of 119885 at 119883 =

0

00 40 80 120

02 04 06 08 10 12Sensitivity (J)

minus20

minus60

minus100

minus120 minus80 minus40

X (m)

Z(m

)

Figure 3 Modified sensitivity distribution in119883119885 section

original sensitivity in order to increase the resolution ofdeep structure Figures 3 and 4 show the modified sensitivitydistribution The synthetic model shows that this modifiedsensitive distribution works well to recover the true model(Figure 5)

4 Numerical Examples

The following synthetic model studies are based on thedipole-dipole array

41 Simple Model For the simple case we have two differentmodelsModel 1 (Figure 6(a)) is a 4mtimes 4mtimes 4m conductive

International Journal of Geophysics 5

body buried 3meters underground with the resistivity of10Ωm The resistivity of wall rock is 100Ωm Both thereceiver spacing and line spacing are 1meter There are 25points for injecting currents in each profile The backgroundresistivity is approximated from forward modeling and it isused for the initial model in the inversion

Model 2 (Figure 6(b)) is almost the same as the previousone Based on the previous model we added some topogra-phy to this model The highest elevation of the topography is4meters The slope in Figure 6(b) is 30 degrees which leadsthe length of slope to be 8meters

The inversion results belong the profile at119884 = 0 are shownhere Figures 6(c) and 6(d) are the contourline of apparentresistivity at 119884 = 0 for models 1 and 2 Figure 6(c) shows anapparent resistivity distribution for a standard dipole-dipoleanomaly The resistivity of wall rock below and above theanomaly is well recovered and the value is close to 100ΩmFor Figure 6(d) the apparent resistivity is distorted by theartificial anomaly caused by topography We can see that theresistivity of wall rock is almost 3 times as the true valueand the anomaly caused by the 3D low resistivity structureis unclear

Figures 6(e) and 6(f) show one of vertical sections ofinversion results at 119884 = 0 We can see that the 3D conductiveanomaly is well recovered either for the flat surface model orthemodel with topographywithout the redundant structures

42 Complex Model The illustration of this model is shownin Figure 7(a) with two anomalous bodies below some valleyThe first anomalous body is a highly resistive body (500Ωm)buried 2meters underground with the size of 4m times 4m times

4m The second anomaly is a conductive body (10Ωm)burried 4meters below the surface with the size of 4m times

4m times 6m The horizontal distance between these twoanomalous bodies is 4meters The resistivity of the wall rock(background resistivity) is 100ΩmThe observation profile isoriented to 119909 direction with observation spacing of 1meterThere are 31 points for injecting current along each profileThe survey is conducted along five profiles with the spacingof 1meterThe initial model is obtained in a similar way to theone we described in Section 41

Figure 7(b) shows some slices of the apparent resistivitydistribution for a standard dipole-dipole anomalyThe appar-ent resistivity is distorted by the artificial anomaly caused bytopography

After 11 iterations which take 4115 seconds the nor-malized misfit decreased from 1263 to 497 which isaround the noise level of the data The inversion result isshown in Figure 7(c) From the figure we can see that thetwo anomalous bodies are well recovered and the artificialanomaly caused by the topography is eliminated Figure 8shows that the predicted data is very similar to the observeddata on section 119884 = 0

5 Case Study

The primary object of the survey is to estimate the basementdepth which can be used as some information for building

08

24

16

12

20

00 minus20 minus60 minus80minus40 minus100

Sens

itivi

ty (J

)

Z (m)

Figure 4Themodified sensitivity changes as a function of119885 at119883 =

0

00

00

40 80

minus40

minus80

minus120

minus80 minus40

X (m)

Z(m

)

(a)

00

minus40

minus80

minus120

00 40 80minus80 minus40

X (m)

Z(m

)

(b)

Figure 5 A comparison of inversion between original (a) andmodified (b) sensitivity at section 119884 = 0

house The survey site is on the river bed where the topogra-phy is quite flat We employed the multielectrode resistivitymethod with dipole-dipole array configurationThree surveylines are conducted with line distance of 30m and electrodedistance of 15m totally we have 60 electrodes The surveyis conducted in the winter season and the frozen soil withthickness of 1mndash15m caused the high resistivity layer nearthe surface According to geological information the surfacelayer is quaternary alluvium which mostly consisted of loess

6 International Journal of Geophysics

YX

Z

100Ωm 10Ωm

3m

4m 4m4m

Y = 3Y = 0

Y = minus3

(a) 3D low resistivity body without topography

402000

0000 40

4080

80

minus8080minus40

minus40

Y = 3

Y = 0

Y = minus3

X (m)

Z(m

)

Y (m)

(b) 3D low resistivity body model with topography

80 85 90 95 100 105 110 115 120

minus20

minus40

minus60

00 40

10080

80minus80 minus40

X (m)

Z(m

)

120588 (Ωm)

(c) Apparent resistivity at 119884 = 0 for model 1

00

20

20 60 100 140 180 220 260 300 340

minus20

minus40

00 40

100

100200

80minus80 minus40

X (m)

Z(m

)120588 (Ωm)

(d) Apparent resistivity at 119884 = 0 for model 2

00

40 50 60 70 80 90 100 110 120

00 40 80

minus40

minus80

minus120

minus80 minus40

X (m)

Z(m

)

120588 (Ωm)

(e) Vertical section of inversion result at 119884 = 0 for model 1

00

40

10 20 30 40 50 60 70 80 90 100 110 120

minus40

minus80

00 40 80minus80 minus40

X (m)

Z(m

)

120588 (Ωm)

(f) Vertical section of inversion result at 119884 = 0 for model 2

Figure 6 A comparison of inversion result for the model without and with topography

silt and gravel The lower layer is Jurassic red sandstone andas such it is a relative stable layer

Based on the observed data for our survey configurationwe discretized our subsurface into 80 times 60 times 40 cells Wechoose 100Ωm as the resistivity of the wall rock After 10iterations the normalizedmisfit reaches 125The inversionresult is shown in Figure 9 In 55m along profile 135 drillingshows that the first 5m consisted of frozen soil sand andgravel from 5ndash98m we observed water-bearing ooze below98m is red sandstone Our inversion result shows very goodfitting with the information that we get from drilling

6 Conclusion

In this paper we implemented the 3DDC resistivity inversionbased on regularized conjugate gradient method We usedfinite element method with triangular prism discretizationfor forward modeling of apparent resistivity Synthetic modelstudies show that the inversion algorithm works well torecover the true model with complex topography The inver-sion process is very stable and computation speed is fast

The electric potential at each node caused by differentsource is precomputed in the forward modeling and saved

International Journal of Geophysics 7

00 40 80 120

(b)

40 8000

30 50 70 90 110 130 150 170 190 50 90 130 170 210

(c)

minus40minus40 minus80minus80minus120

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

Y=2

Y=3

Y=2

Y=1

Y=0

Y=0

Y=1

Y=3

Y=minus3

Y=minus2

Y=minus1

Y=minus3

Y=minus2

Y=minus1

00

0040

80

00 40 80

(a)

minus40

minus40

minus80minus80

minus40

minus20

minus60

minus80

minus100

minus120

Y = 3Y = 0Y = minus3

100Ωm

500Ωm10Ωm

X (m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)

Y(m)

X (m)X (m)

120588 (Ωm) 120588 (Ωm)

Figure 7 The inversion result for a complex model (a) A model with two bodies under a valley (b) Slices view of apparent resistivity (c)Slices view of 3D inversion result

8 International Journal of Geophysics

30 50 70 90 110 130 150 170

00 40 80 120

minus20

minus60

minus100

minus120 minus80 minus40

120588 (Ωm)

X (m)

Z(m

)

(a)

30 50 70 90 110 130 150 170

00 40 80 120

minus20

minus60

minus100

minus120 minus80 minus40

120588 (Ωm)

X (m)

Z(m

)

(b)

Figure 8 A comparison between observed apparent resistivity (a) and predicted apparent resistivity (b) for profile 119884 = 0

(a) Survey configuration

0

07

10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

09 11 13 15 17 19 21 23 25 27 29

minus10

minus5

0

minus10

minus5

0

minus10

minus5

Line 133

Line 135

Line 137

log(120588)

X (m)

Z(m

)Z

(m)

Z(m

)

(b) Inversion result

Figure 9 The inversion of real data

for the inversion stage to compute Jacobi matrix By doingthis the computation cost can be reduced dramatically in theinversion process Only one forward modeling is required inorder to compute the sensitivity matrix

We conducted a sensitivity analysis and introduced aweighting function to optimize sensitivity matrix By apply-ing this proper model weighting the vertical location of theanomalous body can be well recovered

Acknowledgments

The authors acknowledge National Natural Science Foun-dation of China Grants nos 40974076 41174104 ChineseAcademy Geological Sciences (Sinoprobe-03-02-04) and theKey Laboratory of Metallogenic Prediction of Nonferrous

Metals of Ministry of Education for the support of thisproject

References

[1] J-K Qiang and Y-Z Luo ldquoThe resistivity FEM numericalmodeling on 3-D undulating topographyrdquo Chinese Journal ofGeophysics vol 50 no 5 pp 1606ndash1613 2007 (Chinese)

[2] A C Tripp G W Hohmann and C M Swift Jr ldquoTwo-dimen-sional resistivity inversionrdquoGeophysics vol 49 no 10 pp 1708ndash1717 1984

[3] S K Park and G P Van ldquoInversion of pole-pole data for 3-Dresistivity structure beneath arrays of electrodesrdquo Geophysicsvol 56 no 7 pp 951ndash960 1991

[4] Y Sasaki ldquo3-D resistivity inversion using the finite-elementmethodrdquo Geophysics vol 59 no 12 pp 1839ndash1848 1994

International Journal of Geophysics 9

[5] J Zhang R L Mackie and T R Madden ldquo3-D resistivity for-ward modeling and inversion using conjugate gradientsrdquo Geo-physics vol 60 no 5 pp 1313ndash1325 1995

[6] M H Loke and R D Barker ldquoPractical techniques for 3D resis-tivity surveys and data inversionrdquo Geophysical Prospecting vol44 no 3 pp 499ndash523 1996

[7] X P Wu and G M Xu ldquoDerivation and analysis of partialderivative matrix in resistivity 3-D inversionrdquo Oil GeophysicalProspecting vol 34 pp 363ndash372 1999

[8] X-P Wu and G-M Xu ldquoStudy on 3-D resistivity inversionusing conjugate gradient methodrdquo Chinese Journal of Geophys-ics vol 43 no 3 pp 420ndash427 2000

[9] X-P Wu ldquo3-D resistivity inversion under the condition ofuneven terrainrdquoChinese Journal of Geophysics vol 48 no 4 pp932ndash936 2005

[10] H-F Liu J-X Liu R-W Guo X-K Deng and B-Y RuanldquoEfficient inversion of 3D IP data for continuous model withcomplex geometryrdquo Journal of Jilin University vol 41 no 4 pp1212ndash1218 2011

[11] N G Papadopoulos M-J Yi J-H Kim P Tsourlos and GN Tsokas ldquoGeophysical investigation of tumuli by means ofsurface 3D electrical resistivity tomographyrdquo Journal of AppliedGeophysics vol 70 no 3 pp 192ndash205 2010

[12] P I Tsourlos and R D Ogilvy ldquoAn algorithm for the 3-Dinversion of tomographic resistivity and induced polarisationdata preliminary resultsrdquo Journal of the Balkan GeophysicalSociety vol 2 pp 30ndash45 1999

[13] J G Huang B Y Ruan and G S Bao ldquoResistivity inversion on3-D section based on FEMrdquo Journal of Central South Universityof Technology vol 35 pp 295ndash299 2004

[14] TGunther C Rucker andK Spitzer ldquoThree-dimensionalmod-eling and inversion of dc resistivity data incorporating topogra-phymdashII inversionrdquo Geophysical Journal International vol 166no 2 pp 506ndash5517 2006

[15] G A Oldenborger and P S Routh ldquoThe point-spread functionmeasure of resolution for the 3-D electrical resistivity experi-mentrdquoGeophysical Journal International vol 176 no 2 pp 405ndash414 2009

[16] A Dey and H F Morrison ldquoResistivity modeling for arbitrarilyshaped three-dimensional structuresrdquoGeophysics vol 44 no 4pp 753ndash760 1979

[17] R G Ellis and D W Oldenburg ldquoThe pole-pole 3-D DC-re-sistivity inverse problem a conjugate- gradient approachrdquo Geo-physical Journal International vol 119 no 1 pp 187ndash194 1994

[18] D J LaBrecque M Miletto W Daily A Ramirez and EOwen ldquoThe effects of noise on Occamrsquos inversion of resistivitytomography datardquo Geophysics vol 61 no 2 pp 538ndash548 1996

[19] M-J Yi J-H Kim Y Song S-J Cho S-H Chung and J-H Suh ldquoThree-dimensional imaging of subsurface structuresusing resistivity datardquoGeophysical Prospecting vol 49 no 4 pp483ndash497 2001

[20] C C Pain J V Herwanger M H Worthington and C R E deOliveira ldquoEffectivemultidimensional resistivity inversion usingfinite-element techniquesrdquo Geophysical Journal Internationalvol 151 no 3 pp 710ndash728 2002

[21] A Pidlisecky E Haber and R Knight ldquoRESINVM3D a 3Dresistivity inversion packagerdquo Geophysics vol 72 no 2 pp H1ndashH10 2007

[22] L Marescot S P Lopes S Rigobert and A G Green ldquoNon-linear inversion of geoelectric data acquired across 3D objectsusing a finite-element approachrdquo Geophysics vol 73 no 3 ppF121ndashF133 2008

[23] M S Zhdanov Geophysical Inverse Theory and RegularizationProblems Elsevier New York NY USA 2002

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Geology Advances in

2 International Journal of Geophysics

FEM algorithm they get reasonable result Huang et al[13] increased the accuracy of inversion by introducing thevolume factor and switching from global inversion to localinversion However this modified inversion process can onlybe applied to flat surface Gunther et al [14] achieved 3Dresistivity inversion for arbitrary topography based on finiteelement with unstructured mesh Oldenborger and Routh[15] also introduced 3D resistivity inversion with the assis-tance of point spread function

In this paper we will focus on the study of fast 3D resis-tivity inversion problem for complex topography based onweighted regularized conjugate gradient method The mod-ified sensitivity matrix can be used to recover relatively deepanomalous body

2 3D DC Resistivity Forward Modeling andInversion Theory

21 3119863 Resistivity Forward Modeling with Complex Topogra-phy The forwardmodelingmethod is based onfinitemethodwith triangular prism discretization [1] which can simulatevariable topography We approximated the unknown electricpotential within each element by dual-linear function andmixed boundary condition [16] is implemented By assem-bling each element equation we can formulate a global linearsystem of equations as follows

