A geostatistically based inverse model for electrical resistivity

surveys and its applications to vadose zone hydrology

T.-C. Jim Yeh,1 S. Liu,1 R. J. Glass,2 K. Baker,3 J. R. Brainard,2 D. Alumbaugh,4

and D. LaBrecque,5

Received 24 January 2001; revised 16 May 2002; accepted 6 June 2002; published 6 December 2002.

[1] A sequential, geostatistical inverse approach was developed for electrical resistivitytomography (ERT). Unlike most ERT inverse approaches, this new approach allowsinclusion of our prior knowledge of general geological structures of an area and pointelectrical resistivity measurements to constrain the estimate of the electrical resistivityfield. This approach also permits sequential inclusion of different data sets, mimicking theERT data collection scheme commonly employed in the field survey. Furthermore, usingthe conditional variance concept, the inverse model quantifies uncertainty of the estimatecaused by spatial variability and measurement errors. Using this approach, numericalexperiments were conducted to demonstrate the effects of bedding orientation on ERTsurveys and to show both the usefulness and uncertainty associated with the inverseapproach for delineating the electrical resistivity distribution using down-hole ERT arrays.A statistical analysis was subsequently undertaken to explore the effects of spatialvariability of the electrical resistivity-moisture relation on the interpretation of the changein water content in the vadose zone, using the change in electrical resistivity. Core sampleswere collected from a field site to investigate the spatial variability of the electricalresistivity-moisture relation. Numerical experiments were subsequently conducted toillustrate how the spatially varying relations affect the level of uncertainty in theinterpretation of change of moisture content based on the estimated change in electricalresistivity. Other possible complications are also discussed. INDEX TERMS: 0903 Exploration

Geophysics: Computational methods, potential fields; 1869 Hydrology: Stochastic processes; 1875

Hydrology: Unsaturated zone; 1866 Hydrology: Soil moisture; 3260 Mathematical Geophysics: Inverse

theory; KEYWORDS: geostatistical inverse model, electrical resistivity tomography, vadose zone, resistivity-

moisture relation, spatial variability, sequential/successive linear estimator

Citation: Yeh, T.-C. J., S. Liu, R. J. Glass, K. Baker, J. R. Brainard, D. L. Alumbaugh, and D. LaBrecque, A geostatistically based

inverse model for electrical resistivity surveys and its applications to vadose zone hydrology, Water Resour. Res., 38(12), 1278,

doi:10.1029/2001WR001204, 2002.

1. Introduction

[2] The DC resistivity survey is an inexpensive andwidely used technique for investigation of near-surfaceresistivity anomalies. It recently has become popular forthe investigation of subsurface pollution problems [NationalResearch Council (NRC ), 2000]. In principle, it measuresthe voltage generated by a transmission of current betweenelectrodes implanted at the ground surface. Apparent (bulkor effective) electrical resistivity is then calculated and usedto interpret subsurface anomalies.[3] Classical formulas for determining apparent electrical

resistivity assume homogeneity, and the potential field is

smooth because of its highly diffusive nature. Consequently,conventional interpretations of electrical resistivity surveydata have been virtually ineffective for environmentalapplications, where electrical resistivity anomalies aresubtle, complex, and multiscale. To overcome these diffi-culties, a contemporary electrical resistivity survey has beendesigned to collect extensive electric current and electricpotential data sets in multi-dimensions. Without assumingsubsurface homogeneity, a mathematical computer model isemployed to invert the data sets to estimate the resistivityfield, using a regularized optimization approach [e.g., Dailyet al., 1992; Ellis and Oldenburg, 1994; Li and Oldenburg,1994; Zhang et al., 1995]. However, the general uniquenessand resolution of the three-dimensional electrical resistivityinversion have not been investigated sufficiently thus far[Carle et al., 1999; NRC, 2000].[4] While the physical process is different, the governing

equation for electric currents and potential fields created inthe electrical resistivity survey is analogous to that forsteady flow in saturated porous media. The mathematicalsolution to the inversion of an electrical resistivity survey istherefore similar to that of a groundwater hydrologicalsurvey. Groundwater hydrologists and reservoir engineers

1Department of Hydrology and Water Resources, University of Arizona,Tucson, Arizona, USA.

2Sandia National Laboratory, Los Alamos, New Mexico, USA.3Idaho National Engineering and Environmental Laboratory, Idaho Falls,

Idaho, USA.4Department of Civil and Environmental Engineering, University of

Wisconsin-Madison, Madison, Wisconsin, USA.5MultiPhase Technologies, Sparks, Nevada, USA.

Copyright 2002 by the American Geophysical Union.0043-1397/02/2001WR001204$09.00

14 - 1

WATER RESOURCES RESEARCH, VOL. 38, NO. 12, 1278, doi:10.1029/2001WR001204, 2002

have attempted to solve the inverse problem of flow throughmultidimensional, heterogeneous porous media for the lastfew decades [e.g., Gavalas et al., 1976]. Extensive reviewson the inverse problem of subsurface hydrology and varioussolution techniques are given by Yeh [1986], Sun [1994],and McLaughlin and Townley [1996]. They concluded thatprior information on geological structure, and some pointmeasurements of parameters to be estimated are necessaryto better constrain the solution of the inverse problem. Asimilar finding was also reported by Oldenburg and Li[1999] and Li and Oldenburg [2000] for the inverse prob-lems in geophysics.[5] Groundwater hydrologists also have used a multi-

component linear estimator (cokriging) to estimate thehydraulic conductivity field from scattered measurementsof pressure head and hydraulic conductivity in saturatedflow problems [Kitanidis and Vomvoris, 1983; Hoeksemaand Kitanidis, 1984]. The popularity of cokriging is attrib-uted to its ability to incorporate spatial statistics, pointmeasurements of hydraulic conductivity, and hydraulic headinto the estimation and to yield conditional mean estimates.Cokriging is also capable of quantifying the uncertaintyassociated with its estimate due to limited information andheterogeneity. Kitanidis [1997] articulated the differencesbetween cokriging and the classical inverse methods insubsurface hydrology. Nevertheless, cokriging is a linearestimator and it is limited to mildly nonlinear systems, suchas aquifers of mild heterogeneity, where the variance of thenatural logarithm of hydraulic conductivity, slnK

