WATER RESOURCES RESEARCH, VOL. 28, NO. 9, PAGES 2387-2398, SEPTEMBER 1992

Inversion of the Borden Tracer Experiment Data' Investigation of Stochastic Moment Models

ALLAN D. WOODBURY

Department of Geological Engineering, University of Manitoba, Winnipeg, Manitoba, Canada

E. A. SUDICKY

Waterloo Centre for Groundwater Research, University of Waterloo, Waterloo, Ontario, Canada

Inversion of Dagan's two- and three-dimensional stochastic models using Freyberg's (1986), Rajaram and'Gelhar's (1988, 1991), and Barry and Sposito's (1990) moment data from the Borden experiment is carried out to examine (1) the validity of the two-dimensional (Dagan, 1982) and three-dimensional (Dagan, 1988) models and (2) the reduction in uncertainty of the spatial moments over nonconditioned estimates. A direct application of Bayesian statistical inference, in conjunction with Monte Carlo integration, is used to produce posterior probability density functions for the parameters. The parameter ranges from all methods show horizontal integral scales A between 1.85 and 4.04 m, vertical integral scales A z between 0.144 and 0.459 m, and log hydraulic conductivities In (K) between 0.120 and 0.197. These results compare well to the earlier estimates of Woodbury and Sudicky (1991) and Robin eta!. (1991). We show that moment prediction uncertainty is substantially reduced when both tracer moment data and prior estimates of the In (K) geostatistical parameters are incorporated into theoretical formulae based on stochastic dispersion theory.

INTRODUCTION

Sudicky [ 1986] described the results of a sampling program in which a large number of hydraulic conductivity measure- ments were taken along two transects at the site of an elaborate tracer test performed in the Borden aquifer in Ontario, Canada. These measurements, combined with a detailed evaluation of the dispersion characteristics of the injected tracer cloud [Freyberg, 1986], provided a unique opportunity to examine the validity of stochastic theories of contaminant transport that have emerged over the past decade. Based on the field data and subsequent geostatistical inferences, Sudicky [1986] computed mean values, vari- ances, and integral scales for the underlying log conductivity distribution of the Borden aquifer. Then, using these quan- tities as input to the two-dimensional stochastic moment model of Dagan [1982, 1987] and the asymptotic approach of Gelbar and Axness [1983], the predicted field-scale flow and dispersion parameters were shown to be reasonably consis- tent with the observed evolution of the tracer plume as interpreted by Freyberg [!986]. Subsequent comments on Sudicky's [1986] work are given by Kemblowski [1988], White [1988], Molz and Giiven [!988], and Sudicky [1988]. Other discussions centering on the applicability of various competing theories to explain the observed tracer behavior are given by Naff et al. [1988, 1989], Barry [1990], Dagan [1989a, 1990a], Neuman and Zhang [1990], and Zhang and Neurnan [1990].

To our knowledge, none of the existing three-dimensional ensemble mean second-moment solutions for the spread of a tracer cloud have been rigorously validated through detailed Monte Carlo numerical experiments. Some activity in these directions has nevertheless been initiated with numerical

Copyright 1992 by the American Geophysical Union. Paper number 92WR0!005. 0043-1397/92/92WR.0 ! 005505.00

verification of Dagan's [1982] two-dimensional model [e.g., Quinodoz and Valocchi, 1990]. In three dimensions, numer- ical experiments aimed at validating the asymptotic disper- sion theory by Gelhar and Axness [1983] and the time- dependent theory of Dagan [1988] have been limited to simulations involving one or a few aquifer realizations because of the massive computer resources required for such an undertaking [Thomson and Ge!har, 1990; Dasinger and Gelhar, 1991]. There is also the need to perform detailed field tracer tests in well-characterized sandy aquifers of different geologic origin for the purpose of validating sto- chastic dispersion theories. The tracer test described by LeBlanc et al. [1991] and Garabedian et al. [1991] at the Cape Cod site is an excellent illustration of how this need is being met.

Data from the 1983-1986 Borden experiment and Su- dicicy's [1986] analysis of the spatial variability of In (K) at this site are still yielding insight into mass transport pro- cesses. Recently, Woodbury and Sudicky [1991] furthered Sudicky's [1986] analysis by systematically reevaluating the geostatistical parameters of the hydraulic conductivity field at the Borden site. These authors provided a lengthy discus- sion on their choice of an exponential model with the "nugget" as the form of the spatial correlation of the In (K) field. With this choice of model, nonlinear parameter esti- mation by Woodbury and Sudicky [1991] produced a vari- ance of 0.25, nugget of 0.07, and integral scales of 5.1 m horizontal and 0.21 m vertical on transect A-A'. For transect B-B' these values were 0.37, 0.11, 8.3 m, and 0.34 m, respectively.

A recent, separate study of the spatial distribution of In (K) at the Borden site was performed by Robin et al. [1991] using spectral methods. Their analyses included an investi- gation of six power-spectral density functions (equivalent to six models of covafiance) and concluded that the exponen- tial model with nugget, such as that used by Woodbury and

2387

2388 WOODBURY AND SUDICKY: INVERSION OF BORDEN TRACER DATA

Sudicky [1991], gave the best fit to the data, based on a residual sum of squares criterion. Spatial parameters com- puted by these two groups compared well, considering the very different approaches utilized.

Woodbury and Sudicky [1991] also evaluated a condi- tioned form of Dagan's [1982, 1987] two-dimensional sto- chastic equations that relate geostatistical parameters to a tracer cloud's expected spreading moments and the vari- ances of the spreading moments. The equations were eval- uated using the parameter estimates and covariances deter- mined from line A-A' as input, with a velocity of 8.6 cm/d. The results were compared with actual values determined from the field test, as evaluated by Freyberg [1986] and Rajaram and Gelhar [1988]. The geostatistical parameters developed in Woodbury and Sudicky's [1991] study pro- duced an excellent fit to both sets of calculated moments.

Variations in the calculated moments about the expected values from the model were well within estimated error

envelopes. Our purpose in this study is to invert Dagan's two- and

three-dimensional stochastic models using moment data from the Borden experiment as computed by Freyberg [1986], Rajararn and Gelhar [1988, 1991], and Barry and Sposito [ 1990]. We compare the resulting parameter sets for integral scale, variance, and steady velocity with our earlier findings based on a purely geostatistical analysis.

Some attempts at inversion of the Borden plume data have been carried out. For example, Freyberg [1986] carried out an informal calibration of Dagan's [1982] two-dimensional model to the results of the Borden tracer plume experiment in order to estimate In (K) variance and its integral scales. His study showed reasonable agreement between Sudicky's original geostatistical parameters and those estimated by calibration. Zhang and Neuman [1990] made comparisons of their three-dimensional model to the Borden tracer data and

noted some discrepancy between theory and the experimen- tal moment data. Barry and Sposito [1990] applied a more formal approach by estimating the In (K) variance by non- linear least squares. Their parameter estimation, however, held all other parameters fixed in the analysis. Their best estimate showed a In (K) variance of 0.22, in good agree- ment with the geostatistical estimates of Woodbury and Sudicky [ 1991].

