Bayesian methodology to stochastic capture zone determination: Conditioning on transmissivity measurements

Transcription

1 WATER RESOURCES RESEARCH, VOL. 38, NO. 9, 1164, doi: /2001wr000950, 2002 Bayesian methodology to stochastic capture zone determination: Conditioning on transmissivity measurements L. Feyen, 1 P. J. Ribeiro Jr., 2,3 F. De Smedt, 1 and P. J. Diggle 3 Received 18 September 2001; revised 20 February 2002; accepted 27 February 2002; published 6 September [1] A methodology to determine the uncertainty associated with the delineation of well capture zones in heterogeneous aquifers is presented. The log transmissivity field is modeled as a random space function and the Bayesian paradigm accounts for the uncertainty that stems from the imperfect knowledge about the parameters of the stochastic model. Unknown parameters are treated as random quantities and characterized by a prior probability distribution. Log transmissivity measurements are incorporated into Bayes theorem, updating the prior distribution and yielding posterior estimates of the mean value and the covariance parameters of the log transmissivity. Conditional simulations of the log transmissivity field are generated using samples from the posterior distribution of the parameters, yielding samples from the predictive distribution of the log transmissivity field. The uncertainty in the model parameters is propagated to the predictive uncertainty of the capture zone by solving numerically the groundwater flow equation, followed by a semianalytical particle-tracking algorithm. The method is applied to a set of hypothetical flow fields for various sampling densities and assuming different levels of parameter uncertainty. Simulation results for all the sampling densities show no univocal relation between the predictive uncertainty of the capture zones and the level of parameter uncertainty. However, in general, the predictive uncertainty increases when parameter uncertainty is taken into account. INDEX TERMS: 1829 Hydrology: Groundwater hydrology; 1832 Hydrology: Groundwater transport; 1869 Hydrology: Stochastic processes; KEYWORDS: groundwater, capture zone, spatial stochastic approach, Bayesian Citation: Feyen, L., P. J. Ribeiro Jr., F. De Smedt, and P. J. Diggle, Bayesian methodology to stochastic capture zone determination: Conditioning on transmissivity measurements, Water Resour. Res., 38(9), 1164, doi: /2001wr000950, Department of Hydrology and Hydraulic Engineering, Free University Brussels, Brussels, Belgium. 2 Departamento de Estatística, Universidade Federal do Paraná, Paraná, Brazil. 3 Department of Mathematics and Statistics, Lancaster University, Lancaster, UK. Copyright 2002 by the American Geophysical Union /02/2001WR Introduction [2] Groundwater is an indispensable resource in agriculture, industry, and public water supply. Groundwater resources are vulnerable to contamination and protection of the quality of groundwater supplies is of major concern. Groundwater is easily gained from water-bearing aquifers using abstraction wells. A time-related capture zone of an abstraction well delineates the area from which water is captured by the well within the specified time interval. Delineation of well capture zones is therefore a necessary step in the protection of water supplies from accidental contamination and provides a proactive approach toward the protection of groundwater supplies that is far more costeffective than postcontamination aquifer cleanup procedures. [3] The location of the capture zone depends on the hydrogeological conditions of the system and the properties of the well. The latter are usually known but large uncertainty often exists about the system geometry, the dominating flow processes, the boundary conditions and the hydraulic properties of the aquifer. In this paper we only consider uncertainty in the hydraulic properties of the system, and in particular we focus on the transmissivity T(x) (L 2 T 1 ) since it is considered to be one of the most influential parameters in the delineation of capture zones. [4] Aquifers, as a rule, are highly heterogeneous and the values of transmissivity within an aquifer can vary substantially over very short distances. Moreover, our ability to measure directly and in detail the spatial distributions of the hydraulic properties is inherently limited, either economically or physically. The inability to describe the spatial distribution of the properties deterministically has led to the use of the theory of random space functions (RSF) in a attempt to characterize the spatial variability and uncertainty of groundwater flow parameters, as first outlined by Delhomme [1978]. The technique was first applied to the capture zone problem by Varljen and Shafer [1991] using a Monte Carlo approach. Since then the stochastic Monte Carlo (MC) approach has been applied by several authors to quantify the predictive uncertainty of capture zones [Franzetti and Guadagnini, 1996; Guadagnini and Franzetti, 1999; van Leeuwen et al., 1998; Riva et al., 1999; and van Leeuwen et al., 2000]. [5] The MC approach is based on the generation of equiprobable realizations of the transmissivity field which are subsequently used to solve the groundwater flow and transport equation, resulting in a predictive distribution of the capture zone or any other variable of interest. Typically

2 3-2 FEYEN ET AL.: BAYESIAN METHODOLOGY TO CAPTURE ZONE DELINEATION this method, as well as other geostatistical methods, does not take into account the uncertainty in parameter estimates when generating realizations of the stochastic field. Instead, the model parameters used to generate the simulations are often estimated from a limited number of data and plugged into the prediction equations, as if they were the true values. We acknowledge that in practice, the prediction uncertainty has at least two components: (1) the inherent uncertainty in the true value of the random variable when the stochastic mechanism that generates the data is known; (2) the additional uncertainty when the generating mechanism is unknown. The second is often ignored in conventional geostatistics and in particular in the applications of the MC approach to capture zone delineation mentioned above. [6] In this paper we shall describe a Bayesian approach which accounts for the parameter uncertainty in the capture