An Approach to Parameter Estimation and Stochastic Control in Water Resources With an Application to Reservoir Operation

This paper presents an algorithm for the estimation of parameters in state‐space models that represent hydrologic processes in which the state variables are observed with error. The algorithm is based on the maximization of the conditional expectation of the likelihood function of the state equation. The estimation algorithm is numerically stable and guarantees local convergence under mild conditions. It is also shown that the estimation algorithm can be coupled with an optimal control method to yield a combined control estimation technique that can be easily implemented. An application of the theory and methods developed herein is given for flood routing via reservoir operation. Copyright 1985 by the American Geophysical Union

The inverse problem for confined aquifer flow: Identification and estimation with extensions

The contributions of this work are twofold. First, a methodology for estimating the elements of parameter matrices in the governing equation of flow in a confined aquifer is developed. The estimation techniques for the distributed‐parameter inverse problem pertain to linear least squares and generalized least squares methods. The linear relationship among the known heads and unknown parameters of the flow equation provides the background for developing criteria for determining the identifiability status of unknown parameters. Under conditions of exact or overidentification it is possible to develop statistically consistent parameter estimators and their asymptotic distributions. The estimation techniques, namely, two‐stage least squares and three stage least squares, are applied to a specific groundwater inverse problem and compared between themselves and with an ordinary least squares estimator. The three‐stage estimator provides the closer approximation to the actual parameter values, but it also shows relatively large standard errors as compared to the ordinary and two‐stage estimators. The estimation techniques provide the parameter matrices required to simulate the unsteady groundwater flow equation. Second, a nonlinear maximum likelihood estimation approach to the inverse problem is presented. The statistical properties of maximum likelihood estimators are derived, and a procedure to construct confidence intervals and do hypothesis testing is given. The relative merits of the linear and maximum likelihood estimators are analyzed. Other topics relevant to the identification and estimation methodologies, i.e., a continuous‐time solution to the flow equation, coping with noise‐corrupted head measurements, and extension of the developed theory to nonlinear cases are also discussed. A simulation study is used to evaluate the methods developed in this study. Copyright 1987 by the American Geophysical Union

On solution of the inverse problem for confined aquifer flow via maximum likelihood

Joint estimation of transmissivity (T) and storativity (S) in a confined aquifer is done via maximum likelihood (ML). The differential equation of groundwater flow is discretized by the finite-element method, leading to equation ψφ{symbol}t+Γxt=ut. Elements of matrices ψ and Γ, as well as estimated covariance matrix of noise term ut, are functions of T and S. By minimizing the negative loglikelihood function corresponding to discretized groundwater flow equation with respect to T and S, ML estimators are obtained. The ML approach is found to yield accurate estimates of T and S (within 9 and 10% of their actual values, respectively) and showed quadratic convergence in Newton's search technique. Prediction of aquifer response, using ML estimators, results in estimated piezometric heads accurate to ±0.5 m from their actual, exact values. Statistical properties of ML estimators are derived and some basic results for statistical inference are given. © 1986 Plenum Publishing Corporation

Error Analysis and Stochastic Differentiability in Subsurface Flow Modeling

In the stochastic analysis of steady aquifer flow, the log hydraulic conductivity is the random “input” and the piezometric potential is the random “output.” Their joint behavior is governed by a differential equation that, in view of the random nature of its dependent (piezometric potential) and independent (log hydraulic conductivity) variables, represents a stochastic differential equation. The analysis of the distributional properties of the piezometric potential field involves the product of the gradients of log hydraulic conductivity and of the piezometric potential. Previous research in this field assumes that such product of gradients is small in some sense. This paper derives a closed‐form expression for the standard deviation, and hence the order of magnitude, of the product of the random gradients of log hydraulic conductivity and of piezometric potential. It was found in this research that for statistically homogeneous log hydraulic conductivity fields (1) the product of the random gradients may or may not have a zero mean, depending on whether the specific discharge is a constant or a random quantity, respectively; (2) under joint normality of the log hydraulic conductivity and the piezometric potential fields, their random gradients are statistically independent if the specific discharge is constant but are dependent when the specific discharge is random; (3) the standard deviation of the product of random gradients is proportional to the variance of log‐hydraulic conductivity times a term involving three quantities: the covariance of the piezometric potential, the covariance of the log hydraulic conductivity, and the cross covariance of the latter two fields; (4) a necessary and sufficient condition for the smallness of the product of random gradients is that the second derivatives of the covariance of the log hydraulic conductivity, the covariance of the piezometric potential, and the cross covariance of the latter two random fields be finite and that the variance of log hydraulic conductivity be much less than one. This paper also reviews some fundamental principles on the stochastic analysis of random fields and their importance to the modeling of log hydraulic conductivity fields and to the analysis of subsurface flow. Specifically, the paper highlights the role of Gaussian distributional assumptions in deriving key results of stochastic groundwater flow via perturbation analysis. Copyright 1990 by the American Geophysical Union