Specifically, the following topics are examined in this analysis' (1) a discussion of the various moment data sets available for analysis, (2) inverse methods and the effects of assuming different likelihood functions on the inverse re- sults, (3) limitations of the moment models as applied to the Borden data sets, (4) the effects of supplying prior informa- tion (in the form of estimates derived from geostatistics), and (5) results of individual inversions of the spatial moment data describing the bromide and chloride tracers.

Our analysis begins with a discussion of the various data sets available for analysis and then examines the above- mentioned points in detail.

BROMIDE AND CHLORIDE DATA SETS

The Borden tracer test experiment was conducted in 1983 and was reported by Mackay et al. [1986]. The experiment involved injection of seven tracers into the aquifer, followed by a 3-year monitoring program of over 5000 measurements points in three dimensions. Freyberg [1986] selected two

tracers, bromide and chloride, for detailed moment studies. Freyberg's study proceeded along the following path. First, the tracer data were vertically integrated over uniform elevations above and below each plume. Second, the verti- cally integrated data were interpolated on a regular square grid by a weighted, inverse distance method. Third and finally, the interpolated horizontal data were used in an areal quadrature scheme to produce zeroth, first-, and second- order horizontal moments at each time sampling.

Freyberg's original analysis was repeated by Barry et at. [1988]. Their analysis included two different approaches to the vertical integration, followed by surface interpolation using the DISSPLA package [Integrated Systems So3'hvare Corporation, 1985], a bicubic spline fit to the horizontal surface, and finally numerical integration. They presented their data as graphical representations focusing on the br0. mide tracer.

Rajaram and Gelhat [1988] calculated a three-dimensional set of moment estimates utilizing a modification of the approach developed by Garabedian et al. [1988] for the Cape Code tracer experiments. The former authors performed a vertical integration of the data by trapezoidal quadrature, constructed a triangular grid in the horizontal plane, and then integrated in this plane using linear interpolation. Their analysis included second-moment generation for both bro- mide and chloride tracers.

Barry and Sposito [1990] also presented a three- dimensional analysis. They first interpolated the scattered data set to a regular grid in three dimensions. The choice of interpolator was made by a cross-validation technique. Jack- knifed estimates of the three-dimensional moments, as well as confidence intervals, were generated. In this analysis, only bromide moments were calculated, due to a concern about background levels of chloride in portions of the site, although Barry et ai. [1988, p. 1812] concluded that there was "little or no evidence of substantial mixing between the CI- plume and background solute."

Rajaram and Gelhat [1991] provided a more formal ac- counting of their original work [Rajaram and Gelhar, 1988], compared their three-dimensional results to those of Barry and Sposito [1990], and included an interpretation of the spatial moments with respect to current stochastic theories of mass transport. Unfortunately, the chloride tracer results presented in their earlier work were absent from this publi- cation.

Results from Freyberg [1986], Rajaram and Gelhar [1988], and Barry and Sposito [1990] are listed in Tables la and lb. In order to be consistent with Freyberg [ 1986] and Barry and Sposito [1990], the values listed for Rajaram and Gelhar [1988] are the components of the spatial moment tensor in the direction of mean flow. Also note that the values for

Barry and Sposito [1990] are based on scaling of their Figure 3, as the actual numerical values were not available. Gener- ally, good agreement can be seen between Freyberg's and Rajaram and Gelhar's computations. Mean square statistics for the difference between the second-order spatial moment data calculated by Freyberg [ 1986] and Rajaram and Gelbar [1988] are 3.78 for Xll and 0.57 for X22 (bromide), and 20.99 for X•i and 0.75 for X22 (chloride). Here, Xll and X22 are the major and minor axes, respectively, of the spatial second moment tensor. Differences are more pronounced between Rajaram and Ge!har's and Freyberg's estimates for the chloride tracer at later sampling sessions. Rajaram and

WOODBURY AND SUDICKY: INVERSION OF BORDEN TRACER DATA

TABLE la. Moment Calculations: Bromide Tracer

Time, days RG X•l F Xi• SB Xii RG X22 F X22 SB X22 1. !.9 1.8 2.7 2.3 2.0 2.9 9. 1.8 2.3 2.2 2.6 2.8 2.7

16. 2.0 1.9 1.8 2.6 2.8 2.7 29. 2.5 2.4 2.7 2.4 2.5 2.5 43. 3.8 3.6 3.6 2.5 2.5 2.5 63. 5.0 4.9 5.4 2.5 2.7 2.5 85. 5.4 5.2 4.5 3.0 2.6 5.3

259. 15.9 16.7 !3.6 4.1 4.6 4.1 381. 18.7 20.8 30.0 5.4 5.0 6.9 429. 25.8 22.0 28.2 5.5 7.3 4.6 462. 31.9 30.1 34.6 5.3 6.6 5.7 647. 54.7 50.1 62.7 8.0 6.9 5.0

1038. 74.9 83.3 80.0 12.3 18.8 !9.7

RG refers to.Rajaram and Gelhat' [1988], SB to Barry and Sposito [1990], and F to Freyberg [1986]. Values are in square meters for X i• and X22. Values for Rajaram and Gelhar's data refer to components of the moment tensor in the direction of mean flow.

2389

Gelhar [1991] noticed that their estimates at late time had lower overall mass recovery than did Freyberg's, a result these authors attributed to plume truncation. Mean square deviations between Rajaram/Gelhar and Barry/Sposito are 17.4 for Xl! and 1.5 for X22 (bromide).

Differences between second-moment estimates for the bromide and chloride tracers most likely indicate random errors due to differences in analytical precision in measure- ment and possibly different transport behavior for each tracer. Moment calculation differences among the various teams of researchers appear to be due to the selection of interpolation and integration schemes [see Rajaram and Gelhat, 1991, Table 2]. Figures 1 and 2 show the differences between second moments computed for bromide and chlo- ride for Rajaram/Gelhar and Freyberg. As can be seen, the Rajaram and GeIhar [1988] computations appear to show an apparent trend for the X]i data, with larger residuals noted for increased time values. Less apparent trend is shown for X22. The equivalent plot is shown on Figure 2 for moment data calculated by Freyberg. Both moments appear to scat- ter randomly about a zero mean; however, none of the scatter in any of the above plots is probably statistically significant. Correlation in these residuals is difficult to ascer- tain due to the low numbers of data pairs present. Mean square deviations between second-moment results for the

TABLE lb. Moment Calculations: Chloride Tracer

Time, days RG Xi! F X!• RG X22 F X22

1. 1.8 2.! 2.2 2.4 9. 1.7 1.7 2.5 2.4

16. 2.2 2.3 2.6 2.8 29. 2.6 2.5 2.4 2.6 43. 3.9 4.4 2.5 2.7 63. 5.1 4.4 2.6 2.4 85. 4.9 5.7 3.0 3.3

259. !8.! 17.8 4.7 4.4 381. 21.3 20.6 5.9 4.4 429. 31.3 24.3 6.4 6.0 462. 37.3 27.8 6.2 5.5 647. 61.9 5!.5 7.8 5.5

RG refers to Rajaram and Gelhar[1988] and F to Freyberg [1986]. Value for X• and X22 are in square meters. Values for Rajaram and Gelhar's data refer to components of the moment tensor in the direction of mean flow.

bromide and chloride tracers are 1.41 for X• and 0.56 for X22 for Freyberg and 11.05 for Xl• and 0.2! for X22 for Rajaram and Gelhar [1991]. Notice that the mean square deviations between the results calculated by the different authors (and methods of moment calculation; see previous paragraph) are in most cases higher than the mean square differences between the bromide and chloride tracers. This observation lends support for the use of both tracers in the subsequent analyses in this paper.

