The traditional water-supply planning problem is characterized by two main steps: (1) project future water requirements based on present rates of economic growth, 'and (2) schedule water development projects to be introduced into the system on time to meet these predicted requirements. If alternative projects are thought to exist, the one thought to cost the least amount is selected. As project costs rise and actual new water availabilities become less, there is a growing awareness that more new water is not necessarily the only answer. Increased efficiency in water use through conservation, reuse, transfer to less consumptive and higher valued applications, and improved management techniques are becoming practical alternatives. These alternatives lead to a need for a restatement of water-supply planning objectives in more precise forms than have heretofore been put forth. The various water- supply planning objective functions including the traditional one are all expressions which maximize the difference between gains and los se s involved with water development. They can be expressed mathematically and differentiated on the basis of how these gains and losses are defined. In the traditional sense, gains derived from meeting projected requirements are assumed to be infinite, and losses are taken to be actual project costs; therefore, maximization of net gains is accomplished by minimizing project costs and gains do not even have to be expressed. Consideration of alternatives, however, requires that gains be expressed quantitatively as benefits to individuals, communities, or regions, i. e. , primary, secondary, or tertiary benefits. The same thing holds for the expression of total costs. An objective function used to express the water-supply problem in the Tucson Basin, Arizona, considers gains as cash revenue to a hypothetical central water-control agency which sells water to the users within the basin. Losses are considered as marginal costs to the agency for producing, treating, and distributing water. The concept of economic demand is used to estimate the amount of water that municipal, agricultural, and industrial users will purchase at different prices. The possible sources of supply considered are groundwater from within the basin, groundwater from the neighboring Avra Valley Basin, reclaimed waste water, and Central Arizona Project water from the Colorado River. Constraints are formulated in order to determine optimal allocations of water under different conditions. The model used is referred to as a pricing model and is optimized by first decomposing the objective function into component parts, each part representing terms involving only one source of water. Then in instances involving inequality constraints, quadratic programming is used. In other instances where equality constraints or unconstrained conditions exist, Lagrangian multipliers and the calculus are used. The se latter conditions arise when it is determined at which point certain constraints become inactive. In the completely general case, this type of decomposition is not possible, but it appears that in many specific uses objective functions of this nature can be profitably decomposed. and optima determined much more conveniently than otherwise possible.

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The traditional metropolitan water -supply planning problem is characterized by two main steps: (a) project future water requirements based on present rates of economic growth,, and (b) schedule water development projects to be introduced into the system on time to meet these predicted requirements. The City of Tucson plans its water supply essentially in this manner. The prime objective of this phase of our research was to formally review the above problem and to formulate it in terms of concepts of management science. Implied commitments to accept Colorado River water and gradual changes in quality of Tucson's groundwater force serious consideration of the economic tradeoffs between alternative sources and uses of water. These alternatives lead to a need for a restatement of water - supply planning objectives in more precise forms than have heretofore been put forth. The doctoral dissertation by G. Clausen addresses itself to the above restatement with actual data on the Tucson basin. The various water -supply planning objective functions including the traditional one are all expressions which maximize the difference between gains and losses involved with water development. They can be expressed mathematically and differentiated on the basis of how these gains and losses are defined. In the traditional sense, gains derived from meeting projected requirements are assumed to be infinite, and losses are taken to be actual project costs and not social costs associated with undesirable economic growth. Therefore, maximization of net gains is accomplished by minimizing project costs, and gains do not even have to be expressed. Consideration of alternatives, however, requires that gains be expressed quantitatively as benefits to individuals, communities, or regions, i.e., primary, secondary, or tertiary benefits. The same logic holds for the expression of total costs. An objective function, used to express the water- supply problem in the Tucson Basin, considers gains as cash revenue to a hypothetical central water - control agency which sells water to the users within the basin. Losses are considered as marginal costs to the agency for producing, treating, and distributing water. The concept of economic demand is used to estimate the amount of water that municipal, industrial, and agricultural users will purchase at different prices. Linear demand functions are postulated. The possible sources of supply considered are groundwater from within the basin, groundwater from the neighboring Avra Valley Basin, reclaimed waste water, and Central Arizona Project water from the Colorado River. Constraints are formulated to allow for limits on water availability, for social limits on water prices, and for minimal requirements of each user over a specified time period; these permit a determination of optimal allocations of water under different conditions to answer "what if' questions, given the assumptions of the model. The resulting static model is termed a pricing model and is optimized by first decomposing the objective function into component parts with each part representing terms involving only one source of water. In instances involving inequality constraints, quadratic programming is used. In other instances where equality constraints or unconstrained conditions exist, Lagrange multipliers and calculus methods are used. These latter conditions arise when it is determined at which point certain constraints become inactive. In the completely general case, this type of decomposition is not possible, but it appears that in many specific uses objective functions of this nature can be profitably decomposed and optima determined much more conveniently than otherwise possible. The model clearly identifies the opportunity costs associated with the required use of Colorado River water in lieu of the cheaper Tucson groundwater.

