A complete reexamination of Sudicky's (1986) field experiment for the geostatistical characterization of hydraulic conductivity at the Borden aquifer in Ontario, Canada is performed. The sampled data reveal that a number of outliers {low In (K) valuest are present in the data base. These low values cause difficulties in both variogram estimation and determining population statistics. The analysis shows that assuming either a normal distribution or exponential distribution for log conductivity is appropriate. The classical, Cressie/Hawkins and squared median of the absolute deviations {SMAD) estimators are used to compute experimental variograms. None of these estimators provides completely satisfactory variograms for the Borden data with the exception of the classical estimator with outliers removed from the data set. Theoretical exponential variogram parameters are determined from nonlinear (NL) estimation. Differences are obtained between NL fits and those of Sudicky (1986). For the classical-screened estimated variogram, NL fits produce an In (K) variance of 0.24, nugget of 0.07, and integral scales of 5.1 m horizontal and 0.21 m vertical along A-A'. For B-B' these values are 0.37, 0.11, 8.3 and 0.34. The fitted parameter set for B-B' data {horizontal and vertical) is statistically different than the parameter set determined for A-A'. We 'also evaluate a probabilistic form of Dagan's (1982, 1987) equations relating geostatistical parameters to a tracer cloud's spreading moments. The equations are evaluated using the parameter estimates and covariances determined from line A-A' as input, with a velocity equal to 9.0 cm/day. The results are compared with actual values determined from the field test, but evaluated by both Freyberg (1986) and Rajaram and Gelhar (1988). The geostatistical parameters developed from this study produce an excellent fit to both sets of calculated plume moments when combined with Dagan's stochastic theory for predicting the spread of a tracer cloud. INTRODUCTION Sudick2y [ 1986] described the results of a sampling program in which a large number of hydraulic conductivity measurements were taken along two transects at the site of an daborate tracer test performed in the Borden aquifer in Ontario, Canada. These measurements, combined with a derailed evaluation of the dispersion characteristics of the injected tracer cloud [Freyberg, 1986], provided a unique opportunity to examine the validity of modern stochastic theories of contaminant transport that have emerged over the past decade. Based on the field data and subsequent geostatistica! inferences, Sudiclo, [1986] computed mean values, variances and integral scales for the underlying log conductivity distribution of the Borden aquifer. Then, by using these quantities as input to stochastic transport theories by Dagan [1982, 1987] and Gelhar and Axness [1983], the predicted field-scale flow and dispersion parameters were shown to be consistent with the observed evolution of the tracer plume as interpreted by Freyberg [1986]. The geostati...

In this paper we show that given prior information in terms of a lower and upper bound, a prior bias, and constraints in terms of measured data, minimum relative entropy (MRE) yields exact expressions for the posterior probability density function (pdf) and the expected value of the linear inverse problem. In addition, we are able to produce any desired confidence intervals. In numerical simulations, we use the MRE approach to recover the release and evolution histories of plume in a one‐dimensional, constant known velocity and dispersivity system. For noise‐free data, we find that the reconstructed plume evolution history is indistinguishable from the true history. An exact match to the observed data is evident. Two methods are chosen for dissociating signal from a noisy data set. The first uses a modification of MRE for uncertain data. The second method uses “presmoothing” by fast Fourier transforms and Butterworth filters to attempt to remove noise from the signal before the “noise‐free” variant of MRE inversion is used. Both methods appear to work very well in recovering the true signal, and qualitatively appear superior to that of Skaggs and Kabala [1994]. We also solve for a degenerate case with a very high standard deviation in the noise. The recovered model indicates that the MRE inverse method did manage to recover the salient features of the source history. Once the plume source history has been developed, future behavior of a plume can then be cast in a probabilistic framework. For an example simulation, the MRE approach not only was able to resolve the source function from noisy data but also was able to correctly predict future behavior.

[1] The subsurface temperature field beneath Winnipeg, Canada, is significantly different from that of the surrounding rural areas. Downward heat flow to depths as great as 130 m has been noted in some areas beneath the city and groundwater temperatures in a regional aquifer have risen by as much as 5°C in some areas. Numerical simulation of heat transport supports the conjecture that these temperature changes can be largely attributed to heat loss from buildings and the temperature at any given point is sensitive to the distance from and the age of any buildings. The effect is most noticable when buildings are closely spaced, which is typical of urban areas. Temperature measurements in areas more than a few hundred meters away from any heated structure were only a few tenths of a degree Celsius greater than those observed outside the city, suggesting that other reasons for increases in subsurface temperature, such as changes in surface cover or climate change, may be responsible for some of the some of the observed increase in temperatures. These sources of additional heat to the subsurface make it difficult to resolve information on past climates from temperatures measured in boreholes and monitoring wells. In some areas, the temperature increases may also have an impact on geothermal energy resources. This impact might be in the form of an increase in heat pump efficiency or in the case of the Winnipeg area, a decrease in the efficiency of direct use of groundwater for cooling.

The urban heat island effect has received significant attention in recent years due to the possible effect on long‐term meteorological records. Recent studies of this phenomenon have suggested that this may not be important to estimates of regional climate change once data are properly corrected. However, surface air temperatures within urban environments have significant variation, making correction difficult. In the current study, we examine subsurface temperatures in an urban environment and the surrounding rural area to help characterize the nature of this variability. The results of our study indicate that subsurface temperatures are linked to land‐use and supports previous work indicating that the urban heat island effect has significant and complex spatial variability. In most situations, the relationship between subsurface and surface processes cannot be easily determined, indicating that previous studies that relying on such a linkage may require further examination.