zone delineation problem. This approach begins with the specification of a stochastic model for the log transmissivity field and the measurements. Once the model has been specified, we use general statistical principles to incorporate the log transmissivity measurements and to make predictions. In particular, we use the Bayesian paradigm as by Kitanidis [1986] and Handcock and Stein [1993], so that the predictive distribution of the capture zone makes proper allowance for the uncertainty inherent to the estimation of model parameter values from the data. The availability of feasible Monte Carlo sampling algorithms for a very wide range of statistical models has been an essential step in the development of practical Bayesian inference. A different method to account for the parameter uncertainty in the delineation of capture zones was presented by Feyen et al. [2001]. In contrast to what is presented here, these authors focused on conditioning on head observations within the generalized likelihood uncertainty estimation framework and assumed that no transmissivity measurements were available. 2. Bayesian Methodology to Capture Zone Delineation [7] Bayesian inference starts with the formulation of a model that is thought to be adequate to describe the situation of interest. We use the notation [ ] for the distribution of the quantity within the square brackets, the notation p( ) for the probability of the quantity within the brackets, and a vertical bar to indicate conditioning. Furthermore, we use the term posterior in reference to the conditional distribution [Q Y] of the model parameters resulting from the Bayesian theorem. The term predictive is used to refer to the conditional distribution of the log transmissivity [S(x) Y] or capture zone [CAP(x, t) Y], integrated over the unknown parameters. Based on findings of Freeze [1975] and Hoeksema and Kitanidis [1985], it is reasonable to assume that S(x) = log T(x) is a Gaussian process, where x =(x, y) isthe 2-dimensional spatial coordinate. Our stochastic model for S(x) is that the log transmissivity field is a realization of a stationary Gaussian RSF with mean m, variance s 2 and correlation function r(x 1 x 2 ) = Corr{S(x 1 ), S(x 2 )}. In what follows, we shall assume that S(x) is isotropic, so that r(x 1 x 2 )=r(u) where u is the Euclidean distance between the spatial locations x 1 and x 2. Furthermore, we shall specify r(.) to be an exponential correlation function defined as r(u; j) = exp( u/j), where j is the integral scale of the process. [8] In order to apply this model for prediction, the mean and covariance parameters need to be inferred from measurements. In hydrological applications the number of measurements is often limited. This introduces uncertainty in the estimation of the parameters that should be accounted for. The uncertainty associated with parameter estimation based on the sample variogram and the selection of an appropriate model has been addressed by several authors [e.g., Russo and Jury, 1987; Shafer and Varljen, 1990; Zheng and Silliman, 2000]. However, in geostatistical applications to hydrogeology, more specific in the delineation of capture zones, this source of uncertainty is still often neglected. The methodology adopted here acknowledges uncertainty in the three model parameters denoted by Q =(m, s 2, j). [9] Within the Bayesian approach the parameters of the specified model are considered to be random quantities characterized by a distribution. A prior distribution is specified for the vector of unknown parameters of the stochastic model. This distribution is meant to represent the belief about the parameters before the data have been considered. After observing the data Y, Bayes rule below (equation (1)) is applied to obtain the posterior distribution [Q Y ] for the unknown parameters, which combines the prior information and the information in the data expressed by the likelihood function L[Q Y ] [Y Q]. ½QjY Š / ½QŠ½YjQŠ ð1þ The likelihood is a function of the parameter vector Q and, for the model assumed here, has an expression given by the equation of a multivariate normal distribution. The posterior distribution for the parameters reflects the uncertainty about them after the data have been observed and does not necessarily define a standard probability distribution. Therefore inference by numerical simulation using a Monte Carlo sampling algorithm as in the work by Diggle and Ribeiro [2002] and Diggle et al. [2002] is adopted. [10] Spatial prediction is based on the predictive distribution of the log transmissivity field, which is obtained by Z ½SðxÞjYŠ ¼ ½SðxÞjY; QŠ½QjY ŠdQ: ð2þ Samples from this distribution are conditional realizations of the log transmissivity field and can be obtained through conditioning by kriging [Chilès and Delfiner, 1999, p ], using the parameter sets sampled from the posterior distribution. From equation (2) it can be seen that the Bayesian predictive distribution incorporates the parameter uncertainty by averaging the conditional distribution [S(x) Y, Q] over the parameter space, with weights given by the posterior distribution of the model parameters [Q Y ]. Typically the effect of the averaging in equation (2) is that the predictions tend to be more conservative, in the sense that the variance of [S(x) Y] will usually be larger than that of the distribution SðxÞjY; ˆQ, obtained by plugging an estimate ˆQ of Q into the classical predictive distribution [S(x) Y, Q]. [11] Each transmissivity field obtained by simulating from the predictive distribution (equation (2)), is subse-

