An Integrative Approach for Monitoring Water...

Rep

rodu

ced

from

Vad

ose

Zon

e Jo

urna

l. P

ublis

hed

by S

oil S

cien

ce S

ocie

ty o

f Am

eric

a. A

ll co

pyrig

hts

rese

rved

.

An Integrative Approach for Monitoring Water Movement in the Vadose Zone

Shuyun Liu and Tian-Chyi J. Yeh*

ABSTRACT While ERT surveys are useful, Yeh et al. (2002) showedthat uncertainties in tracking moisture movement in theElectrical resistivity tomography (ERT), during the past few years,vadose zone on the basis of ERT surveys alone canhas emerged as a potentially cost-effective, noninvasive tool for im-

aging changes of moisture content in the vadose zone. The accuracy be quite significant. They reported that both inverseof ERT surveys, however, has been the subject of debate because of modeling of the ERT surveys and the spatial variabilityits nonunique inverse solution and spatial variability in the constitutive of the parameters in the constitutive resistivity–moisturerelation between resistivity and moisture content. In this paper, an content relation contribute to the overall uncertainties.integrative inverse approach for ERT, based on a stochastic informa- Depending on the number of parameters to be invertedtion fusion concept of Yeh and Simunek, was developed to derive the and the quantity of data available (Yeh and Simunek,best unbiased estimate of the moisture content distribution. Unlike

2002), the ERT survey inverse problem is often ill posed,classical ERT inversion approaches, this new approach assimilatesand may have no unique solution. For ill-posed prob-prior information about the geological and moisture content structureslems, most ERT inversion approaches employ optimi-in a given geological medium, as well as sparse point measurements ofzation algorithms with some type of regularization (e.g.,the moisture content, electrical resistivity, and electric potential. Using

these types of data and considering the spatial variability of the resisti- Tikhonov, 1963). While the regularization algorithm yieldsvity–moisture content relation, the new approach directly estimates smooth estimates, there is no guarantee that it will pro-three-dimensional moisture content distributions instead of simply duce the best unbiased estimate of the resistivity field,changes in moisture content in the vadose zone. Numerical experi- which reflects flow processes and underlying geologi-ments were conducted to investigate the effect of uncertainties in the cal structures. Neither does the regularization provideprior information on the estimate. The effects of spatial variability in a meaningful way to quantify the uncertainty associatedthe constitutive relation were then examined on the interpretation of

with the spatial variability. Additionally, when convert-the change in moisture content, based on the change in electricaling changes in resistivity to changes in moisture content,resistivity from the ERT survey. Finally, the ability of the integrativeit is often assumed that the parameters in the constitu-approach was tested by directly estimating moisture distributions intive relation (e.g., Archie’s Law) are constant across thethree-dimensional, heterogeneous vadose zones. Results show that the

integrative approach can produce accurate estimates of the moisture entire domain. Recent studies by Baker (2001) and Yehcontent distributions and that incorporating some measurements of et al. (2002) reported pronounced spatial variability ofthe moisture content is essential to improve the estimate. these parameters in the field. Neglecting this spatial vari-

ability and using a single resistivity–moisture calibrationcurve can add to the level of uncertainty in the final in-

Electrical resistivity surveys are increasingly used terpreted change in moisture content (in an unquanti-to collect extensive electric current and potential fiable way).

data in multiple dimensions to image subsurface electri- Besides the uncertainties inherent in the interpretationcal resistivity distribution (Daily et al., 1992; Ellis and of ERT surveys, most current three-dimensional inverseOldenburg, 1994; Li and Oldenburg, 1994; Zhang et al., models demand significant computational resources to1995). Recently, ERT surveys have found their way into process the large data sets typically collected during a sur-subsurface hydrology applications. This is attributed to vey. Furthermore, a change in moisture content only pro-the fact that knowledge of the spatial distribution of the vides qualitative information about water movement inelectrical properties of subsurface media can provide the vadose zone. The actual moisture content values atvaluable information for characterizing waste sites and each point, which are vital to hydrological investigationsmonitoring flow and contaminant movement in the va- or hydrological inverse modeling (e.g., Yeh and Simu-dose zone. For example, during an infiltration event, the nek, 2002), remain unknown. For instance, since the un-moisture content of a geological medium is generally saturated hydraulic conductivity is a nonlinear functionthe only factor that undergoes dramatic changes, and the of moisture content, changes in the moisture content alonechanges in resistivity can be related to changes in mois- do not provide enough information to uniquely charac-ture content. Tracking the changes in resistivity through terize the unsaturated hydraulic properties of the vadosetime, therefore, has been found to be useful for detecting zone. As a consequence, an inverse approach is neededtemporal changes in the moisture content of the vadose that can account for spatial variability of the resistivity–zone (Daily et al., 1992; Zhou et al., 2001). moisture content relation, efficiently process the large

number of data sets, and produce detailed moisturecontent distributions with the least uncertainty.S. Liu, Burgess & Niple, 5025 E. Washington St., Phoenix, AZ 85034;

T.-C.J. Yeh, Dep. of Hydrology and Water Resour., John W. Harsh- While the physical process of electric current flow isbarger Bldg., The Univ. of Arizona, Tucson, AZ 85721. Received 18 different from that of groundwater flow, the governingJuly 2003. Original Research Paper. *Corresponding author (yeh@ equation of the electric current and the potential fieldshwr.arizona.edu).

created during ERT surveys is analogous to that of steadyPublished in Vadose Zone Journal 3:681–692 (2004). Soil Science Society of America677 S. Segoe Rd., Madison, WI 53711 USA Abbreviations: ERT, electrical resistivity tomography.

681

Rep

rodu

ced

from

Vad

ose

Zon

e Jo

urna

l. P

ublis

hed

by S

oil S

cien

ce S

ocie

ty o

f Am

eric

a. A

ll co

pyrig

hts

rese

rved

.