2 , is lessthan 0.1. When the degree of aquifer heterogeneity is large(slnK

2 > 1), the linear assumption becomes inadequate.Therefore cokriging cannot take full advantage of thehydraulic head information to obtain a good estimate ofhydraulic properties [Yeh et al., 1996].[6] To overcome this shortcoming, Yeh et al. [1995,

1996], Gutjahr et al. [1994], and Zhang and Yeh [1997]developed an iterative geostatistical technique, referred to asa successive linear estimator (SLE). In this technique, alinear estimator was used successively to incorporate thenonlinear relation between hydraulic properties and thehydraulic head. This method also employs a conditionalcovariance concept to quantify reductions in uncertainty dueto the incorporation of subsequent information. Yeh et al.[1995, 1996] and Zhang and Yeh [1997] demonstrated thatwith the same amount of information, the SLE methodrevealed a more detailed hydraulic conductivity field thancokriging. Hughson and Yeh [1998, 2000] extended the SLEmethod to the inverse problem in three-dimensional, var-iably saturated, heterogeneous porous media. On the basisof the SLE algorithm, Yeh and Liu [2000] developed asequential SLE technique for hydraulic tomography toprocess the large amount of data created by the tomography,and subsequently characterize aquifer heterogeneity. Theyinvestigated the effect of monitoring intervals, pumpingintervals, and the number of pumping locations on the finalestimate of hydraulic conductivity, and they establishedguidelines for a design of a hydraulic tomography test.[7] In section 2 of this paper, we introduce the concept of

stochastic representation of electrical resistivity tomography(ERT) inverse problems. In sections 3 and 4 we describe thedevelopment of a geostatistically based sequential SLEmethodology for ERT inversion problems. Section 5 offers

numerical examples that illustrate the usefulness of the newinversion approach, and describes the effects of geologicalstructures on the layout of electrical resistivity surveys. Therelation between the electrical resistivity and the moisturecontent of 25 soil cores were measured and analyzed forspatial variability in section 6. Impacts of spatial variabilityon the estimated changes in moisture content in the vadosezone, using ERT surveys were explored and discussed insection 7.

2. Stochastic Conceptualization of ERTInverse Problems

[8] Assume that in a geological formation, the electriccurrent flow induced by an electrical resistivity survey canbe described by

r � x xð Þrf xð Þð Þ þ I xð Þ ¼ 0; ð1Þ

where x is location, f is electric potential [V], I representsthe electric current source per volume [A/m3], and x is theelectrical conductivity [S/m], a reciprocal of the electricalresistivity, r [ohm m], which is assumed to be locallyisotropic. The boundary conditions associated with (1) are

fjG1¼ f* x xð Þrf � njG2

¼ q; ð2Þ

where f* is the electric potential specified at boundary G1,q denotes the prescribed electric current per unit area, and nis the unit vector normal to the boundary G2.[9] The electrical conductivity or resistivity of geological

media varies spatially due to inherent heterogeneous geo-logical processes [Sharma, 1997]. One way to describe thespatial variability of the electrical conductivity is the sto-chastic representation approach, similar to that used in geo-hydrology for the variability of hydraulic properties ofaquifers and vadose zones [see Gelhar, 1993; Yeh, 1998].Specifically, the natural logarithm of the electrical conduc-tivity, lnx(x), of a geological formation is to be considered asa stochastic process. The process is described in terms of anunconditional mean, hlnx(x)i = � (h i denotes the expectedvalue) and perturbations, c(x), which have an infinitenumber of possible realizations, characterized by a jointprobability distribution. Assuming that the perturbation is asecond-order stationary stochastic process, its joint proba-bility distribution can then be adequately represented by itsunconditional covariance function, Rcc. The covariancefunction essentially depicts the average spatial correlationstructure (pattern) of the electrical conductivity of a field inthe statistical sense. The use of the natural logarithm trans-formation is merely a mathematical convenience. Similarly,the electric potential field induced during an ERT survey canbe considered as a stochastic process and presented by f(x) =V(x) + v(x), where V(x) = hf(x)i and v(x) are the uncondi-tional perturbations of the electric potential.[10] Suppose that there are electrical conductivity meas-

urements (referred to as the primary variable or primaryinformation) from borehole electrical resistivity surveys,c*i = (lnx(xi) �) where I = 1, 2,. . .nc, and nc is the totalnumber of electrical conductivity measurements. Fromthese measurements, we have estimated the mean andcovariance function of the electrical conductivity field. An

14 - 2 YEH ET AL.: GEOSTATISTICAL MODEL FOR ERT

ERT survey is then conducted, and we have collected m setsof nv electric potential (perturbation) values, vj*, where j =nc + 1, nc + 2,.. . ., nc + m � nv. Hereinafter, they arereferred to as secondary information. We then seek aninverse model that can produce the electric potential andelectrical conductivity fields that preserve the observedelectrical potential and electrical conductivity values atsample locations. In addition, the fields must possess thestatistics (i.e., the mean and covariance function) describingtheir spatial variability, and also satisfy underlying physicalprocesses (i.e., the governing electric potential equation). Ina conditional probability concept, such an electric potentialor electrical conductivity field is a conditional realization off(x) field or lnx(x) field, respectively, among many possiblerealizations of the ensemble. The conditional electricalconductivity fields of the ensemble can be expressed asthe sum of conditional mean electrical conductivity and itsconditional perturbation, i.e., lnxc (x) = �c(x) + cc(x). Thesubscript c denotes the state of being conditioned. Similarly,the conditional potential fields can be written as fc(x) =Vc(x) + vc (x). While many possible realizations of suchconditional lnx(x) and f(x) fields exist, the conditional meanfields (i.e., �c(x) and Vc(x)) are unique, although notnecessarily exactly reflective of the true fields.[11] One way to derive these conditional mean fields is to

