Mills, William Carlisle; Department of Hydrology & Water Resources, The University of Arizona (Department of Hydrology and Water Resources, University of Arizona (Tucson, AZ), 1980-06)
      Many planning decisions related to the land phase of the hydrologic cycle involve uncertainty due to stochasticity of rainfall inputs and uncertainty in state and knowledge of hydrologic processes. Consideration of this uncertainty in planning requires quantification in the form of probability distributions. Needed probability distributions, for many cases, must be obtained by transforming distributions of rainfall input and hydrologic state through deterministic models of hydrologic processes. Probability generating functions are used to derive a recursive technique that provides the necessary probability transformation for situations where the hydrologic output of interest is the cumulative effect of a random number of stochastic inputs. The derived recursive technique is observed to be quite accurate from a comparison of probability distributions obtained independently by the recursive technique and an exact analytic method for a simple problem that can be solved with the analytic method. The assumption of Poisson occurrence of rainfall events, which is inherent in derivation of the recursive technique, is examined and found reasonable for practical application. Application of the derived technique is demonstrated with two important hydrology- related problems. It is first demonstrated for computing probability distributions of annual direct runoff from a watershed, using the USDA Soil Conservation Service (SCS direct runoff model and stochastic models for rainfall event depth and watershed state. The technique is also demonstrated for obtaining probability distributions of annual sediment yield. For this demonstration, the-deterministic transform model consists of a parametric event -based sediment yield model and the SCS models for direct runoff volume and peak flow rate. The stochastic rainfall model consists of a marginal Weibull distribution for rainfall event duration and a conditional log -normal distribution for rainfall event depth, given duration. The stochastic state model is the same as used for the direct runoff application. Probability distributions obtained with the recursive technique for both the direct runoff and sediment yield demonstration examples appear to be reasonable when compared to available data. It is, therefore, concluded that the recursive technique, derived from probability generating functions, is a feasible transform method that can be useful for coupling stochastic models of rainfall input and state to deterministic models of hydrologic processes to obtain probability distributions of outputs where these outputs are cumulative effects of random numbers of stochastic inputs.

      Lovell, Robert Edmund,1921-; Department of Hydrology & Water Resources, The University of Arizona (Department of Hydrology and Water Resources, University of Arizona (Tucson, AZ), 1975-06)
      The problem of selecting appropriate mathematical models for use in studying hydrological phenomena has created a situation in which the choice of suitable models by hydrologic practitioners has become exceedingly complex. The extensive comments in the literature indicate that neither the traditional system of technical journals nor the more modern computer -based retrieval schemes have really solved the problem. Further examination shows that similar problems have arisen in many fields, hence a well organized attack on the specific problem of hydrologic model choice can have a more general application. The present problem is identified as a requirement to codify and make accessible to users information in a more directly user oriented format. The problem of model choice arises at several levels, ranging from decision on what fundamental structure to use, to choice of parameters, and on to model calibration and validation. This paper is focused on a scheme to aid in model structure choice. The essential ingredients of model structure choice, and indeed of many choice processes, are extracted and embedded in a generalized set theoretic mathematical notational framework in order to give some insight into the nature of the problem. Within this framework the specialized features of the model choice problem are analyzed, and a specialized model is developed for assisting in model choice and all problems similarly situated. These considerations lead to the development of a finite vector of objective statements with codified responses prepared by a panel of qualified researchers who are willing and able to construct the essential information in a user oriented format. It is required that the panel not only couch their information in objective oriented terms but that they also generate value judgments for the individual components. In this way, those using the system can take advantage of the expert opinions embedded in the model while, at the same time, tailoring the choice to meet their own specific needs and aspirations. This results in what is defined as a mathematical CHOICEMODEL. The implementation of a system for interactive computation of the CHOICEMODEL is described in detail, and the associated computer programs are presented in appendices. A detailed instruction manual is given, and the implementation of the method is illustrated by an easily understood model of the ingredients of the problem of selecting an 8 -track stereo tape deck for home use. The plan is outlined whereby hydrologic choice models can be developed within the CHOICEMODEL system by a selected panel of expert EVALUATORs.