682 VADOSE ZONE J., VOL. 3, MAY 2004

flow in a confined aquifer. The mathematical treatment � � �0��m [4]of the inversion of an ERT survey is therefore similar

where � is bulk electrical resistivity, �0 is a fitting param-to that used in hydraulic tomography (Yeh and Liu,eter that is related to the electrical resistivity of pore2000; Bohling et al., 2002). Using this similarity, thewater, m is a dimensionless fitting parameter, and � de-sequential, successive linear estimator approach for hy-notes volumetric moisture content. We assume that �0draulic tomography (Yeh and Liu, 2000), which has beendoes not change during an infiltration event in a field.validated with sandbox experiments (Liu et al., 2002),Using Eq. [4], the linear relation between log resistivitywas extended to ERT surveys by Yeh et al. (2002).before and after an infiltration can be expressed asHowever, the work by Yeh et al. (2002) did not directly

estimate moisture content distributions, nor consider the �ln�(x) � �m(x)�ln�(x) [5]spatial variability in the constitutive resistivity–moisture

According to this equation, the change of log resistivitycontent relation.(�ln�) is linearly proportional to the change in log mois-We developed an integrative algorithm based on a sto-ture content (�ln�). If m is spatially invariant, the pat-chastic information fusion concept (Yeh and Simunek,tern of �ln�(x) directly corresponds to the pattern of2002) to provide the best unbiased estimate of moisture�ln�(x) in the entire field. On the other hand, if m is spa-content distributions in the vadose zone by fusing bothtially variable and independent of �ln�(x), then the pat-hydrological and geophysical information. The hydro-tern of change seen in ln�(x) does not directly corre-logical information includes point measurements of thespond to the pattern of �ln� (x).moisture content and prior information about moisture

Furthermore, Eq. [5] is derived with the assumptioncontent distributions (i.e., mean, variance, and correla-that during an infiltration event �0 remains constant,tion structure). The geophysical information consists ofwhich may not always be valid. The resistivity of a po-point measurements of electric potentials, parametersrous medium can be highly variable, depending on theof the resistivity–moisture content relation, and priordegree of saturation and the type of ions present in theinformation about spatial variability of these param-pore water (Sharma, 1997). The spatial variability of �0eters. Two-dimensional numerical experiments wereand m may directly correspond to the pore water chemis-first conducted to evaluate the effect of uncertaintiestry. Since silica, which comprises most mineral grainsassociated with the prior information. Subsequently, nu-(except metallic ores and clays), is an insulator, the ob-merical experiments were used to demonstrate the ro-served electrical conduction in porous media is mainlybustness of the integrative inverse approach for delineat-through interstitial pore water. When clay minerals areing transient moisture content distributions during apresent, a relatively large number of ions may flow intononuniform infiltration event in a three-dimensionalor out of solution, through ion exchange, thus signifi-heterogeneous vadose zone.cantly changing the electrical conductivity of the fluid.During an infiltration event, other chemical reactions orTHEORYprocesses may occur due to differences in water chemis-

Governing Equation for the Electric Potential try in the infiltrated water, thus altering the compositionof ions present in pores and changing the nature ofIn a geological formation, the electric current flowthe pore electrolytes. This adds another level of spatialinduced by an electrical resistivity survey in general canvariability to the resistivity distribution.be described by

Baker (2001) measured electrical resistivity as a func-� · [�(x)�φ(x)] � I(x) � 0 [1] tion of moisture content for core samples collected from

where φ is electric potential (V), I represents the electric a bore hole at the Sandia-Tech Vadose Zone infiltrationcurrent source density per volume (A m�3), and � is the field site in Socorro, NM. A total of 25 samples were col-electrical conductivity (S m�1). Electrical conductivity, lected from eight 1.52-m (5-foot) continuous cores. The�, is the reciprocal of the electrical resistivity, � (�m), electrical resistivity values of the samples at severalwhich is assumed to be locally isotropic. The boundary moisture contents were determined using an impedanceconditions associated with Eq. [1] are analyzer. Equation [4] was subsequently fitted to the

measured resistivity and moisture data to determine theφ|1� φ* [2]

values for �0 and m. On the basis of analysis of the dataand set, Yeh et al. (2002) reported that both ln�0 and lnm

were approximately normally distributed. The geomet-�(x)�φ ·n|2� i [3]

ric mean of �0 was 7.036 �m and the variance, standardwhere φ* is the electric potential specified at boundary deviation, and coefficient of variation for ln�0 were1, i denotes the electrical current density per unit area 0.633, 0.796, and 40.8%, respectively. For m, the geomet-(A m�2), and n is the unit vector normal to the boun- ric mean was 1.336 (dimensionless), while variance, stan-dary 2 . dard deviation, and coefficient of variation for lnm were

0.034, 0.185, and 63.7%, respectively. In addition, theyConstitutive Resistivity and Moisture found that both parameters are not entirely disorderedContent Relation in space but correlated over short distances. For ln�0,an exponential variogram model was used to describe itsIn this study, a power law relation was used to relate

resistivity to moisture content (e.g., Yeh et al., 2002): spatial variability with sill, range, and nugget values of

Rep

rodu

ced

from

Vad

ose

Zon

e Jo

urna

l. P

ublis

hed

by S

oil S

cien

ce S

ocie

ty o

f Am

eric

a. A

ll co

pyrig

hts

rese

rved

.

www.vadosezonejournal.org 683

0.8, 3.5 m, and 0.08, respectively. Similarly, an exponen- respect to the parameters, and are computed using anadjoint state method. Details of the derivation of thesetial variogram model was used for lnm. The sill, range,

and nugget values for lnm were 0.043, 3.5 m, and 0.01, sensitivities can be found in Sun and Yeh (1992), Li andYeh (1998) and Hughson and Yeh (2000). The sensi-respectively. No significant correlation between ln�0 and

lnm was reported. tivity of the electric potential at location i to a pertur-bation in a parameter at location k can be generallyThe observed spatial variability of ln�0 and lnm im-

plies that equivalent changes in moisture content at expressed asdifferent locations in the medium may lead to differentchanges in the measured electrical resistivity. As a re- �φi

�k

� ��k

��

��φ� ·��d� [7]

sult, the pattern of change in resistivity, detected byERT surveys in a field, may not necessarily reflect the

Specifically, the sensitivities for the parameters in thistrue pattern of change in the moisture content. To over-study are:come this difficulty in interpreting ERT field surveys,

we developed an integrative inverse algorithm that is �φi

�fk

� ��k

��φ� ·��d� [8]based on the concept of stochastic information fusiondeveloped by Yeh and Simunek (2002).