119870(119909119910119911120590)119880(119909119910119911120590) = 119875(119909119904119910119904119911119904120591Ω) (1)

where 119909 119910 and 119911 are the coordinate for the node 120590 is theconductivity for the element Ω and 120591 indicate the boundaryat infinity and the earthrsquos surface 119870 is the element equationmatrix which is related to the location of nodes and elementconductivity 119880 is the electric potential which is a functionof node location and conductivity in the node 119875 containsthe information of source location and boundary conditionsEquation (1) can be solved using incomplete Cholesey con-jugate gradient method (ICCG) By decomposing the matrix119870 and substituting back to (1) we can compute the electricpotential 119880 (these values need to be stored to formulate theJacobi matrix in inversion) at every node corresponding to allpoint electric sources Apparent resistivity can be computedfor different source-receiver array configuration

The algorithm is implemented by Fortran language on PCmachine We adopted triangular prism discretization of thesubsurface in order to simulate the topography and complexanomalous bodies underground

22 AdaptiveWeighed Regularization Inversion Algorithm Ina compact form the forward modeling process describedpreviously can be written as follows

119889119894 = 119860 (119898119895) 119894 = 1 2 119898 119895 = 1 2 119899 (2)

where 119889119894 is the observed data 119898119895 is the model parameter(conductivity in the element) and 119860 is a nonlinear operatorwhich is implemented using finite element method Theinverse problem in (2) to recover model parameter from

observed data is an ill-posed problem with nonunique solu-tion

To obtain stable solution for this ill-posed problem oneneeds to consider regularization theory The regularized the-ory has been applied to 3D DC resistivity inversion for a longtime [11 17ndash22] In our study we consider a minimization ofTikhonov parametric functional as follows

119875120572(119889119898) = 119860 (119898) minus 119889

2+ 120572119904 (119898119898apr) = min (3)

where 120572 is regularization parameter (Lagrange parameter)119904(119898) is a nonnegative stabilizing functional of model param-eter with the property of monotonic decreasing and 119898apr issome a priori information of model parameters

It was shown by Zhdanov [23] that the regularizationparameter can be selected using an adaptive scheme as fol-lows

120572119896 = 1205720119902119896 119896 = 1 2 3 119899 0 lt 119902 lt 1 (4)

where 119902 is a real number smaller than 1 and 1205720 is the initialvalue for regularization parameter which is selected in sucha way to balance the misfit functional and the stabilizer asfollows

1205720 =119860(119898) minus 119889

2

10038171003817100381710038171003817119898 minus 119898apr

10038171003817100381710038171003817

2 (5)

The quality and amplitude of different data vary in geo-physical applicationDataweightingmatrix is preferred in theinversion process in order to increase the contribution of datawith good quality without suppressing geophysical informa-tion from data with low amplitude After taking into accountof data weighting the Tikhonov parametric functional in (3)can be modified as following

119875120572(119889119898) = (119882119889119860 (119898) minus119882119889119889)

119879(119882119889119860 (119898) minus119882119889119889)

+ 120572(119882119898119898 minus119882119898119898apr)119879(119882119898119898 minus119882119898119898apr)

= min(6)

where119882119889 is the data weighting matrix which can be selectedaccording to the amplitude and quality of different data thesuperscript 119879 indicates the complex transpose 119882119898 is themodel weighting matrix which results in practically equalresolution of the inversion with respect to different param-eters of the model It was shown that the model weightingmatrix can be computed as follows in such equal resolutioncriteria

119882119898 = diag (119865lowast119865)18 (7)

where 119865 is the Frechet derivative matrix corresponding tothe nonlinear forward modeling operator 119860 in (2) 119865lowast is thetranspose of 119865

International Journal of Geophysics 3

We used regularized conjugated gradient method [23] forthe minimization of (6)

r119899 = 119860 (119898119899) minus d

l120572119899 = 119865lowast1198822

119889 r119899 + 1205721198822

119898 (119898119899 minus 119898apr)

120573120572

119899 =

1003817100381710038171003817l120572119899

1003817100381710038171003817

2

1003817100381710038171003817l120572119899minus11003817100381710038171003817

2

l120572119899 = l120572119899 + 120573120572

119899 l120572

119899minus1

119896120572

119899 =

(l120572119899 l120572119899)

(10038171003817100381710038171003817119882119889119865119898

119899

l11989910038171003817100381710038171003817

2+ 120572

10038171003817100381710038171003817119882119898 l119899

10038171003817100381710038171003817

2)

119898119899+1 = 119898119899 minus 119896120572

119899 l120572

119899

(8)

where r119899 is the residual vector between the observed data andpredicted data

Based on the algorithm in (8) we implemented 3D regu-larized inversion for DC resistivity with complex topography

23The Formulation of Frechet DerivativeMatrix Weusuallyfix the size of cells for model parameter in the DC resistivityinversion and assume that the apparent resistivity on theearthrsquos surface is only function of the conductivity distribu-tion in the cells As such the Frechet derivative matrix iscomputed as follows

119865 =120597120588si120597120590119895

(9)

where 120588si is the measured apparent resistivity data and 120590119895 isthe conductivity value for the 119895th cell

As we know the measured apparent resistivity on theearthrsquos surface is related to the electric potential in the obser-vation position Take the dipole-dipole array as an example

120597120588119860119861119872119873119878

120597120590=

minus119866119860119861119872119873

119868119860119861

(120597119880119860119872

120597120590minus120597119880119860119873

120597120590minus120597119880119861119872

120597120590+120597119880119861119873

120597120590)

(10)

where 119866119860119861119872119873 is array coefficient which is a function of

electrode spacing 119868119860119861 is the injected current and 119880119860119872 is the

electric potential at 119872 when current is injected from elec-trode 119860 Equation (10) indicates that the derivative of appar-ent resistivity to the model conductivity can be transformedto the derivative of electric potential to the conductivity ineach cell The electric potential on the earthrsquos surface can becomputed from (1) By considering the first order derivativeof both sides of (1) to the model parameter we can obtain thefollowing formula

K120597U120597120590

+120597K120597120590

U = 0 (11)

which can be written as follows

K120597U120597120590

= minus120597K120597120590

U (12)

Equation (12) is in a similar to that of as (1)The right sideof (12) can be treated as an auxiliary currentWe can solve thelinear system of equations in (1) to obtain the derivative of theelectric potential on the earthrsquos surface to the conductivity ineach cell This derivative can also be computed by using thefollowing equations

120597119880119860119872

120597120590119890

= minus 1198801198601 1198801198602 1198801198603 1198801198604 1198801198605 1198801198606

sdot

[[[[[[[[[[[[[[[[[[[[[[[[[[

[

12059711989611

120597120590

12059711989612

120597120590

12059711989613

120597120590

12059711989614

120597120590

12059711989615

120597120590

12059711989616

120597120590

12059711989621

120597120590

12059711989622

120597120590

12059711989623

120597120590

12059711989624

120597120590

12059711989625

120597120590

12059711989626

120597120590

12059711989631

120597120590

12059711989632

120597120590

12059711989633

120597120590

12059711989634

120597120590

12059711989635

120597120590

12059711989636

120597120590

12059711989641

120597120590

12059711989642

120597120590

12059711989643

120597120590

12059711989644

120597120590

12059711989645

120597120590

12059711989646

120597120590

12059711989651

120597120590

12059711989652

120597120590

12059711989653

120597120590

12059711989654

120597120590

12059711989655

120597120590

12059711989656

120597120590

12059711989661

120597120590

12059711989662

120597120590

12059711989663

120597120590

12059711989664

120597120590

12059711989665

120597120590

12059711989666

120597120590

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

sdot

1198801198601

1198801198602

1198801198603

1198801198604

1198801198605

1198801198606

(13)

As we can see the computation cost in formulating the Jacobimatrix is trivial since the electric potential in all the nodescorresponding to different source location already precom-puted at the stage of forward modeling and the stiffnessmatrix is also computed in forward modeling process Thecomputation of Jacobi matrix is only a transformation whichutilizes the forward modeling result In real application weonly need the forward modeling result on the nodes whichlocate on the earthrsquos surface instead of the electric potentialon all nodes Also one needs to notice that only one forwardmodeling is required to obtain Jacobi matrix As such thecomputation cost for inversion is reduced dramatically andit makes the inversion be implemented on a PC machine

4 International Journal of Geophysics

3 Discussions for Inversion

31 Stability Issue for Inversion In 3D resistivity inversionthe misfit for inversion will stay in a relative large level ifthe initial model is not properly chosen which will cause thebackground conductivity to be either too small or too largeIn this study we implemented some computation to calculatethemaximum andminimumvalue of apparent resistivity andselect reasonable background conductivity from this infor-mation to be our initial model for inversion

Another main issue is that the inversion will be unstableif the variation of inversion parameter is either too large ortoo small In order to solve this problem we will take thelogarithmof themodel parameter and do the inversion in thisnew space As a result the variation of model parameter isin control Moreover the upper and lower limits for modelparameter can be set in the inversion by applying thismethod

32 Speed for Inversion The modern PC machine is alreadyvery powerful now with the development of CPU andincrease of memory However it can still take hours and evendays to run an inversion if the number of nodes increases dra-matically especially in 3D problem As a result it is veryimportant to choose proper method of data storage and fastsolver for linear system of equations Incomplete Choleseyconjugate gradient method (ICCG) works well for DC resis-tivitymethodwithmanymoving electric source By using thismethod we only need to decompose the stiffnessmatrix oncewhich takes approximately several minutes and the substitu-tion is also very fast The computation of Jacobi matrix is atime-consuming process In this paper we used the methodof electric potential combination to form the Jacobi matrixwhich can reduce the computation time dramaticallyWe runthe inversion in a PC machine and the model discretizationis 44 times 22 times 17 times 2 which generates 32912 cells The numberof observation is 1505 with 5 profiles and 155 points to injectcurrents The forward modeling takes around 5 minutes andeach iteration of inversion takes approximately 6 minutesThe formulation of Jacobimatrix takes around 30 seconds andit takes other 30 seconds to update the model The total timefor inversion is around 1 hour if the number of iteration is setto be 10

33 Sensitivity Analysis The sensitivity we mentioned heredescribes how sensitive the measured apparent resistivity tothe change of conductivity in each element is Aswe know theelectric potential decreases as the distance between the obser-vation point and the source increasesThe apparent resistivityalso decreases as the observation point moves away from thesource since the apparent resistivity is proportional to theelectric potential Figure 1 shows a sensitivity distribution in119883119885 vertical section As we can see from Figure 1 the datais more sensitive to shallow structure than deep structureFigure 2 shows the sensitivity changes as a function of depthat 119883 = 0 The curve in Figure 2 concaves upward and thesensitivity decreases dramatically as depth increases As aresult the inverted anomaly will move upward if we usethe sensitivity shown in Figures 1 and 2 We can modifythe sensitivity distribution to make it decay slower than

00 40 80 120

02 04 06 08 10 12Sensitivity (J)

minus20

minus60

minus100

minus120 minus80 minus40

X (m)

Z(m

)

Figure 1 Original sensitivity distribution in119883119885 section

0000

16

08

04

12

minus20 minus60 minus80minus40 minus100

Sens

itivi

ty (J

)

Z (m)

Figure 2 The original sensitivity changes as a function of 119885 at 119883 =

0

00 40 80 120

02 04 06 08 10 12Sensitivity (J)

minus20

minus60

minus100

minus120 minus80 minus40

X (m)

Z(m

)

Figure 3 Modified sensitivity distribution in119883119885 section

original sensitivity in order to increase the resolution ofdeep structure Figures 3 and 4 show the modified sensitivitydistribution The synthetic model shows that this modifiedsensitive distribution works well to recover the true model(Figure 5)

4 Numerical Examples

The following synthetic model studies are based on thedipole-dipole array

41 Simple Model For the simple case we have two differentmodelsModel 1 (Figure 6(a)) is a 4mtimes 4mtimes 4m conductive

International Journal of Geophysics 5

body buried 3meters underground with the resistivity of10Ωm The resistivity of wall rock is 100Ωm Both thereceiver spacing and line spacing are 1meter There are 25points for injecting currents in each profile The backgroundresistivity is approximated from forward modeling and it isused for the initial model in the inversion

Model 2 (Figure 6(b)) is almost the same as the previousone Based on the previous model we added some topogra-phy to this model The highest elevation of the topography is4meters The slope in Figure 6(b) is 30 degrees which leadsthe length of slope to be 8meters

The inversion results belong the profile at119884 = 0 are shownhere Figures 6(c) and 6(d) are the contourline of apparentresistivity at 119884 = 0 for models 1 and 2 Figure 6(c) shows anapparent resistivity distribution for a standard dipole-dipoleanomaly The resistivity of wall rock below and above theanomaly is well recovered and the value is close to 100ΩmFor Figure 6(d) the apparent resistivity is distorted by theartificial anomaly caused by topography We can see that theresistivity of wall rock is almost 3 times as the true valueand the anomaly caused by the 3D low resistivity structureis unclear

Figures 6(e) and 6(f) show one of vertical sections ofinversion results at 119884 = 0 We can see that the 3D conductiveanomaly is well recovered either for the flat surface model orthemodel with topographywithout the redundant structures

42 Complex Model The illustration of this model is shownin Figure 7(a) with two anomalous bodies below some valleyThe first anomalous body is a highly resistive body (500Ωm)buried 2meters underground with the size of 4m times 4m times

4m The second anomaly is a conductive body (10Ωm)burried 4meters below the surface with the size of 4m times

4m times 6m The horizontal distance between these twoanomalous bodies is 4meters The resistivity of the wall rock(background resistivity) is 100ΩmThe observation profile isoriented to 119909 direction with observation spacing of 1meterThere are 31 points for injecting current along each profileThe survey is conducted along five profiles with the spacingof 1meterThe initial model is obtained in a similar way to theone we described in Section 41

Figure 7(b) shows some slices of the apparent resistivitydistribution for a standard dipole-dipole anomalyThe appar-ent resistivity is distorted by the artificial anomaly caused bytopography

After 11 iterations which take 4115 seconds the nor-malized misfit decreased from 1263 to 497 which isaround the noise level of the data The inversion result isshown in Figure 7(c) From the figure we can see that thetwo anomalous bodies are well recovered and the artificialanomaly caused by the topography is eliminated Figure 8shows that the predicted data is very similar to the observeddata on section 119884 = 0

5 Case Study

The primary object of the survey is to estimate the basementdepth which can be used as some information for building

08

24

16

12

20

00 minus20 minus60 minus80minus40 minus100

Sens

itivi

ty (J

)

Z (m)

Figure 4Themodified sensitivity changes as a function of119885 at119883 =

0

00

00

40 80

minus40

minus80

minus120

minus80 minus40

X (m)

Z(m

)

(a)

00

minus40

minus80

minus120

00 40 80minus80 minus40

X (m)