INVERSE METHODOLOGY

The comparison of field data with results from stochastic models raises some important issues [Sposito et al., 1986]. For example, a stochastic contaminant transport model predicts ensemble mean concentrations and, as clearly dem- onstrated by the numerical simulations of Graham and McLaughlin [1989a], will most likely not produce the

rv' 6o

-8.0

RAJARAM AND GELHAR (1988, 1991) BROMIDE & CHLORIDE X•t AND X•.z RESIDUALS

• X•

X X.-X • X2 2 iiiii illilllllFiiii iiiiilllllllll ii i111 I

0 200 40• 60• BOO

Fig. 1. Differences between bromide and chloride tracers, as shown by X• and X22 data [from Rajaram and Gelhar, 1988, 1991].

2390 WOODBURY AND SUDICKY: INVERSION OF BORDEN TRACER DATA

FREYBERG (1986) 4.0q BROMIDE & CHLORIDE

• X• AND Xaa RESIDUALS

f¾' - ***** X•

-4. M - x• X= .

-

-

-

.

.

(DAS) Fig. 2. Differences between bromide and chloride tracers, as

shown by X•i and X22 data [from Freyberg, 1986].

irregular plumes observed in the field. This situation is unlike that in other fields, such as turbulent diffusion (see Dagan [!990b] for discussion), where it is common to assume that the concentration measured in any realization is close to its ensemble average. Another difficulty in the use of stochastic models is simply that the statistical parameters in these models are unknown and have to be estimated. One way to reduce the effects of parameter uncertainty is to condition a stochastic model to observed variables such as hydraulic conductivity, hydraulic heads, concentrations, and the like [Graham and McLaughlin, 1989b] with a subsequent loss of stationarity.

There are also stochastic models that predict ensemble average spatial moments about a spreading tracer cloud (i.e., center of mass, second spatial moment, etc.). Dagan [1990b] points out that these low-order moments may be less affected by uncertainty than the contaminant transport models mentioned above. Dagan [1990b] also investigated the variance of ensemble mean second-order moments, examining under what circumstances variances approach zero and hence when ergodic conditions are realized.

How does one then use field measurements of plume moments to estimate parameters for a stochastic model? Compounding the difficulties in answering this question is the recognition that the field data (measured and interpo- lated) are uncertain. That is, although "true" data exist, we cannot determine what they are. Hence the "measured" data are considered to be random variables themselves.

The approach taken to address some of the above points is similar in philosophy to that of Graham and McLaughlin [1989b]. That is, the ensemble second moments are re- stricted to functions which reproduce the measured plume moments. In this manner the conditional statistics, such as predicted ensemble moments at times when no measure- ments were taken, will account for the properties measured at a specific time.

Several stochastic theories exist that predict the low-order spatial moments of a tracer plume, and these models are

functions of several parameters. An application of a standard inverse approach, such as nonlinear least squares, basically assumes the residuals •i(t) between predicted and "mea. sured" plume second-order moments to be normally distrib. uted and effectively minimizes the variance of the second moment of the predicted plume. The method of least squares is valid if the underlying distribution for the residuals is Gaussian, but in the above case, poor field estimates and possibly nonergodic behavior could cause some of the fi to be outliers in a classic Gaussian sense. Therefore it is desirable to have an inverse technique that allows assump- tions related to the distribution of •' to be modified.

The Bayesian methodology readily lends itself to the problem of updating prior probabilities based on uncertain field measurements. For this reason the background to the inversion reported in this paper rests with an application of Bayesian statistical inference. Bayesian inference supposes that an observer can define a personal prior probability density function (pdf) about some random variable m. This pdf, p(m), can be defined on the basis of personal experience or judgement. Bayes' rule [e.g., Press, 1989] quantifies how this personal pdf can be changed on the basis of measure- ments. Simply stated, Bayes' rule is

posterior cr likelihood x prior

Consider an observed random variable d*. If the conditional pdf of d*, given m, is given by then Bayes' rule states that

(d*lm)p(m) q>(mld*) = (1)

•(d * rn)p(rn) drn ß (mld*) is the conditional pdf of m, given d*, and (I)(d*lm) represents the conditional pdf from forward modeling. This equation can be seen as a general solution of the inverse problem. If a likelihood function can be defined (i.e., the forward model exists), and there is a compatibility between observed results and a prior understanding of model param- eters (i.e., •(d*lm) > 0 for some m where p(m) > 0), then Bayes' rule implies that the resulting posterior pdf exists and is unique [Tarantola, 1987, p. 53]. However, the posterior pdf may have more than one maximum likelihood point. Indeed, rI)(mld*) may be pathological, as a result o• infinite variance, or nonnormalizable (the denominator in (1)is unbounded). Pathological states simply reflect the nature of the information on the model provided by the data and prior information.

In many problems in groundwater analysis the forward modeling problem can be written as

a = g(m)

where d is a set of theoretically predicted data and g represents a nonlinear transformation from model to data space. If the combination of forward modeling and measure- ment errors are assumed Gaussian, then [Tarantola, 1987, p. 68]

ß (a* m) = [(2 rr)"•lc r

'[a* - g(m)]rC ' ß exp {-• • [a* - g(m)]} (2)

WOODBURY AND SUDICKY: INVERSION OF BORDEN TRACER DATA 2391

where C r is the covariance matrix describing forward mod- eling uncertainties resulting from bo.th measurement and modeling error (that error describes imperfect knowledge of the physics of the forward model) and nd is the length of vector d*. If the prior distribution of the model is also assumed to be Gaussian, then

p(m) -- [(2rr)•mICM ]1/2

I TC•i *) ß exp [--5 (m--m*) (m-m ] (3)

Here, m* is the prior model and CM is the covariance matrix describing variance and correlation of the parameters. Taranto!a and Va!ette [1982] illustrate the important result that if the forward modeling is linear, that is, if d = Gm, and if the above likelihood and prior information are both Gaus- sian, then the posterior density of m is Gaussian. This result can be verified by performing the integration in (1) analyti- cally [see Backus, !988]. The resulting posterior pdf •(mld*) is Gaussian, and the first two moments of this pdf are given by

(m) = m* + C•tG r(GC•tG r + Cr)-l(d. _ Gm*)

CM, = C•t - C•tG r(GC•tG r + Cr)-IGCM

where (m) and CM, are expected value and covariance of the posterior pdf. These results are well known.