3 FEYEN ET AL.: BAYESIAN METHODOLOGY TO CAPTURE ZONE DELINEATION 3-3 quently used to solve the steady state groundwater Tðx; @y Tðx; yþ@h þ R ¼ 0; where h is the hydraulic head (L), and R a general sink/ source term (L T 1 ), which here is limited to pumping. This provides the corresponding realization of the head field. Then, a semianalytical particle-tracking analysis is performed for the flow domain to determine the capture zone (CAP) associated with the head field. The procedure is repeated for every simulation of the transmissivity field. Finally, statistical processing of the ensemble of capture zones results in the predictive distribution of the capture zone [CAP(x, t) Y ], where t is the time coordinate. This distribution is conditioned on the data Y only and thus accounts for both the natural variability of the transmissivity field and the parameter uncertainty. Figure 1. Plan view of the hypothetical field (not to scale). 3. Application [12] When performing an uncertainty analysis, it is useful to have a reference field for which the true conditions are known. Therefore to represent the reality, a hypothetical flow field was constructed numerically [see Feyen et al., 2001]. We adopt dimensionless space (x 0, y 0 ) and time (t 0 ) coordinates resulting from the following transformations [Bear and Jacobs, 1965]: x 0 ¼ 2pq 0 Q x; y0 ¼ 2pq 0 Q y; t0 ¼ 2pq2 0 nq t; where Q (L 2 T 1 ) is the extraction rate per unit thickness of aquifer, q 0 = T G j 0 /D (L T 1 ) is the Darcy background flow velocity, T G is the geometric mean transmissivity, D is the aquifer thickness, j 0 is the background hydraulic gradient, and n is the effective aquifer porosity. The integral scale of the process j is expressed in dimensionless terms and is given by j 0 =2pq 0 j/q. [13] The flow domain is shown in Figure 1. The abstraction well is located at the origin (x 0, y 0 ) = (0, 0) of a Cartesian coordinate system. Neumann conditions are imposed at the north ( y 0 = 119; x 0 = 117,126) and south boundaries ( y 0 = 119; x 0 = 117,126), with the specified flux being zero. Dirichlet conditions are specified at the east (x 0 = 117; y 0 = 119,119) and west (x 0 = 126; y 0 = 119,119) sides of the domain, inducing a mean background gradient. The stochastic transmissivity fields are generated for the central part of the domain ( 6 <x 0 < 15; 8<y 0 < 8), using a regular grid with spacing x 0 = 0.1. The surrounding part of the domain consists of 3 rows with increasing spacing (x 0 =10 0,10 1 and 10 2 ) and one additional row (x 0 = 0.1) to impose the boundary conditions. This aims to minimize the interference of the boundaries on inferences about the probabilistic capture zones. The transmissivity in this part of the flow domain is equal to the geometric mean transmissivity T G of the central part. A discussion on the effect of grid discretization and boundary conditions on the convergence of the numerical solution is given by Franzetti and Guadagnini [1996]. [14] In order to get more general results, the analysis is performed for a set of 50 hypothetical transmissivity fields, ð4þ all of them characterized by the parameter vector Q =(m, s 2, j 0 ) = (1.5, 1.0, 1.5). The hypothetical fields are generated with the sequential Gaussian simulation algorithm of GSLIB [Deutsch and Journel, 1998] using 50 randomly sampled seed numbers. For each hypothetical field we consider 5 sets of transmissivity measurements with increasing sampling density and positioned in a regular pattern within the area of interest ( 3<x 0 < 15; 8<y 0 < 8). These sets of measurements are defined such that the smaller sets are subsets of the larger ones, aiming to circumvent the effect of varying measurement locations between the sampling densities. The numbers of measurements considered here are 4, 13, 25, 41 and 81. The measurements update the prior distribution for the model parameters, yielding the posterior distribution from which we sample parameter sets and generate conditional simulations, i.e. realizations of the transmissivity fields which honor the data at the measurement locations. We have used the package geor [Ribeiro and Diggle, 2001] to update the priors and sample from the posterior distribution of the model parameters. [15] After solving the flow equation for every transmissivity field a particle-tracking analysis is performed for the inner area ( 3 <x 0 < 12; 5 <y 0 < 5) of the central part, by releasing a particle at each grid cell and using forward tracking. For every realization, the isochrone (x 0, y 0, t 1 0 )is defined as the boundary of the capture zone CAP(x 0, y 0, t 1 0 ). The isochrone encloses all the starting locations (x 0, y 0 )of the particles such that t 0 t 1 0, where t 0 is the travel time of a particle toward the well. Finally, statistical processing of the ensemble of capture zones results in the numerical approximation of the predictive capture zone distribution [CAP(x 0, y 0, t 1 0 ) Y ], defined by: CAP x 0 1 ; y0 1 ; 1 t0 1 jy ¼ m X m i¼1 I x 0 1 ; y0 1 ; t0 1 jy i ; where m is the number of simulated transmissitivity fields. The terms inside the summation on the right hand side of equation (5) equal one if the particle released in the point (x 1 0, y 1 0 ) is captured by the well within the time interval t 0, and zero otherwise. This distribution defines at a point (x 1 0, y 1 0 ), for a given time t 1 0, the probability p(cap(x 1 0, y 1 0, t 1 0 ) Y ) ð5þ

4 3-4 FEYEN ET AL.: BAYESIAN METHODOLOGY TO CAPTURE ZONE DELINEATION that an inert particle released from this point will reach the well within the specified time span t 1 0. In the analysis presented here, the capture zones are determined for 8 time intervals, from t 0 = 0.5 to t 0 = 4.0 with steps of 0.5. The zone of uncertainty is defined as the area where 0 < p(cap(x 0, y 0, t 0 ) Y) < 1, and the 95% uncertainty interval is defined as the area enclosed by the 2.5 and 97.5 percentiles. The median 0.5 (x 0, y 0, t 0 ) of the predictive distribution is the line for which p(cap(x 0, y 0, t 0 ) Y ) = 0.5. [16] In order to quantify the effect of conditioning and parameter uncertainty on the predictive distribution of capture zones, we adopt the following two measures of performance defined by van Leeuwen et al. [2000]: Z w s ¼ f s ðx 0 ; t 0 Þdx 0 ; x 0 where f s ðx 0 ; t 0 Þ ¼ 1 if 0:025 < p ð CAP ð x0 ; t 0 ÞjYÞ < 0:975 0 otherwise; Z w m ¼ f m ðx 0 ; t 0 Þdx 0 ; x 0 where f m ðx 0 ; t 0 Þ ¼ 8 < 1 ifðcapðx 0 ; t 0 ÞjYÞ > 0:5 and HYPOðx 0 ; t 0 Þ¼0 1 ifðcapðx 0 ; t 0 ÞjYÞ < 0:5 and HYPOðx 0 ; t 0 Þ¼1 : 0 otherwise; ð6þ where HYPO(x 0,t 0 ) is the true value associated with the hypothetical capture zone, which is either one or zero. The first measure quantifies the spatial extent of the 95% uncertainty zone for a given time interval t 0 and represents the spread of the predictive uncertainty. The second performance measure reflects the proximity of 0.5 (x 0, y 0, t 0 ) to the true hypothetical capture zone. In the remainder of the paper we report the averaged values hw s i and hw m i over the 50 fields, and their respective variance, s 2 w s and s 2 w m. 4. Analysis [17] The analysis presented here was performed for different