An Integrative Approach for Monitoring Water Movement in the Vadose Zone

Transcription

1 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) showed Electrical resistivity tomography (ERT), during the past few years, that uncertainties in tracking moisture movement in the has emerged as a potentially cost-effective, noninvasive tool for imaging vadose zone on the basis of ERT surveys alone can changes of moisture content in the vadose zone. The accuracy be quite significant. They reported that both inverse of ERT surveys, however, has been the subject of debate because of modeling of the ERT surveys and the spatial variability its nonunique inverse solution and spatial variability in the constitutive of the parameters in the constitutive resistivity moisture relation 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 inverted tion fusion concept of Yeh and Šimůnek, was developed to derive the and the quantity of data available (Yeh and Šimůnek, best unbiased estimate of the moisture content distribution. Unlike classical ERT inversion approaches, this new approach assimilates 2002), the ERT survey inverse problem is often ill posed, prior information about the geological and moisture content structures and may have no unique solution. For ill-posed probin a given geological medium, as well as sparse point measurements of lems, most ERT inversion approaches employ optimi- the moisture content, electrical resistivity, and electric potential. Using zation algorithms with some type of regularization (e.g., these types of data and considering the spatial variability of the resisti- Tikhonov, 1963). While the regularization algorithm yields vity moisture content relation, the new approach directly estimates smooth estimates, there is no guarantee that it will prothree-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 geologiments were conducted to investigate the effect of uncertainties in the cal structures. Neither does the regularization provide prior information on the estimate. The effects of spatial variability in a meaningful way to quantify the uncertainty associated the constitutive relation were then examined on the interpretation of the change in moisture content, based on the change in electrical with the spatial variability. Additionally, when convert- resistivity from the ERT survey. Finally, the ability of the integrative ing changes in resistivity to changes in moisture content, approach was tested by directly estimating moisture distributions in it is often assumed that the parameters in the constitu- three-dimensional, heterogeneous vadose zones. Results show that the tive relation (e.g., Archie s Law) are constant across the integrative approach can produce accurate estimates of the moisture entire domain. Recent studies by Baker (2001) and Yeh content distributions and that incorporating some measurements of et al. (2002) reported pronounced spatial variability of the moisture content is essential to improve the estimate. these parameters in the field. Neglecting this spatial variability and using a single resistivity moisture calibration E curve can add to the level of uncertainty in the final inlectrical resistivity surveys are increasingly used terpreted change in moisture content (in an unquantito collect extensive electric current and potential fiable way). data in multiple dimensions to image subsurface electri- Besides the uncertainties inherent in the interpretation cal resistivity distribution (Daily et al., 1992; Ellis and of ERT surveys, most current three-dimensional inverse Oldenburg, 1994; Li and Oldenburg, 1994; Zhang et al., models demand significant computational resources to 1995). Recently, ERT surveys have found their way into process the large data sets typically collected during a sursubsurface hydrology applications. This is attributed to vey. Furthermore, a change in moisture content only prothe fact that knowledge of the spatial distribution of the vides qualitative information about water movement in electrical properties of subsurface media can provide the vadose zone. The actual moisture content values at valuable information for characterizing waste sites and each point, which are vital to hydrological investigations monitoring flow and contaminant movement in the va- or hydrological inverse modeling (e.g., Yeh and Šimůdose zone. For example, during an infiltration event, the nek, 2002), remain unknown. For instance, since the unmoisture content of a geological medium is generally saturated hydraulic conductivity is a nonlinear function the only factor that undergoes dramatic changes, and the of moisture content, changes in the moisture content alone changes in resistivity can be related to changes in mois- do not provide enough information to uniquely characture content. Tracking the changes in resistivity through terize the unsaturated hydraulic properties of the vadose time, therefore, has been found to be useful for detecting zone. As a consequence, an inverse approach is needed temporal 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 moisture S. Liu, Burgess & Niple, 5025 E. Washington St., Phoenix, AZ 85034; content distributions with the least uncertainty. T.-C.J. Yeh, Dep. of Hydrology and Water Resour., John W. Harsh- While the physical process of electric current flow is barger Bldg., The Univ. of Arizona, Tucson, AZ Received 18 different from that of groundwater flow, the governing July Original Research Paper. *Corresponding author (yeh@ equation of the electric current and the potential fields hwr.arizona.edu). created during ERT surveys is analogous to that of steady Published in Vadose Zone Journal 3: (2004). Soil Science Society of America 677 S. Segoe Rd., Madison, WI USA Abbreviations: ERT, electrical resistivity tomography. 681

