Abstract
Stochastic models applicable to hydrologic time series and extreme events are investigated in earlier chapters. Such models are important for the simulation of complex water resource systems and for flood estimation purposes. However, in the design of individual reservoirs where one of the main criteria is the probability of failure, a more direct approach can be adopted if the inflow data are independent or have a Markov type of dependence. The choice between alternatives is usually made so that this probability does not exceed a stipulated value, which depends on the purposes served. Also important is the average number of times a reservoir will spill or empty during a given period. Another interesting outcome is the probability of first time emptiness.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Benjamin, J. R., and Cornell, C. A. (1970). Probability, Statistics and Decision for Civil Engineers, McGraw-Hill, New York
Billingsley, P. (1961). Statistical Inference for Markov Processes, University of Chicago Press, Chicago, Illinois
Brooks, C. E. P., and Carruthers, N. C. (1953). Handbook of Statistical Methods in Meteorology, H. M. Stationary Office, London, pp. 281–339
Chin, E. H. (1977). Modelling daily precipitation occurrence process with Markov chain. Water Resour. Res., 13, 949–56
Chung, K. L. (1967). Markov Chains with Stationary Transition Probabilities, 2nd edn, Springer, Berlin
Cole, J. A. (1966). Application of two season statistics to reservoir yield calculations. Proceedings of the International Association of Scientific Hydrology Symposium, Garda, Italy, pp. 590–1
Cox, D. R., and Miller, H. D. (1965). The Theory of Stochastic Processes, Methuen, London
Cox, D. R., and Smith, W. L. (1961). Queues, Monographs on Applied Probability and Statistics, Methuen, London
Crovelli, R. A. (1973). Stochastic models for precipitation. Proceedings of the International Symposium on Uncertainties in Hydrologic and Water Resource Systems, vol. 1, 11–14 December 1972, University of Arizona, Tucson, pp. 284–98
David, F. N., and Barton, D. E. (1962). Combinatorial Chance, Griffin, London
Doran, D. G. (1975). An efficient transition definition for discrete state reservoir analysis: the divided interval technique. Water Resour. Res., 11, 867–73
Dyck, S., and Schramm, M. (1966). Queuing theory and multipurpose reservoir design. Proceedings of the International Association of Scientific Hydrology Symposium, Garda, Italy, paper 2 (71), pp. 707–10
Encyclopaedia Britannica (1977), vol. 14, 15th edn, William Benton, Chicago, Illinois, pp. 1113–15
Farmer, E. E., and Homeyer, J. W. (1974). The probability of consecutive rainless days. Water Resour. Bull., 10, 914–24
Feller, W. (1968). Introduction to Probability Theory and its Applications, vol. 1, 3rd edn, Wiley, New York
Fiering, M. B. (1962). Queuing theory and simulation in reservoir design. Trans. Am. Soc. Civ. Eng., 127, part I, paper 3367, 1114–44
Gabriel, K. R., and Neumann, J. (1962). A Markov chain model for daily rainfall occurrence at Tel Aviv. Q. J. R. Meteorol. Soc., 88, 90–5
Gani, J. (1957). Problems in the probability theory of storage systems. J. R. Statist. Soc. B, 19, 181–206
Gani, J. (1965). Flooding models. Proceedings of the Reservoir Yield Symposium,21–23 September 1965, Water Research Association, Medmenham, paper 4, pp. 4–1–4–16
Gillett, B. E. (1976). Introduction to Operations Research, A Computer Oriented Algorithmic Approach, McGraw-Hill, New York
Gnedenko, B. V. (1968). The Theory of Probability, 4th edn (translated from Russian by B. D. Seckler), Chelsea, New York
Gould, B. W. (1961). Statistical methods for estimating the design capacity of dams. J. Inst. Eng. Aust., 33, 405–16
Gould, B. W. (1966). Communication on `Probability of reservoir yield failure using Moran’s steady state method and Gould’s probability writing method’ by R. A. Harris. J. Inst. Water Eng., 20, 141–6
Green, J. R. (1964). A model for rainfall occurrence. J. R. Statist. Soc. B., 26, 345–53
Green, J. R. (1970). A generalized probability model for sequences of wet and dry days. Mon. Weath. Rev., 98, 238–41
Gross, D., and Harris, C. M. (1974). Fundamentals of Queuing Theory, Wiley, New York