Z(m

)

(b)

Figure 5 A comparison of inversion between original (a) andmodified (b) sensitivity at section 119884 = 0

house The survey site is on the river bed where the topogra-phy is quite flat We employed the multielectrode resistivitymethod with dipole-dipole array configurationThree surveylines are conducted with line distance of 30m and electrodedistance of 15m totally we have 60 electrodes The surveyis conducted in the winter season and the frozen soil withthickness of 1mndash15m caused the high resistivity layer nearthe surface According to geological information the surfacelayer is quaternary alluvium which mostly consisted of loess

6 International Journal of Geophysics

YX

Z

100Ωm 10Ωm

3m

4m 4m4m

Y = 3Y = 0

Y = minus3

(a) 3D low resistivity body without topography

402000

0000 40

4080

80

minus8080minus40

minus40

Y = 3

Y = 0

Y = minus3

X (m)

Z(m

)

Y (m)

(b) 3D low resistivity body model with topography

80 85 90 95 100 105 110 115 120

minus20

minus40

minus60

00 40

10080

80minus80 minus40

X (m)

Z(m

)

120588 (Ωm)

(c) Apparent resistivity at 119884 = 0 for model 1

00

20

20 60 100 140 180 220 260 300 340

minus20

minus40

00 40

100

100200

80minus80 minus40

X (m)

Z(m

)120588 (Ωm)

(d) Apparent resistivity at 119884 = 0 for model 2

00

40 50 60 70 80 90 100 110 120

00 40 80

minus40

minus80

minus120

minus80 minus40

X (m)

Z(m

)

120588 (Ωm)

(e) Vertical section of inversion result at 119884 = 0 for model 1

00

40

10 20 30 40 50 60 70 80 90 100 110 120

minus40

minus80

00 40 80minus80 minus40

X (m)

Z(m

)

120588 (Ωm)

(f) Vertical section of inversion result at 119884 = 0 for model 2

Figure 6 A comparison of inversion result for the model without and with topography

silt and gravel The lower layer is Jurassic red sandstone andas such it is a relative stable layer

Based on the observed data for our survey configurationwe discretized our subsurface into 80 times 60 times 40 cells Wechoose 100Ωm as the resistivity of the wall rock After 10iterations the normalizedmisfit reaches 125The inversionresult is shown in Figure 9 In 55m along profile 135 drillingshows that the first 5m consisted of frozen soil sand andgravel from 5ndash98m we observed water-bearing ooze below98m is red sandstone Our inversion result shows very goodfitting with the information that we get from drilling

6 Conclusion

In this paper we implemented the 3DDC resistivity inversionbased on regularized conjugate gradient method We usedfinite element method with triangular prism discretizationfor forward modeling of apparent resistivity Synthetic modelstudies show that the inversion algorithm works well torecover the true model with complex topography The inver-sion process is very stable and computation speed is fast

The electric potential at each node caused by differentsource is precomputed in the forward modeling and saved

International Journal of Geophysics 7

00 40 80 120

(b)

40 8000

30 50 70 90 110 130 150 170 190 50 90 130 170 210

(c)

minus40minus40 minus80minus80minus120

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

Y=2

Y=3

Y=2

Y=1

Y=0

Y=0

Y=1

Y=3

Y=minus3

Y=minus2

Y=minus1

Y=minus3

Y=minus2

Y=minus1

00

0040

80

00 40 80

(a)

minus40

minus40

minus80minus80

minus40

minus20

minus60

minus80

minus100

minus120

Y = 3Y = 0Y = minus3

100Ωm

500Ωm10Ωm

X (m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)

Y(m)

X (m)X (m)

120588 (Ωm) 120588 (Ωm)

Figure 7 The inversion result for a complex model (a) A model with two bodies under a valley (b) Slices view of apparent resistivity (c)Slices view of 3D inversion result

8 International Journal of Geophysics

30 50 70 90 110 130 150 170

00 40 80 120

minus20

minus60

minus100

minus120 minus80 minus40

120588 (Ωm)

X (m)

Z(m

)

(a)

30 50 70 90 110 130 150 170

00 40 80 120

minus20

minus60

minus100

minus120 minus80 minus40

120588 (Ωm)

X (m)

Z(m

)

(b)

Figure 8 A comparison between observed apparent resistivity (a) and predicted apparent resistivity (b) for profile 119884 = 0

(a) Survey configuration

0

07

10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

09 11 13 15 17 19 21 23 25 27 29

minus10

minus5

0

minus10

minus5

0

minus10

minus5

Line 133

Line 135

Line 137

log(120588)

X (m)

Z(m

)Z

(m)

Z(m

)

(b) Inversion result

Figure 9 The inversion of real data

for the inversion stage to compute Jacobi matrix By doingthis the computation cost can be reduced dramatically in theinversion process Only one forward modeling is required inorder to compute the sensitivity matrix

We conducted a sensitivity analysis and introduced aweighting function to optimize sensitivity matrix By apply-ing this proper model weighting the vertical location of theanomalous body can be well recovered

Acknowledgments

The authors acknowledge National Natural Science Foun-dation of China Grants nos 40974076 41174104 ChineseAcademy Geological Sciences (Sinoprobe-03-02-04) and theKey Laboratory of Metallogenic Prediction of Nonferrous

Metals of Ministry of Education for the support of thisproject

References

[1] J-K Qiang and Y-Z Luo ldquoThe resistivity FEM numericalmodeling on 3-D undulating topographyrdquo Chinese Journal ofGeophysics vol 50 no 5 pp 1606ndash1613 2007 (Chinese)

[2] A C Tripp G W Hohmann and C M Swift Jr ldquoTwo-dimen-sional resistivity inversionrdquoGeophysics vol 49 no 10 pp 1708ndash1717 1984

[3] S K Park and G P Van ldquoInversion of pole-pole data for 3-Dresistivity structure beneath arrays of electrodesrdquo Geophysicsvol 56 no 7 pp 951ndash960 1991

[4] Y Sasaki ldquo3-D resistivity inversion using the finite-elementmethodrdquo Geophysics vol 59 no 12 pp 1839ndash1848 1994

International Journal of Geophysics 9

[5] J Zhang R L Mackie and T R Madden ldquo3-D resistivity for-ward modeling and inversion using conjugate gradientsrdquo Geo-physics vol 60 no 5 pp 1313ndash1325 1995

[6] M H Loke and R D Barker ldquoPractical techniques for 3D resis-tivity surveys and data inversionrdquo Geophysical Prospecting vol44 no 3 pp 499ndash523 1996

[7] X P Wu and G M Xu ldquoDerivation and analysis of partialderivative matrix in resistivity 3-D inversionrdquo Oil GeophysicalProspecting vol 34 pp 363ndash372 1999

[8] X-P Wu and G-M Xu ldquoStudy on 3-D resistivity inversionusing conjugate gradient methodrdquo Chinese Journal of Geophys-ics vol 43 no 3 pp 420ndash427 2000

[9] X-P Wu ldquo3-D resistivity inversion under the condition ofuneven terrainrdquoChinese Journal of Geophysics vol 48 no 4 pp932ndash936 2005

[10] H-F Liu J-X Liu R-W Guo X-K Deng and B-Y RuanldquoEfficient inversion of 3D IP data for continuous model withcomplex geometryrdquo Journal of Jilin University vol 41 no 4 pp1212ndash1218 2011

[11] N G Papadopoulos M-J Yi J-H Kim P Tsourlos and GN Tsokas ldquoGeophysical investigation of tumuli by means ofsurface 3D electrical resistivity tomographyrdquo Journal of AppliedGeophysics vol 70 no 3 pp 192ndash205 2010

[12] P I Tsourlos and R D Ogilvy ldquoAn algorithm for the 3-Dinversion of tomographic resistivity and induced polarisationdata preliminary resultsrdquo Journal of the Balkan GeophysicalSociety vol 2 pp 30ndash45 1999

[13] J G Huang B Y Ruan and G S Bao ldquoResistivity inversion on3-D section based on FEMrdquo Journal of Central South Universityof Technology vol 35 pp 295ndash299 2004

[14] TGunther C Rucker andK Spitzer ldquoThree-dimensionalmod-eling and inversion of dc resistivity data incorporating topogra-phymdashII inversionrdquo Geophysical Journal International vol 166no 2 pp 506ndash5517 2006

[15] G A Oldenborger and P S Routh ldquoThe point-spread functionmeasure of resolution for the 3-D electrical resistivity experi-mentrdquoGeophysical Journal International vol 176 no 2 pp 405ndash414 2009

[16] A Dey and H F Morrison ldquoResistivity modeling for arbitrarilyshaped three-dimensional structuresrdquoGeophysics vol 44 no 4pp 753ndash760 1979

[17] R G Ellis and D W Oldenburg ldquoThe pole-pole 3-D DC-re-sistivity inverse problem a conjugate- gradient approachrdquo Geo-physical Journal International vol 119 no 1 pp 187ndash194 1994

[18] D J LaBrecque M Miletto W Daily A Ramirez and EOwen ldquoThe effects of noise on Occamrsquos inversion of resistivitytomography datardquo Geophysics vol 61 no 2 pp 538ndash548 1996

[19] M-J Yi J-H Kim Y Song S-J Cho S-H Chung and J-H Suh ldquoThree-dimensional imaging of subsurface structuresusing resistivity datardquoGeophysical Prospecting vol 49 no 4 pp483ndash497 2001

[20] C C Pain J V Herwanger M H Worthington and C R E deOliveira ldquoEffectivemultidimensional resistivity inversion usingfinite-element techniquesrdquo Geophysical Journal Internationalvol 151 no 3 pp 710ndash728 2002

[21] A Pidlisecky E Haber and R Knight ldquoRESINVM3D a 3Dresistivity inversion packagerdquo Geophysics vol 72 no 2 pp H1ndashH10 2007

[22] L Marescot S P Lopes S Rigobert and A G Green ldquoNon-linear inversion of geoelectric data acquired across 3D objectsusing a finite-element approachrdquo Geophysics vol 73 no 3 ppF121ndashF133 2008

[23] M S Zhdanov Geophysical Inverse Theory and RegularizationProblems Elsevier New York NY USA 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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EcologyInternational Journal of

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EarthquakesJournal of

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Applied ampEnvironmentalSoil Science

Volume 2014

Mining

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Journal of

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International Journal of

Geophysics

OceanographyInternational Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

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MineralogyInternational Journal of

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MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

International Journal of Geophysics 3

We used regularized conjugated gradient method [23] forthe minimization of (6)

r119899 = 119860 (119898119899) minus d

l120572119899 = 119865lowast1198822

119889 r119899 + 1205721198822

119898 (119898119899 minus 119898apr)

120573120572

119899 =

1003817100381710038171003817l120572119899

1003817100381710038171003817

2

1003817100381710038171003817l120572119899minus11003817100381710038171003817

2

l120572119899 = l120572119899 + 120573120572

119899 l120572

119899minus1

119896120572

119899 =

(l120572119899 l120572119899)

(10038171003817100381710038171003817119882119889119865119898

119899

l11989910038171003817100381710038171003817

2+ 120572

10038171003817100381710038171003817119882119898 l119899

10038171003817100381710038171003817

2)

119898119899+1 = 119898119899 minus 119896120572

119899 l120572

119899

(8)

where r119899 is the residual vector between the observed data andpredicted data

Based on the algorithm in (8) we implemented 3D regu-larized inversion for DC resistivity with complex topography

23The Formulation of Frechet DerivativeMatrix Weusuallyfix the size of cells for model parameter in the DC resistivityinversion and assume that the apparent resistivity on theearthrsquos surface is only function of the conductivity distribu-tion in the cells As such the Frechet derivative matrix iscomputed as follows

119865 =120597120588si120597120590119895

(9)

where 120588si is the measured apparent resistivity data and 120590119895 isthe conductivity value for the 119895th cell

As we know the measured apparent resistivity on theearthrsquos surface is related to the electric potential in the obser-vation position Take the dipole-dipole array as an example

120597120588119860119861119872119873119878

120597120590=

minus119866119860119861119872119873

119868119860119861

(120597119880119860119872

120597120590minus120597119880119860119873

120597120590minus120597119880119861119872

120597120590+120597119880119861119873

120597120590)

(10)

where 119866119860119861119872119873 is array coefficient which is a function of

electrode spacing 119868119860119861 is the injected current and 119880119860119872 is the

electric potential at 119872 when current is injected from elec-trode 119860 Equation (10) indicates that the derivative of appar-ent resistivity to the model conductivity can be transformedto the derivative of electric potential to the conductivity ineach cell The electric potential on the earthrsquos surface can becomputed from (1) By considering the first order derivativeof both sides of (1) to the model parameter we can obtain thefollowing formula

K120597U120597120590

+120597K120597120590

U = 0 (11)

which can be written as follows

K120597U120597120590

= minus120597K120597120590

U (12)

Equation (12) is in a similar to that of as (1)The right sideof (12) can be treated as an auxiliary currentWe can solve thelinear system of equations in (1) to obtain the derivative of theelectric potential on the earthrsquos surface to the conductivity ineach cell This derivative can also be computed by using thefollowing equations

120597119880119860119872

120597120590119890

= minus 1198801198601 1198801198602 1198801198603 1198801198604 1198801198605 1198801198606

sdot

[[[[[[[[[[[[[[[[[[[[[[[[[[

[

12059711989611

120597120590

12059711989612

120597120590

12059711989613

120597120590

12059711989614

120597120590

12059711989615

120597120590

12059711989616

120597120590

12059711989621

120597120590

12059711989622

120597120590

12059711989623

120597120590

12059711989624

120597120590

12059711989625

120597120590

12059711989626

120597120590

12059711989631

120597120590

12059711989632

120597120590

12059711989633

120597120590

12059711989634

120597120590

12059711989635

120597120590

12059711989636

120597120590

12059711989641

120597120590

12059711989642

120597120590

12059711989643

120597120590

12059711989644

120597120590

12059711989645

120597120590

12059711989646

120597120590

12059711989651

120597120590

12059711989652

120597120590

12059711989653

120597120590

12059711989654

120597120590

12059711989655

120597120590

12059711989656

120597120590

12059711989661

120597120590

12059711989662

120597120590

12059711989663

120597120590

12059711989664

120597120590

12059711989665

120597120590

12059711989666

120597120590

]]]]]]]]]]]]]]]]]]]]]]]]]]

]

sdot

1198801198601

1198801198602

1198801198603

1198801198604

1198801198605

1198801198606

(13)

As we can see the computation cost in formulating the Jacobimatrix is trivial since the electric potential in all the nodescorresponding to different source location already precom-puted at the stage of forward modeling and the stiffnessmatrix is also computed in forward modeling process Thecomputation of Jacobi matrix is only a transformation whichutilizes the forward modeling result In real application weonly need the forward modeling result on the nodes whichlocate on the earthrsquos surface instead of the electric potentialon all nodes Also one needs to notice that only one forwardmodeling is required to obtain Jacobi matrix As such thecomputation cost for inversion is reduced dramatically andit makes the inversion be implemented on a PC machine

