Random Variables vs. Uncertain Values: Stochastic Modeling and Design

Recent decades have seen great progress in the use of stochastic methods to model aspects of water resource problems. However, the design implications of stochastic modeling have been relatively overlooked. In particular, there is little explicit realization that the stochastic modeling of single-valued, yet uncertain phenomena yields qualitatively different information than the stochastic modeling of multi-valued and uncertain phenomena in terms of the role of the modeled uncertainty in design decision-making. This paper explores this issue by way of some simple examples. INTRODUCTION Stochastic modeling has become a well accepted, but still controversial set of methods in water resources. The water resources engineer now has a plethora of analytical, approximate, and Monte Carlo techniques available for propagating probability distributions for input variables and model parameters through a set of model equations to estimate the probability distribution of model output variables (Burges and Lettenmaier 1975; Loucks et al. 1981). A large number of techniques are also available for incorporating random variables into optimization models (Tung 1986; Wagner 1975). These methods have greatly increased the sophistication of how uncertainty is considered in engineering design. However, at times applications of stochastic modeling and optimization seem somewhat automatic and ill-considered for a particular design context. This paper argues that stochastic modeling of physically deterministic and single-valued phenomena with imperfectly known characteristics yields a qualitatively different form of information than stochastic modeling of phenomena which are physically time-varying and random. This difference is illustrated for two hypothetical examples: the selection of a water supply source and the design of a wastewater treatment plant. These examples are not intended to necessarily illustrate optimal design methodologies for such facilities. Rather, for each example, decision theoretic formulations reveal the distinction between these two types of uncertainty and the potential importance of this distinction. UNCERTAINTY: RANDOM VERSUS IMPERFECTLY KNOWN VALUES Uncertainty generally implies that one is unsure of the particular value a variable will take on. A variable's value may be uncertain both if the variable is single-valued, deterministic, and constant, but has an imperfectly known value, or if the variable's value is constantly fluctuating with a random pattern. In both cases, a variable's uncertainty may be expressed by a probability distribution. In the first case, the singlevalued variable's value will often become increasingly well-known through experimentation and experience. In the second case, the value of a multi-valued fluctuating variable, such as streamflow or precipitation, may become no better known as experience with the variable accumulates. This practical difference in types of uncertain variables mirrors the difference between frequentist and Bayesian approaches to probability theory (Jaynes 1986). Before a reservoir is built, its cost is always an uncertain estimate. But upon completion, the cost, which is a single-valued variable, is well known. However, the streamflow entering the reservoir is likely to remain almost as uncertain after the reservoir's construction as before. This distinction between variables with imperfectly known values and those with randomly fluctuating values can be crucial for engineering design or alternative evaluation problems. This is proven through some simple examples. In each case, important engineering distinctions arise: 1) because some designs allow more flexibility as more is learned about the project's environment while other alternatives are tailored more specifically for a narrow range of