VOL. 13, NO. 4 WATER RESOURCES RESEARCH AUGUST 1977 Bayes Designof a Reservoir Under Random Sediment Yield LUCIEN DUCKSTEIN Departments of Systems and IndustrialEngineering and Hydrologyand Water Resources University of Arizona, Tucson, Arizona 85721 FERENC SZIDAROVSZKY Department of Numerical Methods, 1088 Budapest VIII, Muzeum krt. 6-8 Eotvos University, Hungary SIDNEY YAKOWITZ Department of Systems and IndustrialEngineering, University of Arizona, Tucson, Arizona 85721 The designof a reservoirsubject to long-range sediment accumulation stemming from the sum of a random number of random sedimentation events is investigated. The event-based simulationmethod, which is applied to a case studyin southern Arizona, involves generating synthetic sequences of Poisson inputsinto the modifieduniversal soil loss equation.The stochastic inputsresult from a fitted bivariate distributionof runoff-producing precipitation events (representing the amount and duration of such precipitation)and an independent fitted exponential distributionof interarrivaltime between events. The simulatedsequences of sediment yield events thus obtained are usedto calculateaccumulated sediment yield and costof a givendesign for each sequence. The optimum design and corresponding Bayes risk are evaluated in four cases: (1) undernatural uncertainty, (2) undernatural uncertainty and uncertainty in the bivariate rainfall distributionparameters, (3) under natural uncertaintyand uncertaintyin the Poisson counting distributionparameter, and (4) under all three types of uncertainty. The effectof rainfall record length is ascertained by further computerexperiments, but only a partial Bayesian analysis is provided because of the complexity created by a three-dimensional parameteruncertainty. The optimum reservoir capacity and corresponding Bayesrisk are shown to increase substantially (up to 20 and 90%, respec- tively) as more uncertainties are incorporated into the model. INTRODUCTION The purpose of this paper is to provide a Bayesian method- ology for designing a reservoir subject to random sedimenta- tion. More specifically, the approach is meantto provide(1) a probabilistic forecast of long-range sediment yield applicable to an ungagedwatershed, (2) an optimum decision under natural and parameter uncertainty, and (3) a Bayesian eval- uation of the worth of hydrologic record length used in the decision; this evaluation is only partially provided in this study because of the complex nature of the uncertainty. The ap- proach, which is illustrated by the case of the Charleston watershed in southern Arizona, extends the research described by Szidarovszky et al. [1976],in which no decision analysis was performed. Furthermore, the elementsof the presentmodel are similar to those in the model of Smith et al. [1974, 1977], but the methodology is different, as will be explained later. The designof a storage reservoir subjectto sedimentation must include a 'dead storage'volume to accountfor sediment that accumulates during the useful lifetime of the structure [Jacobi, 1971]. An underdesign of the reservoir leads to a shorteningof the lifetime [McHenry, 1974] or to additional removal costs,while an overdesign entails unnecessary con- structioncost. There is thus a strong need for obtaining an accurate forecast of the sedimentation to take placeduringthe decision horizon. Sediment yield is most commonly estimated by a linear regression of mean annual yield on watershedand climatic characteristics [Flaxman, 1974;Nordin and Sabol, 1973; Jansen and Painter, 1974; Guymon,1974; McPherson, 1975]. The res- ervoir is designed on the basis of these estimated annualquan- Copyright¸ 1977by the AmericanGeophysical Union. tities; however, as was pointed out by Weber et al. [1976], serious questions may arise about the validity of suchregres- sion resultswhen the wrong transformation is used or when the assumptions of the model are violated. This is especially true in regions with intermittent or ephemeral flow, where sediment accumulated over one or more seasons may be mod- eled as the sumS of a random number of sediment yield events [Woolhiser and Todorovic, 1974].Then, for a Bayesian analysis accountingfor the uncertainty on S the distribution function (DF) of S, Fs(s), must be estimated.Rare are the cases when available time series data of sediment yield events are sta- tionary and long enoughto estimate Fs(s); on the other hand, since precipitationrecords are usually adequate,it is possible to use models transforming rainfall into sediment yield to estimatethe DF of S. At this point it may be noted that the term 'ungaged watershed'refers to a watershedin which no sediment yield measurements are available. Should rainfall records be unavailable aswell, precipitationdata mustthen be transferred from a neighboring watershed, as has beendone in the presentcasestudy. Sediment produced by an individualprecipitation eventmay be estimatedby the use of a physical model [Bennett, 1974; Renardand Laursen,1975]or an empiricalmodel such as the universal soil loss equation [Wischmeier and Smith, 1965; On- stad and Foster, 1975; Williams, 1975]. The latter equation is in fact used here and by Smith et al. [1974, 1977], with the following fundamental difference:whereas Smith et al. calcu- lated the DF of S by a transformation of random variables, a simulation approachis usedin the present investigation. The next section describes the methodology, which is then applied to the Charlestonwatershed in the subsequent 'application' section. Paper number 7W0350. 713