4 International Journal of Geophysics

3 Discussions for Inversion

31 Stability Issue for Inversion In 3D resistivity inversionthe misfit for inversion will stay in a relative large level ifthe initial model is not properly chosen which will cause thebackground conductivity to be either too small or too largeIn this study we implemented some computation to calculatethemaximum andminimumvalue of apparent resistivity andselect reasonable background conductivity from this infor-mation to be our initial model for inversion

Another main issue is that the inversion will be unstableif the variation of inversion parameter is either too large ortoo small In order to solve this problem we will take thelogarithmof themodel parameter and do the inversion in thisnew space As a result the variation of model parameter isin control Moreover the upper and lower limits for modelparameter can be set in the inversion by applying thismethod

32 Speed for Inversion The modern PC machine is alreadyvery powerful now with the development of CPU andincrease of memory However it can still take hours and evendays to run an inversion if the number of nodes increases dra-matically especially in 3D problem As a result it is veryimportant to choose proper method of data storage and fastsolver for linear system of equations Incomplete Choleseyconjugate gradient method (ICCG) works well for DC resis-tivitymethodwithmanymoving electric source By using thismethod we only need to decompose the stiffnessmatrix oncewhich takes approximately several minutes and the substitu-tion is also very fast The computation of Jacobi matrix is atime-consuming process In this paper we used the methodof electric potential combination to form the Jacobi matrixwhich can reduce the computation time dramaticallyWe runthe inversion in a PC machine and the model discretizationis 44 times 22 times 17 times 2 which generates 32912 cells The numberof observation is 1505 with 5 profiles and 155 points to injectcurrents The forward modeling takes around 5 minutes andeach iteration of inversion takes approximately 6 minutesThe formulation of Jacobimatrix takes around 30 seconds andit takes other 30 seconds to update the model The total timefor inversion is around 1 hour if the number of iteration is setto be 10

33 Sensitivity Analysis The sensitivity we mentioned heredescribes how sensitive the measured apparent resistivity tothe change of conductivity in each element is Aswe know theelectric potential decreases as the distance between the obser-vation point and the source increasesThe apparent resistivityalso decreases as the observation point moves away from thesource since the apparent resistivity is proportional to theelectric potential Figure 1 shows a sensitivity distribution in119883119885 vertical section As we can see from Figure 1 the datais more sensitive to shallow structure than deep structureFigure 2 shows the sensitivity changes as a function of depthat 119883 = 0 The curve in Figure 2 concaves upward and thesensitivity decreases dramatically as depth increases As aresult the inverted anomaly will move upward if we usethe sensitivity shown in Figures 1 and 2 We can modifythe sensitivity distribution to make it decay slower than

00 40 80 120

02 04 06 08 10 12Sensitivity (J)

minus20

minus60

minus100

minus120 minus80 minus40

X (m)

Z(m

)

Figure 1 Original sensitivity distribution in119883119885 section

0000

16

08

04

12

minus20 minus60 minus80minus40 minus100

Sens

itivi

ty (J

)

Z (m)

Figure 2 The original sensitivity changes as a function of 119885 at 119883 =

0

00 40 80 120

02 04 06 08 10 12Sensitivity (J)

minus20

minus60

minus100

minus120 minus80 minus40

X (m)

Z(m

)

Figure 3 Modified sensitivity distribution in119883119885 section

original sensitivity in order to increase the resolution ofdeep structure Figures 3 and 4 show the modified sensitivitydistribution The synthetic model shows that this modifiedsensitive distribution works well to recover the true model(Figure 5)

4 Numerical Examples

The following synthetic model studies are based on thedipole-dipole array

41 Simple Model For the simple case we have two differentmodelsModel 1 (Figure 6(a)) is a 4mtimes 4mtimes 4m conductive

International Journal of Geophysics 5

body buried 3meters underground with the resistivity of10Ωm The resistivity of wall rock is 100Ωm Both thereceiver spacing and line spacing are 1meter There are 25points for injecting currents in each profile The backgroundresistivity is approximated from forward modeling and it isused for the initial model in the inversion

Model 2 (Figure 6(b)) is almost the same as the previousone Based on the previous model we added some topogra-phy to this model The highest elevation of the topography is4meters The slope in Figure 6(b) is 30 degrees which leadsthe length of slope to be 8meters

The inversion results belong the profile at119884 = 0 are shownhere Figures 6(c) and 6(d) are the contourline of apparentresistivity at 119884 = 0 for models 1 and 2 Figure 6(c) shows anapparent resistivity distribution for a standard dipole-dipoleanomaly The resistivity of wall rock below and above theanomaly is well recovered and the value is close to 100ΩmFor Figure 6(d) the apparent resistivity is distorted by theartificial anomaly caused by topography We can see that theresistivity of wall rock is almost 3 times as the true valueand the anomaly caused by the 3D low resistivity structureis unclear

Figures 6(e) and 6(f) show one of vertical sections ofinversion results at 119884 = 0 We can see that the 3D conductiveanomaly is well recovered either for the flat surface model orthemodel with topographywithout the redundant structures

42 Complex Model The illustration of this model is shownin Figure 7(a) with two anomalous bodies below some valleyThe first anomalous body is a highly resistive body (500Ωm)buried 2meters underground with the size of 4m times 4m times

4m The second anomaly is a conductive body (10Ωm)burried 4meters below the surface with the size of 4m times

4m times 6m The horizontal distance between these twoanomalous bodies is 4meters The resistivity of the wall rock(background resistivity) is 100ΩmThe observation profile isoriented to 119909 direction with observation spacing of 1meterThere are 31 points for injecting current along each profileThe survey is conducted along five profiles with the spacingof 1meterThe initial model is obtained in a similar way to theone we described in Section 41

Figure 7(b) shows some slices of the apparent resistivitydistribution for a standard dipole-dipole anomalyThe appar-ent resistivity is distorted by the artificial anomaly caused bytopography

After 11 iterations which take 4115 seconds the nor-malized misfit decreased from 1263 to 497 which isaround the noise level of the data The inversion result isshown in Figure 7(c) From the figure we can see that thetwo anomalous bodies are well recovered and the artificialanomaly caused by the topography is eliminated Figure 8shows that the predicted data is very similar to the observeddata on section 119884 = 0

5 Case Study

The primary object of the survey is to estimate the basementdepth which can be used as some information for building

08

24

16

12

20

00 minus20 minus60 minus80minus40 minus100

Sens

itivi

ty (J

)

Z (m)

Figure 4Themodified sensitivity changes as a function of119885 at119883 =

0

00

00

40 80

minus40

minus80

minus120

minus80 minus40

X (m)

Z(m

)

(a)

00

minus40

minus80

minus120

00 40 80minus80 minus40

X (m)

Z(m

)

(b)

Figure 5 A comparison of inversion between original (a) andmodified (b) sensitivity at section 119884 = 0

house The survey site is on the river bed where the topogra-phy is quite flat We employed the multielectrode resistivitymethod with dipole-dipole array configurationThree surveylines are conducted with line distance of 30m and electrodedistance of 15m totally we have 60 electrodes The surveyis conducted in the winter season and the frozen soil withthickness of 1mndash15m caused the high resistivity layer nearthe surface According to geological information the surfacelayer is quaternary alluvium which mostly consisted of loess

6 International Journal of Geophysics

YX

Z

100Ωm 10Ωm

3m

4m 4m4m

Y = 3Y = 0

Y = minus3

(a) 3D low resistivity body without topography

402000

0000 40

4080

80

minus8080minus40

minus40

Y = 3

Y = 0

Y = minus3

X (m)

Z(m

)

Y (m)

(b) 3D low resistivity body model with topography

80 85 90 95 100 105 110 115 120

minus20

minus40

minus60

00 40

10080

80minus80 minus40

X (m)

Z(m

)

120588 (Ωm)

(c) Apparent resistivity at 119884 = 0 for model 1

00

20

20 60 100 140 180 220 260 300 340

minus20

minus40

00 40

100

100200

80minus80 minus40

X (m)

Z(m

)120588 (Ωm)

(d) Apparent resistivity at 119884 = 0 for model 2

00

40 50 60 70 80 90 100 110 120

00 40 80

minus40

minus80

minus120

minus80 minus40

X (m)

Z(m

)

120588 (Ωm)

(e) Vertical section of inversion result at 119884 = 0 for model 1

00

40

10 20 30 40 50 60 70 80 90 100 110 120

minus40

minus80

00 40 80minus80 minus40

X (m)

Z(m

)

120588 (Ωm)

(f) Vertical section of inversion result at 119884 = 0 for model 2

Figure 6 A comparison of inversion result for the model without and with topography

silt and gravel The lower layer is Jurassic red sandstone andas such it is a relative stable layer

Based on the observed data for our survey configurationwe discretized our subsurface into 80 times 60 times 40 cells Wechoose 100Ωm as the resistivity of the wall rock After 10iterations the normalizedmisfit reaches 125The inversionresult is shown in Figure 9 In 55m along profile 135 drillingshows that the first 5m consisted of frozen soil sand andgravel from 5ndash98m we observed water-bearing ooze below98m is red sandstone Our inversion result shows very goodfitting with the information that we get from drilling

6 Conclusion

In this paper we implemented the 3DDC resistivity inversionbased on regularized conjugate gradient method We usedfinite element method with triangular prism discretizationfor forward modeling of apparent resistivity Synthetic modelstudies show that the inversion algorithm works well torecover the true model with complex topography The inver-sion process is very stable and computation speed is fast

The electric potential at each node caused by differentsource is precomputed in the forward modeling and saved

International Journal of Geophysics 7

00 40 80 120

(b)

40 8000

30 50 70 90 110 130 150 170 190 50 90 130 170 210

(c)

minus40minus40 minus80minus80minus120

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

Y=2

Y=3

Y=2

Y=1

Y=0

Y=0

Y=1

Y=3

Y=minus3

Y=minus2

Y=minus1

Y=minus3

Y=minus2

Y=minus1

00

0040

80

00 40 80

(a)

minus40

minus40

minus80minus80

minus40

minus20

minus60

minus80

minus100

minus120

Y = 3Y = 0Y = minus3

100Ωm

500Ωm10Ωm

X (m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)

Y(m)

X (m)X (m)

120588 (Ωm) 120588 (Ωm)

Figure 7 The inversion result for a complex model (a) A model with two bodies under a valley (b) Slices view of apparent resistivity (c)Slices view of 3D inversion result

8 International Journal of Geophysics

30 50 70 90 110 130 150 170

00 40 80 120

minus20

minus60

minus100

minus120 minus80 minus40

120588 (Ωm)

X (m)

Z(m

)

(a)

30 50 70 90 110 130 150 170

00 40 80 120

minus20

minus60

minus100

minus120 minus80 minus40

120588 (Ωm)

X (m)

Z(m

)

(b)

Figure 8 A comparison between observed apparent resistivity (a) and predicted apparent resistivity (b) for profile 119884 = 0

(a) Survey configuration

0

07

10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

09 11 13 15 17 19 21 23 25 27 29

minus10

minus5

0

minus10

minus5

0

minus10

minus5

Line 133

Line 135

Line 137

log(120588)

X (m)

Z(m

)Z

(m)

Z(m

)

(b) Inversion result

Figure 9 The inversion of real data

for the inversion stage to compute Jacobi matrix By doingthis the computation cost can be reduced dramatically in theinversion process Only one forward modeling is required inorder to compute the sensitivity matrix

We conducted a sensitivity analysis and introduced aweighting function to optimize sensitivity matrix By apply-ing this proper model weighting the vertical location of theanomalous body can be well recovered

Acknowledgments

The authors acknowledge National Natural Science Foun-dation of China Grants nos 40974076 41174104 ChineseAcademy Geological Sciences (Sinoprobe-03-02-04) and theKey Laboratory of Metallogenic Prediction of Nonferrous

Metals of Ministry of Education for the support of thisproject

References

[1] J-K Qiang and Y-Z Luo ldquoThe resistivity FEM numericalmodeling on 3-D undulating topographyrdquo Chinese Journal ofGeophysics vol 50 no 5 pp 1606ndash1613 2007 (Chinese)

[2] A C Tripp G W Hohmann and C M Swift Jr ldquoTwo-dimen-sional resistivity inversionrdquoGeophysics vol 49 no 10 pp 1708ndash1717 1984

[3] S K Park and G P Van ldquoInversion of pole-pole data for 3-Dresistivity structure beneath arrays of electrodesrdquo Geophysicsvol 56 no 7 pp 951ndash960 1991

[4] Y Sasaki ldquo3-D resistivity inversion using the finite-elementmethodrdquo Geophysics vol 59 no 12 pp 1839ndash1848 1994

International Journal of Geophysics 9

[5] J Zhang R L Mackie and T R Madden ldquo3-D resistivity for-ward modeling and inversion using conjugate gradientsrdquo Geo-physics vol 60 no 5 pp 1313ndash1325 1995

[6] M H Loke and R D Barker ldquoPractical techniques for 3D resis-tivity surveys and data inversionrdquo Geophysical Prospecting vol44 no 3 pp 499ndash523 1996

[7] X P Wu and G M Xu ldquoDerivation and analysis of partialderivative matrix in resistivity 3-D inversionrdquo Oil GeophysicalProspecting vol 34 pp 363ndash372 1999

[8] X-P Wu and G-M Xu ldquoStudy on 3-D resistivity inversionusing conjugate gradient methodrdquo Chinese Journal of Geophys-ics vol 43 no 3 pp 420ndash427 2000

[9] X-P Wu ldquo3-D resistivity inversion under the condition ofuneven terrainrdquoChinese Journal of Geophysics vol 48 no 4 pp932ndash936 2005

[10] H-F Liu J-X Liu R-W Guo X-K Deng and B-Y RuanldquoEfficient inversion of 3D IP data for continuous model withcomplex geometryrdquo Journal of Jilin University vol 41 no 4 pp1212ndash1218 2011

[11] N G Papadopoulos M-J Yi J-H Kim P Tsourlos and GN Tsokas ldquoGeophysical investigation of tumuli by means ofsurface 3D electrical resistivity tomographyrdquo Journal of AppliedGeophysics vol 70 no 3 pp 192ndash205 2010

[12] P I Tsourlos and R D Ogilvy ldquoAn algorithm for the 3-Dinversion of tomographic resistivity and induced polarisationdata preliminary resultsrdquo Journal of the Balkan GeophysicalSociety vol 2 pp 30ndash45 1999

[13] J G Huang B Y Ruan and G S Bao ldquoResistivity inversion on3-D section based on FEMrdquo Journal of Central South Universityof Technology vol 35 pp 295ndash299 2004

