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The central intention of this survey is to review dynamic programming models for water resource problems and to examine computational techniques which have been used to obtain solutions to these problems. Problem areas surveyed here include aqueduct design, irrigation system control, project development, water quality maintenance, and reservoir operations analysis. Computational considerations impose severe limitation on the scale of dynamic programming problems which can be solved. Inventive numerical techniques for implementing dynamic programmihg have been applied to water resource problems. Discrete dynamic programming, differential dynamic programming, state incremental dynamic programming, and Howard's policy iteration method are among the techniques reviewed. Attempts have been made to delineate the successful applications, and speculative ideas are offered toward attacking problems which have not been solved satisfactorily.

This paper describes a modification of differential dynamic programming (DDP) which makes that technique applicable to certain constrained sequential decision problems such as multireservoir control problems discussed in the hydrology literature. The authors contend that the method proffered here is superior to available alternatives. This belief is supported by analysis (wherein it transpires that constrained DDP does not suffer the ‘curse of dimensionality’ and requires no discretization) and computational experimentation (wherein DDP is found to quickly locate solutions of 4‐reservoir problems introduced by other investigations as well as the solution of a 10‐reservoir problem thought to be beyond the capability of alternative methods).

A new statistically based approach to the problem of estimating spatially varying aquifer transmissivities on the basis of steady state water level data is presented. The method involves solving a family of generalized nonlinear regression problems and then selecting one particular solution from this family by means of a comparative analysis of residuals. A linearized error analysis of the solution is included. This analysis allows one to estimate the covariance of the transmissivity estimates as well as the square error of the estimates of hydraulic head. In addition to the explicitly statistical orientation of the method, it has an additional feature of permitting the user to incorporate a priori information about the transmissivities. This information may be based on actual field data such as pumping tests, or on statistical data accumulated from similar aquifers elsewhere in the world. A highly efficient explicit numerical scheme for solving the inverse problem in an approximate manner when errors in water level data are sufficiently small is also described. When these errors are large, the explicit scheme may still be useful for obtaining a rapid initial idea about the approximate location of the optimum solution. Paper 1 presents the theory and illustrates it by a theoretical example. The purpose of this example is to demonstrate the effectiveness of our method in dealing with noisy data obtained from a known model. Application of the method to real data will be described in paper 2.

The nearest-neighbour method, because of its intuitively appealing nature and competitive theoretical properties, deserves consideration in time-series applications akin to attention it has received lately in the i.i.d. case. Here it is shown that as a nonparametric regression device, like the kernel method, under the G 2 mixing assumption, it converges in quadratic mean at the Stone-optimal rate. In the closing sections, our methodology is extended to a broader pattern-recognition context, and applied to hydrologic data.

The setting is a stationary, ergodic time series. The challenge is to construct a sequence of functions, each based on only finite segments of the past, which together provide a strongly consistent estimator for the conditional probability of the next observation, given the infinite past. Ornstein gave such a construction for the case that the values are from a finite set, and recently Algoet extended the scheme to time series with coordinates in a Polish space.The present study relates a different solution to the challenge. The algorithm is simple and its verification is fairly transparent. Some extensions to regression, pattern recognition, and on-line forecasting are mentioned.

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This study concerns problems of time-series forecasting under the weakest of assumptions. Related results are surveyed and are points of departure for the developments here, some of which are new and others are new derivations of previous findings.The contributions in this study are all negative, showing that various plausible prediction problems are unsolvable, or in other cases, are not solvable by predictors which are known to be consistent when mixing conditions hold.