Unfortunately, when the forward model is nonlinear, then the integrations posed by (1) are intractable analytically and one cannot compute marginal densities and moments of the posterior distribution in terms of closed form expressions. However, the integral in (1) can be evaluated using Monte Carlo integration techniques, especially if the size of model space is small and the functional relationship defined by d = g(m) can be computed relatively easily. Let the posterior pdf be written as

1

•(mld* ) = - x p(m)L(m) (4)

For example, let

i r C L(m) = exp {-•[d* - g(m)] •l[d* - g(m)]}

be the "likelihood" function. The normalizing constant t, in (4) is defined as

• = •M L(m)p(m) dm (5)

The first two moments of •(mld*) are (1) the expected value

(m)- •a• m•(mld*) dm (6)

and (2) the covariance

CM, = ••t mm r•(mld*) dm- (m){m} r (7)

The fundamental problem posed is one of integrating each of the above three expressions (5)-(7). This can be accom- plished using the method of Monte Carlo integration and the

concept of importance sampling [e.g., Shreider, 1966, pp. 100-102]. The integral posed by (5) can be evaluated by generating a series of N random model vectors using a multivariate random number generator with p(m) as the pdf. Note from (5) that vis the expected value of L(m). Therefore an unbiased estimator of v can be written as

N

t, = • •'• L(mi) i--1

where the m i are generated out of a random population with pdf p(m). Note that the likelihood function is evaluated for each model vector. The mean and covariance can then be calculated as

N

(m k) = • • rn ikL(mi) i=1

N

Ckj = • E mi•:mijL(mi ) - <m•}{mj} i--I

The generation of random points can be terminated when a criterion of relative precision of the integral (5) is satisfied.

The maximum likelihood point can be determined by examining the set of randomly generated models for that point which maximizes (2). The maximum likelihood point of a nonlinear inverse problem may not be the same as the expected value. Because Gaussian distributions are symmet- ric, the maximum likelihood point coincides with the mean value. For an arbitrary nonlinear surface (e.g., multimodal, skewed, platykurtic) the maximum likelihood point can be far from the mean value.

The Bayesian technique used in this paper essentially samples the posterior pdf surface close to its maximum to find an average value of the model parameters. These values may be more appropriate than a single estimate. If the maximum likelihood point is reasonably close to (m), then one can place a certain amount of confidence that the posterior pdf is not significantly skewed or multimodal. Higher-order statistics such as skewness and kurtosis can also be generated by the approach, and various tests can be made to determine if the resulting pdf is near Gaussian in form.

In this study we also determine the maximum likelihood parameter estimate by using a constrained maximum likeli- hood approach [Woodbury et al., 1987]. This method essen- tially minimizes the X 2 misfit between observed and calcu- lated quantities. This scheme easily handles hard constraints, does not require partial derivatives of the objec- tive function with respect to the parameters, and can be employed to minimize complicated nonlinear systems. This method is used as an independent check on final solutions. In this way, assessment of .the effects of prior information can be examined, as the above approach does not require a prior estimate of the parameters.

The covariance of the parameter estimates is a useful end product of an inverse scheme. However, because the distri- bution for m may be non-Gaussian, the covariance of the estimated model parameters may be difficult to interpret, especially in terms of confidence intervals. In such cases the covariance of the parameters can be interpreted as the

2392 WOODBURY AND $UDICKY: INVERSION OF BORDEN TRACER DATA

covariances from an equivalent linear problem, which may or may not be relevant. In these cases the best understanding may be achieved by determining the probability of a model parameter lying within a certain range. For example,

P(a <- m -< b) = •(mld*) dm

which can be readily determined as part of the numerical integration of (5).

second-order moments of the Borden tracer plume. Averag. ing In (K) in the vertical direction over an integral scale results in a reduction of the variance of the averaged In (K) from cr•r to 0.74try. The equations for Xil(t) and X22(t) for a horizontally isotropic medium are given by Dagan [1987] and Sudicky [1986, equations 14a and 14b]. The model requires cr•r, Xh = Xx = Ay, and I = Ut, where U is the average groundwater velocity. U is approximately given by

K6'J! U-

LIKELIHOOD FUNCTIONS

if the statistical inference of a parameter set is insensitive to the assumptions about the data and the model represent- ing the data, it exhibits criteria robustness [Box and Tiao, 1973]. This sensitivity is low for large quantities of data but becomes more pronounced for small and moderate data sets. If posterior inferences are insensitive to different likelihoods or different priors, Box and Tiao [ 1973] refer to this property as inference robustness. The most common likelihood func-

tion to assume at the outset of a least squares exercise is that of (2), even though C r is often unknown, is assumed to be diagonal, and is estimated from data residuals. Tests per- formed on modeling residuals given next in our results section indicate that residuals can be reasonably assumed Gaussian.

An equally appropriate form of the likelihood function is similar to a double-exponential pdf [see Menke, 1984, p. 133]. For example,

L(m) = exp - i=l ø'i

Here, tr i refers to the average deviation of the ith data point. In the analysis used in this paper both likelihoods are used. Viewed in the context of constructing models that fit the data, maximizing different likelihoods amounts to minimiz- ing different norms, that is, L I and L 2 norms. Woodbury et al. [1987] show that if the data are free of gross errors, a difference in two constructed models based on different

norms suggests an incorrect parameterization.

INVERSION OF BORDEN AQUIFER DATA

Available Stochastic Models

Sudicky and Huyakorn [1991] reviewed the impact of spatial variations in hydraulic conductivity on groundwater flow and mass transport. Their review focused on various stochastic-analytic approaches. For the purposes of under- standing the spatial spreading of a tracer plume as a function of plume residence time, the works of Dagan [1982, 1987, !988] are particularly relevant. These works include devel- opment of both vertically averaged and full three- dimensional moment models for statistically anisotropic aquifers. All models assume steady flow in one direction. The reader should note that alternative theory such as that applied by Zhang and Neuman [1990] and Naff[1990] exists.

Sudicky [1986] and Freyberg [1986] found reasonable success in the deterministic application of theoretical formu- lae described by Dagan [ 1982] for prediction of the longitu- dinal Xl 1 and horizontal-transverse X22 ensemble average

where J• is the hydraulic gradient, n is the porosity, and K 0 is the geometric mean hydraulic conductivity.

It is important to recognize that Dagan's expressions for Xii are approximate in the sense that they reflect two- dimensional transport in a vertically integrated In (K) field, and vertical tracer movement is restricted on account of primarily horizontal groundwater flow or presumed thin "impervious" beds [e.g., Dagan, 1988, 1989a]. Current research, particularly that of Naffet al. [1989] and Rajaram and Gelhat I1991], questions the applicability of the Dagan two-dimensional model to the Borden experiment; specifi- cally, the reduction of log conductivity variance, estimated by Sudicky [1986], related to (without strong geologic evi- dence) an arbitrary choice of averaging depth.