levels of uncertainty by progressively increasing the number of uncertain parameters in order to evaluate the effect of each one individually. Since the vector parameter Q = (m, s 2, j 0 ) consists of three parameters, we have considered 4 different situations, ranging from all parameters assumed to be known to all unknown. Hereafter we use the subscript ( * ) to indicate when a parameter is assumed to be known and fixed to the corresponding value in the parameter vector (m *, s * 2, j * 0 ) = (1.5, 1.0, 1.5) used to generate the hypothetical fields Specification of Priors [18] Within the Bayesian framework a prior distribution is specified for the random parameters. The following independent priors have been used for the cases where the respective parameters are unknown: (1) for the mean parameter m a flat prior, corresponding to a conjugate Gaussian prior with arbitrarily large variance; (2) for the variance a prior proportional to 1/s 2, which corresponds to a flat prior for log(s 2 ); (3) for the dimensionless correlation parameter a reciprocal discrete prior between 0 and 10, with the upper limit chosen in order to limit effects of ergodicity. The choice of priors is inherently subjective and the main objection to Bayesian inference is that the conclusions will depend on the specific choice of a prior distribution. The priors defined here can be interpreted as expressions of ignorance, as we assumed that no information about the hydrogeological parameters was present before the data are collected. Using these priors, we expect that the information in the data, expressed by the likelihood function, should dominate the form of the resulting posterior distribution. However, in real applications some knowledge about the process and/or model parameters may be available and informative priors can be used to express this a priori information Natural Variability [19] When all parameters are assumed to be known, the capture zone predictive distribution is given by [CAP(x 0, t 0 ) Y, m *, s * 2, j * 0 ], i.e., conditioned on the data and the fixed values for the parameters. In their work, Franzetti and Guadagnini [1996], Guadagnini and Franzetti [1999], and van Leeuwen et al. [1998, 2000] sample from this distribution to estimate the predictive uncertainty of capture zones, considering various degrees of domain heterogeneity. However, this reflects only the uncertainty associated with the lack of ability to predict exactly the transmissivity at unsampled locations. This uncertainty can be seen as the basic uncertainty, since the parameters of the stochastic model are assumed to be perfectly known. Therefore this predictive distribution only reflects the uncertainty resulting from the kriging variance. This basic uncertainty is not necessarily the minimum uncertainty, as the variance of the predictive distribution [CAP(x 0, t 0 ) Y, m *, s * 2, j * 0 ] is not necessarily smaller than the equivalent variance for the situation where one or more parameters are unknown. Conceptually it reflects the minimal uncertainty, but in absolute terms this is not always true, depending on the data and prior distributions assumed, as will be shown further herein Uncertainty in the Mean Parameter [20] Consider the mean parameter m to be unknown while the covariance parameters are fixed at their true values, i.e., Q =(m, s * 2, j * 0 ). Since the likelihood function L[m Y, s * 2, j * 0 ]is given by the equation of a normal distribution and the assumed prior is flat, the posterior distribution for m is also a Gaussian distribution. The predictive distribution for the log transmissivity is obtained by averaging over the unknown parameter m: h i Z Sðx 0 ÞjY; s 2 * ; j0 * ¼ Z ¼ h Sðx 0 Þ; mjy; s 2 * ; j0 * h Sðx 0 ÞjY; m; s 2 * ; j0 * i dm i ½mjYŠdm: ð7þ The first probability distribution inside the last integral is the conditional distribution with m assumed known. Under the assumed model this distribution is Gaussian with mean and variance given by the ordinary kriging estimator [Journel and Huijbregts, 1978]. This result establishes the basic relation between Bayesian and kriging estimators as

5 FEYEN ET AL.: BAYESIAN METHODOLOGY TO CAPTURE ZONE DELINEATION 3-5 discussed by Kitanidis [1986] and Omre and Halvorsen [1989]. The second term is the posterior distribution for m which is also a normal distribution. Therefore the term inside the integral corresponds to a bivariate Gaussian density, and it follows that the resulting predictive distribution is also Gaussian. Samples from the capture zone predictive distribution [CAP(x 0, t 0 ) Y, s * 2, j * 0 ] are obtained numerically by transforming the log transmissivity fields sampled from the predictive distribution, where the transformation is given by the groundwater flow model and particle-tracking algorithm. The resulting capture zone predictive distribution reflects both the natural variability and the uncertainty in estimating the mean parameter from the measurements Uncertainty in the Mean and Variance Parameter [21] Consider now uncertainty in the mean and sill parameters, i.e., Q =(m, s 2, j * ). The posterior is a normal scaled inverse c 2 distribution given by the product of a Gaussian and a scaled inverse c 2 density function [see Gelman et al., 1995, Appendix A]. The predictive distribution for S(x 0 ) is obtained by h i ZZ h i Sðx 0 ÞjY; j 0 * ¼ Sðx 0 Þ; m; s 2 jy; j 0 * dm ds 2 ZZ h i h i ¼ Sðx 0 Þ; mjy; s 2 ; j 0 * dm s 2 jy; j 0 * ds 2 Z h ih i ¼ Sðx 0 ÞjY; s 2 ; j 0 * s 2 jy; s 0 * ds 2 : The first term inside the last integral is the predictive distribution given by equation (7) and the second is the posterior for s 2. For our choice of priors, the predictive distribution is a multivariate t distribution. The capture zone predictive distribution [CAP (x 0, t 0 ) Y, j * 0 ] now reflects the natural variability and the uncertainty in estimating the mean and sill from the measurements Uncertainty in All Model Parameters [22] For the case where all the parameters of the stochastic model are unknown, it is convenient to factor the posterior distribution as follows: s 2 jy; j 0 m; s 2 ; j 0 jy ¼ mjy; s 2 ; j 0 ð8þ ½ j 0 jyš; ð9þ where [m Y, s 2, j 0 ] is Gaussian, [s 2 Y, j 0 ] follows a scaled inverse c 2 distribution and [j 0 Y] does not follow any standard probability distribution but is such that: j 0 ½ ½ jyš / m; s2 ; j 0 Š½Yjm; s 2 ; j 