David E. Dougherty

Characterization of aquifer properties using artificial neural networks: Neural kriging

A method for pattern completion based on the application of artificial neural networks and possessing many operational objectives of the ordinary kriging approach, neural kriging, is developed. A neural kriging (NK) network is described, implemented in a parallelizing algorithm, and applied to develop maps of discrete spatially distributed fields (e.g., log hydraulic conductivity). NK is, in the case of two discrete field values, similar to indicator kriging. It uses a feed-forward counterpropagation training approach because field observations are available and because fast yet reliable results are obtained. NK is data-driven and requires no estimate of a covariance function. The optimal design of the NK network is found to depend on the number of hidden units in a more complex way than expected. The quality of the estimate of each pixel of the NK maps can be presented as well, as in kriging, to help identify areas in which additional information will be most beneficial. A comparison with a reference field shows that the NK network produces unbiased errors relative to sample bias and reproduces the variogram of a quantized random field with reasonable accuracy. Ordinary kriging (OK) followed by quantization can also perform well; however, estimation errors in the variogram selected for use in OK (in this case the range cofficient in particular) must be carefully examined and treated. The NK method can provide multiple realizations of the estimated field, all of which respect observations; hence conditional simulation is demonstrably possible. The combination of simplicity, interpolation, reasonably accurate prediction statistics, ability to provide conditional simulations, and computational speed suggest that artificial neural networks can be useful tools in geohydrology when applied to specific well-defined problems for which they are well suited, such as aquifer characterization.

Application of particle methods to reliable identification of groundwater pollution sources

An alternative strategy for identifying sources of contamination in groundwater systems is presented. Under the assumption that the remediation cost is affected by the level of contamination, the proposed scheme provides probabilistic estimates of source locations and spill-time histories. Moreover, the method successfully assesses the relative importance of each potential source.

Optimal groundwater management: 2. Application of simulated annealing to a field‐scale contamination site

Simulated annealing is used to analyze alternate design strategies for groundwater remediation at a contaminated field site. The simulated annealing optimization algorithm is combined with a field-scale flow and transport simulation model for an unconfined aquifer to determine nearly optimal pumping schedules for a pump-and-treat remediation system at a proposed Superfund site in central California. A series of demonstration problems is presented using two different optimization formulations. The results of these experiments indicate that the method can be applied to realistic groundwater management problems. The computational expense of simulated annealing is large yet comparable to other nonlinear optimization techniques. A practical empirically based strategy is provided for selecting and adjusting the parameters necessary for successful optimization.

Design Optimization for Multiple Management Period Groundwater Remediation

A technique for obtaining a (nearly) optimal scheme using multiple management periods has been developed. The method has been developed for very large scale combinatorial optimization problems. Simulated annealing has been extended to this problem. An importance function is developed to accelerate the search for good solutions. These tools have been applied to groundwater remediation problems at Lawrence Livermore National Laboratory (LLNL). A deterministic site-specific engineering-type flow and transport model (based on the public domain code SUTRA) is combined with the heuristic optimization technique. The objective is to obtain the time-varying optimal locations of the remediation wells that will reduce concentration levels of volatile organic chemicals in groundwater below a given threshold at specified areas on the LLNL site within a certain time frame and subject to a variety of realistic complicating factors. The cost function incorporates construction costs, operation and maintenance costs for injection and extraction wells, costs associated with piping and treatment facilities, and a performance penalty for well configurations that generate flow and transport simulations that exceed maximum concentration levels at specified locations. The resulting application reported here comprises a huge optimization problem. The importance function detailed in this paper has led to rapid convergence to solutions. The performance penalty allows different goals to be imposed on different geographical regions of the site; in this example, short-term off-site plume containment and long-term on-site cleanup are imposed. The performance of the optimization scheme and the effects of various trade-offs in management objectives are explored through examples using the LLNL site.

Simultaneous estimation of transmissivity values and zonation

The extended Kalmanfilter (EKF) has long been recognized as a powerful, yet computationally intensive, methodology for stochastic parameter estimation. Three improvements to traditional algorithms are presented and applied to heterogeneous transmissivity estimation. First, the costly EKF covariance updates are replaced by more efficient approximations. Second, the zonation structure of the distributed parameterfield being estimated is dynamically determined and refined using a partitional clustering algorithm. Third, a new method of mergingfirst and second moments of randomfields that have heterogeneous statistics is introduced. We apply this method, called random field union, as an alternative to conventional randomfield averaging for the systematic shrinking of covariance matrices as the dimensionality of the parameter space is reduced. The effects of these three improvements are examined. In applications to steady state groundwaterflow test problems, we show that thefirst and second improvements reduce the computational time requirements dramatically, while the second and third can improve the accuracy and stability of the results. The resulting integrated method is successfully applied to a larger, more realistic calibration test case under steady and cyclostationary flow conditions (similar to regular seasonalfluctuations). Whenflow is steady, the method can be viewed as iterative; whenflow is transient, the method is fully recursive.