Another issue related to the general problem of the appli- cability of stochastic solutions is ergodicity. A plume that is small relative to the log conductivity integral scales, partic- ularly at early time and if the injection volume is small, will effectively only experience a partial sampling of the ground- water velocity population. Therefore the applicability of a model that assumes ergodicity cannot be guaranteed. At later times a plume should effectively spread over enough integral scales to ensure a convergence to satisfying ergodic requirements, at least from a pragmatic point of view [Dagan, 1990b].

Woodbury and Sudicky [1991] had reasonable success in fitting the two-dimensional vertically averaged form of

,

Dagan's [1987] model to the moment data using the original In (K) geostatistical parameters of Sudicky [1986]. However, attempts at applying three-dimensional theories by Wood- bury and Sudicky [1991]. Naff et al. [1988], and Zhang and Neuman [!990] produced less than satisfactory results, particularly for the tracer spread in the transverse directions. The rather marked temporal variations in the flow direction that were observed over the course of the experiment (see Figure 2 of $udicky [ 1986]) and ignored in the analyses could account for the discrepancy in the transverse horizontal direction. Naff et al. [1989] also raised this issue of the influence of flow transients on the transverse spreading of the Borden tracer plume. Work by Rehfeldt [1988], based on approximations of the hydraulic gradient flow angle varia- tions with time at the Borden site showed an enhanced macrodispersivity that closely matched the value computed by Freyberg [1986] from the field data. Rehfeldt [1988] als0 predicted that flow transients in the horizontal transverse direction have a minor effect on longitudinal and vertical spreading.

Based on the above arguments we have adopted an approach of only using the longitudinal X l• and vertical X33 (where available) moment data. This action effectively re- duces the number of degrees of freedom available in the

WOODBURY AND SUDICKY: INVERSION OF BORDEN TRACER DATA 2393

analysis but represents a reasonable course of action to follow until such time as the flow transients issue at Borden has been resolved.

Estimation of Modeling Errors

As mentioned in a previous section, the matrix Cr is the covariance matrix that describes the combination of mea- surement errors and forward modeling errors. C r is often estimated from the data residuals and is assumed to be a diagonal matrix equal to tr,•l. This action in effect distorts model error, which can be correlated and heteroscedastic. In the inverse problem posed in this paper, C r represents a temporal correlation matrix, and when the sampling interval is constant, the structure of this matrix can be represented by a lag one autoregressive model with a single parameter [Carrera and Neuman, 1986]. Unfortunately, the Borden tracers were not sampled at regular intervals; hence C r should be specified as a known matrix. With the low number of actual data points present, estimation of the correlation matrix using a repetitive type of procedure like that of Neuman and Jacobson [!984] would be difficult and would probably not lead to significant differences from the ap- proach we have adopted here. To reduce computational effort we make the assumption that the temporal correlation is weak so that we treat Cr as a diagonal matrix. Recall that an unbiased estimator of rr3 is

N

i•• Ida- di] 2 s•=(N-M) '= Here, we adopt the following approach. First, it is acknowl- edged that the magnitude of forward modeling errors is unknown. Second, the experimental errors are estimated by computing the mean square errors between the chloride and bromide tracers for each set of moments X• and/or X33. The mean square errors of the difference in the chloride and bromide tracer plume moments for the Rajaram and Gelhar [1988] data set are computed to be 9.65 m 4 for Xli and 0.13 m 4 for X33. Third, these mean square errors are used as estimates of cr• for each .set of moments for both the maximum likelihood and Bayesian inverses. Fourth, and finally, for either method the variance of unit weight is computed:

1 rru 2 = d* 2(d*-d)'(d- )

(N - M)cr a

This variable should be statistically close to unity for a theoretically consistent model and therefore indicates that the prior estimate cr,• is acceptable. If the mean square residual errors computed from the inverse are less than the mean square errors of the tracers, then it is concluded that the model fit is more precise than experimental errors; however, if the converse is true, it indicates that the mod- eling errors are greater than experimental.

Three-Dimensional Inverse Results: XI• Moments Only

In this section the predicted second moments X• (t) are given by Dagan [1988, equation (23)]. The parameter vector m consists of X•(t = 0), tr2r, A = Ax = Ay, and X z. The groundwater velocity U is also treated as a parameter, but it

BORDEN AQUIFER RAJARAM AND GELHAR BROMIDE

/

/

/

/

/

/

/

/

/

/

/

/

/

/

,

E;• .... 1060' ' '"' '1E;•O

Fig. 3. Linear regression results for first-moment estimates, based on bromide plume moments, as estimated by Rajaram and Gelhat [1988].

has a very small prior variance so that its posterior value is almost identically the same. The reader should recall at the outset of this analysis that Woodbury and Sudicky's [1991] estimates for the various parameters (based on a geostatis- tical analysis of the In (K) data) are tr• = 0.172, An -- 5.14 m, and A z = 0.209 m for A-A'. Bromide data collected at time equal to 1038 days are included in this study; however, questions have been raised in the past as to the suitability of these data, particularly for the X22 moments. The reader is referred to Freyberg [1986], Sudicky [1988], and Rajaram and Gelhar [1991] on the nature of the plume spreading at large time. An independent estimate of the mean velocity is obtained by linear regression using the plume displacements of Rajaram and Gelhat [1991]. By constraining the first moment to have a zero value at zero time, the calculation gives a velocity equal to 0.093 + .00065 m/d and confirms that the position of the center of mass of the bromide plume is anomalous at time equals 1038 days (see Figure 3). Nevertheless, the inverse analyses provided below indicate that including the time equals 1038 days (for bromide) does not appreciably affect results.

Tables 2 and 3 summarize the results of the maximum

likelihood estimates of the model parameters. L 2 norm refers to the Gaussian likelihood, while L I indicates an exponential likelihood. Inconsistent results between norms suggest either an incorrect parameterization or gross errors [Woodbury et aI., 1987]. Table 2 is based on Freyberg's [1986] data, while Table 3 is taken from the analysis of Rajaram and Gelhat [1988, 1991]. In Table 3 some differ- ences in the bromide data are noted between the 1988 and

1991 calculations, and we have elected to use the 1991 values for the bromide tracer. Notice that in both tables there is

good agreement between the estimates of the integral scale for bromide. These results are lower than that estimated by Woodbury and Sudicky [1991] but within their confidence intervals. Notice that when comparing L• and L2 results in

2394 WOODBURY AND SUDICKY: INVERSION OF BORDEN TRACER DATA

TABLE 2. Inverse Results From Maximum Likelihood Method

for Bromide/Chloride Tracers, L2 Norm, Using Freyberg [1986] Moment Estimates

Parameter [m]o [m],.