solve the inverse problem to obtain all possible conditionalrealizations of the electrical resistivity field. An average ofthe possible realizations will yield the conditional-meanelectrical resistivity field (see Hanna and Yeh [1998] andothers for geohydrology applications). An alternative to theabove is to solve the inverse problem in terms of theconditional mean equation.[12] By substituting the conditional stochastic variables

into the governing electric potential equation (1) and takingthe expected value, the conditional-mean equation takes theform

r � �c xð ÞrVc xð Þ½ þ r � cc xð Þrvc xð Þh i þ I xð Þ ¼ 0: ð3Þ

In equation (3), the current source, I(x), is considereddeterministic. Notice that the true conditional mean �c(x)and Vc(x) fields do not satisfy the continuity equation (3)unless the second term involving the product of perturba-tions is zero. This term represents the uncertainty because ofa lack of information of the two variables at locations wheremeasurements are not available. The uncertainty will vanishunder two conditions, namely, (1) all the electrical conduc-tivity values in the domain are specified (i.e., cc(x) = 0), or(2) all the electric potential values in the domain are known(i.e., vc(x) = 0). In practice, these two conditions will neverbe met, and evaluation of this term is intractable at thismoment. Consequently, in the subsequent analysis we willassume this term is proportional to the conditional meanelectric potential gradient such that we can rewrite the meanequation as

r � �ceff xð ÞrVc xð Þ½ þ I xð Þ ¼ 0: ð4Þ

This conditional mean equation has the same form asequation (1) but variables are expressed as the conditionaleffective electrical conductivity, �ceff (x), and conditionalmean electric potential field, Vc(x). The conditional effective

electrical conductivity thus is a parameter field thatcombines the conditional mean electrical conductivity�c(x) and hcc(x)rvc(x)i(rVc(x))

1. According to thisconcept, the conditional effective electrical conductivity isa parameter field that agrees with the electrical conductivitymeasurements at sample locations, and it yields a condi-tional mean electric potential field that preserves values ofelectric potential measurements when it is employed in theforward model (4), subject to boundary conditions (2). Onthe basis of this concept, an optimal inverse solution toequation (4) seeks the conditional effective electricalconductivity field. The successive linear estimator (SLE)approach is appropriate for this purpose.

3. Geostatistically Based SLE Inversion for ERT

[13] Below we present a brief description of the SLEalgorithm. A detailed discussion of the algorithm is givenby Yeh and Liu [2000]. The SLE algorithm, in general,consists of seven steps. Step 1 starts with a linear estimatorusing primary and secondary information to estimate thevalue of the primary variable at j locations where noinformation is available about the variable:

C ¼ LTCC*þ LT

v v*; ð5Þ

where C is a j � 1 vector of the estimated primary variable,c, (i.e., the estimated cc) at j locations and c*, and v* areavailable information about the primary and secondaryvariables (i.e., electrical conductivity, and potential measure-ments, respectively) at nd (nd = nc + nv) sample locations.The nd � j matrix, lc, and nv � j matrix, lv, are thecokriging weights applied to the primary and the secondaryinformation. They are related to the spatial covariancefunction of the primary and the secondary variable (i.e., Rccand Rvv, respectively) as well as the cross-covariancebetween the primary and secondary variables, Rcv. The cov-ariance function, Rcc, of the primary variable is prescribed apriori, and Rvv , and Rcv are calculated using a first-orderanalysis and the given Rcc [see Yeh and Liu, 2000]. Toinclude the uncertainty due to measurement errors associatedwith primary and secondary information, additional var-iances can be added to the diagonals of Rcc and Rvv. Theweights are then obtained from the solution of a cokrigingsystem of equations. In step 2, the covariance of the primaryvariable is modified to reflect effects of the availableinformation. That is,

R rþ1ð ÞCC ¼ R rð Þ

CC �R rð ÞCCLC R rð Þ

vCLv ð6Þ

where �RCC(r) is a j � nd subset of Rcc

(r) , covariance of c. Thesuperscript in parentheses is the iteration index and r = 0 atthis time. In step 3, the newly estimated variable field fromequation (5) is used to simulate the electric potential field, v,using the forward model (4). In step 4, the conditionalcovariances, Rvv

(r + 1) and their cross-covariance, Rvc(r + 1) are

updated using a first-order analysis. For step 5, these newlyevaluated covariances and cross-covariances compute newweights, lv, using

R rð Þvv Lv ¼ R T

vC : ð7Þ

YEH ET AL.: GEOSTATISTICAL MODEL FOR ERT 14 - 3

Step 6 is where the new weights, along with the differencebetween simulated v(r) and observed v*, are used to improvethe estimate of the primary variable. That is,

C rþ1ð Þ ¼ C rð Þ þ L rð ÞTv v* v rð Þ

� �: ð8Þ

In equation (8), C rð Þ represents the conditional primaryvariable estimate at iteration r. In step 7, the weights are thenused to update conditional covariances for the next iterationusing

R rþ1ð Þcc ¼ R rð Þ

cc R rð ÞvcLv: ð9Þ

This newly updated primary variable field (8) and the newconditional covariance (9) are used again in steps (3) and (4),followed by steps (5) through (7). Steps (3) through (7) arerepeated until no improvement in the estimate of the primaryvariable is found (i.e., when the variance of the estimatedprimary variable stabilizes). Notice that parallel computingprocedures can be implemented in many steps of SLE.

