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Abstract

Despite remarkable new developments in stochastic hydrology and adaptations of advanced methods from operations research, stochastic control, and artificial intelligence, solutions of complex real-world problems in hydrogeology have been quite limited. The main reason is the ultimate reliance on first-principle models that lead to complex, distributed-parameter partial differential equations (PDE) on a given scale. While the addition of uncertainty, and hence, stochasticity or randomness has increased insight and highlighted important relationships between uncertainty, reliability, risk, and their effect on the cost function, it has also (a) introduced additional complexity that results in prohibitive computer power even for just a single uncertain/random parameter; and (b) led to the recognition in our inability to assess the full uncertainty even when including all uncertain parameters. A paradigm shift is introduced: an adaptation of new methods of intelligent control that will relax the dependency on rigid, computer-intensive, stochastic PDE, and will shift the emphasis to a goal-oriented, flexible, adaptive, multiresolutional decision support system (MRDS) with strong unsupervised learning (oriented towards anticipation rather than prediction) and highly efficient optimization capability, which could provide the needed solutions of real-world aquifer management problems. The article highlights the links between past developments and future optimization/planning/control of hydrogeologic systems.

Introduction

Advances in optimal aquifer management over the last few decades have spanned from "decision making" to "optimization", "planning", and later, "control" techniques, all of which could be unified as "planning-control- optimization".

Planning/control/optimization in hydrogeology is a domain of intersection between complex natural systems, human industrial practice, and theoretical knowledge, which is extremely difficult to analyze and further develop; this is because, from a knowledge organization perspective, this area of practical knowledge has been in disarray. Knowledge related to subsurface operations (groundwater remediation, water resources utilization, insitu leaching, heap leaching, oil reservoir management) relates to many scales of representation, yet, this fact has not been taken into account in an organized manner for solving aquifer management problems. Fortunately, the rapid development of aquifer management optimization and control in recent years has laid the foundation for such integration. These advances can be roughly divided into three major categories:

(2) Control theories
- Deterministic control-deterministic feedback (DF), differential dynamic programming (DDP), (Jones et al. 1987; Jones 1992).
- Stochastic control-dynamic dual control or differential control, combined with differential calculus, small perturbations, and Kalman filter or other stochastic inverse models (Andricevic and Kitanidis 1990; Lee and Kitanidis 1991, 1996; Georgakakos and Vlatsa 1991; Culver and Shoemaker 1992, 1993; Whiffen and Shoemaker 1993; Andricevic 1993; Philbrick and Kitanidis 1998, 2000).

(3) Artificial intelligence
- (AI; also called "soft computing" and "machine learning") and Search (optimization) algorithms, particularly artificial neural networks (ANN; e.g., Boger and Guterman 1997; Boger 2002; Rogers and Dowla 1994; Bhattacharya et al. 2003), genetic algorithms (GA; e.g., Rogers and Dowla 1994; McKinney et al. 1994; Maskey et al. 2000, 2002), fuzzy logic (FL; e.g., Dou et al. 1997; Wong et al. 2002; Lobbrecht and Solomatine 2002), simulated annealing (Dougherty and Marryott 1991), tabu search (Zheng and Wang 1999), and kriging interpolation (of response surfaces in the state space, generated by sensitivity coefficients; e.g., Landa and Gu"yagu"ler 2003).

It should be mentioned that the distinction between decision analysis and optimization/control seems rather arbitrary. Freeze and Gorelick (1999) suggested that "the fundamental difference between these two approaches lies in the fact that decision analysis considers a broad suit of technological strategies from which one of many predetermined design alternatives is selected as the best, while stochastic optimization determines the optimal pump-andtreat design but considers only one technological strategy at a time". In other words, decision analysis is a crude (low-resolution) process of searching for a discrete set of values for the decision variables by calculating the objective function for every discrete design alternative. Later in the text, a framework that unifies these two approaches is suggested.

On another front, the recent extension of uncertainty theories to the conceptual models themselves (or model structure) by Neuman (2003); and Ye et al. (2004) is an important milestone that enables the next step in stochastic optimization/control in hydrogeology and other areas. In the following review, recent developments in optimal watershed and surface water are not included, but merely touch on a few recent advances in optimal aquifer management, with some parallels in oil reservoir management, in order to highlight the links between past developments and future optimization-planning-control of hydrogeologic systems.

Optimization under uncertainty

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President and CEO, Shlomo Orr, PhD, Peng, has over 33 years of extensive consulting, research, and project management experience in the field of Hydrology and Water Resources. He earned a PhD in Hydrology and Water Resources with a minor in Soil and Water Sciences, and BSc and MSC in Civil Engineering. Dr. Orr's background includes modeling, planning, and controlling complex subsurface flow and transport phenomena. He has a broad background in conceptual and computational-numerical modeling of fluid flow and solute transport in saturated and unsaturated porous and fractured formations, including stochastic models that account for uncertainties and provide the basis for risk assessment.

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