For groundwater management problems, the presence of model uncertainty means that the control strategy must be overdesigned to ensure reliability, with the cost of this overdesign increasing with increasing model uncertainty. A number of recent studies have focused on reducing model uncertainty as a means for reducing costs. One approach involves sequentially solving the groundwater management and parameter estimation problems, with the parameter estimates and management policy updated as new data are acquired. The basic premise of this approach is that as new information is acquired the model uncertainty is reduced and the cost of optimal groundwater management will decrease.
Jones [1992] presents an approach which sequentially applies deterministic groundwater hydraulic management and parameter estimation. The problem studied is optimal stage-wise management of aquifer hydraulics in the presence of an uncertain transmissivity field. The methodology first solves a deterministic management model---based on the best available estimates of transmissivity---to identify the well pumping rates required to meet the target head at the end of a pumping stage. During each stage, hydraulic head measurements are collected, and at the end of each stage the transmissivity estimates are updated to reflect the information in the hydraulic head data. The management design is then revised on the basis of the updated parameter estimates.
Georgakakos and Vlatsa [1991] study the problem of groundwater hydraulic management under consideration of spatially random transmissivity and uncertain boundary conditions. In this case, the management model has a composite objective function that seeks to identify the pumping policy that minimizes the weighted expected-value deviations of pumping costs, hydraulic heads, and pumping rates from their target values. They use a technique of stochastic control, termed open-loop feedback control, to identify the optimal time-varying pumping policy for differently-weighted variations of the composite objective. In addition to being a stochastic formulation, this model differs from the sequential approach of Jones [1992] in that it considers the effect of management decisions for the present stage on model predictions in present and future stages. Although they do not demonstrate sequential parameter estimation and management, Georgakakos and Vlatsa recognize the importance of data collection for reducing model uncertainty and they suggest a sequential approach to handle the stochastic management problem.
In the sequential approaches cited above, the parameter estimation and groundwater management models are decoupled. The management model is solved sequentially, accounting for all available information but recognizing neither the information to be obtained during future data collection efforts nor the ability of the control policy to influence that information. Lee and Kitanidis [1991] [see also Andricevic and Kitanidis, 1990] describe a methodology that links the parameter identification and groundwater management models. The approach is based on the concept of dual control where the optimal management policy is obtained with two objectives in mind: (1) reducing costs based on the present level of information and (2) reducing model uncertainty through groundwater monitoring with the goal of reducing future management costs. The methodology uses an extended Kalman filter in conjunction with differential dynamic programming to determine how the pumping policy should be altered as new data are collected and how the pumping policy can be devised to maximize the information to be obtained from future data collection efforts. The authors present a hypothetical example for aquifer remediation design. A comparison of the dual control formulation with a sequential formulation finds that the dual control model is more efficient in terms of its ability to estimate the unknown transmissivities and minimize the cost of remediation. Examples demonstrate that the optimal management policy might be one that selects pumping rates in the early stages to perturb the system (as in a pumping or tracer test) in order to reduce model uncertainty in future stages.
Whiffen and Shoemaker [1993] present a two-stage method of nonlinear weighted feedback to design aquifer remediation strategies under model uncertainty. The first stage uses a differential dynamic programming algorithm to define a deterministic optimal pumping policy for the `best-estimate' parameter values. In stage two, the deterministic optimal policy from stage one is implemented, and hydraulic head and contaminant concentration are monitored. As one would expect, the heads and concentrations observed in the field will differ from those predicted under the `best-estimate' parameter values. Feedback laws determine how the pumping policy should be altered to compensate for the observed deviations. The authors demonstrate the weighted feedback approach for a hypothetical problem of pump-and-treat aquifer remediation design. The objective function includes costs for pumping and treating water, and penalty `costs' for violation of the pumping and concentration constraints. For their examples, they show that it is possible to identify feedback laws that are robust with respect to a range of parameter bias and uncertainty. It should be noted that the method of weighted feedback does not explicitly involve parameter estimation or parameter estimate uncertainty analysis; the feedback law alters the deterministic pumping policy based on the observation deviations and the weight given those observations. However, reliability of a feedback law can only be evaluated with respect to a defined state of model uncertainty, so there is a need to characterize this uncertainty at the outset in order to ensure that the feedback law is reliable.
In the studies cited above, the monitoring strategy is specified at the outset and there is no attempt to `optimize' the network design. Andricevic [1993] takes the concept of dual control one step further by explicitly addressing the problem of sampling design using a sequential development of the sampling and management strategies. The sampling-network design and the withdrawal-strategy design are linked through the network design criterion, which is to reduce model prediction uncertainty such that uncertainty in the objective function is minimized. The methodology is demonstrated for a hypothetical problem of groundwater supply. The dual control algorithm determines the optimal, time-varying pumping strategy to minimize a penalty-type objective that compromises between meeting withdrawal demands and target head levels. The sampling network design algorithm sequentially selects head measurement locations based on the uncertainty in head at that location and the sensitivity of the objective function to that uncertainty. It is shown that if a sampling strategy were designed solely on the basis of reducing uncertainty in hydraulic heads, it would not necessarily result in the best strategy for minimizing the uncertainty associated with the dual objectives of minimizing costs and maximizing future sample information.
The studies cited above do not directly address the economic worth of hydrogeologic information, i.e. they do not consider the trade-off between the cost of ground-water sampling and the value of information gained. An example of such an analysis is presented by Tucciarelli and Pinder [1991] who study the problem of optimal data acquisition for stochastic aquifer remediation design. Their methodology is based on a chance-constrained formulation of the management model, relating the concentration uncertainty to uncertainty in transmissivity estimates. The data collection strategy is determined by defining the optimal compromise between increasing costs due to data collection and decreasing pumping costs due to reduced parameter uncertainty. For each potential sampling strategy, the total costs (pumping costs plus measurement costs) are compared and the least expensive pumping/measurement strategy selected. A hypothetical example demonstrates that sampling programs can be unacceptable because the cost of data collection is greater than the associated reduction in pumping costs.
James and Gorelick [1994] use Bayesian decision analysis to evaluate the worth of groundwater monitoring in the context of groundwater management. The problem studied is optimal groundwater remediation design in the presence of uncertainty about the location and history of the source and the heterogeneity of the hydraulic conductivity field. Here sampling groundwater quality serves to reduce the uncertainty about the location and extent of the plume, thereby reducing the cost of hydraulic containment. The goal is to identify the optimum monitoring locations to minimize the expected cost of remediation plus sampling. Results indicate that the number of samples and the total cost of management and monitoring are most sensitive to the hydraulic conductivity mean. An important result of this work is that the locations with greatest uncertainty about plume presence were on average poor candidates for sampling, which demonstrates again the importance of considering the overall cost of management and monitoring when designing a combined groundwater monitoring and management policy.