4. Sequential Inversion of ERT Surveys

[14] The above discussion describes the SLE method forthe secondary information collected during one excitation(i.e., one current source location) in an ERT survey. Themethod can simultaneously include all the secondary infor-mation collected during all the excitations in an ERTsurvey; however, the system of equations, (7), canbecome extremely large and ill conditioned. Thereforestable solutions to the equations can become difficult toobtain [Hughson and Yeh, 2000]. To avoid this problem, thesecondary information collected from excitations at differ-ent locations is used sequentially. Specifically, our sequen-tial method starts the inversion with the secondaryinformation collected from the excitation at one selectedlocation. Once the estimated field converges to the givencriterion, the newly estimated conditional effective primaryfield and its conditional covariance are used as priorinformation for the inversion of the next excitation. Thatis, the conditional effective c is used to evaluate both theconditional mean v, and sensitivity matrix associated withthe excitation at the new location. Using the first-orderanalysis, the sensitivity matrix and conditional covariance,Rcc, yield the v covariance and its cross-covariance with c,which are subsequently employed to derive the newweights. With the conditional mean of v, the new weights,and the observed v*, equation (8) then produces a newestimate of the conditional effective c, representing theestimate based on the information from the excitation at thenew location. The iterative process (similar to steps (3)through (7)) is then used to include the nonlinear relationbetween v and c. Once the solution converges and iterationstops, the same procedure is applied to the information ofthe next excitation. This sequential process continues untilthe data sets created from all the excitations are all usedin the inversion.[15] Our sequential approach uses the estimated electrical

conductivity field and covariance, conditioned on previoussets of potential measurements, as prior information for thenext estimation based on a new set of current source data.

The conditional moments are propagated sequentially untilall the data sets are fully utilized. Such a sequentialapproach thus allows accumulation of the vast amount ofsecondary information obtained from an ERT survey, whilemaintaining the system of equations to be solved at amanageable size and with the least numerical difficulties.Vargas-Guzman and Yeh [1999] provided a theoretical proofto show that such a sequential approach is identical to thesimultaneous approach for linear systems. This sequentialalgorithm for hydraulic tomography was also tested andverified in sandbox experiments [Liu et al., 2002].

5. Numerical Examples

[16] To demonstrate the ability of our inverse method forERT, a vertical profile of a hypothetical vadose zone (200cm � 10 cm � 200 cm) was created and discretized into 20� 1 � 20 elements of 1000 cm3. A stochastic random fieldgenerator [Gutjahr, 1989] was used to assign an electricalconductivity value to each element. The electrical conduc-tivity field was assumed to have a geometric mean of0.01261 [S/m] and a covariance with an exponential corre-lation structure and a variance of 0.5 for the naturallogarithm of the electrical conductivity field. The correla-tion structure was anisotropic with a horizontal correlationof 240 cm and a vertical correlation scale of 20 cm. Thisanisotropic structure yielded a heterogeneous and stratifiedelectrical conductivity distribution shown in Figure 1a.[17] Two cases were examined to illustrate effects of

stratification on an ERT array layout. In cases I, an ERTsurface array is deployed on the hypothetical stratifiedformation, representing an ERT array parallel to stratifica-tion. In cases II, the same surface array is deployed on thesurface of a formation (Figure 1b) identical to the hypo-thetical vadose zone in cases I but rotated counterclock-wise 90�, representing an ERT array perpendicular tostratification.[18] During both ERT surveys, a pole-pole array was

used. Nineteen potential electrodes (triangles in Figures1c, 1d, 1e, and 1f ) and one current electrode (circle) wereused during each survey. In this example, the currentelectrode was moved 4 times from one position to anotherto create four voltage/current data sets, in which each setconsists of 19 potential measurements and one currentmeasurement. Note that the graphic depicts four circles:Only one current electrode was used in each survey, but itwas positioned at four different locations for the foursurveys. The ERT surveys were simulated numerically usingequation (1), assuming the surface to be a boundary with noelectric current flux and the remaining three sides of thevadose zone to be prescribed electric potential boundaries.[19] Once the voltage/current data sets became available

and one electrical conductivity measurement was taken(square in Figures 1c and 1d), the sequential inverseapproach was employed to estimate the electrical conduc-tivity field. Figures 1c and 1d depict the estimated condi-tional effective c fields for cases I and II, respectively. Acomparison between Figures 1c and 1d shows that the samesurface array results in better estimates at depths in case IIthan in case I. Similarly, Figures 1e and 1f show that theconditional variance (uncertainty of estimate) is smaller atdepths in case II than in case I. The conditional variance at alocation indicates the uncertainty of the estimate at the

14 - 4 YEH ET AL.: GEOSTATISTICAL MODEL FOR ERT

location. For example, if the conductivity at a location isknown exactly, the conditional variance at that location iszero. Otherwise, the conditional variance is equal to thevariance of the electrical conductivity field. Alternately, theconditional variance is smaller than the variance because ofincorporation of the secondary information during theinverse modeling. Therefore the smaller the conditionalvariance at a given location, the better the estimate. Figure1e shows that the surface ERT yields small conditionalvariances only near the land surface where electric potentialmeasurements were taken. The variance increases rapidlywith depth, indicating that the effectiveness of the surveydecreases because of the stratification of the electricalconductivity field of the medium. Notice that during thegeneration of the electrical conductivity field, the stratifica-tion is denoted statistically by a long correlation scale in onedirection and a short correlation scale in the other. Noticealso that during the inversion of the ERT survey data sets forthe two cases, the information about the covariance functioncorresponding to each case was prescribed, implying thedifference in the inverse results comes solely from thenature of the inverse problem. As a consequence, Figures1a–1f manifest the fact that orientation of stratification canaffect the effectiveness of an ERT array layout. That is,when the array is perpendicular to the stratification (case II),

potentials at measurement locations possess informationover greater extents (depths in this study) than when thearray is parallel to the stratification. Because of this fact andbecause most of geological formations are horizontallystratified, a down-hole ERT array is generally more effectivein depicting the electrical conductivity field over a greatervolume than a surface ERT array.[20] In addition to the first two cases, case III and case IV