0 Š ½mjY; s 2 ; j 0 Š½s 2 jy; j 0 Š : ð10þ To circumvent this we adopt inference through Monte Carlo sampling [Tanner, 1996] and predictions are based on the resulting empirical distributions. The proposed algorithm works as follows [Diggle and Ribeiro, 2001]: (1) discretize the prior for the integral scale over the chosen interval; (2) use equation (10) to compute the posterior distribution over this discrete set; (3) sample from the resulting distribution; (4) attach the sampled value to [m, s 2 Y, j 0 ], which is a normal scaled inverse c 2 distribution, and sample from this distribution; (5) repeat steps 3 and 4 as many times as required to obtain a stable estimation of the target distribution. The resulting triplets (m, s 2, j 0 ) are samples from the posterior distribution of the model parameters. [23] The predictive distribution for S(x 0 ) is obtained by ZZZ ½Sðx 0 ÞjYŠ ¼ Sðx 0 Þ; m; s 2 ; j 0 jy dm ds 2 dj 0 ZZZ ¼ Sðx 0 Þ; m; s 2 jy; j 0 dm ds 2 ½j 0 jyšdj 0 Z ¼ ½Sðx 0 ÞjY; j 0 Š½j 0 jyšdj 0 ; ð11þ where the first term in the last integral is the predictive distribution given by equation (8) and the second term the marginal posterior distribution of the integral scale j 0 given by equation (10). Finally, the samples from this distribution are transformed, as described in section 4.3, to obtain the capture zone predictive distribution [CAP(x 0, t 0 ) Y ], which reflects the total uncertainty in the prediction of the capture zones, under the assumed stochastic model. 5. Results and Discussion [24] Samples from the capture zone predictive distribution were generated for the 4 uncertainty scenarios described in the previous section. Since the variability among generated transmissivity fields is larger when parameter uncertainty is accounted for, the number of samples from the posterior and predictive distributions increased with the number of unknown parameters, ranging from m = 1000 for Q =(m *, s * 2, j * 0 )tom = 5000 for Q =(m, s 2, j 0 ). This strategy ensured stable estimates of the predictive capture zone distribution for all scenarios considered and resulted in a total of simulations (5 data sets 50 fields ( ) realizations). [25] Figure 2 shows the predictive distribution [CAP(x 0, t 0 = 3.0) Y ] for one selected reference field and for 13 (Figure 2a) and 81 (Figure 2b) transmissivity measurements. The light line indicates the location of the true capture zone for this reference field. The comparison of the contours on both panels shows that for an increasing number of T measurements the zone of uncertainty shrinks and the shape of the capture zone predictive distribution approaches the reference capture zone. Van Leeuwen et al. [2000] report similar findings but the reduction in variance they observed resulted only from the increasing number of conditioning points for prediction and parameter uncertainty was not taken into account. Here, aside from the effect of the kriging variance, the reduction in the predictive uncertainty is also explained by the reduction of the variance of the posterior distribution [Q Y ] in particular when more data are available. Thus incorporating more data in the analysis tends to decrease both the effects of parameter uncertainty and natural spatial variability. [26] The reduction in the spread of the predictive uncertainty and the better correspondence between 0.5 and the true capture zone as more data are included can also be observed in Figure 3. In Figure 3, hw s i and hw m i are plotted against the number of measurements for the scenario where all the parameters are unknown. Both trends are initially

6 3-6 FEYEN ET AL.: BAYESIAN METHODOLOGY TO CAPTURE ZONE DELINEATION Figure 2. s 2, j 0 ). Capture zone predictive distribution for t 0 = 3; (a) 13 T and (b) 81 T measurements for Q =(m, more pronounced and decline as more data are included. This can be explained by the screening effect due to partial data redundancy [van Leeuwen et al., 2000] combined with a decline in the rate of reduction of the posterior distribution variance when more data are available. Figure 3 also shows that the predictive uncertainty expands with increasing time, as the length of the particle pathways toward the well become longer and more heterogeneity is encountered by the particles. The patterns in Figures 2 and 3 were observed for all reference fields and all degrees of parameter uncertainty. [27] Figure 4 shows the effect of the degree of parameter uncertainty on the spread of the capture zone predictive distribution. Graphs of hw s i and s 2 w s versus the degree of uncertainty are presented for the various sampling densities. The numbers in the x axis correspond to the degrees of uncertainty as described in section 4. Comparison of the plots for different numbers of log T measurements shows a reduction of hw s i and s 2 w s with an increasing number of measurements. As for hw s i in Figure 3, the reduction in s 2 w s is initially more pronounced and decays as more measurements are introduced. [28] The values of hw s i and s 2 w s in situation 1, when all parameters are known, reflect the spread of the predictive uncertainty and its respective variance resulting from the natural variability of the field. Incorporating the uncertainty in m (situation 2) results in higher values for hw s i and s 2 w s for all sampling densities. Fields generated in this scenario are characterized by the same variability but a different geometric mean transmissivity. Fields with a low geometric mean have smaller capture zones than the ones with a higher sampled value for the mean. This leads to the expansion of the 95% credibility interval for most of the reference fields and thus higher values for hw s i. [29] Moving to situation 3, further in the x axis, uncertainty in s 2 is also taken into account. In comparison to situation 2, we observe a reduction in the spread of the predictive distribution and a growth of the variance of the spread. This is valid for all sampling densities. The reduction of hw s i is due to the asymmetry of the marginal posterior distribution of the sill parameter. This effect is more pronounced for the small data sets. Many fields with little variability are generated resulting in less spread of the capture zones and Figure 3. Measures of conditioning performance (a) w s and (b) w m versus the number of T measurements for Q =(m, s 2, j 0 ).