�φi

�ak

� ��k

� m��φ� ·��d� [9]Inverse (or Estimation) Algorithm

Linear Estimator �φi

�nk

� ��k

� m ln(�)��φ� ·��d� [10]Since the parameters, �0, �, and m, vary spatially, andtheir specification at every point in space is practicallyimpossible, they will be treated as random fields (e.g., where �k is the domain of the element containing nodeYeh, 1992 and 1998), which are characterized by their k if a finite element approach is used and � representsstatistical moments (i.e., means, variances, and correla- the adjoint state variable, which can be solved for usingtion scales). Notice that the fields can be either station- the adjoint state equation:ary or nonstationary stochastic processes. Accordingly,

� · [�(x)��] � �(x � xk) [11]� and the electric potential φ are also regarded as sto-chastic processes. In the following analysis, it is assumed where � is the Dirac delta function and xk is the measure-that ln �0 � F � f, ln� � A � a, lnm � N � n and φ � ment location of the electric potential. Notice that theH � h, where F, A, N, and H are the mean values, and mean electric potential, �φ�, is needed to evaluate thef, a, n, and h are the perturbations. The primary goal sensitivities (see Eq. [8], [9], and [10]). The mean electricof the inverse (or estimation) algorithm is to estimate potential is derived by solving a mean equation that is� and � at any point in the three-dimensional geologic of the same form as the Eq. [1] (Yeh et al., 2002), with amedium, although it can be used to estimate other pa- mean electrical resistivity and moisture content relationrameters (i.e., �0, m, and φ) as well. The estimation al- that is the same as Eq. [4], but with the parameters setgorithm integrates point parameter measurements at to their mean values.several locations (including �0, m, and �) and ERT mea- Once the mean electric potential field is derived, thesurements that consist of φ and the transmitted current above sensitivity equations are used to calculate covari-density, I. Prior information about the means and co- ance of h and the cross covariance between h and .variances of �0, m, and � is also included; this information Rewriting Eq. [6] in a matrix form in terms of perturba-could be estimated from core samples, geological well tions yieldslogs, or outcrops (Yeh et al., 2002). The effects of uncer-tainty in the prior information will be studied below in {h} � �

Jh�H, �X�

{} [12]the Effects of Uncertainties in the Prior Informationsection.

where {} indicates the vector of the discretized variable,Using a first-order analysis, a state variable, such asJh is a jacobian matrix representing the derivativesφ, can be expanded in a Taylor series about the meanof the potential with respect to the parameters (i.e., �φ/values of parameters. Neglecting the second- and higher-�i), which can be obtained using Eq. [8] through [10],order terms of the Taylor series leads to a linear relationand has dimensions of nh nelem, where the numberbetween the state variable and the parameters, i: of voltage measurement locations is given as nh, andnelem is the total number of elements in the domain.φ � �φ� � �

i

i�φ�i

��Xi��φ�

[6] Multiplying Eq. [12] by the transpose of {�} (i.e., {f},{a}, {n}) and {h}, and then taking the expectations onboth sides, yieldsFor ERT surveys, i � i � �i� represents the zero mean

perturbation of a natural log transformed parameter, Rh � JhRsuch as f � ln�0 � F, a � ln� � A, and n � lnm � N;Rhh � �

JhRJTh [13]the zero mean perturbation of the state variable (i.e.,

electric potential) is h � φ � �φ� � φ � H; �φ/�i repre-sent the sensitivity derivatives of electric potential with where superscript T denotes transpose; Rh represents

Rep

rodu

ced

from

Vad

ose

Zon

e Jo

urna

l. P

ublis

hed

by S

oil S

cien

ce S

ocie

ty o

f Am

eric

a. A

ll co

pyrig

hts

rese

rved

.


the cross-covariance functions between h and f, h, and and Yeh (2000), and Vargas-Guzman and Yeh (2002)is employed. That is,a, or h and n, with dimensions of nh nelem; R denotes

the covariance functions of f, a, and n with dimensions(r�1) � (r) � �(r)T

h �φ* � φ(r)� [16]of nelem nelem, and they are given a priori. A nuggetcan be added to the covariance to represent measure- where (r�1) and (r) represent the parameter estimatedment errors, or variations within the sample scale if they at iteration r � 1 and r, φr is the electric potential atare known. Because little information is available about the measurement locations calculated from the forwardthe parameters f, a, and n in the field, these parameters simulation using parameters estimated at iteration r,are assumed to be independent of each other in our and �h

(r) is the weight at iteration r, which is determinedstudy. Such an assumption merely represents the worst from the following:scenario in which knowledge of one variable does not

εrhh�

rh � ε(r)

h [17]provide any information about the others. Rhh in Eq.[13] represents the covariance function of h, which has The solution to Eq. [17] requires knowledge of εhh

(r) anddimensions of nh nh. Similarly, a nugget representing εh

(r); they are estimated using the following approxima-errors or variability due to scale disparity in potential tions at each iteration:measurements can be added to this covariance function.

ε(r)hh � �

J(r)h ε(r)

JT(r)hUsing these cross-covariance and covariance functions,

a first-order estimate of the perturbations of the log-ε(r)

h � J(r)h ε(r)

[18]transformed parameters is obtained by using the condi-tional expectation given observed primary information where Jh is the sensitivity matrix of nh nelem at* (i.e., �0, m, and �) and secondary information h* (i.e., iteration r, and superscript T stands for the transpose.φ) collected during an ERT survey (Priestley, 1989; Yeh At iteration r � 0, ε is given byand Zhang, 1996):

ε1 � R � R� � Rh�h [19]

� �T * � �T

h h* [14]where R is a subset of R. For r � 1, the residual

where is a nelem long vector of the estimated perturba- covariances are evaluated according totion of parameters, �0, m, and �; * and h* are perturba-

ε(r�1) � ε(r)

� ε(r)h�

(r)h [20]tions of measured parameters and electrical potential,

respectively; � and �h are weights (or cokriging weights Notice that these residual covariances represent first-in geostatistics) for the measurements, * and h*, re- order approximates of the conditional covariances.spectively. They are evaluated as follows: Once all three parameters, f, a, and n, are estimated

by fully utilizing the potential data and direct measure-�C Ch

Ch Chh ��

�h � �C

Ch [15] ment of the parameters (if any), the electrical conductiv-

ity is then estimated using Eq. [4]. Afterwards, the meanwhere C represents the covariance of between mea- electric potential equation is solved again with the newlysurement locations; Chh represents the covariance of h estimated electrical conductivity for a new electric po-between measurement locations; and Ch denotes cross tential field. Then, the maximum change of �

2 (the vari-covariance between h and at measurement locations. ance of the estimated parameters of f, a, and n) andThe right-hand side of Eq. [15] consists of the covariance the change in the largest potential misfit among all mea-of between measurement locations and the estimate surement locations between two successive iterations areposition, and the cross covariance of h at measurement evaluated. If both changes are smaller than prescribedlocations and at the estimate location. In other words, tolerances, the iteration stops. If not, new values for εh

the weights used in the estimation are not only directly and εhh are evaluated using Eq. [18]. Equation 17 is thenrelated to spatial correlation structures of parameters solved to obtain a new set of weights that are used inand state variables at measurement locations, but also Eq. [16] with [φ* � φ(r)] to obtain a new estimate of thethe cross correlations of the parameters and state vari- parameters. A theoretical proof of the convergence ofables, and correlation and cross correlation between the successive linear estimator is given in Vargas-Guz-measurements and the location where the parameter is man and Yeh (2002).to be estimated. These matrices all are subsets of thecovariance and cross covariance matrices in Eq. [13]. Sequential Estimation Approach

The previous section describes the ERT inversion al-Successive Linear Estimator gorithm for only one set of primary and secondary infor-

mation obtained in one DC transmission. This algorithmEquation [14] approximates the nonlinear relationbetween the parameter to be estimated and the mea- can simultaneously include all potential measurements

collected during all DC transmissions in an ERT survey.sured electric potential by means of a linear first-orderapproximation. Thus, the equation cannot fully exploit However, the system of equations (Eq. [15] and [17])

can become extremely large and ill conditioned, inelectric potential measurements. To circumvent this prob-lem, a successive linear estimator similar to that used which case stable solutions to the equations are difficult

to obtain (Hughson and Yeh, 2000). To avoid numericalby Yeh et al. (1996), Zhang and Yeh (1997), Hanna andYeh (1998), Vargas-Guzman and Yeh (1999), Hughson difficulties in solving the large system of equations, the

Rep

rodu

ced

from

Vad

ose

Zon

e Jo

urna

l. P

ublis

hed

by S

oil S

cien

ce S

ocie

ty o

f Am

eric

a. A

ll co

pyrig

hts

rese

rved

.


Table 1. Hydrological and statistical parameters used in two-voltage data sets are included sequentially. The sequen-dimensional analysis.tial algorithm used is similar to the one developed for

Parameter Mean Variance �x �z Covariance modeluse in hydraulic tomography inversion (Yeh and Liu,2000). In essence, the proposed sequential approach uses cmthe estimated electrical conductivity field, the �0, m, and Ks (cm s�1) 0.0063 0.1 240 20 exponential

� (1/cm) 0.028 0.1 240 20 exponential� fields, and their covariances and cross covariances, as� 2.0 0.01 240 20 exponentialprior information for the next estimation using new sets

of current–voltage data from DC transmissions at dif-ferent locations. Vargas-Guzman and Yeh (1999) and of uncertainties in the mean and covariance functions ofYeh and Simunek (2002) gave an illustrative example � on the estimates of the � distribution. In these experi-of the sequential approach and explained the necessity ments, the other two parameters, �0 and m, are spatialfor updating the covariances and cross covariances. The variables but their spatial distributions are here assumedsequential inclusion of data sets from different DC trans- to be known precisely. This assumption is necessary tomissions continues until all data sets have been utilized. reduce the number of cases to be analyzed. The resultsAll data sets are fully processed by propagating the of this investigation should be applicable to cases whereconditional first and second moments from one data set the prior information of these two parameters also in-to another. Such a sequential approach allows accumu- volve uncertainty.lation of high-density secondary information obtained The numerical experiments were conducted for a hy-from an ERT survey, while maintaining the covariance pothetical, two-dimensional, vertical vadose zone withmatrix at a manageable size that can be solved with mini- dimensions of 200 by 200 cm, discretized into 10 hori-

zontal and 20 vertical elements, 200 cm2 each. The unsat-mal numerical difficulties.urated hydraulic properties of each element were as-Inversions of ERT surveys for environmental applica-sumed to be described by the Mualem–van Genuchtentions are generally ill posed since the number of param-model (van Genuchten, 1980):eters to be estimated is often much greater than the num-

ber of measurements of the state variable (see Yeh andK(�) � Ks1 � (|��|)(��1)[1 � (|��|)�]��2/[1 � (|��|)�]�/2

Simunek, 2002 for a discussion about necessary and suf-ficient conditions for a well-posed inverse problem). An �(�) � (�s � �r)�1 � (|��|)��(��)

� �r [21]ill-posed problem has an infinite number of global minima

where K(�) is the unsaturated hydraulic conductivityand solutions. Classical inverse algorithms (e.g., regular-as a function of the pressure head, �; Ks is the saturatedized least-squares approach, Tikhonov, 1963) can only de-hydraulic conductivity; � and � are shape factors; � �rive an estimated parameter field that produces an elec-(1 � 1/�); �s is the saturated moisture content; and �rtric potential field honoring measurements at samplingis the residual moisture content. To represent heteroge-locations and a smooth estimate at other locations. Thisneity, the parameters of Eq. [21] were assumed to besmooth field, however, does not necessarily honor thestochastic processes. Since spatial variations in �s andcharacteristics of the spatial variability of the true pa-�r are generally negligible, both were treated as deter-rameters (e.g., mean, variance, and correlation struc-ministic constants with values of 0.366 and 0.029, respec-ture). On the other hand, our sequential–successive lin-tively. The parameters Ks, �, and � for each element inear estimator estimates conditional effective parametersthe simulation domain were generated using a method(Yeh et al., 1996; Hanna and Yeh, 1998; Yeh, 1998; Yehby Gutjahr (1989) with specified means, variances, andand Simunek, 2002). It aims to yield a parameter fieldcorrelation structures as listed in Table 1. These gener-that produces not only parameter values and state vari-ated random fields are then denoted as the true distri-ables observed at measurement locations, but also con-butions of the hydraulic parameters for the hypotheti-ditional effective parameter values at locations wherecal site.no measurements are available. The effective param-

The initial condition of the site was assumed to be hy-eters are our estimates based on the spatial statistics ofdrostatic. Specifically, no-flux boundary conditions werethe parameter fields and their cross correlations withspecified on the two sides, a water table was prescribedstate variables. These cross correlations are evaluatedat the bottom of the site, and a constant pressure headusing the governing equation for the electric potentialboundary of �200 cm was assigned to the top boundary.and fluid flow processes. Discussions of advantages ofAn infiltration event was created by changing the pre-the approach can be found in Yeh and Simunek (2002).scribed pressure head from �200 to �80 cm at thecenter (from x � 80 to 120 cm) of the top boundary(z � 200 cm). Subject to these boundary conditions,NUMERICAL EXPERIMENTSthe Richards equation for variably saturated flow wasEffects of Uncertainties in the Prior Information solved using a finite element model (Srivastava and Yeh,

The proposed inverse approach requires input for the 1992) to derive a steady-state moisture content distri-prior information such as the mean, variance, and corre- bution under nonuniform infiltration. Two cases werelation structures of the parameters to be estimated (i.e., simulated with the same initial and boundary conditions:�0, m, and �). In practice, these inputs have to be all es- (i) infiltration into a homogeneous geological formationtimated and involve uncertainty. The following numeri- whose hydraulic properties are specified using their

mean values (Table 1) and (ii) infiltration into a hetero-cal experiments are devoted to investigating the effect

Rep

rodu

ced

from

Vad

ose

Zon

e Jo

urna

l. P

ublis

hed

by S

oil S

cien

ce S

ocie

ty o

f Am

eric

a. A

ll co

pyrig

hts

rese

rved

.