were also investigated. In case III, two down-hole ERTarrays (19 potential electrodes and 1 current electrode) wereused. For case IV, a combination of the down-hole arrays (19potential electrodes and 1 current electrode) and one surfacearray (20 potential electrodes) were deployed in the hypo-thetical electrical conductivity field. Figure 2a illustrates theestimated conditional effective electrical conductivity fieldfor case III. A combination of the surface and the down-holeelectrode arrays (case IV) yields a higher-resolution image ofthe electrical conductivity field near the surface (Figure 2b).This is attributed to the additional surface-monitoring array.This conclusion is also evident in Figures 2c and 2d, wherethe corresponding conditional variance distribution for eachcase is shown.[21] While the conditional variance provides uncertainty

of the estimate, it is an ensemble statistic, which may not beappropriate for the single realization described in these

Figure 1. (a and b) True electrical conductivity perturbation fields, c, for cases I (horizontally stratifiedmedia) and II (vertically stratified media), respectively. (c and d) Estimated conditional effectiveconductivity perturbation fields for cases I and II, respectively. (e and f ) Conditional variances for cases Iand II, respectively.

YEH ET AL.: GEOSTATISTICAL MODEL FOR ERT 14 - 5

examples. Better criteria to compare the estimated conduc-tivity field with the true field are the average absolute errornorm, L1 and the mean square error norm, L2, which aredefined as

L1 ¼ 1

N

XNi¼1

ci cij j L2 ¼ 1

N

XNi¼1

ci cið Þ2; ð10Þ

where ci and ci represent the true and the estimatedperturbation of the conditional effective electrical con-ductivity after a natural logarithm transformation, respec-tively, i indicates the element number, and N is the totalnumber of elements in the hypothetical vadose zone. Thesmaller the L1 and the L2 value, the better the estimate.Figures 2e and 2f show plots of the true c versusestimated conditional effective c fields and the L1 and L2values associated with the two cases. The results areconsistent with those based on the conditional variancecriterion. Also illustrated in Figures 2e and 2f is thediscrepancy between the true and the estimate (scatteringaround the 45� line) caused by the limited measurementsof the electric potential field.[22] In the above inversion examples, the voltage/current

measurements were assumed to be error-free and statisticalparameters such as mean, variance, and correlation scales

required for the inversion were assumed to be known orestimated beforehand. For hydraulic tomography, both theeffects of error in measurements, and uncertainty of thestatistical parameters on the estimate were investigatednumerically by Yeh and Liu [2000] and using sandboxexperiments [Liu et al., 2002]. In short, they found thatuncertainty of the statistical parameters does not influencethe estimate significantly if sufficient and accurate secon-dary information is available. However, errors in pointmeasurements of hydraulic conductivity or hydraulic headcan have significant impacts on the estimate. They reportedthat the benefit of hydraulic tomography vanishes rapidly ifthe hydraulic head monitoring locations remain the same.In addition, network design issues were explored, such asthe sample interval for head measurements and the locationof pumping in terms of the correlation scale of theheterogeneity. We believe their results also hold for theERT.

6. Translation of Electrical ResistivityDistribution to Water Content Distributionfor Hydrological Applications

[23] Electrical resistivity tomography has been used tomonitor spatial and temporal variation of soil water content

Figure 2. (a) Estimated conditional effective electrical conductivity perturbation field for case III(down-hole ERT arrays) and (b) for case IV (a combination of the surface and down-hole arrays). (c andd) Corresponding conditional variances. (e and f ) Corresponding scatterplots.

14 - 6 YEH ET AL.: GEOSTATISTICAL MODEL FOR ERT

[e.g., Daily et al., 1992; Zhou et al., 2001; Brainard et al.,2001]. During an infiltration event, the water content ofgeological media is generally assumed to be the onlyelement that undergoes dramatic changes. Therefore track-ing the change in the electrical resistivity has often beenregarded as a useful means to delineate the change of thewater content in the vadose zone. Specifically, an ERTsurvey is conducted on a site before an infiltration event inorder to obtain the background distribution of the electricalresistivity. After the infiltration event, an ERT survey isundertaken again to acquire the electrical resistivity distri-bution of the wetted soils. Next, the difference between thetwo electrical resistivity distributions is used to interpretinfiltration and the movement of the water plume, assu-ming a relation between the electrical resistivity and watercontent.[24] In order to relate the water content to resistivity, a

power law has been used:

r ¼ roqm: ð11Þ

In (11), r is bulk electrical resistivity, ro is a fitting para-meter that is related to the electrical resistivity of porewater, m is a fitting parameter, and q denotes water content.Using (11), the difference between the natural log of theelectrical resistivity before and after infiltration thenbecomes

�lnr ¼ m� ln q: ð12Þ

According to this equation, if m is constant and knownprecisely, then the change of lnr is linearly proportional tothe change of lnq. However, the change in lnr may notdirectly correspond to the change of lnq if m exhibitssignificant spatial variability. This implies that the sameamount of change in moisture content may lead to differentamounts of change in the electrical resistivity in differentpart of a medium. Notice the variability of ro does notappear in equation (12). A statistical analysis based onequation (12), assuming independence between m and q,leads to an expression for the variability in change in lnr:

var � ln r½ ¼ M2 var � ln q½ þ�2 var m½ ; ð13Þ

where var[�lnq], var[�lnr], and var(m) are variances of�lnq, �lnr and m, respectively. The change in mean lnq isdenoted by �, and M is the mean value of m. According toequation (13), the variability of �lnr depends on not onlythe variance of �lnq but also the variance (spatial varia-bility) of m, M, and �.[25] To investigate the spatial variability of ro and m in