7 FEYEN ET AL.: BAYESIAN METHODOLOGY TO CAPTURE ZONE DELINEATION 3-7 Figure 4. Plots of (a e) hw s i and (f j) s 2 w s versus degree of uncertainty for different sampling sets. smaller values of hw s i. However, the variability in estimating the sill of the transmissivity field from the data is larger for the smaller data sets. This is reflected in the pronounced rise in the values of s 2 w s, an effect that is gradually attenuated with the introduction of more data. [30] When all parameters are considered unknown (scenario 4, far right on x axis in Figure 4) a further reduction in the spread of the capture zone predictive distribution is observed for the smaller data sets. With 25 or more measurements, the capture zone predictive uncertainty increases when compared with situation 3. This can be explained by the fact that it is difficult to obtain reliable estimates of the integral scale with small data sets, which results in simulated fields with very small or zero corre-

8 3-8 FEYEN ET AL.: BAYESIAN METHODOLOGY TO CAPTURE ZONE DELINEATION Figure 4. (continued) lation lengths. Fields with little or no correlation lack zones of connected high transmissivity values, leading to smaller capture zones and a reduction in the spread of the capture zone predictive distribution. When more measurements are available fewer fields with small correlation lengths are simulated from the posterior. Combined with the heavy right tail often observed in the posterior of j 0, this results in an expansion of the predictive uncertainty about the capture zones in comparison with situation 3, where j 0 is fixed to the true value. A similar behavior is observed for s 2 w s, with a reduction in its value for the smaller sample sets, contrasting with a rise when more data are introduced. Again, the reduction in s 2 w s when few data are available is a result of the absence of or the little correlation in many of the simulated fields. Some of these fields are characterized by a high value for the sill but, given the weak correlation, the extreme values in these fields are poorly connected, resulting in a reduction of s 2 w s. When more data are available higher values of the correlation length are obtained from the posterior and high and low transmissivity values are better connected, which results in a rise of s 2 w s. [31] Figure 5 shows the effect of the parameter uncertainty on the proximity of the median isochrone 0.5 to the true capture zone. The values of hw m i and s 2 w m are plotted against the degree of uncertainty for the small, intermediate and large data sets. In accordance to what is observed for hw m i in Figure 3, s 2 w m also decreases with increasing sampling density. The amount of reduction declines as more measurements are introduced. The values of hw m i rise for the intermediate and large data sets, indicating a worse correspondence of 0.5 to the true capture zone as more parameters are considered uncertain. For the two smallest data sets hw m i decreases again when moving from situation 3 to situation 4 as a result of the many low values obtained when sampling from [j 0 Y ], which results in simulated fields with little variability as explained before. When more measurements become available, the correlation scale is better identified and hw m i increases again. The behavior of s 2 w m and hw m i is similar, although a decrease of s 2 w m is observed for sampling densities up to 25 transmissivity measurements when j 0 is assumed unknown. The variability of s 2 w m with respect to the degree of uncertainty is much less pronounced than what is observed for s 2 w s. [32] Figure 6 illustrates the relative effect of parameter uncertainty on the performance measures. Graphs of sc hw s i and sc hw m i versus the degree of uncertainty are plotted for t 0 = 4.0. The subscript sc indicates that the property is scaled by its respective value for the situation where all model parameters are known. Therefore these scaled quantities represent the relative value of the property with respect to the scenario accounting only for the natural variability in the field (scenario 1). [33] The graph for sc hw s i shows that the relative effect of parameter uncertainty on the average spread of the capture zone is more pronounced for higher sampling densities. For increasing number of measurements the high rate of reduction of the kriging variance (see Figure 4) tends to outweigh the effect of the improvement in estimating the geometric mean transmissivity on the predictive uncertainty. This explains the inverse relation between the number of measurements and the value of sc hw s i for situation 2. The high uncertainty when estimating the mean with only 4 measurements could explain the higher value of sc hw s i for this case. As explained above, the reduction observed in the average spread of the capture zone predictive uncertainty when

9 FEYEN ET AL.: BAYESIAN METHODOLOGY TO CAPTURE ZONE DELINEATION 3-9 Figure 5. Plots of (a c) hw m i and (d f ) s 2 w m versus degree of uncertainty for 4, 25, and 81 T measurements. accounting for the uncertainty in the sill parameter results from the asymmetry of the marginal posterior [s 2 Y ]. This is more pronounced for small data sets, which explains the steepest decrease with 4 measurements. When j 0 is also considered unknown the values of sc hw s i strongly diverge as a result of the better identification of j 0 for higher sampling densities. For the smallest conditioning set a reduction in the average spread to a level lower than that of natural variability scenario is observed. For the highest sampling density the average spread of the capture zone predictive distribution expands 30% when parameter uncertainty is accounted for. [34] The graph for sc hw m i in Figure 6 also reveals that the effect of parameter uncertainty on this performance measure is more pronounced for higher sampling densities. How-