A h 3.71 1.85 o-3 0.120' 0.175 A z 0.229 0.207 MSE(X•i) 14.67 15.11

Here [m]t,,c is the maximum likelihood estimate for the parame- ters using bromide and chloride data' rr2r is the variance, A/, is the horizontal integral scale, and A z is the vertical integral scale. MSE values refer to the mean square errors of the deviations between the observed and computed values of Xi•.

*The a priori bound reached.

TABLE 4. Inverse Results of Bayesian Approach for Bromide Tracer, L•, L 2 Norms, Using Rajaram and Gelhar [1991]

Moment Estimates

L• Norm L2 Norm

Parameter (m) SD MPL (m) SD MPL

A h !.97 0.268 2.46 1.96 0.215 1.95 try, 0.191 0.018 0.158 0.190 0.017 0.186 A z 0.144 0.156 0.354 0.190 0.155 0.459 MSE(X•) 8.81 8.37

Here (m) refers to the expected value of the parameter, M?L the maximum of the posterior distribution tr 2 is refe. rs to Ah is the horizontal integral scale, and i z is'theSerticfll• variance, .

integral scale. MSE values refer to the mean square errors of the deviations between the observed and computed values of X•l.

either Table 2 or 3, An is increased at the expense of lowering rr2r. These results clearly show the intimate link between these two parameters and possibly how difficult it is to obtain good resolution of them.

Tables 4 and 5 show Bayesian estimates of the model parameters based upon prior estimates obtained by Wood- bury and Sudicky [1991] and using the Rajaram and Gelhat [1988, 1991] moment estimates. Prior information is assumed to be of Gaussian form (equation (2)) and is developed from the geostatistical analysis of Woodbury and Sudicky [1991]. MPL in each table refers to the maximum of the posterior distribution. SD refers to square roots of the diagonals of the parameter variance-covariance matrix. Notice that the in- verse results compare favorably between different tracers. One can see that the differences in parameter estimates are lower than estimation errors. Mean square errors for the bromide tracer (8.37) are about the same as the mean square differences between the chloride and bromide tracers (8.25).

Table 6 shows both Bayesian and maximum likelihood estimates using Barry and Sposito's [1990] moment calcula- tions for the bromide tracer. One advantage of their proce- dure is that each "measured" moment has an uncertainty associated with it as a result of their interpolation procedure. Uncertainties computed by these authors are generally lower for moment calculations at early times. Results in Table 6 are based on Barry and Sposito's Figure 3, as the actual numer- ical values were not available. Mean square deviations between bromide and chloride tracers are used as common weights in a procedure identical to that described in the previous paragraphs for the other data sets. Results of this latter data set (Table 6) confirm the findings from the previous two analyses, although higher mean square errors are noted. An attempt is made to use the covariances determined by Barry and Sposito [!990] as individual vari- ances for the moment data in the inversion. This approach

TABLE 3. Inverse Results From Maximum Likelihood Method for Bromide/Chloride Tracers, L2 Norm, Using Rajaram and

Gelhat [1988, 1991] Moment Estimates

Parameter [m]t, [m]c

Ah 3.60 2.62 rr2r 0.120' 0.156 A z 0.185 0.193 MSE(X • • ) 6.96 22.79

leads to inconsistent results and poor standard deviations of unit weight. While the jacknife procedure employed by Barry and Sposito [ 1990] is beneficial to an assessment of the reliability of the individual moment calculations, the vari- ances produced from the procedure cannot account for all sources of model error (for example, the differences in moments computed from different interpolation schemes) and cannot be used to help condition the inverse. Earlier times, when calculation uncertainties are small, are precisely the times when the modeling errors (nonergodic conditions) are largest.

The parameter estimates from all data sets show good consistency between different likelihoods and different trac- ers. With the exception of the chloride data, the computed mean square errors are lower than our estimate of experi- mental uncertainty. The parameter ranges from all methods (Tables 2-5)show 1.85 < Ah < 4.04, 0.144 < A z < 0.459, and 0.120 < cr2y < 0.197, which are statistically comparable to the earlier estimates of Woodbury and Sudicky [1991]. We conclude that the model representing the data shows infer- ence robustness.

Three-Dimensional Inverse Results: Xll and X33 Moments

In this phase of the analysis the conditioning effect of including the X'33 moment data in the analysis are examined. Here, the parameter vector m consists of and A z. The data vector d* in this analysis consists of both chloride and bromide data for X ll and X33 appended into a single vector. Only the data set by Rajaram and Gelhat [1988, 1991] can be used in this phase.

X• •(t) is given by Dagan [1988, equation (23)], and X33(t) is given by Barry [1990, equation (5)] as an alternative to

TABLE 5. Inverse Results of Bayesian Approach for Chloride Tracer, L•, L2 Norms, Using Rajararn and Gelhat [1988]

Moment Estimates

L• Norm L2 Norm

Parameter (m) SD MPL (m) SD MPL

A h 2.43 0.455 2.96 2.45 0.319 3.27 rr2• 0.!85 0.019 0.141 0.182 0.019 0.145 A z 0.152 0.153 0.348 0.175 0.149 0.443 MSE(X•) !8.38 17.96

See description for Table 2. See description for Table 4.

WOODBURY AND SUDICKY: INVERSION OF BORDEN TRACER DATA 2395

TABLE 6. Inverse Results of Bayesian/Maximum Likelihood Approches for Bromide Tracer, L i, L 2 Norms, Using Barry and Sposito [1990] Moment Estimates

L • Norm L2 Norm

Parameter (m) SD MPL ML (m) SD MPL ML

A h 2.57 0.386 3.32 2.26 2.65 0.344 3.93 4.04 o'2}, 0.183 0.019 0.140 0.197 0.179 0.019 0.123 0.120' MSE(Xi•) 17.43 21.29 15.89 17.28

Here (m) refers to the expected value of the parameter, MPL refers to the maximum of the posterior distribution. ML is the maximum likelihood solution, o'2 r is the variance, and An is the horizontal integral scale. MSE values refer to the mean square errors of the deviations between the observed and computed Xl• values.

Dagan [1988, equation (37)]. The same prior parameter estimates are used as in the previous section. Results are presented in Table 7.

Figure 4 shows the expected rates of plume spreading in the longitudinal direction (X i i) based on Woodbury and Sudicky's [1991] geostatistical analysis. Also shown are the envelopes given by the approximate 95% confidence limits ((Xli) +- 2o.x,,). These confidence limits assume that the posterior variances are from a Gaussian process. Because L = Ut, the predictions are alternatively expressed as a function of the plume residence time t, and the values have been augmented by (X•(t = 0)). These predictions are based on a variogram model arrived at using the A-A' data (tr•, = 0.244, o'02 = 0.072, An = 5.14 m, and A z = 0.209 m). Also provided in these figures are field estimates of X•l along the flow direction, based on the calculations performed by Freyberg [1986], Barry and Sposito [1990], and Rajaratn and Gelhar [1988, 1991] for both the bromide and chloride tracers. The field data fall below (X•) at all levels of time, and most of the data points fall outside of the (X•) +- 2Crx,, uncertainty envelopes, indicating a possible bias in the X• prediction.