[14] TGunther C Rucker andK Spitzer ldquoThree-dimensionalmod-eling and inversion of dc resistivity data incorporating topogra-phymdashII inversionrdquo Geophysical Journal International vol 166no 2 pp 506ndash5517 2006

[15] G A Oldenborger and P S Routh ldquoThe point-spread functionmeasure of resolution for the 3-D electrical resistivity experi-mentrdquoGeophysical Journal International vol 176 no 2 pp 405ndash414 2009

[16] A Dey and H F Morrison ldquoResistivity modeling for arbitrarilyshaped three-dimensional structuresrdquoGeophysics vol 44 no 4pp 753ndash760 1979

[17] R G Ellis and D W Oldenburg ldquoThe pole-pole 3-D DC-re-sistivity inverse problem a conjugate- gradient approachrdquo Geo-physical Journal International vol 119 no 1 pp 187ndash194 1994

[18] D J LaBrecque M Miletto W Daily A Ramirez and EOwen ldquoThe effects of noise on Occamrsquos inversion of resistivitytomography datardquo Geophysics vol 61 no 2 pp 538ndash548 1996

[19] M-J Yi J-H Kim Y Song S-J Cho S-H Chung and J-H Suh ldquoThree-dimensional imaging of subsurface structuresusing resistivity datardquoGeophysical Prospecting vol 49 no 4 pp483ndash497 2001

[20] C C Pain J V Herwanger M H Worthington and C R E deOliveira ldquoEffectivemultidimensional resistivity inversion usingfinite-element techniquesrdquo Geophysical Journal Internationalvol 151 no 3 pp 710ndash728 2002

[21] A Pidlisecky E Haber and R Knight ldquoRESINVM3D a 3Dresistivity inversion packagerdquo Geophysics vol 72 no 2 pp H1ndashH10 2007

[22] L Marescot S P Lopes S Rigobert and A G Green ldquoNon-linear inversion of geoelectric data acquired across 3D objectsusing a finite-element approachrdquo Geophysics vol 73 no 3 ppF121ndashF133 2008

[23] M S Zhdanov Geophysical Inverse Theory and RegularizationProblems Elsevier New York NY USA 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

4 International Journal of Geophysics

3 Discussions for Inversion

31 Stability Issue for Inversion In 3D resistivity inversionthe misfit for inversion will stay in a relative large level ifthe initial model is not properly chosen which will cause thebackground conductivity to be either too small or too largeIn this study we implemented some computation to calculatethemaximum andminimumvalue of apparent resistivity andselect reasonable background conductivity from this infor-mation to be our initial model for inversion

Another main issue is that the inversion will be unstableif the variation of inversion parameter is either too large ortoo small In order to solve this problem we will take thelogarithmof themodel parameter and do the inversion in thisnew space As a result the variation of model parameter isin control Moreover the upper and lower limits for modelparameter can be set in the inversion by applying thismethod

32 Speed for Inversion The modern PC machine is alreadyvery powerful now with the development of CPU andincrease of memory However it can still take hours and evendays to run an inversion if the number of nodes increases dra-matically especially in 3D problem As a result it is veryimportant to choose proper method of data storage and fastsolver for linear system of equations Incomplete Choleseyconjugate gradient method (ICCG) works well for DC resis-tivitymethodwithmanymoving electric source By using thismethod we only need to decompose the stiffnessmatrix oncewhich takes approximately several minutes and the substitu-tion is also very fast The computation of Jacobi matrix is atime-consuming process In this paper we used the methodof electric potential combination to form the Jacobi matrixwhich can reduce the computation time dramaticallyWe runthe inversion in a PC machine and the model discretizationis 44 times 22 times 17 times 2 which generates 32912 cells The numberof observation is 1505 with 5 profiles and 155 points to injectcurrents The forward modeling takes around 5 minutes andeach iteration of inversion takes approximately 6 minutesThe formulation of Jacobimatrix takes around 30 seconds andit takes other 30 seconds to update the model The total timefor inversion is around 1 hour if the number of iteration is setto be 10

33 Sensitivity Analysis The sensitivity we mentioned heredescribes how sensitive the measured apparent resistivity tothe change of conductivity in each element is Aswe know theelectric potential decreases as the distance between the obser-vation point and the source increasesThe apparent resistivityalso decreases as the observation point moves away from thesource since the apparent resistivity is proportional to theelectric potential Figure 1 shows a sensitivity distribution in119883119885 vertical section As we can see from Figure 1 the datais more sensitive to shallow structure than deep structureFigure 2 shows the sensitivity changes as a function of depthat 119883 = 0 The curve in Figure 2 concaves upward and thesensitivity decreases dramatically as depth increases As aresult the inverted anomaly will move upward if we usethe sensitivity shown in Figures 1 and 2 We can modifythe sensitivity distribution to make it decay slower than

00 40 80 120

02 04 06 08 10 12Sensitivity (J)

minus20

minus60

minus100

minus120 minus80 minus40

X (m)

Z(m

)

Figure 1 Original sensitivity distribution in119883119885 section

0000

16

08

04

12

minus20 minus60 minus80minus40 minus100

Sens

itivi

ty (J

)

Z (m)

Figure 2 The original sensitivity changes as a function of 119885 at 119883 =

0

00 40 80 120

02 04 06 08 10 12Sensitivity (J)

minus20

minus60

minus100

minus120 minus80 minus40

X (m)

Z(m

)

Figure 3 Modified sensitivity distribution in119883119885 section

original sensitivity in order to increase the resolution ofdeep structure Figures 3 and 4 show the modified sensitivitydistribution The synthetic model shows that this modifiedsensitive distribution works well to recover the true model(Figure 5)

4 Numerical Examples

The following synthetic model studies are based on thedipole-dipole array

41 Simple Model For the simple case we have two differentmodelsModel 1 (Figure 6(a)) is a 4mtimes 4mtimes 4m conductive

International Journal of Geophysics 5

body buried 3meters underground with the resistivity of10Ωm The resistivity of wall rock is 100Ωm Both thereceiver spacing and line spacing are 1meter There are 25points for injecting currents in each profile The backgroundresistivity is approximated from forward modeling and it isused for the initial model in the inversion

Model 2 (Figure 6(b)) is almost the same as the previousone Based on the previous model we added some topogra-phy to this model The highest elevation of the topography is4meters The slope in Figure 6(b) is 30 degrees which leadsthe length of slope to be 8meters

The inversion results belong the profile at119884 = 0 are shownhere Figures 6(c) and 6(d) are the contourline of apparentresistivity at 119884 = 0 for models 1 and 2 Figure 6(c) shows anapparent resistivity distribution for a standard dipole-dipoleanomaly The resistivity of wall rock below and above theanomaly is well recovered and the value is close to 100ΩmFor Figure 6(d) the apparent resistivity is distorted by theartificial anomaly caused by topography We can see that theresistivity of wall rock is almost 3 times as the true valueand the anomaly caused by the 3D low resistivity structureis unclear

Figures 6(e) and 6(f) show one of vertical sections ofinversion results at 119884 = 0 We can see that the 3D conductiveanomaly is well recovered either for the flat surface model orthemodel with topographywithout the redundant structures

42 Complex Model The illustration of this model is shownin Figure 7(a) with two anomalous bodies below some valleyThe first anomalous body is a highly resistive body (500Ωm)buried 2meters underground with the size of 4m times 4m times

4m The second anomaly is a conductive body (10Ωm)burried 4meters below the surface with the size of 4m times

4m times 6m The horizontal distance between these twoanomalous bodies is 4meters The resistivity of the wall rock(background resistivity) is 100ΩmThe observation profile isoriented to 119909 direction with observation spacing of 1meterThere are 31 points for injecting current along each profileThe survey is conducted along five profiles with the spacingof 1meterThe initial model is obtained in a similar way to theone we described in Section 41

Figure 7(b) shows some slices of the apparent resistivitydistribution for a standard dipole-dipole anomalyThe appar-ent resistivity is distorted by the artificial anomaly caused bytopography

After 11 iterations which take 4115 seconds the nor-malized misfit decreased from 1263 to 497 which isaround the noise level of the data The inversion result isshown in Figure 7(c) From the figure we can see that thetwo anomalous bodies are well recovered and the artificialanomaly caused by the topography is eliminated Figure 8shows that the predicted data is very similar to the observeddata on section 119884 = 0

5 Case Study

The primary object of the survey is to estimate the basementdepth which can be used as some information for building

08

24

16

12

20

00 minus20 minus60 minus80minus40 minus100

Sens

itivi

ty (J

)

Z (m)

Figure 4Themodified sensitivity changes as a function of119885 at119883 =

0

00

00

40 80

minus40

minus80

minus120

minus80 minus40

X (m)

Z(m

)

(a)

00

minus40

minus80

minus120

00 40 80minus80 minus40

X (m)

Z(m

)

(b)

Figure 5 A comparison of inversion between original (a) andmodified (b) sensitivity at section 119884 = 0

house The survey site is on the river bed where the topogra-phy is quite flat We employed the multielectrode resistivitymethod with dipole-dipole array configurationThree surveylines are conducted with line distance of 30m and electrodedistance of 15m totally we have 60 electrodes The surveyis conducted in the winter season and the frozen soil withthickness of 1mndash15m caused the high resistivity layer nearthe surface According to geological information the surfacelayer is quaternary alluvium which mostly consisted of loess

6 International Journal of Geophysics

YX

Z

100Ωm 10Ωm

3m

4m 4m4m

Y = 3Y = 0

Y = minus3

(a) 3D low resistivity body without topography

402000

0000 40

4080

80

minus8080minus40

minus40

Y = 3

Y = 0

Y = minus3

X (m)

Z(m

)

Y (m)

(b) 3D low resistivity body model with topography

80 85 90 95 100 105 110 115 120

minus20

minus40

minus60

00 40

10080

80minus80 minus40

X (m)

Z(m

)

120588 (Ωm)

(c) Apparent resistivity at 119884 = 0 for model 1

00

20

20 60 100 140 180 220 260 300 340

minus20

minus40

00 40

100

100200

80minus80 minus40

X (m)

Z(m

)120588 (Ωm)

(d) Apparent resistivity at 119884 = 0 for model 2

00

40 50 60 70 80 90 100 110 120

00 40 80

minus40

minus80

minus120

minus80 minus40

X (m)

Z(m

)

120588 (Ωm)

(e) Vertical section of inversion result at 119884 = 0 for model 1

00

40

10 20 30 40 50 60 70 80 90 100 110 120

minus40

minus80

00 40 80minus80 minus40

X (m)

Z(m

)

120588 (Ωm)

(f) Vertical section of inversion result at 119884 = 0 for model 2

Figure 6 A comparison of inversion result for the model without and with topography

silt and gravel The lower layer is Jurassic red sandstone andas such it is a relative stable layer

Based on the observed data for our survey configurationwe discretized our subsurface into 80 times 60 times 40 cells Wechoose 100Ωm as the resistivity of the wall rock After 10iterations the normalizedmisfit reaches 125The inversionresult is shown in Figure 9 In 55m along profile 135 drillingshows that the first 5m consisted of frozen soil sand andgravel from 5ndash98m we observed water-bearing ooze below98m is red sandstone Our inversion result shows very goodfitting with the information that we get from drilling

6 Conclusion

In this paper we implemented the 3DDC resistivity inversionbased on regularized conjugate gradient method We usedfinite element method with triangular prism discretizationfor forward modeling of apparent resistivity Synthetic modelstudies show that the inversion algorithm works well torecover the true model with complex topography The inver-sion process is very stable and computation speed is fast

The electric potential at each node caused by differentsource is precomputed in the forward modeling and saved

International Journal of Geophysics 7

00 40 80 120

(b)

40 8000

30 50 70 90 110 130 150 170 190 50 90 130 170 210

(c)

minus40minus40 minus80minus80minus120

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

Y=2

Y=3

Y=2

Y=1

Y=0

Y=0

Y=1

Y=3

Y=minus3

Y=minus2

Y=minus1

Y=minus3

Y=minus2

Y=minus1

00

0040

80

00 40 80

(a)

minus40

minus40

minus80minus80

minus40

minus20

minus60

minus80

minus100

minus120

Y = 3Y = 0Y = minus3

100Ωm

500Ωm10Ωm

X (m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)

Y(m)

X (m)X (m)

120588 (Ωm) 120588 (Ωm)

Figure 7 The inversion result for a complex model (a) A model with two bodies under a valley (b) Slices view of apparent resistivity (c)Slices view of 3D inversion result

8 International Journal of Geophysics

30 50 70 90 110 130 150 170

00 40 80 120

minus20

minus60

minus100

minus120 minus80 minus40

120588 (Ωm)

X (m)

Z(m

)

(a)

30 50 70 90 110 130 150 170

00 40 80 120

minus20

minus60

minus100

minus120 minus80 minus40

120588 (Ωm)

X (m)

Z(m

)

(b)

Figure 8 A comparison between observed apparent resistivity (a) and predicted apparent resistivity (b) for profile 119884 = 0

(a) Survey configuration

0

07

10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

09 11 13 15 17 19 21 23 25 27 29

minus10

minus5

0

minus10

minus5

0

minus10

minus5

Line 133

Line 135

Line 137

log(120588)

X (m)

Z(m

)Z

(m)

Z(m

)

(b) Inversion result

Figure 9 The inversion of real data

for the inversion stage to compute Jacobi matrix By doingthis the computation cost can be reduced dramatically in theinversion process Only one forward modeling is required inorder to compute the sensitivity matrix

We conducted a sensitivity analysis and introduced aweighting function to optimize sensitivity matrix By apply-ing this proper model weighting the vertical location of theanomalous body can be well recovered

Acknowledgments

The authors acknowledge National Natural Science Foun-dation of China Grants nos 40974076 41174104 ChineseAcademy Geological Sciences (Sinoprobe-03-02-04) and theKey Laboratory of Metallogenic Prediction of Nonferrous

Metals of Ministry of Education for the support of thisproject

References

[1] J-K Qiang and Y-Z Luo ldquoThe resistivity FEM numericalmodeling on 3-D undulating topographyrdquo Chinese Journal ofGeophysics vol 50 no 5 pp 1606ndash1613 2007 (Chinese)

[2] A C Tripp G W Hohmann and C M Swift Jr ldquoTwo-dimen-sional resistivity inversionrdquoGeophysics vol 49 no 10 pp 1708ndash1717 1984

[3] S K Park and G P Van ldquoInversion of pole-pole data for 3-Dresistivity structure beneath arrays of electrodesrdquo Geophysicsvol 56 no 7 pp 951ndash960 1991

[4] Y Sasaki ldquo3-D resistivity inversion using the finite-elementmethodrdquo Geophysics vol 59 no 12 pp 1839ndash1848 1994