2 682 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 to that used in hydraulic tomography (Yeh and Liu, where is bulk electrical resistivity, 0 is a fitting param- 2000; Bohling et al., 2002). Using this similarity, the eter that is related to the electrical resistivity of pore sequential, successive linear estimator approach for hynotes volumetric moisture content. We assume that 0 water, m is a dimensionless fitting parameter, and de- draulic tomography (Yeh and Liu, 2000), which has been validated with sandbox experiments (Liu et al., 2002), does not change during an infiltration event in a field. was extended to ERT surveys by Yeh et al. (2002). Using Eq. [4], the linear relation between log resistivity However, the work by Yeh et al. (2002) did not directly before and after an infiltration can be expressed as estimate moisture content distributions, nor consider the ln (x) m(x) ln (x) [5] spatial variability in the constitutive resistivity moisture content relation. According to this equation, the change of log resistivity We developed an integrative algorithm based on a stoture content ( ln ). If m is spatially invariant, the pat- ( ln ) is linearly proportional to the change in log moischastic information fusion concept (Yeh and Šimůnek, 2002) to provide the best unbiased estimate of moisture tern of ln (x) directly corresponds to the pattern of content distributions in the vadose zone by fusing both ln (x) in the entire field. On the other hand, if m is spahydrological and geophysical information. The hydrotern of change seen in ln (x) does not directly corre- tially variable and independent of ln (x), then the patlogical information includes point measurements of the moisture content and prior information about moisture spond to the pattern of ln (x). content distributions (i.e., mean, variance, and correlathat during an infiltration event 0 remains constant, Furthermore, Eq. [5] is derived with the assumption tion structure). The geophysical information consists of point measurements of electric potentials, parameters which may not always be valid. The resistivity of a po- of the resistivity moisture content relation, and prior rous medium can be highly variable, depending on the information about spatial variability of these parampore water (Sharma, 1997). The spatial variability of 0 degree of saturation and the type of ions present in the eters. Two-dimensional numerical experiments were first conducted to evaluate the effect of uncertainties and m may directly correspond to the pore water chemis- associated with the prior information. Subsequently, nu- try. Since silica, which comprises most mineral grains merical experiments were used to demonstrate the roserved electrical conduction in porous media is mainly (except metallic ores and clays), is an insulator, the ob- bustness of the integrative inverse approach for delineatthrough interstitial pore water. When clay minerals are ing transient moisture content distributions during a nonuniform infiltration event in a three-dimensional present, a relatively large number of ions may flow into heterogeneous vadose zone. or out of solution, through ion exchange, thus signifi- cantly changing the electrical conductivity of the fluid. THEORY During an infiltration event, other chemical reactions or processes may occur due to differences in water chemis- Governing Equation for the Electric Potential try in the infiltrated water, thus altering the composition In a geological formation, the electric current flow of ions present in pores and changing the nature of induced by an electrical resistivity survey in general can the pore electrolytes. This adds another level of spatial be described by variability to the resistivity distribution. 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 infiltration current source density per volume (A m 3 ), and is the field site in Socorro, NM. A total of 25 samples were colelectrical 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 several which is assumed to be locally isotropic. The boundary moisture contents were determined using an impedance conditions associated with Eq. [1] are analyzer. Equation [4] was subsequently fitted to the φ 1 φ* [2] measured resistivity and moisture data to determine the values for 0 and m. On the basis of analysis of the data and set, Yeh et al. (2002) reported that both ln 0 and lnm (x) φ n 2 i [3] were approximately normally distributed. The geometric mean of 0 was m and the variance, standard where φ* is the electric potential specified at boundary deviation, and coefficient of variation for ln 0 were 1, 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 (dimensionless), while variance, standary 2. dard deviation, and coefficient of variation for lnm were 0.034, 0.185, and 63.7%, respectively. In addition, they Constitutive Resistivity and Moisture found that both parameters are not entirely disordered Content Relation in space but correlated over short distances. For ln 0, In this study, a power law relation was used to relate an exponential variogram model was used to describe its resistivity to moisture content (e.g., Yeh et al., 2002): spatial variability with sill, range, and nugget values of