10 3-10 FEYEN ET AL.: BAYESIAN METHODOLOGY TO CAPTURE ZONE DELINEATION Figure 6. Plots of sc hw s i (a) and (b) sc hw m i versus degree of uncertainty for t 0 =4. ever, the graph shows that all values are larger than one, which indicates that the average correspondence between the median isochrone 0.5 and the reference capture zone decreases when parameter uncertainty is accounted for. This is valid for all sampling densities. Thus, although the average spread of the capture zone predictive distribution can be reduced for some data sets, the predictive median capture zone is consistently a poorer estimator of the reference capture zone when accounting for parameter uncertainty. Uncertainty in m has the biggest impact on the position of 0.5, as the geometric mean transmissivity determines the Darcy background flow velocity in the field. [35] The graphs in Figure 6 indicate that in order to obtain a better estimate of the location of the true capture zone, by means of the median 0.5 of the predictive distribution, estimation of the mean parameter seems to be of most importance. For the spread of the predictive variance, the integral scale also seems to have an important influence on the results. These conclusions hold for the hypothetical fields analyzed. They can not be generalized as they depend on other factors such as the true values of the parameters of the underlying stochastic field, e.g., the importance of determining the variance parameter is expected to increase for higher values of the variance of the true transmissivity field. [36] The results reported here show that the conceptual basic predictive uncertainty, corresponding to the situation when only natural variability is accounted for, does not necessarily coincide with the minimum level of the predictive uncertainty. In reality, the true parameter values are unknown and estimated from the available data by conventional methods. In general these estimates will be closer to the Bayesian point estimates than to the true values. As the Bayesian approach accounts for parameter uncertainty it should result in an increase in the capture zone predictive uncertainty when compared with the predictive uncertainty obtained by just plugging in the estimates. 6. Summary and Conclusions [37] The problem of parameter uncertainty and its effect on the predictive uncertainty of capture zones has been examined within the Bayesian framework. The Bayesian method acknowledges that there are at least two components of the predictive uncertainty when adopting the spatial stochastic approach for prediction: (1) the uncertainty that stems from the inability to predict exactly the transmissivity at unsampled locations when the stochastic process is known; (2) the additional uncertainty given the fact that the parameters of the stochastic model are typically unknown. The parameters of the stochastic model for the log transmissivity are treated as random quantities and are characterized by a prior probability distribution, which reflects what is known about them. Prior information is combined with the information contained in the data by applying the Bayes theorem. Samples from the predictive distribution of the log transmissivity field are generated using conditioning by kriging. For each simulated field the flow equation is solved and a particle-tracking analysis is performed. This results in the predictive distribution for the capture zone, which accounts not only for the randomness of the log transmissivity field but also for the parameter uncertainty. [38] The methodology has been applied to a set of hypothetical fields considering various sampling densities and assuming different levels of parameter uncertainty. Two measures of performance have been used to assess the results. For all levels of parameter uncertainty and sampling densities the predictive uncertainty of the capture zone expanded with the length of the time interval. A reduction in the predictive uncertainty with increasing sampling density was observed as a consequence of the combined effects of the reduction in the variance of the posterior distribution and the reduction of the kriging variance when more data become available. [39] For all sampling densities the average spread and variance of the capture zone predictive distribution were larger when accounting for uncertainty in the mean parameter in comparison to what was obtained when using the true values of the parameters. Introducing the sill as an additional uncertain parameter resulted in a decrease in the predictive uncertainty as a result of the strong asymmetry in the marginal posterior distribution of this parameter. However, due to the increase in the variance of the parameter estimation, the variability of the spread increased further. Both effects were more pronounced for the smaller data sets. For the scenario where all parameters are considered

11 FEYEN ET AL.: BAYESIAN METHODOLOGY TO CAPTURE ZONE DELINEATION 3-11 unknown a further decrease in the predictive spread of the capture zone and its variance has been observed for the small data sets, switching to an increase when more measurements were introduced. This can be explained by the high concentration of small values in the posterior for the correlation scale when few data are available, which results in realizations with little or no correlation. [40] The average proximity of the median isochrone to the true hypothetical capture zone showed a positive correlation with the number of conditioning data and worsened with increasing lengths of time intervals. The results also showed a deterioration in the proximity of 0.5 to the true capture zone as the number of uncertain parameters increased, except for the smaller data sets with uncertainty in the correlation scale accounted for. The latter may be the result of less variable fields generated for the small data sets when low values for the integral scale are sampled from the posterior. The variance of the proximity measure showed a similar behavior, although a decrease of this property was observed to sample densities up to 25 transmissivity measurements with j 0 assumed unknown. [41] The results showed that the conceptual basic capture zone predictive uncertainty, computed