Figure 5 shows the expected rates of plume spreading in the longitudinal direction (X•) for bromide and chloride, conditioned on the inverse results presented in Table 7. Also shown are the envelopes given by {X•) --+ 2•rx, ,. Again, these confidence limits assume that the posterior variances are from a Gaussian process. As can be seen, the field data are clustered about {Xii ) at all levels of time, and essentially all the data fall within the (X• • ) +_ 2ax,, uncertainty envelopes. A reduction of uncertainty in the prediction of the ensemble second moments over the geostatistically based estimates and a correction in the apparent bias is immediately apparent.

Two-Dimensional Inverse Results

In this section we investigate the inversion of Dagan's [1987] two-dimensional model for prediction of the longitu-

TABLE 7. Inverse Results of Three-Dimensional Model Bayesian Approaches, X i I and X33 Data, for Bromide/Chloride Tracers, L2 Norm, Using

Rajaram and Gelhar Moment Estimates

Parameter (m) SD

A.• 0.252 0.112 o- r 0.182 0.020 Ah 2.07 0.272 MSE(Xil) 6.72

dinal X• second moment. The equation for Xll(t) for a horizontally isotropic medium is given by Sudicky [1986, equation (14a)]. The Dagan model is a function of In (K), o-•,, and Ax = Ay. The model also implicitly depends on an assumed vertical integration depth l, which was originally taken to be equal to the vertical integral scale A z [Sudicky, 1986]. This assumption resulted in a reduction of the vari- ance of the averaged In (K) from o.2 r to 0.74o.2r. The actual expression relating variance reduction in In (K) due to vertical averaging is given by Dagan [1989b, p. 327] as

a•,=2 - I +exp - o.r

The parameter vector m consists of Xl •(t = 0), o.•,, '•h = Ax = A s, A z, U, and the vertical averaging depth 1. The same estimates and prior pdf as in the three-dimensional analyses are used again for the bulk of the parameters, and for the vertical depth ! a boxcar prior pdf is utilized with a lower and upper bound of 0 and 3 m, respectively. This pdf reflects the level of prior uncertainty with respect to the depth parameter.

BROl•IDE / CHLORIDE TRACERS GEOSTATISTICAL PREDICTION

00-

• 4o o

/

ii ,

tl I / / / x

/

/ / _ / _• xxxxx Br. Rajaram/Geihar / / ./* •r•. ***** Br. Freybe. rg

. / it ***** Br . '/ // - *•' •...-. CI. Barry/Sppsito ! - ' Rajaram/Gelhar t///t ]• a ß a ß. Cl. Freyberg

fill[ llll II|'l'll III Ill I Ill ' IllIll I ! I El I ! Ill ll[ I I ! III I 1•"l I I' I ! III 200 4ee 600 BOO 1000 1200

TIME (DAYS) Fig. 4. Borden plume moments X ll from Rajaram and Ge!har

[!988, 1991], Freyberg [1986], and Barry and Sposito [1990]. Solid line is the mean position based on Dagan's [ 1988] three-dimensional model using geostatistical parameters developed from line A-A' and incorporating uncertainty in moment calculations. Dashed lines are the upper and lower confidence intervals at 95%.

2396 WOODBURY AND $UDICKY: INVERSION OF BORDEN TRACER DATA

BROMIDE / CHLORIDE TRACERS !•0t INVERSE PREDICT!ON

_

.

/

/ , /

E // x .

• . '••/ xxxxx 8n R•i•mm/Gelho• . x/• •***, •r: Frayberg • 20 .• • • *•** •r. Ba•/Sposito

/•• ,•,, CI. Raja•am•Gelhar

•' "1'i'"i i i i i i' i j i i'1 i i i i'1 i j i i I i i i"'1 i i i i i i i i i i i i I 1"I I I i i i'"1"1 J I I I I I I I"1"'1' I • 20• 4• 6• 8• 1 • 1 2•

Fig. 5. Borden plume moments X• from Rt•aram and Gelhar [1988, 199!], Freyberg [1986], and Barry and S•osito [1990]. Solid line is the mean position based on Dagan's [ 1988] three-dimensional model conditioned on the inverse results in this paper. Dashed lines are the upper and lower confidence intervals at 95%.

• 8o

Table 8 shows the results of the Bayesian estimation for the combined set of chloride and bromide tracer moments

calculated by Freyberg [1986]. The parameter estimates show good comparison for the variance of In (K) between the two- and three-dimensional models. However, it can be

seen that the resulting horizontal integral scale (5.6 m) is larger than that obtained by the other analyses. The integra- tion depth (0.288 m) is also shorter than the vertical integral scale (0.323 m). Note the posterior standard deviations of l, Ah, and A z are large, and the standard deviation of A h is only slightly less than its prior estimate derived from geostatis- tics. Examination of the complete variance-covariance ma- trix indicates that the length l and the vertical integral scale are highly correlated and poorly resolved. Under these circumstances moment prediction uncertainty, based on conditioning from the inverse, would not be reduced.

Table 9 shows the results of the Bayesian estimation for the combined set of chloride and bromide tracer moments

calculated by Freyberg [1986] utilizing the normal "con- strained" form of Dagan's two-dimensional model. In this simulation the vertical integral scale is not required. The parameter estimates also show good comparison with other results (see Tables 2-8), particularly for In (K) variance. The horizontal integral scale is estimated to be 4.0 m, a value

TABLE 8. Inverse Results of Unconstrained Two-Dimensional

Model Bayesian Approaches for Bromide/Chloride Tracers, L 2 Norm, Using Freyberg [1986] Moment Estimates

Parameter

Az l

MSE(Xll)

<m> SD

5.60 1.03

0.170 0.017 0.323 0.128 0.288 O. 113 5.68

TABLE 9. Inverse Results of Constrained Two-Dimensional Model Bayesian Approaches for Bromide/Chloride Tracers,

L2 Norm, Using Freyberg [1986] Moment Estimates

Parameter (m) SD

Ah 3.98 0.059 cr2• 0.176 0.004 MSE(X•) 6.19

which is very close to the maximum likelihood results for Freyberg's data and the three-dimensional model (see Table 2).

We observe that the Dagan two-dimensional model pre. diets a very similar In (K) variance and horizontal integral scale to the three-dimensional model. However, attempts to resolve the implicit vertical integral scale and integration depth in Dagan's model failed, and it is noted that these model parameters cannot be properly resolved without "stronger" prior information.