3 , 3.5 m, and 0.08, respectively. Similarly, an exponen- respect to the parameters, and are computed using an tial variogram model was used for lnm. The sill, range, adjoint state method. Details of the derivation of these and nugget values for lnm were 0.043, 3.5 m, and 0.01, sensitivities can be found in Sun and Yeh (1992), Li and respectively. No significant correlation between ln 0 and Yeh (1998) and Hughson and Yeh (2000). The sensi- lnm was reported. tivity of the electric potential at location i to a perturbation The observed spatial variability of ln 0 and lnm implies in a parameter at location k can be generally that equivalent changes in moisture content at expressed as different locations in the medium may lead to different φ changes in the measured electrical resistivity. As a re- i φ d [7] sult, the pattern of change in resistivity, detected by k k ERT surveys in a field, may not necessarily reflect the true pattern of change in the moisture content. To overstudy are: Specifically, the sensitivities for the parameters in this come this difficulty in interpreting ERT field surveys, we developed an integrative inverse algorithm that is φ i based on the concept of stochastic information fusion φ d f k developed by Yeh and Šimůnek (2002). k [8] Inverse (or Estimation) Algorithm φ i m φ d a k k [9] Linear Estimator where is the Dirac delta function and x k is the measure- ment location of the electric potential. Notice that the mean electric potential, φ, is needed to evaluate the f, a, n, and h are the perturbations. The primary goal sensitivities (see Eq. [8], [9], and [10]). The mean electric of 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 a medium, although it can be used to estimate other pa- mean electrical resistivity and moisture content relation rameters (i.e., 0, m, and φ) as well. The estimation al- that is the same as Eq. [4], but with the parameters set gorithm 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, the surements that consist of φ and the transmitted current above sensitivity equations are used to calculate covaridensity, 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 perturbacould be estimated from core samples, geological well tions yields logs, or outcrops (Yeh et al., 2002). The effects of uncertainty in the prior information will be studied below in {h} J h { } [12] the Effects of Uncertainties in the Prior Information H, X section. Using a first-order analysis, a state variable, such as where {} indicates the vector of the discretized variable, φ, can be expanded in a Taylor series about the mean J h is a jacobian matrix representing the derivatives values of parameters. Neglecting the second- and higher- of the potential with respect to the parameters (i.e., φ/ order terms of the Taylor series leads to a linear relation i ), which can be obtained using Eq. [8] through [10], between the state variable and the parameters, and has dimensions of n h nelem, where the number i : of voltage measurement locations is given as n h, and φ nelem is the total number of elements in the domain. φ φ i [6] i Since the parameters, 0,, and m, vary spatially, and their specification at every point in space is practically impossible, they will be treated as random fields (e.g., Yeh, 1992 and 1998), which are characterized by their statistical moments (i.e., means, variances, and correlation scales). Notice that the fields can be either stationary or nonstationary stochastic processes. Accordingly, and the electric potential φ are also regarded as stochastic processes. In the following analysis, it is assumed that ln 0 F f, ln A a, lnm N n and φ H h, where F, A, N, and H are the mean values, and φ i n k k m ln( ) φ d [10] where k is the domain of the element containing node k if a finite element approach is used and represents the adjoint state variable, which can be solved for using the adjoint state equation: [ (x) ] (x x k ) [11] i X i φ For ERT surveys, i i i represents the zero mean perturbation of a natural log transformed parameter, such as f ln 0 F, a ln A, and n lnm N; the zero mean perturbation of the state variable (i.e., electric potential) is h φ φ φ H; φ/ i represent the sensitivity derivatives of electric potential with Multiplying Eq. [12] by the transpose of { } (i.e., {f}, {a}, {n}) and {h}, and then taking the expectations on both sides, yields R h J h R R hh J h R Jh T [13] where superscript T denotes transpose; R h represents

4 684 VADOSE ZONE J., VOL. 3, MAY 2004 the cross-covariance functions between h and f, h, and and Yeh (2000), and Vargas-Guzmán and Yeh (2002) a,orhand n, with dimensions of n h nelem; R denotes is employed. That is, the covariance functions of f, a, and n with dimensions ˆ (r 1) ˆ (r) h (r)t φ* φ (r) of nelem nelem, and they are given a priori. A nugget [16] can be added to the covariance to represent measure- where ˆ (r 1) and ˆ (r) represent the parameter estimated ment errors, or variations within the sample scale if they at iteration r 1 and r, φ r is the electric potential at are known. Because little information is available about the measurement locations calculated from the forward the 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 (r) h is the weight at iteration r, which is determined study. Such an assumption merely represents the worst from the following: scenario in which knowledge of one variable does not provide any information about the others. R ε r hh r h ε (r) h [17] hh in Eq. [13] represents the covariance function of h, which has The solution to Eq. [17] requires knowledge of ε hh and dimensions of n h n h. Similarly, a nugget representing ε h ; they are estimated using the following approximaerrors or variability due to scale disparity in potential tions at each iteration: measurements can be added to this covariance function. Using these cross-covariance and covariance functions, ε (r) hh J (r) h ε (r) J T(r) h a first-order estimate of the perturbations of the logε h J (r) h ε (r) [18] transformed parameters is obtained by using the conditional expectation given observed primary information where J h is the sensitivity matrix of n h 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 by and Zhang, 1996): ε 1 R R R h h [19] ˆ T * h T h* [14] where R is a subset of R. For r 1, the residual where ˆ is a nelem long vector of the estimated perturba- covariances are evaluated according to tion of parameters, 0, m, and ; * and h* are perturbaε 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 firstin 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 C h C h C hh h C C