International Journal of Geophysics 9

[5] J Zhang R L Mackie and T R Madden ldquo3-D resistivity for-ward modeling and inversion using conjugate gradientsrdquo Geo-physics vol 60 no 5 pp 1313ndash1325 1995

[6] M H Loke and R D Barker ldquoPractical techniques for 3D resis-tivity surveys and data inversionrdquo Geophysical Prospecting vol44 no 3 pp 499ndash523 1996

[7] X P Wu and G M Xu ldquoDerivation and analysis of partialderivative matrix in resistivity 3-D inversionrdquo Oil GeophysicalProspecting vol 34 pp 363ndash372 1999

[8] X-P Wu and G-M Xu ldquoStudy on 3-D resistivity inversionusing conjugate gradient methodrdquo Chinese Journal of Geophys-ics vol 43 no 3 pp 420ndash427 2000

[9] X-P Wu ldquo3-D resistivity inversion under the condition ofuneven terrainrdquoChinese Journal of Geophysics vol 48 no 4 pp932ndash936 2005

[10] H-F Liu J-X Liu R-W Guo X-K Deng and B-Y RuanldquoEfficient inversion of 3D IP data for continuous model withcomplex geometryrdquo Journal of Jilin University vol 41 no 4 pp1212ndash1218 2011

[11] N G Papadopoulos M-J Yi J-H Kim P Tsourlos and GN Tsokas ldquoGeophysical investigation of tumuli by means ofsurface 3D electrical resistivity tomographyrdquo Journal of AppliedGeophysics vol 70 no 3 pp 192ndash205 2010

[12] P I Tsourlos and R D Ogilvy ldquoAn algorithm for the 3-Dinversion of tomographic resistivity and induced polarisationdata preliminary resultsrdquo Journal of the Balkan GeophysicalSociety vol 2 pp 30ndash45 1999

[13] J G Huang B Y Ruan and G S Bao ldquoResistivity inversion on3-D section based on FEMrdquo Journal of Central South Universityof Technology vol 35 pp 295ndash299 2004

[14] TGunther C Rucker andK Spitzer ldquoThree-dimensionalmod-eling and inversion of dc resistivity data incorporating topogra-phymdashII inversionrdquo Geophysical Journal International vol 166no 2 pp 506ndash5517 2006

[15] G A Oldenborger and P S Routh ldquoThe point-spread functionmeasure of resolution for the 3-D electrical resistivity experi-mentrdquoGeophysical Journal International vol 176 no 2 pp 405ndash414 2009

[16] A Dey and H F Morrison ldquoResistivity modeling for arbitrarilyshaped three-dimensional structuresrdquoGeophysics vol 44 no 4pp 753ndash760 1979

[17] R G Ellis and D W Oldenburg ldquoThe pole-pole 3-D DC-re-sistivity inverse problem a conjugate- gradient approachrdquo Geo-physical Journal International vol 119 no 1 pp 187ndash194 1994

[18] D J LaBrecque M Miletto W Daily A Ramirez and EOwen ldquoThe effects of noise on Occamrsquos inversion of resistivitytomography datardquo Geophysics vol 61 no 2 pp 538ndash548 1996

[19] M-J Yi J-H Kim Y Song S-J Cho S-H Chung and J-H Suh ldquoThree-dimensional imaging of subsurface structuresusing resistivity datardquoGeophysical Prospecting vol 49 no 4 pp483ndash497 2001

[20] C C Pain J V Herwanger M H Worthington and C R E deOliveira ldquoEffectivemultidimensional resistivity inversion usingfinite-element techniquesrdquo Geophysical Journal Internationalvol 151 no 3 pp 710ndash728 2002

[21] A Pidlisecky E Haber and R Knight ldquoRESINVM3D a 3Dresistivity inversion packagerdquo Geophysics vol 72 no 2 pp H1ndashH10 2007

[22] L Marescot S P Lopes S Rigobert and A G Green ldquoNon-linear inversion of geoelectric data acquired across 3D objectsusing a finite-element approachrdquo Geophysics vol 73 no 3 ppF121ndashF133 2008

[23] M S Zhdanov Geophysical Inverse Theory and RegularizationProblems Elsevier New York NY USA 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

International Journal of Geophysics 5

body buried 3meters underground with the resistivity of10Ωm The resistivity of wall rock is 100Ωm Both thereceiver spacing and line spacing are 1meter There are 25points for injecting currents in each profile The backgroundresistivity is approximated from forward modeling and it isused for the initial model in the inversion

Model 2 (Figure 6(b)) is almost the same as the previousone Based on the previous model we added some topogra-phy to this model The highest elevation of the topography is4meters The slope in Figure 6(b) is 30 degrees which leadsthe length of slope to be 8meters

The inversion results belong the profile at119884 = 0 are shownhere Figures 6(c) and 6(d) are the contourline of apparentresistivity at 119884 = 0 for models 1 and 2 Figure 6(c) shows anapparent resistivity distribution for a standard dipole-dipoleanomaly The resistivity of wall rock below and above theanomaly is well recovered and the value is close to 100ΩmFor Figure 6(d) the apparent resistivity is distorted by theartificial anomaly caused by topography We can see that theresistivity of wall rock is almost 3 times as the true valueand the anomaly caused by the 3D low resistivity structureis unclear

Figures 6(e) and 6(f) show one of vertical sections ofinversion results at 119884 = 0 We can see that the 3D conductiveanomaly is well recovered either for the flat surface model orthemodel with topographywithout the redundant structures

42 Complex Model The illustration of this model is shownin Figure 7(a) with two anomalous bodies below some valleyThe first anomalous body is a highly resistive body (500Ωm)buried 2meters underground with the size of 4m times 4m times

4m The second anomaly is a conductive body (10Ωm)burried 4meters below the surface with the size of 4m times

4m times 6m The horizontal distance between these twoanomalous bodies is 4meters The resistivity of the wall rock(background resistivity) is 100ΩmThe observation profile isoriented to 119909 direction with observation spacing of 1meterThere are 31 points for injecting current along each profileThe survey is conducted along five profiles with the spacingof 1meterThe initial model is obtained in a similar way to theone we described in Section 41

Figure 7(b) shows some slices of the apparent resistivitydistribution for a standard dipole-dipole anomalyThe appar-ent resistivity is distorted by the artificial anomaly caused bytopography

After 11 iterations which take 4115 seconds the nor-malized misfit decreased from 1263 to 497 which isaround the noise level of the data The inversion result isshown in Figure 7(c) From the figure we can see that thetwo anomalous bodies are well recovered and the artificialanomaly caused by the topography is eliminated Figure 8shows that the predicted data is very similar to the observeddata on section 119884 = 0

5 Case Study

The primary object of the survey is to estimate the basementdepth which can be used as some information for building

08

24

16

12

20

00 minus20 minus60 minus80minus40 minus100

Sens

itivi

ty (J

)

Z (m)

Figure 4Themodified sensitivity changes as a function of119885 at119883 =

0

00

00

40 80

minus40

minus80

minus120

minus80 minus40

X (m)

Z(m

)

(a)

00

minus40

minus80

minus120

00 40 80minus80 minus40

X (m)

Z(m

)

(b)

Figure 5 A comparison of inversion between original (a) andmodified (b) sensitivity at section 119884 = 0

house The survey site is on the river bed where the topogra-phy is quite flat We employed the multielectrode resistivitymethod with dipole-dipole array configurationThree surveylines are conducted with line distance of 30m and electrodedistance of 15m totally we have 60 electrodes The surveyis conducted in the winter season and the frozen soil withthickness of 1mndash15m caused the high resistivity layer nearthe surface According to geological information the surfacelayer is quaternary alluvium which mostly consisted of loess

6 International Journal of Geophysics

YX

Z

100Ωm 10Ωm

3m

4m 4m4m

Y = 3Y = 0

Y = minus3

(a) 3D low resistivity body without topography

402000

0000 40

4080

80

minus8080minus40

minus40

Y = 3

Y = 0

Y = minus3

X (m)

Z(m

)

Y (m)

(b) 3D low resistivity body model with topography

80 85 90 95 100 105 110 115 120

minus20

minus40

minus60

00 40

10080

80minus80 minus40

X (m)

Z(m

)

120588 (Ωm)

(c) Apparent resistivity at 119884 = 0 for model 1

00

20

20 60 100 140 180 220 260 300 340

minus20

minus40

00 40

100

100200

80minus80 minus40

X (m)

Z(m

)120588 (Ωm)

(d) Apparent resistivity at 119884 = 0 for model 2

00

40 50 60 70 80 90 100 110 120

00 40 80

minus40

minus80

minus120

minus80 minus40

X (m)

Z(m

)

120588 (Ωm)

(e) Vertical section of inversion result at 119884 = 0 for model 1

00

40

10 20 30 40 50 60 70 80 90 100 110 120

minus40

minus80

00 40 80minus80 minus40

X (m)

Z(m

)

120588 (Ωm)

(f) Vertical section of inversion result at 119884 = 0 for model 2

Figure 6 A comparison of inversion result for the model without and with topography

silt and gravel The lower layer is Jurassic red sandstone andas such it is a relative stable layer

Based on the observed data for our survey configurationwe discretized our subsurface into 80 times 60 times 40 cells Wechoose 100Ωm as the resistivity of the wall rock After 10iterations the normalizedmisfit reaches 125The inversionresult is shown in Figure 9 In 55m along profile 135 drillingshows that the first 5m consisted of frozen soil sand andgravel from 5ndash98m we observed water-bearing ooze below98m is red sandstone Our inversion result shows very goodfitting with the information that we get from drilling

6 Conclusion

In this paper we implemented the 3DDC resistivity inversionbased on regularized conjugate gradient method We usedfinite element method with triangular prism discretizationfor forward modeling of apparent resistivity Synthetic modelstudies show that the inversion algorithm works well torecover the true model with complex topography The inver-sion process is very stable and computation speed is fast

The electric potential at each node caused by differentsource is precomputed in the forward modeling and saved

International Journal of Geophysics 7

00 40 80 120

(b)

40 8000

30 50 70 90 110 130 150 170 190 50 90 130 170 210

(c)

minus40minus40 minus80minus80minus120

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

Y=2

Y=3

Y=2

Y=1

Y=0

Y=0

Y=1

Y=3

Y=minus3

Y=minus2

Y=minus1

Y=minus3

Y=minus2

Y=minus1

00

0040

80

00 40 80

(a)

minus40

minus40

minus80minus80

minus40

minus20

minus60

minus80

minus100

minus120

Y = 3Y = 0Y = minus3

100Ωm

500Ωm10Ωm

X (m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)

Y(m)

X (m)X (m)

120588 (Ωm) 120588 (Ωm)

Figure 7 The inversion result for a complex model (a) A model with two bodies under a valley (b) Slices view of apparent resistivity (c)Slices view of 3D inversion result

8 International Journal of Geophysics

30 50 70 90 110 130 150 170

00 40 80 120

minus20

minus60

minus100

minus120 minus80 minus40

120588 (Ωm)

X (m)

Z(m

)

(a)

30 50 70 90 110 130 150 170

00 40 80 120

minus20

minus60

minus100

minus120 minus80 minus40

120588 (Ωm)

X (m)

Z(m

)

(b)

Figure 8 A comparison between observed apparent resistivity (a) and predicted apparent resistivity (b) for profile 119884 = 0

(a) Survey configuration

0

07

10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

09 11 13 15 17 19 21 23 25 27 29

minus10

minus5

0

minus10

minus5

0

minus10

minus5

Line 133

Line 135

Line 137

log(120588)

X (m)

Z(m

)Z

(m)

Z(m

)

(b) Inversion result

Figure 9 The inversion of real data

for the inversion stage to compute Jacobi matrix By doingthis the computation cost can be reduced dramatically in theinversion process Only one forward modeling is required inorder to compute the sensitivity matrix

We conducted a sensitivity analysis and introduced aweighting function to optimize sensitivity matrix By apply-ing this proper model weighting the vertical location of theanomalous body can be well recovered

Acknowledgments

The authors acknowledge National Natural Science Foun-dation of China Grants nos 40974076 41174104 ChineseAcademy Geological Sciences (Sinoprobe-03-02-04) and theKey Laboratory of Metallogenic Prediction of Nonferrous

Metals of Ministry of Education for the support of thisproject

References

[1] J-K Qiang and Y-Z Luo ldquoThe resistivity FEM numericalmodeling on 3-D undulating topographyrdquo Chinese Journal ofGeophysics vol 50 no 5 pp 1606ndash1613 2007 (Chinese)

[2] A C Tripp G W Hohmann and C M Swift Jr ldquoTwo-dimen-sional resistivity inversionrdquoGeophysics vol 49 no 10 pp 1708ndash1717 1984

[3] S K Park and G P Van ldquoInversion of pole-pole data for 3-Dresistivity structure beneath arrays of electrodesrdquo Geophysicsvol 56 no 7 pp 951ndash960 1991

[4] Y Sasaki ldquo3-D resistivity inversion using the finite-elementmethodrdquo Geophysics vol 59 no 12 pp 1839ndash1848 1994

International Journal of Geophysics 9

[5] J Zhang R L Mackie and T R Madden ldquo3-D resistivity for-ward modeling and inversion using conjugate gradientsrdquo Geo-physics vol 60 no 5 pp 1313ndash1325 1995

[6] M H Loke and R D Barker ldquoPractical techniques for 3D resis-tivity surveys and data inversionrdquo Geophysical Prospecting vol44 no 3 pp 499ndash523 1996

[7] X P Wu and G M Xu ldquoDerivation and analysis of partialderivative matrix in resistivity 3-D inversionrdquo Oil GeophysicalProspecting vol 34 pp 363ndash372 1999

[8] X-P Wu and G-M Xu ldquoStudy on 3-D resistivity inversionusing conjugate gradient methodrdquo Chinese Journal of Geophys-ics vol 43 no 3 pp 420ndash427 2000

[9] X-P Wu ldquo3-D resistivity inversion under the condition ofuneven terrainrdquoChinese Journal of Geophysics vol 48 no 4 pp932ndash936 2005

[10] H-F Liu J-X Liu R-W Guo X-K Deng and B-Y RuanldquoEfficient inversion of 3D IP data for continuous model withcomplex geometryrdquo Journal of Jilin University vol 41 no 4 pp1212ndash1218 2011

[11] N G Papadopoulos M-J Yi J-H Kim P Tsourlos and GN Tsokas ldquoGeophysical investigation of tumuli by means ofsurface 3D electrical resistivity tomographyrdquo Journal of AppliedGeophysics vol 70 no 3 pp 192ndash205 2010

[12] P I Tsourlos and R D Ogilvy ldquoAn algorithm for the 3-Dinversion of tomographic resistivity and induced polarisationdata preliminary resultsrdquo Journal of the Balkan GeophysicalSociety vol 2 pp 30ndash45 1999