Discussion

It is instructive to compare both the mean positions of the ensemble second moments (X•l(t)) of the three-dimensional model and sizes of the error envelopes given in Figures 4 and 5. It can be seen that the unconditioned prediction of the X•i moments based on geostatistica! estimates of the !n (K) statistical parameters leads to an apparent bias compared to the field results and relatively large uncertainty. However, the prior pdf of the model parameters (based on geostatis- tics) has a sufficiently large variance structure (see Figure 4) so as to statistically encompass this apparent bias. For example, the probability that the model parameters deter- mined from the L2 inverse of the bromide and chloride data set (Table 7) are randomly drawn from the prior pdf is about 5%. While one might suspect that this result shows an upward bias in the theory, it is not significant at the 95% confidence limit.

Most of the apparent bias could be attributed to a shorter horizontal integral scale than what was estimated by geosta- tistics (5.1 m). However, Robin et al. [1991] estimate an even longer horizontal integral scale (7.5 m). The broad ranges in these statistical parameters illustrate the problems associ- ated with integral scale determination from sampled data.

Another possibility for the apparent bias is that the Dagan [1988] three-dimensional model does in fact overpredict the Xii moments. Dasinger and Gelhar [!991] compared large- scale numerical simulations of three-dimensional In (K) fields to the moment solutions by Dagan [1988]. Although only a few large three-dimensional realizations were per- formed, the results seemed to suggest an upward bias in the theoretical moments. Work in this area is continuing and will be of great interest to future model validation attempts.

Figures 4 and 5 do suggest that moment prediction bias and uncertainty are substantially reduced when both tracer moment data and prior estimates of the In (K) geostatistical parameters are incorporated. It is also evident that combin- ing field estimates of plume moment data with prior esti- mates of the In (K) geostatistics leads to essentially the same plume moment predictions and error envelopes when either the bromide (Table 4) or chloride (Table 5) field data are used independently in the inversion process, thus providing a reassuring measure of robustness.

WOODBURY AND SUDICKY: INVERSION OF BORDEN TRACER DATA 2397

The above discussion leads us to believe that carefully monitored tracer tests performed in the field can be inter- preted to provide reliable estimates of the geostatistical parameters such as In (K) variance and integral scales, along with a measure of the uncertainty in the values of these parameters. Such a use for tracer studies was suggested more than a decade ago by Matheron and de Marsily [1980]. The degree of reliability in the values of the geostatistical parame- ters inferred by fitting theoretical formulae that predicts ensem- ble behavior to observed plume moment data will, however, depend on a theoretically validated model, the travel distance traversed by the plume and its size relative to the In (K) integral scales, and whether or not prior hydraulic conductivity infor- mation is at hand, as demonstrated here.

Perhaps as an alternative to costly, long-term natural gradient tests such as the one performed at Borden, more rapidly performed forced-injection tests offer a workable alternative. This type of test could be similar to the two-well injection-withdrawal test, where a slug of tracer is released into an induced steady flow field. The interpretation of such a test would first necessitate the derivation of theoretical formulae describing ensemble plume moment evolution in nonuniform mean flow fields, which apparently do not presently exist. Some prior hydraulic conductivity informa- tion from cores, "soft" geological information, or perhaps the use of indicator geostatistics for hydrostratigraphic in- terpretation of borehole information [Johnson and Dreiss, 1989] could be used conjunctively for proper selection of well spacings and for providing constraints on the values of the In (K) geostatistical parameters. Further conditioning during the inversion of the tracer moment data could also be provided by taking advantage of hydraulic head information, particularly in regions where the induced hydraulic gradients and head fluctuations are likely to be large. As a first step in evaluating the utility of forced-advection tests to estimate the geostatistical properties of transport parameters in sand and gravel aquifers, detailed numerical experimentation will be useful, but field experimentation will be crucial.

CONCLUSIONS

The primary purpose of this study is to invert Dagan's two- and three-dimensional stochastic models using Frey- berg's [1986], Rajaram and Gelhar's [1988, 1991], and Barry and Sposito's [1990] moment data from the Borden experi- ment to examine (1) the validity of the Dagan [1988] three- dimensional and Dagan [1982] two-dimensional models and (2) the reduction in uncertainty of the spatial moments over nonconditioned estimates. Alternative theory such as that applied by Zhang and Neuman [1990] and Naff[1990] exists but would not lead to results significantly different from those presented here from the perspective of optimal In (K) geostatistical parameter values.

A direct application of Bayesian statistical inference, in conjunction with Monte Carlo integration, is used to produce posterior probability density functions for the In (K) param- eters. This approach allows for usage of different likelihood functions and prior information in any form as long as a prior pdf can be defined. The resulting integrations provide estimates of means, covariances, and higher moments of the parameters, if desired. The approach also permits numerical computation of cumulative probabilities within parameter ranges.

The parameter estimates from all data sets show consis-

tency between different likelihoods and different tracers. Little difference is noted among any of the researcher's moment computations. In al! cases, with the exception of the chloride data (L•), the computed mean square errors are lower than estimates of experimental uncertainty. The pa- rameter ranges from all methods (Tables 2-5) indicate 1.85 < 3, 4.04, 0.144 < 3, z < 0.459, and 0.120 < o-•, < 0.197, which are statistically within the ranges computed earlier by Woodbury and Sudicky [1991] and Robin et al. [1991]. We observe that the Dagan two-dimensional model predicts a very similar In (K) variance and horizontal integral scale to the three-dimensional model; however, attempts to resolve the implicit vertical integral scale and integration depth in Dagan's model failed, and it is noted that these model parameters cannot be properly resolved without "stronger" prior information.

Examination of the three-dimensional inverse results

shows that the unconditioned prediction of the X • l moments based on geostatistical estimates of the In (K) statistical parameters leads to apparent bias compared to the field results and relatively large uncertainty. However, the prior pdf of the model parameters (based on geostatistics) has a sufficiently large variance structure as to statistically encom- pass this apparent bias. Most of the bias could be attributed to a shorter horizontal integral scale than what was esti- mated by geostatistics. The broad ranges in these statistical parameters illustrate the problems associated with integral scale determination from sampled data.

The satisfactory results obtained from the inversion pro- cedure suggest that carefully monitored tracer tests per- formed in the field can be interpreted to provide reliable estimates of the geostatistical parameters such as In (K) variance and integral scales, along with a measure of the uncertainty in the values of these parameters. The degree of reliability in the values of the geostatistical parameters so obtained will, however, depend on the travel distance tra- versed by the plume and its size relative to the In (K) integral scales, for reasons outlined by Dagan [ 1990b], and whether or not prior hydraulic conductivity information is available.

Acknowledgments. This research was funded by grants from the Natural Sciences and Engineering Research Council of Canada (NSERC) to A.D. Woodbury and E. A. Sudicky. The authors have benefited very much from discussions with Gedeon Dagan, Lynn Gelhar, and Shlomo Neuman.

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E. A. Sudicky, Waterloo Centre for Groundwater Research, University of Waterloo, Waterloo, Ontario, Canada N2L 3131.

A.D. Woodbury, Department of Geological Engineering, Univer- sity of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2.

(Received March 22, 1991; revised April 20, 1992;

accepted April 28, 1992.)