h [15] ment of the parameters (if any), the electrical conductivity is then estimated using Eq. [4]. Afterwards, the mean where C represents the covariance of between mea- electric potential equation is solved again with the newly surement locations; C hh represents the covariance of h estimated electrical conductivity for a new electric pobetween measurement locations; and C h denotes cross tential field. Then, the maximum change of 2 (the varicovariance between h and at measurement locations. ance of the estimated parameters of f, a, and n) and The right-hand side of Eq. [15] consists of the covariance the change in the largest potential misfit among all meaof between measurement locations and the estimate surement locations between two successive iterations are position, and the cross covariance of h at measurement evaluated. If both changes are smaller than prescribed locations and at the estimate location. In other words, tolerances, the iteration stops. If not, new values for ε h the weights used in the estimation are not only directly and ε hh are evaluated using Eq. [18]. Equation 17 is then related to spatial correlation structures of parameters solved to obtain a new set of weights that are used in and state variables at measurement locations, but also Eq. [16] with [φ* φ (r) ] to obtain a new estimate of the the cross correlations of the parameters and state vari- parameters. A theoretical proof of the convergence of ables, and correlation and cross correlation between the successive linear estimator is given in Vargas-Guzmeasurements and the location where the parameter is mán and Yeh (2002). to be estimated. These matrices all are subsets of the covariance and cross covariance matrices in Eq. [13]. Sequential Estimation Approach Successive Linear Estimator The previous section describes the ERT inversion algorithm for only one set of primary and secondary information Equation [14] approximates the nonlinear relation obtained in one DC transmission. This algorithm between the parameter to be estimated and the mea- can simultaneously include all potential measurements sured electric potential by means of a linear first-order collected during all DC transmissions in an ERT survey. approximation. Thus, the equation cannot fully exploit However, the system of equations (Eq. [15] and [17]) electric potential measurements. To circumvent this problem, can become extremely large and ill conditioned, in a successive linear estimator similar to that used which case stable solutions to the equations are difficult by Yeh et al. (1996), Zhang and Yeh (1997), Hanna and to obtain (Hughson and Yeh, 2000). To avoid numerical Yeh (1998), Vargas-Guzmán and Yeh (1999), Hughson difficulties in solving the large system of equations, the

5 685 voltage data sets are included sequentially. The sequendimensional analysis. Table 1. Hydrological and statistical parameters used in two- tial algorithm used is similar to the one developed for use in hydraulic tomography inversion (Yeh and Liu, Parameter Mean Variance x z Covariance model 2000). In essence, the proposed sequential approach uses cm the estimated electrical conductivity field, the 0, m, and K s (cm s 1 ) exponential fields, and their covariances and cross covariances, as (1/cm) exponential exponential prior information for the next estimation using new sets of current voltage data from DC transmissions at different locations. Vargas-Guzmán and Yeh (1999) and of uncertainties in the mean and covariance functions of Yeh and Šimůnek (2002) gave an illustrative example on the estimates of the distribution. In these experiof the sequential approach and explained the necessity ments, the other two parameters, 0 and m, are spatial for updating the covariances and cross covariances. The variables but their spatial distributions are here assumed sequential inclusion of data sets from different DC trans- to be known precisely. This assumption is necessary to missions continues until all data sets have been utilized. reduce the number of cases to be analyzed. The results All data sets are fully processed by propagating the of this investigation should be applicable to cases where conditional first and second moments from one data set the prior information of these two parameters also into another. Such a sequential approach allows accumu- volve uncertainty. lation of high-density secondary information obtained The numerical experiments were conducted for a hyfrom an ERT survey, while maintaining the covariance pothetical, two-dimensional, vertical vadose zone with matrix at a manageable size that can be solved with mini- dimensions of 200 by 200 cm, discretized into 10 horizontal mal numerical difficulties. and 20 vertical elements, 200 cm 2 each. The unsatmal Inversions of ERT surveys for environmental applicasumed urated hydraulic properties of each element were astions are generally ill posed since the number of parammodel to be described by the Mualem van Genuchten eters to be estimated is often much greater than the number (van Genuchten, 1980): of measurements of the state variable (see Yeh and Šimůnek, 2002 for a discussion about necessary and sufficient K( ) K s 1 ( ) ( 1) [1 ( ) ] 2 /[1 ( ) ] /2 conditions for a well-posed inverse problem). An ( ) ( s r ) 1 ( ) ( ) r [21] ill-posed problem has an infinite number of global minima and solutions. Classical inverse algorithms (e.g., regularas a function of the pressure head, ; K s is the saturated where K( ) is the unsaturated hydraulic conductivity ized least-squares approach, Tikhonov, 1963) can only dehydraulic conductivity; and are shape factors; rive an estimated parameter field that produces an electric potential field honoring measurements at sampling (1 1/ ); s is the saturated moisture content; and r locations and a smooth estimate at other locations. This is the residual moisture