the field, the electrical resistivity as a function of themoisture content was measured for 25 samples collectedfrom a borehole at the Sandia-Tech Vadose Zone (STVZ)infiltration field site, Socorro, New Mexico. The field sitesediments are part of the Sierra Ladrones Formation, Up-per Santa Fe Group. They consist of fine to coarsegrained, poorly consolidated, ancestral Rio Grande axial-river deposits with intermittent layers of debris flowsediments and sedimentary layers of eolian sands (seeBrainard et al. [2001] for a complete site description). A

total of 25 samples were collected from eight 1.5-mlengths of a continuous core from a borehole at the fieldsite.[26] The samples were highly unconsolidated and easily

disintegrated. They had to be repacked into sample ringsto bulk density values determined from preliminary in situmeasurements. The bulk densities were estimated to be1.53 g cm3 for fine-medium sand, 1.61 g cm3 formedium-coarse sand, and 1.34 g cm3 for clays. Thesamples were then placed in a hanging column apparatusand were allowed to reach moisture equilibrium at tensionsfrom 100 cm down to 0 cm to obtain the main wettingcurve (MWC). Also, we reversed the process by measur-ing the moisture retention for the main drainage curve(MDC) starting at the saturated moisture content. Pressurechambers were used to drain the samples at pressuresgreater than 100 cm. Moisture equilibrium was determinedduring imbibition by weighing the samples daily andobserving changes in moisture content. Equilibrium wasdetermined during drainage by monitoring the water levelin the burette.[27] The electrical resistivity was measured at each mois-

ture equilibrium point for both the MWC and MDC byplacing the sample in an impedance analyzer sample holder,and applying a logarithmic sweep of frequencies across thesample. A Hewlett Packard model 4129A LF [Knight,1991] impedance analyzer was connected to a personalcomputer for automated data acquisition of impedancemeasurements. The impedance value corresponding to thefrequency not affected by polarization at the sample/elec-trode interface was used to calculate the electrical resistivity.The electrical resistivity was calculated by multiplying thesample resistance by the ratio of the sample cross-sectionalarea to the sample length [Knight, 1991]. Equation (11) wasthen fit to the measured electrical resistivity and moisturedata to determine the values for ro and m. A plot of themeasured electrical resistivities at each moisture content andbest fit curves for the 25 cores is shown in Figure 3, where itcan be seen that significant variability exists in the electricalresistivity and moisture content relation.[28] Frequency distributions of the values for lnro and

lnm are shown in Figure 4, and it appears that both lnroand lnm are approximately normally distributed. Assum-ing they follow lognormal distributions, the statistics forthese two parameters of the core samples were deter-mined. The geometric mean of ro is 7.036 [ohm m] andthe variance, standard deviation, and percent of coefficientof variation for lnro are 0.633, 0.796, and 40.8, respec-tively. For the parameter m, the geometric mean is 1.336,and variance, standard deviation, and percent of coeffi-cient of variation for lnm are, 0.034, 0.185, and 63.7,respectively.[29] A spatial statistical analysis was also conducted.

Variograms for the parameters are presented in Figure 5.For lnro, an exponential variogram model was selected. Thesill, range, and nugget values are 0.8, 3.5 m, and 0.08,respectively. Similarly, an exponential variogram model waschosen for m. The sill, range, and nugget values for lnm are0.043, 3.5 m, and 0.01, respectively. Statistics of the twoparameters manifest their spatial variability and spatialcorrelation structures. Figures 6a and 6b are plots of theseparameter values with depth and the lithology along the

YEH ET AL.: GEOSTATISTICAL MODEL FOR ERT 14 - 7

borehole. The side-by-side Figures 6a and 6b show that thespatial variation of these parameter values appears tocorrespond qualitatively to the lithology. Further, we foundthat the relation between the electrical resistivity and the

moisture content did not appear to be hysteretic [Baker,2001].

7. Uncertainty in Hydrologic Interpretation

[30] Our field data demonstrated a significant spatialvariability of the electrical resistivity and moisture contentrelation. In this section, the effects of this variability areinvestigated with regard to moisture movement, which wasmonitored using ERT surveys in the vadose zone.[31] We investigated two scenarios in a hypothetical

vadose zone of 200 cm � 20 cm � 200 cm. In scenario1, the electrical resistivity field before and after infiltrationwas known precisely. For scenario 2, the electrical resistiv-ity fields were estimated from a down-hole ERT survey. Thehypothetical vadose zone was discretized into 200 elements,and each element had a dimension of 20 cm in bothhorizontal directions and 10 cm in the vertical. The unsa-turated hydraulic properties of each element were assumedto be described by the Mualem-van Genuchten model [vanGenuchten, 1980]:

q yð Þ ¼ qs qrð Þ 1þ ayj jð Þa½ bð Þþqr ð14aÞ

K yð Þ ¼ Ks 1 ayj jð Þ a1ð Þ1þ ayj jð Þa½ bð Þ

� �2

= 1þ ayj jð Þa½ b=2ð Þ;

ð14bÞ

where y is the capillary pressure head, Ks is the saturatedhydraulic conductivity, a and a are shape factors, and b =

Figure 3. The electrical resistivity-moisture data and thebest fit curves for samples from STVZ site at Socorro, NewMexico.

Figure 4. Frequency distributions for lnr0 and lnm of theSTVZ data set.

Figure 5. Variograms for (a) lnr0 and (b) lnm data sets atthe STVZ site, showing their spatial structures.