Efficient three-dimensional data inversion: Soil characterization and moisture Monitoring from cross-well ground-penetrating radar at a Vermont Test Site

We extend our methodology for three-dimensional parameter structure and value estimation and apply it to a Vermont test site. Ground-penetrating radar (GPR) cross-well travel times are inverted for estimation of heterogeneous GPR soil velocities before and after a controlled release of salt water in the unsaturated zone. The method, which is based on an approximation of the extended Kaiman filter in conjunction with data-driven zonation, automatically estimates not only distributed zone values but also the number of zones, zone geometry, and zone covariance. Resultant GPR velocity estimates are shown to reduce travel time estimation errors and to be consistent with independent cone penetrometer measurements at all five walls at the site. Comparison of velocity estimates before and after forced injection of salt water is used to detect and visualize soil moisture patterns in three dimensions. By varying the “cluster tolerance criterion” in the data-driven zonation process, the user can obtain a desired resolution of heterogeneity (number of zones used) in the resultant model.

BIOMEDICAL OPTICAL TOMOGRAPHY USING DYNAMIC PARAMETERIZATION AND BAYESIAN CONDITIONING ON PHOTON MIGRATION MEASUREMENTS

Stochastic reconstruction techniques are developed for mapping the interior optical properties of tissues from exterior frequency-domain photon migration measurements at the air-tissue interface. Parameter fields of absorption cross section, fluorescence lifetime, and quantum efficiency are accurately reconstructed from simulated noisy measurements of phase shift and amplitude modulation by use of a recursive, Bayesian, minimum-variance estimator known as the approximate extended Kalman filter. Parameter field updates are followed by data-driven zonation to improve the accuracy, stability, and computational efficiency of the method by moving the system from an underdetermined toward an overdetermined set of equations. These methods were originally developed by Eppstein and Dougherty [Water Resources Res. 32, 3321 (1996)] for applications in geohydrology. Estimates are constrained to within feasible ranges by modeling of parameters as beta-distributed random variables. No arbitrary smoothing, regularization, or interpolation is required. Results are compared with those determined by use of Newton-Raphson-based inversions. The speed and accuracy of these preliminary Bayesian reconstructions suggest the near-future application of this inversion technology to three-dimensional biomedical imaging with frequency-domain photon migration.

Three-dimensional Bayesian optical image reconstruction with domain decomposition

Most current efforts in near-infrared optical tomography are effectively limited to two-dimensional reconstructions due to the computationally intensive nature of full three-dimensional (3-D) data inversion. Previously, we described a new computationally efficient and statistically powerful inversion method APPRIZE (automatic progressive parameter-reducing inverse zonation and estimation). The APPRIZE method computes minimum-variance estimates of parameter values (here, spatially variant absorption due to a fluorescent contrast agent) and covariance, while simultaneously estimating the number of parameters needed as well as the size, shape, and location of the spatial regions that correspond to those parameters. Estimates of measurement and model error are explicitly incorporated into the procedure and implicitly regularize the inversion in a physically based manner. The optimal estimation of parameters is bounds-constrained, precluding infeasible values. In this paper, the APPRIZE method for optical imaging is extended for application to arbitrarily large 3-D domains through the use of domain decomposition. The effect of subdomain size on the performance of the method is examined by assessing the sensitivity for identifying 112 randomly located single-voxel heterogeneities in 58 3-D domains. Also investigated are the effects of unmodeled heterogeneity in background optical properties. The method is tested on simulated frequency-domain photon migration measurements at 100 MHz in order to recover absorption maps owing to fluorescent contrast agent. This study provides a new approach for computationally tractable 3-D optical tomography.

Optimal 3-D traveltime tomography

We propose a practical new method for 3-D traveltime tomography. The method combines an efficient approximation to the extended Kalman filter for rapid, accurate, nonlinear tomography, with the concept of data‐driven zonation, in which the dimensionality and geometry of the parameterization are dynamically determined using cluster analysis and region merging by random field union. The Bayesian filter uses geostatistics as it recursively incorporates measurements in an optimal (minimum‐variance) manner. Geologic knowledge is introduced through a priori estimates of the parameter field and its spatial covariance. Conditional estimates of the parameter number, geometry, value, and covariance are evolved. An initial decomposition of the 3-D domain into 2-D slices, the simplified filter design, and the data‐driven reduction in parameter dimensionality, all contribute to make the method computationally feasible for large 3-D domains. The method is verified by the inversion of crosswell seismic traveltimes to 3-...

Probabilistic Simulation for Reliable Solute Source Identification in Heterogeneous Porous Media

A probabilistic framework to identify solute sources in heterogeneous porous media is the theme of the present paper. Monte Carlo analyses of reversed time solute transport are conducted with the help of stochastically generated hydraulic conductivity fields, kriging, and the random walk particle tracking method. The methodology is capable of pinpointing the most probable solute source and assessing, in a rational manner, the relative liability of each source.