when only random variability is accounted for, did not always coincide with the minimum level of the predictive uncertainty. We observed that for the small data sets, accounting for the parameter uncertainty resulted in a smaller average spread of the predictive capture zone distribution in comparison to the uncertainty resulting from natural variability only. For all sampling densities we observed that the median isochrone becomes a worse predictor of the reference capture zone when parameter uncertainty is accounted for. In reality the true values of the parameters are unknown and need also to be inferred from the available data. For a particular data set, the inferred mean and covariance parameters can be different from the Bayesian estimates, in either direction. However, we would expect these estimates to be closer to the Bayesian estimates than to the hypothetical values that were used here to reflect the uncertainty due to randomness only. We argue that in general predictions based on fitted parameters underestimate the true predictive variance. Predictions based on the Bayesian predictive distribution tend to be more conservative and probably more realistic. Use of the method will therefore have an impact on groundwater management practice in the sense that in general it will lead to stricter safety regulations. [42] Although in this study the method has only been applied to a set of hypothetical fields, the method is applicable to real field cases. Currently the authors are applying the method to a well field in Belgium in which the presented method is combined with the earlier developed GLUE methodology in order to incorporate both transmissivity and hydraulic head data to reduce the uncertainty in the delineation of capture zones. [43] Acknowledgments. The authors wish to thank K. J. Beven and J. Freer for the cooperation and the access to and aid with the parallel system at the Institute of Environmental Sciences at Lancaster University. The first author wishes to acknowledge the Fund for Scientific Research Flanders for providing a Research Assistant Scholarship and a travel grant to the first author. The second author wishes to thank CAPES/Brazil, grant BEX 1676/96-2. The authors also wish to thank two anonymous referees for their comments. References Bear, J., and M. Jacobs, On the movement of water bodies injected into aquifers, J. Hydrol., 3, 37 57, Chilès, J.-P., and P. Delfiner, Geostatistics Modeling Spatial Uncertainty, John Wiley, New York, Delhomme, J. P., Kriging in the hydrosciences, Adv. Water Res., 1(5), , Deutsch, C. V., and A. G. Journel, GSLIB, Geostatistical Software Library and User s Guide, Oxford Univ. Press, New York, Diggle, P. J., and P. J. Ribeiro Jr., Bayesian inference in Gaussian modelbased geostatistics, Geogr. Environ. Modell., in press, Diggle, P. J., P. J. Ribeiro, and O. F. Christensen, An introduction to model based geostatistics, in Spatial Statistics and Computational Methods, Lecture Notes in Statistics, edited by M. B. Hansen and J. Moller, Springer-Verlag, New York, in press, Feyen, L., K. J. Beven, F. De Smedt, and J. Freer, Stochastic capture zone delineation within the GLUE-methodology: Conditioning on head observations, Water Resour. Res., 37(3), , Franzetti, S., and A. Guadagnini, Probabilistic estimation of well catchments in heterogeneous aquifers, J. Hydrol., 174, , Freeze, R. A., A stochastic-conceptual analysis of one-dimensional groundwater flow in nonuniform homogeneous media, Water Resour. Res., 11(5), , Gelman, A., J. B. Carlin, H. S. Stern, and D. B. Rubin, Bayesian Data Analysis, Chapman and Hall, New York, Guadagnini, A., and S. Franzetti, Time-related capture zones for contaminants in randomly heterogeneous formations, Ground Water, 37(2), , Handcock, M., and M. Stein, A Bayesian analysis of kriging, Technometrics, 35(4), , Hoeksema, R. J., and P. K. Kitanidis, Analysis of the spatial structure of properties of selected aquifers, Water Resour. Res., 21(4), , Journel, A. G., and C. J. Huijbregts, Mining Geostatistics, Academic, San Diego, Calif., Kitanidis, P. K., Parameter uncertainty in estimation of spatial functions: Bayesian analysis, Water Resour. Res., 22(4), , Omre, H., and K. B. Halvorsen, The Bayesian bridge between simple and universal kriging, Math. Geol., 21(7), , Ribeiro, P. J., Jr., and P. J. Diggle, GeoR: A package for geostatistical analysis, R News, 1(2), 15 18, Riva, M., A. Guadagnini, and F. Ballio, Time-related capture zones for radial flow in two dimensional randomly heterogeneous media, Stochastic Environ. Res. Risk Assess., 13, , Russo, D., and W. A. Jury, A theoretical study of the estimation of the correlation scale in spatially variable fields, 1, Stationary fields, Water Resour. Res., 23(7), , Shafer, J. M., and M. D. Varljen, Approximation of confidence limits on sample semivariograms from single realizations of spatially correlated random fields, Water Resour. Res., 26(8), , Tanner, M., Tools for Statistical Inference, New York, Springer-Verlag, van Leeuwen, M., C. B. M. te Stroet, A. P. Butler, and J. A. Tompkins, Stochastic determination of well capture zones, Water Resour. Res., 34(9), , van Leeuwen, M., C. B. M. te Stroet, A. P. Butler, and J. A. Tompkins, Stochastic determination of well capture zones conditioned on regular grids of transmissivity measurements, Water Resour. Res., 36(4), , Varljen, M. D., and J. M. Shafer, Assessment of uncertainty in time-related capture zones using conditional simulation of hydraulic conductivity, Ground Water, 29, , Zheng, L., and S. E. Silliman, Estimating the theoretical semivariogram from finite numbers of measurements, Water Resour. Res., 36(1), , F. De Smedt and L. Feyen, Department of Hydrology and Hydraulic Engineering, Free University Brussels, 1050 Brussels, Belgium. (fdesmedt@ vub.ac.be; l.feyen@vub.ac.be) P. J. Diggle, Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, UK. (p.diggle@lancaster.ac.uk) P. J. Ribeiro Jr., Departamento de Estatística, Universidade Federal do Paraná, Paraná, Brazil. (paulojus@gauss.est.ufpr.br)