14 - 8 YEH ET AL.: GEOSTATISTICAL MODEL FOR ERT

11/a. The variability of saturated moisture content, qs, andresidual moisture content, qr, is generally negligible; bothwere treated as deterministic constants with a value of 0.366and 0.029, respectively. The parameters, Ks, a, and a, wereconsidered as random fields with the geometric mean of0.0063 cm s1, 0.028 cm1, and 2.0, respectively. Thevariances of lnKs, lna, and lna were 0.1, 0.1, and 0.01,respectively. It was also assumed that all three parameterspossessed the same exponential covariance function with ahorizontal correlation scale of 240 cm and a vertical correla-tion scale of 20 cm. Following the generation of randomhydraulic parameter fields, a hydrostatic capillary pressurehead distribution, with zero capillary pressure head at thebottom, was assigned to the vadose zone as the initialcondition. The corresponding water content distribution wasconsidered as the background q distribution (Figure 7a).Next, a steady infiltration event was simulated using a finiteelement model for flow and solute transport in variablysaturated media: Modified Method of Characteristics 3(MMOC3) [Srivastava and Yeh, 1992]. The top center ofthe vadose zone (from x = 80 to 120 cm, y = 0 to 20 cm, andz = 200 cm) was treated as a constant head boundary with acapillary pressure head of 80 cm. The remainder of thesurface and the two sides of the domain were considered asno-flux boundaries; the bottom was assumed to be a watertable. Once the simulation of the steady flow field wascompleted, the resulting water content distribution wasdenoted as the q distribution after infiltration (Figure 7b).The change of the lnq distribution before and after theinfiltration was then computed and plotted in Figure 7c.

[32] In order to convert the simulated moisture contentdistribution to an electrical resistivity field, each element ofthe vadose zone was assigned a pair of ro and m valuesusing the random field generator. For the two aforemen-tioned scenarios, three cases, A, B, and C, were considered.The m fields for the three cases were generated with ageometric mean of 1.35. The variances of lnm fields forcases A, B, and C, are 0.0, 0.033, and 0.1, respectively.While the three cases have different m fields, they have anidentical ro field with a geometric mean of 8.5 [ohm m] andvariance of lnr0 equal to 0.1. Again, similar to the hydraulicparameter field, these fields have an exponential covariancestructure with a horizontal correlation scale of 240 cm andthe vertical correlation scale of 20 cm. Notice that althoughhydraulic parameter fields, ro, and m fields are spatiallycorrelated, they are mutually independent.[33] For cases A, B, and C of scenario 1, equation (11)

was used in conjunction with the generated ro, and m fields,and the background q distribution to construct the back-ground electrical resistivity map for this hypothetical site.Similarly, the electrical resistivity distribution was alsoobtained after infiltration, corresponding to the q distribu-tion after infiltration. Then, the change in lnr was derivedby subtracting the lnr after infiltration from the backgroundlnr. The changes in lnr for cases A, B, and C of scenario 1are shown in Figures 8a, 8b, and 8c, respectively, and areplotted against m�lnq of case A along with the values of L1and L2 in Figures 9a, 9b, and 9c.[34] According to Figures 7c, 8a, 8b, and 8c, the change

in lnr reflects the change in lnq only if the electrical

Figure 6. Spatial distributions of (a) r0m, and (b) lithology along a borehole at the STVZ site, Socorro,New Mexico.

YEH ET AL.: GEOSTATISTICAL MODEL FOR ERT 14 - 9

resistivity field is known exactly at every point and m is aconstant (i.e., variance of lnm = 0, Figure 8a). As thevariance of lnm increases, the discrepancy between changein lnr and change in lnq increases (Figures 8b and 8c). Inother words, because of the variability of the parameter m,different parts of a geological medium exhibit differentamounts of change in the electrical resistivity even if theyundergo the same amount of change in water content.[35] In scenario 2, the background electrical resistivity

distribution and the q distribution after infiltration are nolonger known precisely. Instead, both electrical resistivitydistributions were estimated using our sequential inverse

approach to interpret the simulated ERT data collected fromtwo down-hole arrays (circles, triangles, and a square) asindicated in Figures 8d, 8e, and 8f. Specifically, in cases A,B and C of scenario 2, forward simulations of ERT surveyswere conducted in the resistivity fields, created from therandom r0 and m fields and q distributions before and afterinfiltration. The boundary conditions were set to be thesame as in previous cases. These simulations yieldedelectrical potential measurements at specified monitoringlocations of each case, which afterward were used in theinverse model to derive estimated r fields. The changes inlnr for cases A, B, and C were then calculated and are

Figure 7. (a) Simulated moisture distribution (lnq) before infiltration. (b) Simulated moisturedistribution (lnq) after infiltration. (c) Spatial distribution of change in moisture content (�lnq) (i.e.,differences between Figures 7a and 7b).

Figure 8. Distributions of changes in lnr (a, b, and c) for cases A (the variance of m = 0.0), B (thevariance of m = 0.033), and C (the variance of m = 0.1) in scenario 1(no uncertainty in lnr), respectively.Distributions of changes in lnr estimated from ERT surveys (d, e, and f) for cases A, B, and C of scenario2 (uncertainty in lnr from ERT surveys), respectively.

14 - 10 YEH ET AL.: GEOSTATISTICAL MODEL FOR ERT

shown in Figures 8d, 8e, and 8f, respectively. They also areplotted against m�lnq of case A of scenario 1 and areshown in Figures 9d, 9e, and 9f, along with the values of L1and L2. Again, the measurements were considered error-freeand other inputs to the model were assumed to be knownexactly.[36] In this scenario, our estimated electrical resistivity

fields are uncertain because of limited information andspatial variability. A comparison between Figure 7c andFigure 8d demonstrates that even with uncertainty, thechange in lnr in case A of scenario 2 still resembles thechange in lnq when m is constant. The resemblance deteri-orates as the variance of m increases, however (see Figures8e and 8f for cases B and C of scenario 2, respectively).Notice anomalous changes in lnr appear close to thelocations (Figure 8f ) where potential measurements weretaken. This indicates that a great variation in m cansignificantly exacerbate the effect of the limited data seton the interpretation of ERT results. Consequently, cautionmust be taken during interpretation of the change inmoisture content based on the change in the electricalresistivity alone. The accuracy of the interpretation, accord-ing to equation (13), depends on the accuracy of the ERTinversion, the mean value of m, the amount of change inmean lnq, the variability of m, and the change in lnq.