content. To represent heteroge- smooth field, however, does not necessarily honor the neity, the parameters of Eq. [21] were assumed to be characteristics of the spatial variability of the true pa- stochastic processes. Since spatial variations in s and rameters (e.g., mean, variance, and correlation strucministic constants with values of and 0.029, respec- r are generally negligible, both were treated as deterture). On the other hand, our sequential successive lintively. The parameters K s,, and for each element in ear estimator estimates conditional effective parameters (Yeh et al., 1996; Hanna and Yeh, 1998; Yeh, 1998; Yeh the simulation domain were generated using a method and Šimůnek, 2002). It aims to yield a parameter field by Gutjahr (1989) with specified means, variances, and that produces not only parameter values and state variated random fields are then denoted as the true distri- correlation structures as listed in Table 1. These generables observed at measurement locations, but also conbutions of the hydraulic parameters for the hypothetiditional effective parameter values at locations where no measurements are available. The effective param- cal site. eters are our estimates based on the spatial statistics of The initial condition of the site was assumed to be hy- the parameter fields and their cross correlations with drostatic. Specifically, no-flux boundary conditions were state variables. These cross correlations are evaluated specified on the two sides, a water table was prescribed using the governing equation for the electric potential at the bottom of the site, and a constant pressure head and fluid flow processes. Discussions of advantages of boundary of 200 cm was assigned to the top boundary. the approach can be found in Yeh and Šimůnek (2002). An infiltration event was created by changing the pre- scribed pressure head from 200 to 80 cm at the center (from x 80 to 120 cm) of the top boundary NUMERICAL EXPERIMENTS (z 200 cm). Subject to these boundary conditions, the Richards equation for variably saturated flow was Effects 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 distriprior information such as the mean, variance, and corre- bution under nonuniform infiltration. Two cases were lation 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 formation timated and involve uncertainty. The following numeri- whose hydraulic properties are specified using their cal experiments are devoted to investigating the effect mean values (Table 1) and (ii) infiltration into a hetero-

6 686 VADOSE ZONE J., VOL. 3, MAY 2004 Fig. 1. (a) The mean moisture content distribution simulated using homogeneous hydraulic parameters; (b) the true moisture content distribution simulated using heterogeneous hydraulic parameters. geneous formation presented by the generated true hy- process yielded one set of voltage current data. By moving draulic parameter fields. The simulated moisture conthe the transmitter along the bore hole, and repeating tent distributions for the two cases are plotted in Fig. 1a same procedure, four sets of voltage current data and b, and were used for the ERT experiments delated were obtained. In addition, one value from the simu- scribed below. true field was assumed to be measured from a The electrical resistivity,, of the site was assumed to core sample along the bore hole. The locations of the curbe a function of 0, m, and, according to Archie s Law, rent source, voltage, and measurements are displayed Eq. [4]. To derive the electrical resistivity field corre- in Fig. 3. sponding to the simulated distribution at the site, the Besides the voltage and measurements, specifications two parameters, 0 and m, were considered to be random of means, variances, and correlation structures of the fields (stochastic processes) with geometric means of parameters to be estimated are required by the pro- 8.5 m and 1.35, respectively. The variances of ln 0 and posed inverse model. Generally, the statistics of cannot lnm were 0.1 and 0.01, respectively. The two parameters be estimated without uncertainty since the true distriwere assumed to be statistically independent of each bution is unknown. In the following section, three sce- other and both parameters were described with the same narios were examined to investigate the effects of the exponential correlation structure, which had a horiuncertainty in these input spatial statistics. zontal correlation scale of 240 cm and vertical correlaon both sides of the system of equations (i.e., Eq. [15] Notice that the variance of the parameters appears tion scale of 20 cm. Gutjahr s method (1989) was again and [17]) and will be canceled out. Therefore, the value used to generate heterogeneous 0 and m fields as shown of the variance has no effect on the solution of the sysin 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 cornext 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 the the site. Synthetic pole to pole ERT surveys were subse- estimated parameters. quently simulated using the hypothetical domain. First, The first scenario approximates the mean distributwo 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 of x 70 and 130 cm. Ten electrodes were placed 20 cm the field is different from those of the hydraulic propapart 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 same domain. One of the electrodes in the bore hole was (i.e., an exponential model with horizontal and vertical selected as the transmitter with a specified current. The correlation scale of 240 and 20 cm, respectively). Using resultant voltage differences between the reference elec- these approximations, together with other necessary input, trode and the rest of the electrodes were measured; this we estimated perturbations of the distribution, Fig. 2. (a) Generated random field for ln 0 ; (b) generated random field for lnm.