Fig. 1. (a) The mean moisture content distribution simulated using homogeneous hydraulic parameters; (b) the true moisture content distributionsimulated using heterogeneous hydraulic parameters.

geneous formation presented by the generated true hy- process yielded one set of voltage–current data. By mov-ing the transmitter along the bore hole, and repeatingdraulic parameter fields. The simulated moisture con-the same procedure, four sets of voltage–current datatent distributions for the two cases are plotted in Fig. 1awere obtained. In addition, one � value from the simu-and b, and were used for the ERT experiments de-lated true � field was assumed to be measured from ascribed below.core sample along the bore hole. The locations of the cur-The electrical resistivity, �, of the site was assumed torent source, voltage, and � measurements are displayedbe a function of �0, m, and �, according to Archie’s Law,in Fig. 3.Eq. [4]. To derive the electrical resistivity field corre-

Besides the voltage and � measurements, specificationssponding to the simulated � distribution at the site, theof means, variances, and correlation structures of thetwo parameters, �0 and m, were considered to be randomparameters to be estimated are required by the pro-fields (stochastic processes) with geometric means ofposed inverse model. Generally, the statistics of � cannot8.5 �m and 1.35, respectively. The variances of ln�0 andbe estimated without uncertainty since the true � distri-lnm were 0.1 and 0.01, respectively. The two parametersbution is unknown. In the following section, three sce-were assumed to be statistically independent of eachnarios were examined to investigate the effects of theother and both parameters were described with the sameuncertainty in these input spatial statistics.exponential correlation structure, which had a hori-

Notice that the variance of the parameters appearszontal correlation scale of 240 cm and vertical correla-on both sides of the system of equations (i.e., Eq. [15]tion scale of 20 cm. Gutjahr’s method (1989) was again and [17]) and will be canceled out. Therefore, the valueused to generate heterogeneous �0 and m fields as shown of the variance has no effect on the solution of the sys-in Fig. 2a and 2b. tem of equations and in turn, the estimate of parameters.

Using these generated �0, m, and � fields, Eq. [4] was That is, the estimation procedure relies on only the cor-next used to calculate the true resistivity distribution relation and cross correlation. The variance, however,corresponding to the simulated nonuniform � field at does affect the predicted uncertainty associated with thethe site. Synthetic pole to pole ERT surveys were subse- estimated parameters.quently simulated using the hypothetical domain. First, The first scenario approximates the mean � distribu-two bore holes for the ERT survey were assumed to tion using the geometric mean of the true � distribution.penetrate the entire depth of the domain at locations Regardless of the fact that the covariance structure ofx � 70 and 130 cm. Ten electrodes were placed 20 cm the � field is different from those of the hydraulic prop-apart in each bore hole, and the reference receiver and erties due to flow processes (Yeh et al., 1985a, 1985b,transmitter electrodes were assumed to be outside of the 1985c), in this scenario they were assumed to be the samedomain. One of the electrodes in the bore hole was (i.e., an exponential model with horizontal and verticalselected as the transmitter with a specified current. The correlation scale of 240 and 20 cm, respectively). Usingresultant voltage differences between the reference elec- these approximations, together with other necessary in-

put, we estimated perturbations of the � distribution,trode and the rest of the electrodes were measured; this

Fig. 2. (a) Generated random field for ln�0; (b) generated random field for lnm.

Rep

rodu

ced

from

Vad

ose

Zon

e Jo

urna

l. P

ublis

hed

by S

oil S

cien

ce S

ocie

ty o

f Am

eric

a. A

ll co

pyrig

hts

rese

rved

.


Fig. 3. (a) True moisture content distribution; (b) estimated moisture content distribution assuming moisture content has a geometric mean andexponential covariance function; (c) estimated moisture content distribution assuming moisture content has a true mean and exponentialcovariance function; (d) estimated moisture content distribution assuming a true mean and computed covariance function. Circles indicatecurrent source locations, triangles indicate voltage measurement locations, and squares indicate the moisture content measurement location.

which were then added to the geometric mean to derive where R�� is the covariance function of the moisturethe final estimated � distribution (Fig. 3b). A compari- content; R�� denotes the covariance functions of the log-son between the true and estimated � distribution is transformed perturbations of the hydraulic propertiesshown using a scatter plot (Fig. 4a). of Ks, �, and �, which were assumed known; and J�� is the

The covariance function of the � field in the second Jacobian matrix (details of computing J�� can be foundscenario was kept the same as in the first scenario while in Hughson and Yeh, 2000). The final estimated � distri-the ensemble-mean � distribution (Fig. 1a) was used, in- bution and its scatter plot are shown in Fig. 3d and 4c, re-stead of a constant mean value. The ensemble mean � spectively.distribution presents the average of many possible � Comparing Fig. 3b, 3c, and 3d with Fig. 3a reveals thatdistributions due to the specified infiltration in all possi- the estimated � distributions for the three scenariosble heterogeneous sites characterized by the given spa- resemble the general pattern of the true � distribution,tial statistics. Substituting the ensemble mean � field despite the different assumptions used in the prior infor-into the inverse model yields the estimated � perturba- mation. Comparisons of scatter plots (Fig. 4) also leadtion field, which was then added to the ensemble mean to the same conclusion. To quantify this conclusion, the� distribution to derive the final � distribution (Fig. 3c);

L1 and L2 norms of these scenarios were evaluated assee also the corresponding scatter plot (Fig. 4b).follows:The third scenario employed the ensemble mean �

distribution as input, but used a correct covariance func-L1 �

1n �

n

i�1

� �i � �i � [23]tion of � instead of that of the hydraulic properties.The covariance function was computed using the first-order approximation:

L2 �1n �

n

i�1

�i � �i�2

[24]R��

�J��R��JT

�� [22]

Fig. 4. (a) Scatter plot corresponding to Fig. 3b; (b) scatter plot corresponding to Fig. 3c; (c) scatter plot corresponding to Fig. 3d.