Abstract. A full-Bayesian approach to the estimation of transmissivity from hydraulic head and transmissivity measurements is developed for two-dimensional steady state groundwater flow. The approach combines both Bayesian and maximum entropy viewpoints of probability. In the first phase, log transmissivity measurements are incorporated into Bayes' theorem, and the prior probability density function is updated, yielding posterior estimates of the mean value of the log transmissivity field and covariance. The two central moments are generated assuming that the prior mean, variance, and integral scales are "hyperparameters"; that is, they are treated as random variables in themselves which is contrary to classical statistical approaches. The probability density functions (pdfs) of these hyperparameters are, in turn, determined from maximum entropy considerations. In other words, pdfs are chosen for each of the hyperparameters that are maximally uncommitted with respect to unknown information. This methodology is quite general and provides an alternative to kriging for spatial interpolation. The final step consists of updating the conditioned natural logarithm transmissivity (in(T)) field with hydraulic head measurements, utilizing a linearized aquifer equation. It is assumed that the statistical properties of the noise in the hydraulic head measurements are also uncertain. At each step, uncertainties in all pertinent hyperparameters are removed by marginalization. Finally, what is produced is a in(T) field conditioned on measurements of both hydraulic heads and log transmissivity and covariances of the in(T) field. In addition, we can also produce resolution matrices, confidence (credibility) limits, and the like for the In(T) field. It is shown that the application of the methodology yields good estimates of transmissivities, even when hydraulic head measurements are noisy and little or no information is specified on mean values of in(T), variance of in(T), and integral scales. It is common in hydrogeologic practice to utilize nonlinear regression methods for parameter estimation. There are currently several private and public domain software packages that are used for this purpose. In order to achieve unique solutions and avoid unstable behavior in the results, it is necessary to parameterize or zone the problem in such a way as to define a smaller (usually much smaller) number of parameters than data points. As stated above, aquifers are highly heterogeneous, and hydraulic properties can vary significantly over very short distances thus making conceptual models based on large zones questionable in our opinion. Another difficulty lies 2081

It is unclear whether one can (or should) write a tutorial about Bayes. It is a little like writing a tutorial about the sense of humor. However, this tutorial is about the Bayesian approach to the solution of the ubiquitous inverse problem. Inasmuch as it is a tutorial, it has its own special ingredients. The first is that it is an overview; details are omitted for the sake of the grand picture. In fractal language, it is the progenitor of the complex pattern. As such, it is a vision of the whole. The second is that it does, of necessity, assume some ill‐defined knowledge on the part of the reader. Finally, this tutorial presents our view. It may not appeal to, let alone be agreed to, by all.

The pioneering work of Jaynes in Bayesian/maximum entropy methods has been successfully explored in many disciplines. The principle of maximum entropy (PME) is a powerful and versatile tool of inferring a probability distribution from constraints that do not completely characterize that distribution. Minimum relative entropy (MRE) is a method which has all the important attributes of the maximum entropy approach with the advantage that prior information may be easily included. In this paper we use MRE to determine the prior probability density function (pdf) of a set of model parameters based on limited information. The resulting pdf is used in Monte Carlo simulations to provide expected values in field variables such as concentration, and confidence limits. We compare the probabilistic results from a traditional advection-dispersion (ADE) model based on volumetric averaging concepts with that of a model based on the assumption that the hydraulic conductivity is a stationary stochastic process. The results suggest that although NaWs (1990) model satisfies the observed data to a better degree than ADE model, the upper and lower confidence bands about the mean value are larger than the ADE results. This result we attribute to the fact that NaWs (1990) model simply contains more parameters, each of which is unknown and has to be estimated. There is no statistical difference between the expected values of second-spatial moments for the two models. The examples presented in this paper illustrate problems associated with assigning Gaussian pdfs as priors in a probabilistic model. First, such an assumption for the input parameters actually injects more information into the problem than may actually exist, whether consciously or unconsciously. This fact is borne out by comparison with minimum relative entropy theory. Second, the output parameters as suggested from the Monte Carlo analysis cannot be assumed to be Gaussian distributed even when the prior pdf is Gaussian in form. In a practical setting, the significance of this result and the approximation of Gaussian form would depend on the toxicity and environmental standards that apply to the problem. INTRODUCTIONResearch in explicit consideration of parameter variability in groundwater flow models was initiated by Freeze [1975], who considered parameters in his model equations as random variables rather than fixed deterministic quantities. Since that time many papers dealing with various aspects of spatial variability and ensemble averaging have appeared, including the classic work by Gelhat and Axness [1983] (see Sudicky and Huyakorn [1991] for a review). Freeze [1975] adopted the definitions pertaining to random media as given by Clarke [1973] and Greenkorn and Kessler[1969]. Freeze [1975] expressed the general form of a hydrologic model as U t = (I)(Xt_l, Xt_2, ''' ; Ut_l, Ut-2, ' '' ; al, a2,'' ')+ r/t (1) The vector variable xt (t .... , --1, 0, 1, 2,'' ') is the input, and the vector variable ut is the output, with a i being the system parameters. In this termino...

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