7 687 Fig. 3. (a) True moisture content distribution; (b) estimated moisture content distribution assuming moisture content has a geometric mean and exponential covariance function; (c) estimated moisture content distribution assuming moisture content has a true mean and exponential covariance function; (d) estimated moisture content distribution assuming a true mean and computed covariance function. Circles indicate current source locations, triangles indicate voltage measurement locations, and squares indicate the moisture content measurement location. where R is the covariance function of the moisture content; R denotes the covariance functions of the log- transformed perturbations of the hydraulic properties of K s,,and, which were assumed known; and J is the Jacobian matrix (details of computing J can be found in Hughson and Yeh, 2000). The final estimated distri- bution and its scatter plot are shown in Fig. 3d and 4c, re- spectively. Comparing Fig. 3b, 3c, and 3d with Fig. 3a reveals that the estimated distributions for the three scenarios resemble the general pattern of the true distribution, despite the different assumptions used in the prior information. Comparisons of scatter plots (Fig. 4) also lead to the same conclusion. To quantify this conclusion, the L1 and L2 norms of these scenarios were evaluated as follows: which were then added to the geometric mean to derive the final estimated distribution (Fig. 3b). A comparison between the true and estimated distribution is shown using a scatter plot (Fig. 4a). The covariance function of the field in the second scenario was kept the same as in the first scenario while the ensemble-mean distribution (Fig. 1a) was used, instead of a constant mean value. The ensemble mean distribution presents the average of many possible distributions due to the specified infiltration in all possible heterogeneous sites characterized by the given spatial statistics. Substituting the ensemble mean field into the inverse model yields the estimated perturba- tion field, which was then added to the ensemble mean distribution to derive the final distribution (Fig. 3c); see also the corresponding scatter plot (Fig. 4b). The third scenario employed the ensemble mean distribution as input, but used a correct covariance function of instead of that of the hydraulic properties. The covariance function was computed using the firstorder approximation: R J R J T [22] L1 1 ˆ n n i i [23] i 1 L2 1 ˆ n n i i 2 [24] i 1 Fig. 4. (a) Scatter plot corresponding to Fig. 3b; (b) scatter plot corresponding to Fig. 3c; (c) scatter plot corresponding to Fig. 3d.

8 688 VADOSE ZONE J., VOL. 3, MAY 2004 where i and ˆ i represent the true and estimated moisdimensional analysis. Table 2. Hydrological and statistical parameters used in three- ture content, respectively. We found that values of the L1 and L2 norms for the three cases differed only slightly: Parameter Mean Variance x y z Covariance model the norms from the use of the ensemble mean are slightly cm smaller than those from the use of the geometric mean. K s (cm min 1 ) exponential We therefore conclude that choices of mean and covari- (1/cm) exponential exponential ance function of moisture content have only a minor effect on the estimate if a sufficient number of potential measurements are available and the 0 and m distribu- three-dimensional vadose zones. Water movement was tions are known precisely. This result agrees with the simulated in a three-dimensional hypothetical vadose findings by Liu et al., (2002) for hydraulic tomography zone at 1000 and min from the commencement surveys. That is, abundant data sets collected during of an infiltration event. The simulated moisture content tomographic surveys (either ERT or hydraulic tomogradistributions at these two times, 1000 and , were phy surveys) can overcome the uncertainty associated used as the true moisture content fields; and their differwith prior information such as mean, variance, and corence was denoted as the true moisture content change in relation structure. This result is generally true in these synthetic cases with small domains. Under field condifields of 0 and m, the true resistivity fields at 1000 and the following analysis. Following generation of random tions, where a site is three-dimensional and encompasses a much larger domain than in the example, and where min were calculated using Eq. [4] with the gener- the ated 0, m, 1000, and and m fields are unknown, the effects of uncer- fields. tainty in the input parameters are expected to be greater Next, ERT surveys were simulated using these two (Yeh and Šimůnek, 2002). Nevertheless, the reliability resistivity fields. After collecting voltage current data of inverse modeling of tomography surveys increases sets, two different inverse approaches were used to in- with the spatial density of the secondary information. terpret water flow due to the infiltration event. The first A dense deployment of secondary information sensors approach used the Yeh et al. (2002) inverse model based therefore will yield small uncertainty in the result. On on electric potential measurements only to derive re- the other hand, a sparse deployment would require ac- sistivity fields at times of 1000 and min. Next, curate prior information to facilitate statistically unbi- the change in resistivity from 1000 to min is comased estimates. puted. Finally, the estimated change in resistivity is used to interpret the change in moisture content. The second Direct Estimation of Three-Dimensional approach estimated moisture distribution at min Moisture Content Fields directly using our new integrative method. In the following numerical examples, we tested our inverse The hypothetical site was assumed to be a cube, 200 algorithm for transient infiltration and flow through cm on each side, consisting of 2000 elements of 20 by 20 by Fig. 5. (a) Generated lnk s field; (b) generated ln field; (c) generated ln field; (d) true moisture content field at t 1000 min.