Rep

rodu

ced

from

Vad

ose

Zon

e Jo

urna

l. P

ublis

hed

by S

oil S

cien

ce S

ocie

ty o

f Am

eric

a. A

ll co

pyrig

hts

rese

rved

.


Table 2. Hydrological and statistical parameters used in three-where �i and �i represent the true and estimated mois-dimensional analysis.ture content, respectively. We found that values of the

Parameter Mean Variance �x �y �z Covariance modelL1 and L2 norms for the three cases differed only slightly:the norms from the use of the ensemble mean are slightly cmsmaller than those from the use of the geometric mean. Ks (cm min�1) 0.043 0.893 80 80 20 exponential

� (1/cm) 0.067 0.631 80 80 20 exponentialWe therefore conclude that choices of mean and covari-� 1.811 0.015 80 80 20 exponentialance function of moisture content have only a minor ef-

fect on the estimate if a sufficient number of potentialmeasurements are available and the �0 and m distribu- three-dimensional vadose zones. Water movement wastions are known precisely. This result agrees with the simulated in a three-dimensional hypothetical vadosefindings by Liu et al., (2002) for hydraulic tomography zone at 1000 and 50 000 min from the commencementsurveys. That is, abundant data sets collected during of an infiltration event. The simulated moisture contenttomographic surveys (either ERT or hydraulic tomogra- distributions at these two times, �1000 and �50 000, werephy surveys) can overcome the uncertainty associated used as the true moisture content fields; and their differ-with prior information such as mean, variance, and cor- ence was denoted as the true moisture content change inrelation structure. This result is generally true in these

the following analysis. Following generation of randomsynthetic cases with small domains. Under field condi-fields of �0 and m, the true resistivity fields at 1000 andtions, where a site is three-dimensional and encompasses50 000 min were calculated using Eq. [4] with the gener-a much larger domain than in the example, and whereated �0, m, �1000, and �50 000 fields.the �0 and m fields are unknown, the effects of uncer-

Next, ERT surveys were simulated using these twotainty in the input parameters are expected to be greaterresistivity fields. After collecting voltage–current data(Yeh and Simunek, 2002). Nevertheless, the reliabilitysets, two different inverse approaches were used to in-of inverse modeling of tomography surveys increasesterpret water flow due to the infiltration event. The firstwith the spatial density of the secondary information.approach used the Yeh et al. (2002) inverse model basedA dense deployment of secondary information sensorson electric potential measurements only to derive re-therefore will yield small uncertainty in the result. On

the other hand, a sparse deployment would require ac- sistivity fields at times of 1000 and 50 000 min. Next,curate prior information to facilitate statistically unbi- the change in resistivity from 1000 to 50 000 min is com-ased estimates. puted. Finally, the estimated change in resistivity is used

to interpret the change in moisture content. The secondDirect Estimation of Three-Dimensional approach estimated moisture distribution at 50 000 min

Moisture Content Fields directly using our new integrative method.The hypothetical site was assumed to be a cube, 200In the following numerical examples, we tested our in-

verse algorithm for transient infiltration and flow through cm on each side, consisting of 2000 elements of 20 by 20 by

Fig. 5. (a) Generated lnKs field; (b) generated ln� field; (c) generated ln� field; (d) true moisture content field at t � 1000 min.

Rep

rodu

ced

from

Vad

ose

Zon

e Jo

urna

l. P

ublis

hed

by S

oil S

cien

ce S

ocie

ty o

f Am

eric

a. A

ll co

pyrig

hts

rese

rved

.


Fig. 6. (a) Generated true ln�0 field; (b) generated true lnm field.

10 cm. Random values of the heterogeneous hydraulic that both parameters possessed the same exponentialcorrelation structure with a horizontal correlation scaleparameters (Ks, �, and �) were generated and assigned

to each element using the spectral method with the pa- of 80 cm and a vertical correlation scale of 20 cm. Fig-ures 6a and 6b show the generated true fields for theserameters shown in Table 2. Figures 5a, 5b, and 5c show

the generated lnKs, ln�, and ln� fields, respectively. two parameters.Based on these synthetic �0, m, �1000, and �50 000 fields,The initial pressure head condition was assumed to

be hydrostatic. Specifically, the bottom was set at a true synthetic resistivity fields at times of 1000 and 50 000min (�1000 and �50 000, respectively) were calculated usingprescribed pressure head of �50 cm, and the top was

fixed at a pressure head of �250 cm. Infiltration oc- Eq. [4] Electrical resistivity tomography surveys werethen simulated using these two resistivity fields. Figurecurred over an area of 1600 cm2 on the top center of

the cube at a pressure head of �50 cm, while no-flux 8d displays the three-dimensional layout of the ERTsurvey. The design of the ERT survey included fourboundary conditions were assigned to the remainder of

the top and the four sides. This created nonuniform bore holes penetrating the entire depth of the site do-main. The x and y coordinate pairs, in centimeters, ofvertical infiltration fields from a constant source on the

top center of the flow domain. The infiltration process the four bore holes were (50, 50), (150, 50), (50, 150),and (150, 150). Twenty electrodes were installed alongwas simulated using a finite element model (Srivastava

and Yeh, 1992) to obtain moisture distributions. Figure each bore hole. Electrodes were also deployed alongthe surface in four lines with endpoints at the above x–y5d shows the moisture content distribution 1000 min

after infiltration began. Mean moisture content distribu- coordinates. Current sources were installed along theupper right bore hole (150, 150) at the depths of 25, 55,tions at 1000 and 50 000 min were similarly obtained

with effective mean values of the hydraulic parameters. 95, 135, and 175 cm. Using the same collection procedurefor the voltage data as used for the two-dimensionalTrue changes in ln� were computed as the differences

between the true ln� distributions at 1000 and 50 000 numerical examples, five ERT voltage data sets of 111voltage measurements each were obtained. In addition,min (Fig. 7a).