Risk Analysis for Reservoir Operation

The planning of reservoir operation presents decision makers with a trade‐off between competing functions, which are energy production and flood control in this study. To optimally resolve the trade‐off between maximization of energy revenues and minimization of downstream losses, the interaction between the expected value and variance of revenues (accruing from the reservoir operation) is included in a stochastic daily reservoir operation planning model. By parametrically varying the expected value and variance of the objective function, the risk‐averse nature of decision makers is incorporated, resulting in a range of feasible alternative policies that reflect the decision maker's attitude toward revenue maximization and poor performance of the reservoir operation. Copyright 1986 by the American Geophysical Union

Parameter estimation in groundwater: Classical, Bayesian, and deterministic assumptions and their impact on management policies

This work deals with a theoretical analysis of parameter uncertainty in groundwater management models. The importance of adopting classical, Bayesian, or deterministic distribution assumptions on parameters is examined from a mathematical standpoint. In the classical case, the parameters (e.g., hydraulic conductivities or storativities) are assumed fixed (i.e., nonrandom) but unknown. The Bayesian assumption considers the parameters as random entities with some probability distribution. The deterministic case, also called certainty equivalence, assumes that the parameters are fixed and known. Previous work on the inverse problem has emphasized the numerical solution for parameter estimates with the subsequent aim to use them in the simulation of field variables. In this paper, the role of parameter uncertainty (measured by their statistical variability) in groundwater management decisions is investigated. It is shown that the classical, Bayesian, and deterministic assumptions lead to analytically different management solutions. Numerically, the difference between such solutions depends upon the covariance of the parameter estimates. The theoretical analyses of this work show the importance of specifying the proper distributional assumption on groundwater parameters, as well as the need for using efficient and statistically consistent methods to solve the inverse problem. The distributional assumptions on groundwater parameters and the covariance of their sample estimators are shown to be the dominant parameter uncertainty factors affecting groundwater management solutions. An example illustrates the conceptual findings of this work. Copyright 1987 by the American Geophysical Union

SIMULTANEOUS EQUATION SYSTEMS: A CONSISTENT ESTIMATOR FOR UNKNOWN PARAMETERS IN CONFINED AQUIFERS1

ABSTRACT: This paper presents criteria for establishing the identification status of the inverse problem for confined aquifer flow. Three linear estimation methods (ordinary least squares, two‐stage least squares, and three‐stage least squares) and one nonlinear method (maximum likelihood) are used to estimate the matrices of parameters embedded in the partial differential equation characterizing confined flow. Computational experience indicates several advantages of maximum likelihood over the linear methods. Copyright © 1987, Wiley Blackwell. All rights reserve

Quadratic model for reservoir management: Application to the Central Valley Project

A quadratic optimization model is applied to a large‐scale reservoir system to obtain operation schedules. The model has the minimum possible dimensionality, treats spillage and penstock releases as decision variables and takes advantage of system‐dependent features to reduce the size of the decision space. An efficient and stable quadratic programming active set algorithm is used to solve for the optimal release policies. The stability and convergence of the solution algorithm are ensured by the factorization of the reduced Hessian matrix and the accurate computation of the Lagrange multipliers. The quadratic model is compared with a simplified linear model and it is found that optimal release schedules are robust to the choice of model, both yielding an increase of nearly 27% in the total annual energy production with respect to conventional operation procedures, although the quadratic model is more flexible and of general applicability. The adequate fulfillment of other system functions such as flood control and water supply is guaranteed via constraints on storage and spillage variables. Copyright 1985 by the American Geophysical Union