9 689 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 exponential parameters (K s,, and ) were generated and assigned correlation structure with a horizontal correlation scale to each element using the spectral method with the pa- of 80 cm and a vertical correlation scale of 20 cm. Figures rameters shown in Table 2. Figures 5a, 5b, and 5c show 6a and 6b show the generated true fields for these the generated lnk s,ln, and ln fields, respectively. two parameters. The initial pressure head condition was assumed to Based on these synthetic 0, m, 1000,and fields, be hydrostatic. Specifically, the bottom was set at a true synthetic resistivity fields at times of 1000 and prescribed pressure head of 50 cm, and the top was min ( 1000 and , respectively) were calculated using fixed at a pressure head of 250 cm. Infiltration oc- Eq. [4] Electrical resistivity tomography surveys were curred over an area of 1600 cm 2 on the top center of then simulated using these two resistivity fields. Figure the cube at a pressure head of 50 cm, while no-flux 8d displays the three-dimensional layout of the ERT boundary conditions were assigned to the remainder of survey. The design of the ERT survey included four the top and the four sides. This created nonuniform bore holes penetrating the entire depth of the site domain. vertical infiltration fields from a constant source on the The x and y coordinate pairs, in centimeters, of top center of the flow domain. The infiltration process the four bore holes were (50, 50), (150, 50), (50, 150), was simulated using a finite element model (Srivastava and (150, 150). Twenty electrodes were installed along and Yeh, 1992) to obtain moisture distributions. Figure each bore hole. Electrodes were also deployed along 5d shows the moisture content distribution 1000 min the surface in four lines with endpoints at the above x y after infiltration began. Mean moisture content distribu- coordinates. Current sources were installed along the tions at 1000 and min were similarly obtained upper right bore hole (150, 150) at the depths of 25, 55, with effective mean values of the hydraulic parameters. 95, 135, and 175 cm. Using the same collection procedure True changes in ln were computed as the differences for the voltage data as used for the two-dimensional between the true ln distributions at 1000 and numerical examples, five ERT voltage data sets of 111 min (Fig. 7a). voltage measurements each were obtained. In addition, To represent spatial variability in the parameters of 20 0 and m values along each of the four bore holes (a the resistivity and moisture content relation in the field, total of 80 measurements) were assumed to be available the two parameters, 0 and m, were considered as ran- for our analysis, while was sampled at 20 locations dom fields with geometric means of m and 1.336, indicated by squares in Fig. 8d. To investigate the effect respectively. Variances of ln 0 and lnm were assumed of direct measurements on the inverted estimate of to be and 0.034, respectively. We again assumed the moisture content at the site, the moisture content Fig. 7. (a) True ln from 1000 to 5000 min; (b) estimated ln from 1000 to min.

10 690 VADOSE ZONE J., VOL. 3, MAY 2004 Fig. 8. (a) True moisture content at t min; (b) estimated at t min without measurements; (c) estimated at t min with 20 measurements. (d) Schematic diagram for three-dimensional electrical resistivity tomography experiments. Top center square indicates the infiltration area, right triangles indicate voltage measurement locations, circles show current source locations, and squares show the locations of 20 measurements. (e) The scatter plot corresponding to (b); (f) the scatter plot corresponding to (c). distribution at min was estimated without any measurement and compared with the estimate with 20 measurements. Using Resistivity Change to Interpret Water Flow For the purpose of comparison, resistivity changes were computed to reflect water movement in the vadose zone. Forward simulations of ERT surveys based on the previously discussed network layout were conducted using the simulated moisture distributions at 1000 and min. Five voltage data sets were collected and then used in the inverse approach of Yeh et al. (2002) to estimate the resistivity fields at these two specified times. The change in resistivity between 1000 and min was then computed. Figure 7a displays the pattern of the true moisture content change, and Fig. 7b the estimated pattern of resistivity change. A comparison of these two figures indicates that interpretation of water movement based on the pattern of the estimated resistivity change alone led to incorrect estimates of water flow. From 1000 and min, the true water front moved in a southwest direction, while the estimated pat- tern of resistivity change suggests flow in a southeast direction. As reported by Yeh et al. (2002) two factors contribute to this discrepancy: uncertainty in the esti- mated resistivity fields and spatial variability in the parameters of the constitutive relation. Thus, in addi- tion to a dense deployment of voltage sensors, a detailed spatial distribution of the parameter must be known to correctly relate changes in the resistivity to those in the Fig. 9. (a) Conditional variance when no measurements are used at t min; (b) conditional variance when 20 measurements are used at t min. Circles indicate measurement locations.