[13] J G Huang B Y Ruan and G S Bao ldquoResistivity inversion on3-D section based on FEMrdquo Journal of Central South Universityof Technology vol 35 pp 295ndash299 2004

[14] TGunther C Rucker andK Spitzer ldquoThree-dimensionalmod-eling and inversion of dc resistivity data incorporating topogra-phymdashII inversionrdquo Geophysical Journal International vol 166no 2 pp 506ndash5517 2006

[15] G A Oldenborger and P S Routh ldquoThe point-spread functionmeasure of resolution for the 3-D electrical resistivity experi-mentrdquoGeophysical Journal International vol 176 no 2 pp 405ndash414 2009

[16] A Dey and H F Morrison ldquoResistivity modeling for arbitrarilyshaped three-dimensional structuresrdquoGeophysics vol 44 no 4pp 753ndash760 1979

[17] R G Ellis and D W Oldenburg ldquoThe pole-pole 3-D DC-re-sistivity inverse problem a conjugate- gradient approachrdquo Geo-physical Journal International vol 119 no 1 pp 187ndash194 1994

[18] D J LaBrecque M Miletto W Daily A Ramirez and EOwen ldquoThe effects of noise on Occamrsquos inversion of resistivitytomography datardquo Geophysics vol 61 no 2 pp 538ndash548 1996

[19] M-J Yi J-H Kim Y Song S-J Cho S-H Chung and J-H Suh ldquoThree-dimensional imaging of subsurface structuresusing resistivity datardquoGeophysical Prospecting vol 49 no 4 pp483ndash497 2001

[20] C C Pain J V Herwanger M H Worthington and C R E deOliveira ldquoEffectivemultidimensional resistivity inversion usingfinite-element techniquesrdquo Geophysical Journal Internationalvol 151 no 3 pp 710ndash728 2002

[21] A Pidlisecky E Haber and R Knight ldquoRESINVM3D a 3Dresistivity inversion packagerdquo Geophysics vol 72 no 2 pp H1ndashH10 2007

[22] L Marescot S P Lopes S Rigobert and A G Green ldquoNon-linear inversion of geoelectric data acquired across 3D objectsusing a finite-element approachrdquo Geophysics vol 73 no 3 ppF121ndashF133 2008

[23] M S Zhdanov Geophysical Inverse Theory and RegularizationProblems Elsevier New York NY USA 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

6 International Journal of Geophysics

YX

Z

100Ωm 10Ωm

3m

4m 4m4m

Y = 3Y = 0

Y = minus3

(a) 3D low resistivity body without topography

402000

0000 40

4080

80

minus8080minus40

minus40

Y = 3

Y = 0

Y = minus3

X (m)

Z(m

)

Y (m)

(b) 3D low resistivity body model with topography

80 85 90 95 100 105 110 115 120

minus20

minus40

minus60

00 40

10080

80minus80 minus40

X (m)

Z(m

)

120588 (Ωm)

(c) Apparent resistivity at 119884 = 0 for model 1

00

20

20 60 100 140 180 220 260 300 340

minus20

minus40

00 40

100

100200

80minus80 minus40

X (m)

Z(m

)120588 (Ωm)

(d) Apparent resistivity at 119884 = 0 for model 2

00

40 50 60 70 80 90 100 110 120

00 40 80

minus40

minus80

minus120

minus80 minus40

X (m)

Z(m

)

120588 (Ωm)

(e) Vertical section of inversion result at 119884 = 0 for model 1

00

40

10 20 30 40 50 60 70 80 90 100 110 120

minus40

minus80

00 40 80minus80 minus40

X (m)

Z(m

)

120588 (Ωm)

(f) Vertical section of inversion result at 119884 = 0 for model 2

Figure 6 A comparison of inversion result for the model without and with topography

silt and gravel The lower layer is Jurassic red sandstone andas such it is a relative stable layer

Based on the observed data for our survey configurationwe discretized our subsurface into 80 times 60 times 40 cells Wechoose 100Ωm as the resistivity of the wall rock After 10iterations the normalizedmisfit reaches 125The inversionresult is shown in Figure 9 In 55m along profile 135 drillingshows that the first 5m consisted of frozen soil sand andgravel from 5ndash98m we observed water-bearing ooze below98m is red sandstone Our inversion result shows very goodfitting with the information that we get from drilling

6 Conclusion

In this paper we implemented the 3DDC resistivity inversionbased on regularized conjugate gradient method We usedfinite element method with triangular prism discretizationfor forward modeling of apparent resistivity Synthetic modelstudies show that the inversion algorithm works well torecover the true model with complex topography The inver-sion process is very stable and computation speed is fast

The electric potential at each node caused by differentsource is precomputed in the forward modeling and saved

International Journal of Geophysics 7

00 40 80 120

(b)

40 8000

30 50 70 90 110 130 150 170 190 50 90 130 170 210

(c)

minus40minus40 minus80minus80minus120

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

Y=2

Y=3

Y=2

Y=1

Y=0

Y=0

Y=1

Y=3

Y=minus3

Y=minus2

Y=minus1

Y=minus3

Y=minus2

Y=minus1

00

0040

80

00 40 80

(a)

minus40

minus40

minus80minus80

minus40

minus20

minus60

minus80

minus100

minus120

Y = 3Y = 0Y = minus3

100Ωm

500Ωm10Ωm

X (m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)

Y(m)

X (m)X (m)

120588 (Ωm) 120588 (Ωm)

Figure 7 The inversion result for a complex model (a) A model with two bodies under a valley (b) Slices view of apparent resistivity (c)Slices view of 3D inversion result

8 International Journal of Geophysics

30 50 70 90 110 130 150 170

00 40 80 120

minus20

minus60

minus100

minus120 minus80 minus40

120588 (Ωm)

X (m)

Z(m

)

(a)

30 50 70 90 110 130 150 170

00 40 80 120

minus20

minus60

minus100

minus120 minus80 minus40

120588 (Ωm)

X (m)

Z(m

)

(b)

Figure 8 A comparison between observed apparent resistivity (a) and predicted apparent resistivity (b) for profile 119884 = 0

(a) Survey configuration

0

07

10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

09 11 13 15 17 19 21 23 25 27 29

minus10

minus5

0

minus10

minus5

0

minus10

minus5

Line 133

Line 135

Line 137

log(120588)

X (m)

Z(m

)Z

(m)

Z(m

)

(b) Inversion result

Figure 9 The inversion of real data

for the inversion stage to compute Jacobi matrix By doingthis the computation cost can be reduced dramatically in theinversion process Only one forward modeling is required inorder to compute the sensitivity matrix

We conducted a sensitivity analysis and introduced aweighting function to optimize sensitivity matrix By apply-ing this proper model weighting the vertical location of theanomalous body can be well recovered

Acknowledgments

The authors acknowledge National Natural Science Foun-dation of China Grants nos 40974076 41174104 ChineseAcademy Geological Sciences (Sinoprobe-03-02-04) and theKey Laboratory of Metallogenic Prediction of Nonferrous

Metals of Ministry of Education for the support of thisproject

References

[1] J-K Qiang and Y-Z Luo ldquoThe resistivity FEM numericalmodeling on 3-D undulating topographyrdquo Chinese Journal ofGeophysics vol 50 no 5 pp 1606ndash1613 2007 (Chinese)

[2] A C Tripp G W Hohmann and C M Swift Jr ldquoTwo-dimen-sional resistivity inversionrdquoGeophysics vol 49 no 10 pp 1708ndash1717 1984

[3] S K Park and G P Van ldquoInversion of pole-pole data for 3-Dresistivity structure beneath arrays of electrodesrdquo Geophysicsvol 56 no 7 pp 951ndash960 1991

[4] Y Sasaki ldquo3-D resistivity inversion using the finite-elementmethodrdquo Geophysics vol 59 no 12 pp 1839ndash1848 1994

International Journal of Geophysics 9

[5] J Zhang R L Mackie and T R Madden ldquo3-D resistivity for-ward modeling and inversion using conjugate gradientsrdquo Geo-physics vol 60 no 5 pp 1313ndash1325 1995

[6] M H Loke and R D Barker ldquoPractical techniques for 3D resis-tivity surveys and data inversionrdquo Geophysical Prospecting vol44 no 3 pp 499ndash523 1996

[7] X P Wu and G M Xu ldquoDerivation and analysis of partialderivative matrix in resistivity 3-D inversionrdquo Oil GeophysicalProspecting vol 34 pp 363ndash372 1999

[8] X-P Wu and G-M Xu ldquoStudy on 3-D resistivity inversionusing conjugate gradient methodrdquo Chinese Journal of Geophys-ics vol 43 no 3 pp 420ndash427 2000

[9] X-P Wu ldquo3-D resistivity inversion under the condition ofuneven terrainrdquoChinese Journal of Geophysics vol 48 no 4 pp932ndash936 2005

[10] H-F Liu J-X Liu R-W Guo X-K Deng and B-Y RuanldquoEfficient inversion of 3D IP data for continuous model withcomplex geometryrdquo Journal of Jilin University vol 41 no 4 pp1212ndash1218 2011

[11] N G Papadopoulos M-J Yi J-H Kim P Tsourlos and GN Tsokas ldquoGeophysical investigation of tumuli by means ofsurface 3D electrical resistivity tomographyrdquo Journal of AppliedGeophysics vol 70 no 3 pp 192ndash205 2010

[12] P I Tsourlos and R D Ogilvy ldquoAn algorithm for the 3-Dinversion of tomographic resistivity and induced polarisationdata preliminary resultsrdquo Journal of the Balkan GeophysicalSociety vol 2 pp 30ndash45 1999

[13] J G Huang B Y Ruan and G S Bao ldquoResistivity inversion on3-D section based on FEMrdquo Journal of Central South Universityof Technology vol 35 pp 295ndash299 2004

[14] TGunther C Rucker andK Spitzer ldquoThree-dimensionalmod-eling and inversion of dc resistivity data incorporating topogra-phymdashII inversionrdquo Geophysical Journal International vol 166no 2 pp 506ndash5517 2006

[15] G A Oldenborger and P S Routh ldquoThe point-spread functionmeasure of resolution for the 3-D electrical resistivity experi-mentrdquoGeophysical Journal International vol 176 no 2 pp 405ndash414 2009

[16] A Dey and H F Morrison ldquoResistivity modeling for arbitrarilyshaped three-dimensional structuresrdquoGeophysics vol 44 no 4pp 753ndash760 1979

[17] R G Ellis and D W Oldenburg ldquoThe pole-pole 3-D DC-re-sistivity inverse problem a conjugate- gradient approachrdquo Geo-physical Journal International vol 119 no 1 pp 187ndash194 1994

[18] D J LaBrecque M Miletto W Daily A Ramirez and EOwen ldquoThe effects of noise on Occamrsquos inversion of resistivitytomography datardquo Geophysics vol 61 no 2 pp 538ndash548 1996

[19] M-J Yi J-H Kim Y Song S-J Cho S-H Chung and J-H Suh ldquoThree-dimensional imaging of subsurface structuresusing resistivity datardquoGeophysical Prospecting vol 49 no 4 pp483ndash497 2001

[20] C C Pain J V Herwanger M H Worthington and C R E deOliveira ldquoEffectivemultidimensional resistivity inversion usingfinite-element techniquesrdquo Geophysical Journal Internationalvol 151 no 3 pp 710ndash728 2002

[21] A Pidlisecky E Haber and R Knight ldquoRESINVM3D a 3Dresistivity inversion packagerdquo Geophysics vol 72 no 2 pp H1ndashH10 2007

[22] L Marescot S P Lopes S Rigobert and A G Green ldquoNon-linear inversion of geoelectric data acquired across 3D objectsusing a finite-element approachrdquo Geophysics vol 73 no 3 ppF121ndashF133 2008

[23] M S Zhdanov Geophysical Inverse Theory and RegularizationProblems Elsevier New York NY USA 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

International Journal of Geophysics 7

00 40 80 120

(b)

40 8000

30 50 70 90 110 130 150 170 190 50 90 130 170 210

(c)

minus40minus40 minus80minus80minus120

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

minus20

minus60

minus100

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

00

minus40

minus80

minus120

Y=2

Y=3

Y=2

Y=1

Y=0

Y=0

Y=1

Y=3

Y=minus3

Y=minus2

Y=minus1

Y=minus3

Y=minus2

Y=minus1

00

0040

80

00 40 80

(a)

minus40

minus40

minus80minus80

minus40

minus20

minus60

minus80

minus100

minus120

Y = 3Y = 0Y = minus3

100Ωm

500Ωm10Ωm

X (m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)Z

(m)

Z(m

)

Y(m)

X (m)X (m)

120588 (Ωm) 120588 (Ωm)

Figure 7 The inversion result for a complex model (a) A model with two bodies under a valley (b) Slices view of apparent resistivity (c)Slices view of 3D inversion result

8 International Journal of Geophysics

30 50 70 90 110 130 150 170

00 40 80 120

minus20

minus60

minus100

minus120 minus80 minus40

120588 (Ωm)

X (m)

Z(m

)

(a)

30 50 70 90 110 130 150 170

00 40 80 120

minus20

minus60

minus100

minus120 minus80 minus40

120588 (Ωm)

X (m)

Z(m

)

(b)

Figure 8 A comparison between observed apparent resistivity (a) and predicted apparent resistivity (b) for profile 119884 = 0

(a) Survey configuration

0

07

10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

09 11 13 15 17 19 21 23 25 27 29

minus10

minus5

0

minus10

minus5

0

minus10

minus5

Line 133

Line 135

Line 137

log(120588)

X (m)

Z(m

)Z

(m)

Z(m

)

(b) Inversion result

Figure 9 The inversion of real data

for the inversion stage to compute Jacobi matrix By doingthis the computation cost can be reduced dramatically in theinversion process Only one forward modeling is required inorder to compute the sensitivity matrix

We conducted a sensitivity analysis and introduced aweighting function to optimize sensitivity matrix By apply-ing this proper model weighting the vertical location of theanomalous body can be well recovered

Acknowledgments

The authors acknowledge National Natural Science Foun-dation of China Grants nos 40974076 41174104 ChineseAcademy Geological Sciences (Sinoprobe-03-02-04) and theKey Laboratory of Metallogenic Prediction of Nonferrous

Metals of Ministry of Education for the support of thisproject

References

[1] J-K Qiang and Y-Z Luo ldquoThe resistivity FEM numericalmodeling on 3-D undulating topographyrdquo Chinese Journal ofGeophysics vol 50 no 5 pp 1606ndash1613 2007 (Chinese)

[2] A C Tripp G W Hohmann and C M Swift Jr ldquoTwo-dimen-sional resistivity inversionrdquoGeophysics vol 49 no 10 pp 1708ndash1717 1984

[3] S K Park and G P Van ldquoInversion of pole-pole data for 3-Dresistivity structure beneath arrays of electrodesrdquo Geophysicsvol 56 no 7 pp 951ndash960 1991

