A new methodology for solution of the inverse problem in groundwater hydrology is proposed and applied to a site in southeastern New Mexico with extensive hydrogeologic data. The methodology addresses the issue of nonuniqueness of the inverse solutions by generating an ensemble of transmissivity fields considered to be equally likely, each of which is in agreement with the measured transmissivity and pressure data. It consists of generating a selected number of conditionally simulated transmissivity fields and then calibrating each of the fields to match the measured steady state or transient pressures, in a least squares sense. The calibration phase involves an iterative implementation of an automated pilot point approach coupled with conditional simulations. Pilot points are the parameters of calibration. They are synthetic transmissivity data which are added to the transmissivity database to produce a revised conditional simulation during calibration. Coupled kriging and adjoint sensitivity analysis is employed for the optimal location of pilot points, and gradient search methods are used to derive their optimal transmissivities. The pilot point methodology is well suited for characterizing the spatial variability of the transmissivity field in contrast to methods using zonation. Pilot points are located where their potential for minimizing the objective function is the highest. This minimizes the perturbations in the transmissivities which are optimally assigned to the pilot point and results in minimal changes to the covariance structure of the transmissivity field. The calibrated fields honor the transmissivity measurements at their locations, preserve the variogram, and match the measured pressures in a least squares sense. Since there are numerous options in the execution of this methodology, computational experiments have been conducted to identify the most efficient among them. The method has been applied to the Waste Isolation Pilot Plant (WIPP) site, in southeastern New Mexico, where the U.S. Department of Energy is conducting probabilistic system assessment for the permanent disposal of transuranic nuclear waste. The resulting calibrated transmissivity fields are input to a Monte Carlo analysis of the total system performance. The present paper, paper 1 of a two‐paper presentation, describes the methodology. Paper 2, a companion paper, presents the methodology's application to the WIPP site.

Abstract. This paper describes the first major attempt to compare seven different inverse approaches for identifying aquifer transmissivity. The ultimate objective was to determine which of several geostatistical inverse techniques is better suited for making probabilistic forecasts of the potential transport of solutes in an aquifer where spatial variability and uncertainty in hydrogeologic properties are significant. Seven geostatistical methods (fast Fourier transform (FF), fractal simulation (FS), linearized cokriging (LC), linearized semianalytical (LS), maximum likelihood (ML), pilot point (PP), and sequential self-calibration (SS)) were compared on four synthetic data sets. Each data set had specific features meeting (or not) classical assumptions about stationarity, amenability to a geostatistical description, etc. The comparison of the outcome of the methods is based on the prediction of travel times and travel paths taken by conservative solutes migrating in the aquifer for a distance of 5 km. Four of the methods, LS, ML, PP, and SS, were identified as being approximately equivalent for the specific problems considered. The magnitude of the variance of the transmissivity fields, which went as high as 10 times the generally accepted range for linearized approaches, was not a problem for the linearized methods when applied to stationary fields; that is, their inverse solutions and travel time predictions were as accurate as those of the nonlinear methods. Nonstationarity of the "true" transmissivity field, or the presence of "anomalies" such as high-permeability fracture zones was, however, more of a problem for the linearized methods. The importance of the proper selection of the semivariogram of the 1og•0 (T) field (or the ability of the method to optimize this variogram iteratively) was found to have a significant impact on the accuracy and precision of the travel time predictions. Use of additional transient information from pumping tests did not result in major changes in the outcome. While the methods differ in their underlying theory, and the codes developed to implement the theories were limited to varying degrees, the most important factor for achieving a successful solution was the time and experience devoted by the user of the method. •2Stanford University, Stanford, California.•3Duke Engineering and Services, Inc., Austin, Texas.•4University of Arizona, Tucson.•Slnstitut Franqais du Pftrole, Rueil-Malmaison, France.•6University of California, Berkeley.Copyright 1998 by the American Geophysical Union. Paper number 98WR00003.0043-1397/98/98WR-00003509.00 tion, or performance assessment of planned waste disposal projects, it is no longer enough to determine the "best estimate" of the distribution in space of the aquifer parameters. A measure of the uncertainty associated with this estimation is also needed. Geostatistical techniques are ideally suited to filling this role. Basically, geostatistics fits a "structural model" to the data, reflecting their spatial variability. Then, both "best estim...

This paper, the second in a two‐part series, presents the application of a methodology to assess spatial variability of the transmissivities within a regional aquifer in the vicinity of the Waste Isolation Pilot Plant (WIPP), the Culebra dolomite. An innovative aspect of this methodology is the generation of an ensemble of conditional simulations of the transmissivity field which preserve the statistical moments and spatial correlation structure of the measured transmissivity field and honors the measured transmissivity values at their locations. Each simulation is then calibrated, using an iterative procedure, to match an exhaustive set of steady state and transient pressure data. A fully automated inverse algorithm using pilot points as parameters of calibration was employed. The application of this new methodology to the Culebra dolomite flow system produced 70 conditional simulations which were consistent with all the measured transmissivity and head data at the site. Based on an analysis of the calibrated transmissivity fields, the spatial variability of the transmissivity fields was increased as a result of the calibration process. This increase is in part due to the addition of a high‐transmissivity feature to each of the transmissivity fields which is needed to match both steady state and transient state head data.

We review the main stages of the evolution of ideas and methods for solving the inverse problem in hydrogeology; i.e., the identification of the transmissivity field in single-phase flow from piezometric data, in mainly steady-state and, occasionally, transient flow conditions. We first define the data needed to solve an inverse problem in hydrogeology, then describe the numerous approaches that have been developed over the past 40 years to solve it, emphasizing the major contributions made by Shlomo P. Neuman. Finally, we briefly discuss fitting processes that start by defining the unknown field as geological images (generated by Boolean or geostatistical methods).The early attempts at solving the inverse problem were direct, i.e., the transmissivity field was directly determined by using stream lines of the flow and inverting the flow equation along these lines. Faced with the poor results obtained in this manner, hydrogeologists have tried many different ways of minimizing the balance error representing an integral of the mass-balance error for each mesh for a given transmissivity field. These attempts were accompanied by constraints imposed on the transmissivity field in order to avoid instabilities.The idea then emerged that the unknown field should reproduce the local observations of the pressure at the measurement points instead of minimizing a balance error. Second, it should also satisfy a condition of plausibility, which means that the transmissivity field obtained through the inverse solution should not deviate too far from an a priori estimate of the real transmissivity field. This a priori notion led to the inclusion of a Bayesian approach resulting in the search for an optimal solution by maximum likelihood, as expounded later.Simultaneously, the existence of locally measured values in the transmissivity field (obtained by pumping tests) allowed geostatistical methods to be used in the formulation of the problem; the result of this innovation was that three major approaches came into being: (1) the definition of the a priori transmissivity field by kriging; (2) the method of cokriging;(3) the pilot point method. Furthermore, geostatistics made it possible to pose the inverse problem in a stochastic framework and to solve an ensemble of possible and equally probable fields, each of them equally acceptable as a solution.

Fourier-transform-based spectral analysis and filtering techniques, although potentially very useful, have seen little practical application in hydrology. We provide an overview of the Fourier transform and spectral analysis and present examples of how these methods may be applied to practical hydrologic problems: determination of the frequency content of a time series; inference of the physical mechanisms responsible for this frequency content; and evaluation of the performance of a process-based simulation model used for water resource management. In all cases, the methods performed well and were reasonably straightforward to implement, highlighting their general utility.