To represent spatial variability in the parameters of 20 �0 and m values along each of the four bore holes (atotal of 80 measurements) were assumed to be availablethe resistivity and moisture content relation in the field,

the two parameters, �0 and m, were considered as ran- for our analysis, while � was sampled at 20 locationsindicated by squares in Fig. 8d. To investigate the effectdom fields with geometric means of 7.036 �m and 1.336,

respectively. Variances of ln�0 and lnm were assumed of direct � measurements on the inverted estimate ofthe moisture content at the site, the moisture contentto be 0.633 and 0.034, respectively. We again assumed

Fig. 7. (a) True �ln� from 1000 to 5000 min; (b) estimated �ln� from 1000 to 50 000 min.

Rep

rodu

ced

from

Vad

ose

Zon

e Jo

urna

l. P

ublis

hed

by S

oil S

cien

ce S

ocie

ty o

f Am

eric

a. A

ll co

pyrig

hts

rese

rved

.


Fig. 8. (a) True moisture content � at t � 50 000 min; (b) estimated � at t � 50 000 min without � measurements; (c) estimated � at t � 50 000min with 20 � measurements. (d) Schematic diagram for three-dimensional electrical resistivity tomography experiments. Top center squareindicates the infiltration area, right triangles indicate voltage measurement locations, circles show current source locations, and squares showthe locations of 20 � measurements. (e) The scatter plot corresponding to (b); (f) the scatter plot corresponding to (c).

distribution at 50 000 min was estimated without any � of the true moisture content change, and Fig. 7b themeasurement and compared with the estimate with 20 estimated pattern of resistivity change. A comparison of� measurements. these two figures indicates that interpretation of water

movement based on the pattern of the estimated resisti-Using Resistivity Change to Interpret Water Flow vity change alone led to incorrect estimates of water

flow. From 1000 and 50 000 min, the true water frontFor the purpose of comparison, resistivity changesmoved in a southwest direction, while the estimated pat-were computed to reflect water movement in the vadosetern of resistivity change suggests flow in a southeastzone. Forward simulations of ERT surveys based on thedirection. As reported by Yeh et al. (2002) two factorspreviously discussed network layout were conductedcontribute to this discrepancy: uncertainty in the esti-using the simulated moisture distributions at 1000 andmated resistivity fields and spatial variability in the pa-50 000 min. Five voltage data sets were collected andrameters of the �–� constitutive relation. Thus, in addi-then used in the inverse approach of Yeh et al. (2002)tion to a dense deployment of voltage sensors, a detailedto estimate the resistivity fields at these two specifiedspatial distribution of the parameter must be known totimes. The change in resistivity between 1000 and 50 000

min was then computed. Figure 7a displays the pattern correctly relate changes in the resistivity to those in the

Fig. 9. (a) Conditional variance when no � measurements are used at t � 50 000 min; (b) conditional variance when 20 � measurements areused at t � 50 000 min. Circles indicate � measurement locations.

Rep

rodu

ced

from

Vad

ose

Zon

e Jo

urna

l. P

ublis

hed

by S

oil S

cien

ce S

ocie

ty o

f Am

eric

a. A

ll co

pyrig

hts

rese

rved

.


moisture content. Relying only on changes in resistivity merical examples also demonstrate the ability of the in-tegrative ERT survey inverse model for estimating mois-and neglecting spatial variability in the parameters of

the �–� constitutive relation can lead to erroneous flow ture contents directly. The model yields good estimatesat the locations where primary and secondary informa-directions and patterns.tion is measured. Primary information (i.e., the moisture

Direct Estimation of Moisture Content Distribution measurements) contributes significantly to the accuracyof the estimated moisture content distribution. A largeFigures 8a and b show the true moisture content dis-number of potential measurements are useful, but theytribution at 50 000 min and the corresponding estimateddid not dramatically improve the estimate. This furthermoisture content distribution using 111 5 voltage mea-supports the finding by Yeh et al. (2002) that poten-surements, and 80 pairs of �0 and m measurements. Thetial measurements alone are inadequate to characterizeestimated field using the same number of voltage, �0 andwater flow in vadose zone. Finally, we conclude that fus-m measurements, but also including 20 direct � measure-ing geophysical measurements with hydrological infor-ments, is illustrated in Fig. 8c. Comparing the threemation using a stochastic approach is necessary to yieldfigures we find that the proposed inverse algorithm re-hydrologically realistic results under field conditionsproduces the general pattern of the simulated true mois-and to quantify uncertainty associated with the results.ture content distribution, even though the constitutive

relation between resistivity and moisture varies spatially.ACKNOWLEDGMENTSThe inclusion of the 20 � measurements, in particular,

greatly improves the estimates of the � distribution. Fig- This research was funded in part by a DOE EMSP96 grantthrough Sandia National Laboratories (Contract AV-0655#1),ures 8e and 8f are scatter plots corresponding to Fig.a DOE EMSP99 grant through University of Wisconsin8b and 8c, respectively; a 45� line indicates perfect esti-(A019493), and NSF and SERDP grant EAR-0229717. Wemation. The goodness of fit was also evaluated usingalso thank Kris Kuhlman for his technical editing. Many thanksL1 and L2 norms. The reduction in the L1 and L2 normsare extended to two reviewers for providing critical commentsfrom Fig. 8e to 8f suggests that the addition of moistureto improve the paper. In particular, we are greatly indebtedcontent measurements dramatically improves the esti- to Dr. Andreas Kemna, who provided a meticulous, insightful,

mate. The conditional variance of the estimate from our constructive, and open-minded review of our manuscript.stochastic fusion approach can be used to assess the un-certainty associated with the estimate—a smaller condi- REFERENCEStional variance indicates less uncertainty in the estimate.

Baker, K. 2001, Analysis of hydrological and electrical properties atThe conditional variances corresponding to the esti- the Sandia-Tech Vadose Zone Facility. M.S. thesis. New Mexicomates by using zero and 20 � measurements are shown Tech., Socorro, NM.on Fig. 9a and 9b, respectively. Small conditional vari- Bohling, G.C., X. Zhan, J. Butler, Jr., and L. Zheng. 2002. Steady

shape analysis of tomographic pumping tests for characteriza-ances are located close to the four bore holes wheretion of aquifer heterogeneities. Water Resour. Res. 38(12). 1324.secondary information is measured. At locations wheredoi:10.1029/2001WR001176.moisture content measurements were collected, the con- Daily, W., A. Ramirez, D. LaBrecque, and J. Nitao. 1992. Electrical

ditional variance is zero, indicating that these observa- resistivity tomography of vadose water movement. Water Resour.Res. 28:1429–1442.tions are honored in the inverse model and that no

Ellis, R.G., and S.W. Oldenburg. 1994. The pole-pole 3D DC resistivityuncertainty exists.inverse problem: A conjugate gradient approach. Geophys. J. Int.119:187–194.