[37] We emphasize that our illustrations consider onlythe variation of parameters of the simple power law for theelectrical resistivity-moisture relationship in a syntheticvadose zone. Under field conditions, many other factorscan further complicate the interpretation of an ERT survey,and certainly, the validity of the power law deservesfurther exploration regarding the electrical resistivity-mois-ture relation. For instance, while the power law fits ourfield data quite well, it may not be suitable for othergeological media. The electrical resistivity is also wellknown to be sensitive to salt concentration, clay content,ion exchange, temperature [Keller, 1987] and other site-specific attributes.[38] While the electrical resistivity was assumed to be

locally isotropic in this study, in the field it can beanisotropic and measurement-scale dependent. The electri-cal resistivity anisotropy of a medium at a given measure-ment scale, similar to the hydraulic conductivity anisotropy,is an artifact caused by averaging distinct electrical resis-tivity values of layers of material at scales smaller than themeasurement scale. The anisotropy therefore depends on theaverage length and thickness of the layers, and the varianceof each layer’s electrical resistivity, which may vary withthe scale of measurement (or the size of discretization of thedomain used in ERT inversion). In general, the electrical

Figure 9. (a, b, and c) Scatterplots of change in lnr versus change in lnq corresponding to cases A, B,and C of scenario 1 (no uncertainty in lnr) in Figures 8a, 8b, and 8c, respectively. (d, e, and f )Corresponding scatterplots for Figures 8c, 8d, and 8e, respectively, for scenario2 (with uncertainty inlnr).

YEH ET AL.: GEOSTATISTICAL MODEL FOR ERT 14 - 11

resistivity is greatest in the direction perpendicular tolayering and least in the direction parallel to layering.Moreover, because the electrical resistivity of each layercan vary with moisture content and therefore among thelayers, the anisotropy of the averaged bulk electrical resis-tivity is expected to vary with the moisture content. Specif-ically, the electrical resisitivity anisotropy ratio is defined asthe ratio of the bulk electrical resistivity of electric currentflow perpendicular to bedding to the resistivity of electricalcurrent flow parallel to bedding. The anisotropy ratio willincrease as the medium becomes less saturated in a mannersimilar to the moisture-dependent anisotropy in unsaturatedhydraulic conductivity described by Yeh et al. [1985a,1985b, 1985c].[39] These possible complications necessitate further the-

oretical and experimental investigations of the fundamentalelectrical resistivity-moisture relation. Their effects on theinversion of ERT and its hydrological interpretation deservefurther exploration. Finally, while the ability of ERT surveysfor detecting changes in moisture content is attractive, wemust point out that many hydrological analyses demandaccurate measurements of moisture content distributions,not just the monitoring of soil moisture changes.

8. Conclusion

[40] A sequential, geostatistical inverse approach forhydraulic tomography was adapted for electrical resistivitytomography. The sequential inverse approach mimics thesequential ERT data collection scheme commonly em-ployed in a traditional field survey. The inverse methodconstrains the estimate of the electrical resistivity field byincluding borehole measurements of the electrical resistiv-ity, in addition to potential measurements from the ERTsurvey, and information of geological structures through thestatistic spatial covariance. The sequential approach iscomputationally efficient, allows fine-grid discretization ofthe solution domain, and permits sequential inclusion ofdifferent data sets. Furthermore, the conditional variance inthe inverse model quantifies uncertainty in the estimatecaused by the spatial variability of the electrical resistivity,uncertainty in measurement errors, and a limited number ofdata sets.[41] Through numerical experiments based on our

inverse approach, we showed that geological beddingaffects effectiveness of the sampling array of ERT. Sam-pling perpendicular to bedding (down-hole array) increasesthe resolution of the electrical resistivity estimate becauseof the long correlation in the direction parallel to bedding.Conversely, the effectiveness of the surface array (samplingparallel to bedding) is restricted to a shallow depth becauseof the long correlation scale along bedding and the shortcorrelation scale in the direction perpendicular to bedding.Significant variability of the electrical resistivity-moisturerelation was observed in our field samples. Both thetheoretical analyses and numerical experiments suggest thatsuch a spatially varying relation can exacerbate the level ofuncertainty in the interpretation of change of moisturecontent based on the estimated change in the electricalresistivity. These results call for additional studies of theunderlying physics of the electrical resistivity-moisturerelation and its spatial variation. Last, development ofbetter methodologies is needed for incorporating this var-

iability in the interpretation of the ERT survey, such thatERT can be an effective monitoring tool for vadose zoneprocesses.

[42] Acknowledgments. This research is funded in part by a DOEEMSP96 grant through Sandia National Laboratories (contract AV-0655#1)and a DOE EMSP99 grant through University of Wisconsin, A019493. AtSandia National Laboratories, work was supported by DOE EMSP undercontact DE-AC04-94 AL85000. We also gratefully acknowledge ourcolleague, Martha P.L. Whitaker, for proofreading this document in itsvarious states of completion. Many thanks are extended to two reviewersfor providing critical comments to improve the paper. In particular, we aregreatly indebted to Andreas Kemna, who provided a meticulous, insightful,constructive, and open-minded review of our manuscript.

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D. Alumbaugh, Department of Civil and Environmental Engineering,

University of Wisconsin-Madison, 1415 Engineering Drive, Madison, WI53706, USA. (alumbaugh@engr.wisc.edu)

K. Baker, Idaho National Engineering and Environmental Laboratory,P.O. Box 1625, MS 2107, Idaho Falls, ID 83415-2107, USA. (bakeke@inel.gov)

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T.-C. J. Yeh, Department of Hydrology and Water Resources, Universityof Arizona, P.O. Box 210011, Tucson, AZ 85721-0011, USA. (yeh@hwr.arizona.edu)

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