[4] Y Sasaki ldquo3-D resistivity inversion using the finite-elementmethodrdquo Geophysics vol 59 no 12 pp 1839ndash1848 1994

International Journal of Geophysics 9

[5] J Zhang R L Mackie and T R Madden ldquo3-D resistivity for-ward modeling and inversion using conjugate gradientsrdquo Geo-physics vol 60 no 5 pp 1313ndash1325 1995

[6] M H Loke and R D Barker ldquoPractical techniques for 3D resis-tivity surveys and data inversionrdquo Geophysical Prospecting vol44 no 3 pp 499ndash523 1996

[7] X P Wu and G M Xu ldquoDerivation and analysis of partialderivative matrix in resistivity 3-D inversionrdquo Oil GeophysicalProspecting vol 34 pp 363ndash372 1999

[8] X-P Wu and G-M Xu ldquoStudy on 3-D resistivity inversionusing conjugate gradient methodrdquo Chinese Journal of Geophys-ics vol 43 no 3 pp 420ndash427 2000

[9] X-P Wu ldquo3-D resistivity inversion under the condition ofuneven terrainrdquoChinese Journal of Geophysics vol 48 no 4 pp932ndash936 2005

[10] H-F Liu J-X Liu R-W Guo X-K Deng and B-Y RuanldquoEfficient inversion of 3D IP data for continuous model withcomplex geometryrdquo Journal of Jilin University vol 41 no 4 pp1212ndash1218 2011

[11] N G Papadopoulos M-J Yi J-H Kim P Tsourlos and GN Tsokas ldquoGeophysical investigation of tumuli by means ofsurface 3D electrical resistivity tomographyrdquo Journal of AppliedGeophysics vol 70 no 3 pp 192ndash205 2010

[12] P I Tsourlos and R D Ogilvy ldquoAn algorithm for the 3-Dinversion of tomographic resistivity and induced polarisationdata preliminary resultsrdquo Journal of the Balkan GeophysicalSociety vol 2 pp 30ndash45 1999

[13] J G Huang B Y Ruan and G S Bao ldquoResistivity inversion on3-D section based on FEMrdquo Journal of Central South Universityof Technology vol 35 pp 295ndash299 2004

[14] TGunther C Rucker andK Spitzer ldquoThree-dimensionalmod-eling and inversion of dc resistivity data incorporating topogra-phymdashII inversionrdquo Geophysical Journal International vol 166no 2 pp 506ndash5517 2006

[15] G A Oldenborger and P S Routh ldquoThe point-spread functionmeasure of resolution for the 3-D electrical resistivity experi-mentrdquoGeophysical Journal International vol 176 no 2 pp 405ndash414 2009

[16] A Dey and H F Morrison ldquoResistivity modeling for arbitrarilyshaped three-dimensional structuresrdquoGeophysics vol 44 no 4pp 753ndash760 1979

[17] R G Ellis and D W Oldenburg ldquoThe pole-pole 3-D DC-re-sistivity inverse problem a conjugate- gradient approachrdquo Geo-physical Journal International vol 119 no 1 pp 187ndash194 1994

[18] D J LaBrecque M Miletto W Daily A Ramirez and EOwen ldquoThe effects of noise on Occamrsquos inversion of resistivitytomography datardquo Geophysics vol 61 no 2 pp 538ndash548 1996

[19] M-J Yi J-H Kim Y Song S-J Cho S-H Chung and J-H Suh ldquoThree-dimensional imaging of subsurface structuresusing resistivity datardquoGeophysical Prospecting vol 49 no 4 pp483ndash497 2001

[20] C C Pain J V Herwanger M H Worthington and C R E deOliveira ldquoEffectivemultidimensional resistivity inversion usingfinite-element techniquesrdquo Geophysical Journal Internationalvol 151 no 3 pp 710ndash728 2002

[21] A Pidlisecky E Haber and R Knight ldquoRESINVM3D a 3Dresistivity inversion packagerdquo Geophysics vol 72 no 2 pp H1ndashH10 2007

[22] L Marescot S P Lopes S Rigobert and A G Green ldquoNon-linear inversion of geoelectric data acquired across 3D objectsusing a finite-element approachrdquo Geophysics vol 73 no 3 ppF121ndashF133 2008

[23] M S Zhdanov Geophysical Inverse Theory and RegularizationProblems Elsevier New York NY USA 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

8 International Journal of Geophysics

30 50 70 90 110 130 150 170

00 40 80 120

minus20

minus60

minus100

minus120 minus80 minus40

120588 (Ωm)

X (m)

Z(m

)

(a)

30 50 70 90 110 130 150 170

00 40 80 120

minus20

minus60

minus100

minus120 minus80 minus40

120588 (Ωm)

X (m)

Z(m

)

(b)

Figure 8 A comparison between observed apparent resistivity (a) and predicted apparent resistivity (b) for profile 119884 = 0

(a) Survey configuration

0

07

10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

09 11 13 15 17 19 21 23 25 27 29

minus10

minus5

0

minus10

minus5

0

minus10

minus5

Line 133

Line 135

Line 137

log(120588)

X (m)

Z(m

)Z

(m)

Z(m

)

(b) Inversion result

Figure 9 The inversion of real data

for the inversion stage to compute Jacobi matrix By doingthis the computation cost can be reduced dramatically in theinversion process Only one forward modeling is required inorder to compute the sensitivity matrix

We conducted a sensitivity analysis and introduced aweighting function to optimize sensitivity matrix By apply-ing this proper model weighting the vertical location of theanomalous body can be well recovered

Acknowledgments

The authors acknowledge National Natural Science Foun-dation of China Grants nos 40974076 41174104 ChineseAcademy Geological Sciences (Sinoprobe-03-02-04) and theKey Laboratory of Metallogenic Prediction of Nonferrous

Metals of Ministry of Education for the support of thisproject

References

[1] J-K Qiang and Y-Z Luo ldquoThe resistivity FEM numericalmodeling on 3-D undulating topographyrdquo Chinese Journal ofGeophysics vol 50 no 5 pp 1606ndash1613 2007 (Chinese)

[2] A C Tripp G W Hohmann and C M Swift Jr ldquoTwo-dimen-sional resistivity inversionrdquoGeophysics vol 49 no 10 pp 1708ndash1717 1984

[3] S K Park and G P Van ldquoInversion of pole-pole data for 3-Dresistivity structure beneath arrays of electrodesrdquo Geophysicsvol 56 no 7 pp 951ndash960 1991

[4] Y Sasaki ldquo3-D resistivity inversion using the finite-elementmethodrdquo Geophysics vol 59 no 12 pp 1839ndash1848 1994

International Journal of Geophysics 9

[5] J Zhang R L Mackie and T R Madden ldquo3-D resistivity for-ward modeling and inversion using conjugate gradientsrdquo Geo-physics vol 60 no 5 pp 1313ndash1325 1995

[6] M H Loke and R D Barker ldquoPractical techniques for 3D resis-tivity surveys and data inversionrdquo Geophysical Prospecting vol44 no 3 pp 499ndash523 1996

[7] X P Wu and G M Xu ldquoDerivation and analysis of partialderivative matrix in resistivity 3-D inversionrdquo Oil GeophysicalProspecting vol 34 pp 363ndash372 1999

[8] X-P Wu and G-M Xu ldquoStudy on 3-D resistivity inversionusing conjugate gradient methodrdquo Chinese Journal of Geophys-ics vol 43 no 3 pp 420ndash427 2000

[9] X-P Wu ldquo3-D resistivity inversion under the condition ofuneven terrainrdquoChinese Journal of Geophysics vol 48 no 4 pp932ndash936 2005

[10] H-F Liu J-X Liu R-W Guo X-K Deng and B-Y RuanldquoEfficient inversion of 3D IP data for continuous model withcomplex geometryrdquo Journal of Jilin University vol 41 no 4 pp1212ndash1218 2011

[11] N G Papadopoulos M-J Yi J-H Kim P Tsourlos and GN Tsokas ldquoGeophysical investigation of tumuli by means ofsurface 3D electrical resistivity tomographyrdquo Journal of AppliedGeophysics vol 70 no 3 pp 192ndash205 2010

[12] P I Tsourlos and R D Ogilvy ldquoAn algorithm for the 3-Dinversion of tomographic resistivity and induced polarisationdata preliminary resultsrdquo Journal of the Balkan GeophysicalSociety vol 2 pp 30ndash45 1999

[13] J G Huang B Y Ruan and G S Bao ldquoResistivity inversion on3-D section based on FEMrdquo Journal of Central South Universityof Technology vol 35 pp 295ndash299 2004

[14] TGunther C Rucker andK Spitzer ldquoThree-dimensionalmod-eling and inversion of dc resistivity data incorporating topogra-phymdashII inversionrdquo Geophysical Journal International vol 166no 2 pp 506ndash5517 2006

[15] G A Oldenborger and P S Routh ldquoThe point-spread functionmeasure of resolution for the 3-D electrical resistivity experi-mentrdquoGeophysical Journal International vol 176 no 2 pp 405ndash414 2009

[16] A Dey and H F Morrison ldquoResistivity modeling for arbitrarilyshaped three-dimensional structuresrdquoGeophysics vol 44 no 4pp 753ndash760 1979

[17] R G Ellis and D W Oldenburg ldquoThe pole-pole 3-D DC-re-sistivity inverse problem a conjugate- gradient approachrdquo Geo-physical Journal International vol 119 no 1 pp 187ndash194 1994

[18] D J LaBrecque M Miletto W Daily A Ramirez and EOwen ldquoThe effects of noise on Occamrsquos inversion of resistivitytomography datardquo Geophysics vol 61 no 2 pp 538ndash548 1996

[19] M-J Yi J-H Kim Y Song S-J Cho S-H Chung and J-H Suh ldquoThree-dimensional imaging of subsurface structuresusing resistivity datardquoGeophysical Prospecting vol 49 no 4 pp483ndash497 2001

[20] C C Pain J V Herwanger M H Worthington and C R E deOliveira ldquoEffectivemultidimensional resistivity inversion usingfinite-element techniquesrdquo Geophysical Journal Internationalvol 151 no 3 pp 710ndash728 2002

[21] A Pidlisecky E Haber and R Knight ldquoRESINVM3D a 3Dresistivity inversion packagerdquo Geophysics vol 72 no 2 pp H1ndashH10 2007

[22] L Marescot S P Lopes S Rigobert and A G Green ldquoNon-linear inversion of geoelectric data acquired across 3D objectsusing a finite-element approachrdquo Geophysics vol 73 no 3 ppF121ndashF133 2008

[23] M S Zhdanov Geophysical Inverse Theory and RegularizationProblems Elsevier New York NY USA 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

International Journal of Geophysics 9

[5] J Zhang R L Mackie and T R Madden ldquo3-D resistivity for-ward modeling and inversion using conjugate gradientsrdquo Geo-physics vol 60 no 5 pp 1313ndash1325 1995

[6] M H Loke and R D Barker ldquoPractical techniques for 3D resis-tivity surveys and data inversionrdquo Geophysical Prospecting vol44 no 3 pp 499ndash523 1996

[7] X P Wu and G M Xu ldquoDerivation and analysis of partialderivative matrix in resistivity 3-D inversionrdquo Oil GeophysicalProspecting vol 34 pp 363ndash372 1999

[8] X-P Wu and G-M Xu ldquoStudy on 3-D resistivity inversionusing conjugate gradient methodrdquo Chinese Journal of Geophys-ics vol 43 no 3 pp 420ndash427 2000

[9] X-P Wu ldquo3-D resistivity inversion under the condition ofuneven terrainrdquoChinese Journal of Geophysics vol 48 no 4 pp932ndash936 2005

[10] H-F Liu J-X Liu R-W Guo X-K Deng and B-Y RuanldquoEfficient inversion of 3D IP data for continuous model withcomplex geometryrdquo Journal of Jilin University vol 41 no 4 pp1212ndash1218 2011

[11] N G Papadopoulos M-J Yi J-H Kim P Tsourlos and GN Tsokas ldquoGeophysical investigation of tumuli by means ofsurface 3D electrical resistivity tomographyrdquo Journal of AppliedGeophysics vol 70 no 3 pp 192ndash205 2010

[12] P I Tsourlos and R D Ogilvy ldquoAn algorithm for the 3-Dinversion of tomographic resistivity and induced polarisationdata preliminary resultsrdquo Journal of the Balkan GeophysicalSociety vol 2 pp 30ndash45 1999

[13] J G Huang B Y Ruan and G S Bao ldquoResistivity inversion on3-D section based on FEMrdquo Journal of Central South Universityof Technology vol 35 pp 295ndash299 2004

[14] TGunther C Rucker andK Spitzer ldquoThree-dimensionalmod-eling and inversion of dc resistivity data incorporating topogra-phymdashII inversionrdquo Geophysical Journal International vol 166no 2 pp 506ndash5517 2006

[15] G A Oldenborger and P S Routh ldquoThe point-spread functionmeasure of resolution for the 3-D electrical resistivity experi-mentrdquoGeophysical Journal International vol 176 no 2 pp 405ndash414 2009

[16] A Dey and H F Morrison ldquoResistivity modeling for arbitrarilyshaped three-dimensional structuresrdquoGeophysics vol 44 no 4pp 753ndash760 1979

[17] R G Ellis and D W Oldenburg ldquoThe pole-pole 3-D DC-re-sistivity inverse problem a conjugate- gradient approachrdquo Geo-physical Journal International vol 119 no 1 pp 187ndash194 1994

[18] D J LaBrecque M Miletto W Daily A Ramirez and EOwen ldquoThe effects of noise on Occamrsquos inversion of resistivitytomography datardquo Geophysics vol 61 no 2 pp 538ndash548 1996

[19] M-J Yi J-H Kim Y Song S-J Cho S-H Chung and J-H Suh ldquoThree-dimensional imaging of subsurface structuresusing resistivity datardquoGeophysical Prospecting vol 49 no 4 pp483ndash497 2001

[20] C C Pain J V Herwanger M H Worthington and C R E deOliveira ldquoEffectivemultidimensional resistivity inversion usingfinite-element techniquesrdquo Geophysical Journal Internationalvol 151 no 3 pp 710ndash728 2002

[21] A Pidlisecky E Haber and R Knight ldquoRESINVM3D a 3Dresistivity inversion packagerdquo Geophysics vol 72 no 2 pp H1ndashH10 2007

[22] L Marescot S P Lopes S Rigobert and A G Green ldquoNon-linear inversion of geoelectric data acquired across 3D objectsusing a finite-element approachrdquo Geophysics vol 73 no 3 ppF121ndashF133 2008

[23] M S Zhdanov Geophysical Inverse Theory and RegularizationProblems Elsevier New York NY USA 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

EarthquakesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MineralogyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geological ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Geology Advances in