Gutjahr, A. 1989. Fast Fourier transforms for random field generation.CONCLUSIONSProject Rep. Los Alamos Grant 4-R58-2690R. Dep. Mathematics,

Knowledge of detailed moisture content distributions New Mexico Tech, Socorro, NM.Hanna, S., and T.-C.J. Yeh. 1998. Estimation of co-conditional mo-is important to our understanding of vadose zone pro-

ments of transmissivity, hydraulic head, and velocity fields. Adv.cesses and water resources management. An ERT inver-Water Resour 22:87–93.sion algorithm based on the concept of stochastic infor-

Hughson, D.L., and T.-C.J. Yeh. 2000. An inverse model for three-mation fusion is developed to estimate moisture content dimensional flow in variably saturated porous media. Water Re-distributions in three-dimensional vadose zones. The sour. Res. 36:829–839.

Li, B., and T.-C.J. Yeh. 1998. Sensitivity and moment analysis of headapproach integrates point measurements of electric po-in variably saturated regimes. Adv. Water Resour. 21:477–485.tential, parameters of the constitutive relation between

Li, Y., and D.W. Oldenburg. 1994. Inversion of 3D dc-resistivity datamoisture and resistivity, and moisture content. In addi- using an approximate inverse mapping. Geophys. J. Int. 116:tion, the approach includes prior information about the 527–537.

Liu, S., T.-C. J. Yeh, and R. Gardiner. 2002. Effectiveness of hydraulicspatial statistics of the moisture content distribution andtomography: Numerical and sandbox experiments. Water Resour.the parameters of the constitutive relation. Results ofRes. 38(4). doi:10.1029/2001WR000338.this study show that the prior information has effect on

Priestley, M.B. 1989. Spectral analysis and time series. Academicthe estimate if the density of primary and secondary in- Press, San Diego, CA.formation is low, but the effect diminishes as the density Sharma, P.V. 1997. Environmental and engineering geophysics. Cam-

bridge University Press, Cambridge, UK.of primary and secondary information increases.Srivastava, R., and T.-C.J. Yeh. 1992. A three-dimensional numericalNumerical examples illustrate that interpretations of

model for water flow and transport of chemically reactive solutewater movement in the subsurface, based only on the through porous media under variably saturated conditions. Adv.estimated resistivity changes, can be misleading because Water Resour. 15:275–287.

Sun, N.-Z., and W.W.-G. Yeh. 1992. A stochastic inverse solution forof spatial variability in the �–� constitutive relation. Nu-

Rep

rodu

ced

from

Vad

ose

Zon

e Jo

urna

l. P

ublis

hed

by S

oil S

cien

ce S

ocie

ty o

f Am

eric

a. A

ll co

pyrig

hts

rese

rved

.


transient groundwater flow: Parameter identification and reliability of unsaturated flow in heterogeneous soils: C. observations andapplications. Water Resour. Res. 21:465–471.analysis. Water Resour. Res. 28:3269–3280.

Yeh, T.-C.J., M. Jin, and S. Hanna. 1996. An iterative stochasticTikhonov, A.N. 1963. Resolution of ill-posed problems and the regu-inverse method: Conditional effective transmissivity and hydrauliclarization method. (In Russian.) Dokl. Akad. Nauk SSSR 151:head fields. Water Resour. Res. 32:85–92.501–504.

Yeh, T.-C.J., and S.Y. Liu. 2000. Hydraulic tomography: Developmentvan Genuchten, M.Th. 1980. A closed-form equation for predictingof a new aquifer test method. Water Resour. Res. 36:2095–2105.the hydraulic conductivity of unsaturated soils. Soil Sci. Soc.

Yeh, T.-C.J., S. Liu, R.J. Glass, K. Baker, J.R. Brainard, D. Alum-Am. J. 44:892–898. baugh, and D. LaBrecque. 2002. A geostatistically based inverseVargas-Guzman, J.A., and T.-C.J. Yeh. 1999. Sequential kriging and model for electrical resistivity surveys and its applications to va-

cokriging: Two powerful geostatistical approaches. Stochastic Envi- dose zone hydrology. Water Resour. Res. 38(12):1278. doi:10.1029/ron. Res. Risk Assess. 13:416–435. 2001WR001024.

Vargas-Guzman, J.A., and T.-C.J. Yeh. 2002. The successive linear Yeh, T.-C.J., and J. Simunek. 2002. Stochastic fusion of informationestimator: A revisit. Adv. Water Resour. 25:773–781. for characterizing and monitoring the vadose zone. Available at

Yeh, T.-C.J. 1992. Stochastic modeling of groundwater flow and solute www.vadosezonejournal.org. Vadose Zone J. 1:207–221.Yeh, T.-C.J., and J. Zhang. 1996. A geostatistical inverse method fortransport in aquifers. J. Hydrol. Processes 6:369–395.

variably saturated flow in the vadose zone. Water Resour. Res.Yeh, T.-C.J. 1998. Scale issues of heterogeneity in vadose-zone hydrol-32:2757–2766.ogy. In G. Sposito (ed.) Scale dependence and scale invariance in

Zhang, J., R.L. Mackie, and T. Madden. 1995. 3-D resistivity forwardhydrology. Cambridge University Press, Cambridge, UK.modeling and inversion using conjugate gradients. Geophysics 60:Yeh, T.-C.J., L.W. Gelhar, and A.L. Gutjahr. 1985a. Stochastic analy-1313–1325.sis of unsaturated flow in heterogeneous soils: A. statistically iso- Zhang, J., and T.-C.J. Yeh. 1997. An iterative geostatistical inverse

tropic media. Water Resour. Res. 21:447–456. method for steady flow in the vadose zone. Water Resour. Res.Yeh, T.-C.J., L.W. Gelhar, and A.L. Gutjahr. 1985b. Stochastic analy- 33:63–71.

sis of unsaturated flow in heterogeneous soils: B. statistically aniso- Zhou, Q.Y., J. Shimada, and A. Sato. 2001. Three-dimensional spatialtropic media. Water Resour. Res. 21:457–464. and temporal monitoring of soil water content using electrical re-

sistivity tomography. Water Resour. Res. 37:273–285.Yeh, T.-C.J., L.W. Gelhar, and A.L. Gutjahr. 1985c. Stochastic analysis

An Integrative Approach for Monitoring Water...

Documents

Transcript of An Integrative Approach for Monitoring Water...