Inventory-prediction equations describing snowpack water content as functions of readily available or easily obtained inventory variables were developed for use in the ponderosa pine type in Arizona. Although empirical in nature, these equations include parameters assumed to index interception of precipitation inputs, obstruction of direct beam solar radiation, and re-radiation from trees onto the snowpack. Primary consideration was given to forest cover variables in synthesizing the inventory-prediction equations I because currently proposed water improvement programs designed to increase water yield derived from snow consist essentially of vegetative manipulations. Additional independent variables evaluated include potential direct beam solar radiation, elevation, soil, and precipitation inputs. All of the inventory-prediction equations describing a particular snowpack condition were not statistically equivalent in terms of the standard error of estimate or the coefficient of determination. Equations including basal area, bole area I volume, and height-index as expressions of forest cover density were generally better than equations with point density, sum of diameters, and number of trees. Inventory-prediction equations developed to describe snowpack dynamics throughout the accumulation period showed similar statistical form, except as possibly attributable to different precipitation inputs. Equations for characterizing residual snowpacks during spring runoff were statistically weak, possibly because factors other than those considered in this study control the runoff process. The inventory-prediction equations were developed to estimate the mean snowpack water content on a basin, and to describe the trade-off , or the rate of exchange, between snowpack water content and forest-site variables on a decision-making unit. The equations do not necessarily predict changes in recoverable water yield resulting from the implementation of a land management system, however. Nonbiotic characteristics of the land, L e., topographic features, geologic formations, and soil . properties, could conceivably control water yield to the extent that changes predicted by the inventory-prediction equations could be masked. Because of limitations in predicting potential changes in recoverable water yield, it was assumed that a land management system that maximizes snowpack water content on site would also provide the maximum potential for increasing recoverable water yield derived from snow. Management guidelines designed to allow snowpack water content to be maximized on site can be formulated within the framework of the inventory-prediction equations, multiple use management constraints, and forest-based product benefits and costs. Management guidelines indicate that the greatest gain in snowpack water content on site would be realized on decision-making units where the greatest reduction in forest cover density could be prescribed. However, a timber production constraint may limit the array of management possibilities. This constraint was defined as 35 to 40 square feet of basal area or 1,050 to 1,175 cubic feet of volume per acre, depending upon the existing growth percent and the intermingling of tree volumes and size classes. The potential increase in snowpack water content on site will be determined by the magnitude of the reduction in forest cover density and how close management re-direction can approach the timber production constraint. The proportion of the snowpack water content on site converted to recoverable water yield is dependent upon the runoff efficiency.

Project Completion Report, OWRT Project No. A-086-ARIZ / Agreement No. 14-34-0001-8003, Project Dates: 10/01/77-9/30/78 / Acknowledgement: The work upon which this report is based was supported by funds provided by the State of Arizona and the United States Department of Interior, Office of Water Research and Technology as authorized under the Water Resources Act of 1964. / From the Introduction: "The research work under this contract has been conducted by graduate student David B. Dunbar and summarized in his M.S. thesis entitled "Analysis of a Parameter Estimation Technique for Linear Hydrologic Systems Using Monte Carlo Simulation" submitted to the Department of Hydrology and Water Resources, University of Arizona, Tucson, in 1981. The present report is a brief summary of Mr. Dunbar's thesis." David Dunbar's thesis is available at: http://arizona.openrepository.com/arizona/handle/10150/191728 / The primary accomplishment of this research has been demonstrating the power of the deconvolution technique developed by Neuman and de Marsily (1976) in dealing with noisy rainfall- runoff records of short duration. Such records are encountered in arid environments where rainfall often occurs in short isolated bursts and the data are measured with a considerable margin of error. Our research work consisted of superimposing known noise on synthetic rainfall- runoff data and examining the ability of the Neuman -de Marsily deconvolution method to estimate the correct impulse response of the system when the data include only a single storm event. Approximately 50 Monte Carlo simulation runs were performed for each of three different noise models considered in our work. The results clearly demonstrated that the deconvolution model leads to reliable estimates and can be used with confidence in the presence of realistic noise levels. In addition to the Monte Carlo simulation tests and their analysis, certain improvements were introduced into the original deconvolution technique. In particular, the original version of the technique required that the hydrologist exercise subjective judgement in choosing the "best" solution for the deconvolution problem from a large number of admissible solutions. Our new method of selecting the "best" result is based on a comparative analysis of residuals and is more reliable than the earlier subjective approach. The improved method has been applied to real as well as synthetic rainfall -runoff data.

Forecast models are investigated and developed for use in local water management to aid in determining short term water requirements and availability. The forecast models include precipitation occurrence and depth using a Markov chain model, temperature and solar radiation with a multivariate autoregressive model, and streamflow with autoregressive-moving average models. The precipitation, temperature, and solar radiation forecasts are used with a soil moisture model to determine water demands. A state space approach to the Muskingum-Cunge streamflow routing technique is developed. The forecast water demands and streamflow forecasts are used as inputs to this routing model. Forecast model errors and propagation of these errors from one model into the next are investigated.

Abstract not available