11 691 moisture content. Relying only on changes in resistivity merical examples also demonstrate the ability of the integrative ERT survey inverse model for estimating moisand neglecting spatial variability in the parameters of the constitutive relation can lead to erroneous flow ture contents directly. The model yields good estimates directions and patterns. at the locations where primary and secondary informa- tion is measured. Primary information (i.e., the moisture Direct Estimation of Moisture Content Distribution measurements) contributes significantly to the accuracy Figures 8a and b show the true moisture content disnumber of potential measurements are useful, but they of the estimated moisture content distribution. A large tribution at min and the corresponding estimated moisture content distribution using voltage measupports the finding by Yeh et al. (2002) that poten- did not dramatically improve the estimate. This further surements, and 80 pairs of 0 and m measurements. The estimated field using the same number of voltage, tial measurements alone are inadequate to characterize 0 and m measurements, but also including 20 direct measureing geophysical measurements with hydrological infor- water flow in vadose zone. Finally, we conclude that fusments, is illustrated in Fig. 8c. Comparing the three figures we find that the proposed inverse algorithm rehydrologically realistic results under field conditions mation using a stochastic approach is necessary to yield produces the general pattern of the simulated true moisand to quantify uncertainty associated with the results. ture content distribution, even though the constitutive relation between resistivity and moisture varies spatially. The inclusion of the 20 measurements, in particular, ACKNOWLEDGMENTS greatly improves the estimates of the distribution. Fig- This research was funded in part by a DOE EMSP96 grant ures 8e and 8f are scatter plots corresponding to Fig. through Sandia National Laboratories (Contract AV-0655#1), 8b and 8c, respectively; a 45 line indicates perfect esti- a DOE EMSP99 grant through University of Wisconsin mation. The goodness of fit was also evaluated using (A019493), and NSF and SERDP grant EAR We L1 and L2 norms. The reduction in the L1 and L2 norms also thank Kris Kuhlman for his technical editing. Many thanks are extended to two reviewers for providing critical comments from Fig. 8e to 8f suggests that the addition of moisture to improve the paper. In particular, we are greatly indebted content measurements dramatically improves the estito 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 uncertainty associated with the estimate a smaller condi- REFERENCES tional variance indicates less uncertainty in the estimate. Baker, K. 2001, Analysis of hydrological and electrical properties at The conditional variances corresponding to the estithe Sandia-Tech Vadose Zone Facility. M.S. thesis. New Mexico mates 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 Steady ances are located close to the four bore holes where shape analysis of tomographic pumping tests for characteriza- secondary information is measured. At locations where tion of aquifer heterogeneities. Water Resour. Res. 38(12) doi: /2001wr moisture content measurements were collected, the con- Daily, W., A. Ramirez, D. LaBrecque, and J. Nitao Electrical ditional variance is zero, indicating that these observa- resistivity tomography of vadose water movement. Water Resour. tions are honored in the inverse model and that no Res. 28: uncertainty exists. Ellis, R.G., and S.W. Oldenburg The pole-pole 3D DC resistivity inverse problem: A conjugate gradient approach. Geophys. J. Int. 119: CONCLUSIONS Gutjahr, A Fast Fourier transforms for random field generation. Project Rep. Los Alamos Grant 4-R R. Dep. Mathematics, Knowledge of detailed moisture content distributions New Mexico Tech, Socorro, NM. is important to our understanding of vadose zone proments of transmissivity, hydraulic head, and velocity fields. Adv. Hanna, S., and T.-C.J. Yeh Estimation of co-conditional mo- cesses and water resources management. An ERT inversion algorithm based on the concept of stochastic infor- Water Resour 22: Hughson, D.L., and T.-C.J. Yeh An inverse model for threemation fusion is developed to estimate moisture content dimensional flow in variably saturated porous media. Water Redistributions in three-dimensional vadose zones. The sour. Res. 36: approach integrates point measurements of electric poin variably saturated regimes. Adv. Water Resour. 21: Li, B., and T.-C.J. Yeh Sensitivity and moment analysis of head tential, parameters of the constitutive relation between Li, Y., and D.W. Oldenburg Inversion of 3D dc-resistivity data moisture and resistivity, and moisture content. In addi- using an approximate inverse mapping. Geophys. J. Int. 116: tion, the approach includes prior information about the spatial statistics of the moisture content distribution and Liu, S., T.-C. J. Yeh, and R. Gardiner Effectiveness of hydraulic the parameters of the constitutive relation. Results of tomography: Numerical and sandbox experiments. Water Resour. Res. 38(4). doi: /2001wr this study show that the prior information has effect on Priestley, M.B Spectral analysis and time series. Academic the 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 Environmental and engineering geophysics. Cambridge University Press, Cambridge, UK. of primary and secondary information increases. Numerical examples illustrate that interpretations of Srivastava, R., and T.-C.J. Yeh A three-dimensional numerical model for water flow and transport of chemically reactive solute water 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: of spatial variability in the constitutive relation. Nu- Sun, N.-Z., and W.W.-G. Yeh A stochastic inverse solution for