Harris, R. A. (1965). A probability of reservoir yield failure using Moran’s steady state probability method and Gould’s probability routing method. J. Inst. Water Eng., 19, 302–28
Harris, R. A., Dearlove, R. E., and Morgan, M. (1965). Reservoir yield, 2, serially correlated inflows and subsequent attainment of steady state probabilities, Water Res. Assoc., Medmenham, Tech. Paper, No. 45
Hershfield. D. M. (1971). Parameter estimation for wet-dry sequences. Water Resour. Bull., 7, 441–6
Hillier, F. S., and Lieberman, G. J. (1974). Operations Research, 2nd edn, Holden Day, San Francisco, California
Isaacson, D. L., and Madsen, R. W. (1976). Markov Chains—Theory and Applications, Wiley, New York
Jarvis, C. L. (1964). An Application of Moran’s theory of dams to the Ord River project, western Australia. J. Hydrol., 2, 232–47
Joy, C. S., and McMahon, T. A. (1972). Reservoir-yield estimation procedures. Civ. Eng. Trans., Inst. Eng. Aust., 14, 28–36
Kemeny, J. G., and Snell, J. L. (1960). Finite Markov Chains, Van Nostrand, Princeton, New Jersey
Kendall, D. G. (1957). Some problems in the theory of dams. J. R. Statist. Soc. B., 19, 207–33
Kiemeš, V. (1970). A two-step probabilistic model of storage reservoir with correlated inputs. Water Resour. Res., 6, 756–67
Kiemeš, V. (1971). On one difference between the Gould and Moran storage models. Water Resour. Res., 7, 410–4
Kiemeš, V. (1974). Probability distribution of outflow from a linear reservoir. J. Hydrol., 21, 305–14
Kiemeš, V. (1977). Discrete representation of storage for stochastic reservoir optimization. Water Resour. Res., 13, 149–58
Langbein, W. B. (1958). Queuing Theory and Water Storage. J. Hydraul. Div., Am. Soc. Civ. Eng., 84 (HY5), 1811–1–1811–24
Lloyd, E. H. (1963). A probability theory of reservoirs with serially correlated inputs, J. Hydrol., 1, 99–128
Lloyd, E. H. (1967). Stochastic reservoir theory. Adv. Hydrosci., 4, 281–339
Lloyd, E. H. (1974). What is, and what is not, a Markov chain. J. Hydrol., 22, 1–28
Lloyd, E. H., and Odoom, S. (1964). Probability theory of reservoirs with seasdnal input. J. Hydrol., 2, 1–10
Lowry, W. P., and Guthrie, D. (1968). Markov chains of order greater than one. Mon. Weath. Rev., 96, 798–801
Moran, P. A. P. (1954). A probability theory of dams and storage systems. Aust. J. Appl. Sci., 5, 116–24
Moran, P. A. P. (1955). A probability theory of dams and storage systems, modifications of the release rule. Aust. J. Appl. Sci., 6, 117–30
Moran, P. A. P. (1959). The Theory of Storage, Methuen, London
Mosteller, F. (1965). Fifty Challenging Problems in Probability, Addison-Wesley, Reading, Massachusetts
National Bureau of Standards (1959). Tables of the bivariate normal distribution function and related functions. U.S. Dep. Comm., Washington, D.C., Appl. Math. Ser., No. 50
Parzen, E. (1962). Stochastic Processes, Holden Day, San Francisco, California
Pegram, G. G. S. (1974). Factors affecting draft from a Lloyd reservoir. Water Resour. Res., 10, 63–6
Phatarfod, R. M. (1976). Some aspects of stochastic reservoir theory. J. Hydrol., 30, 199–217
Prabhu, N. U. (1958). Some exact results for the finite dam. Ann. Math. Statist., 29, 1234–43
Prabhu, N. U. (1964). Time dependent results in storage theory. J. Appl. Prob.,1, 1–64
Reardon, T. J. (1970). Storage yield and probability from an engineer’s viewpoint (with discussion). Civ. Eng. Trans., Inst. Eng. Aust., 12, 119–24, 168, 169
Revuz, D. (1975). Markov Chains, North-Holland, Amsterdam
Ross, S. M. (1972). Introduction to Probability Models, Academic Press, New York
Saverenskiy, A. D. (1940). Metod rascheta regulirovaniya stoka. Gidrotekhnicheskoe Stroitel’stvo, No. 2, 24–8
Shamblin, J. E., and Stevens, G. T., Jr. (1974). Operations Research, a Fundamental Approach, McGraw-Hill, New York
Strang, G. (1976). Linear Algebra and its Applications, Academic Press, New York
Venetis, C. (1969a). Conditional probabilities of failures in reservoir operation for the Moran–Gould model. Water Resour. Res., 5, 514–18.
Venetis, C.(1970). Comments. Water Resour. Res., 6, 1427–32
Venetis, C. (1969b). On the distribution of the frequency of reservoir deficit, J. Hydrol., 8, 341–6.
Venetis, C. (1970). Correction. J. Hydrol., 10, 103–4. Discussion. J. Hydrol.,10, 199–201
Venetis, C. (1969c). A stochastic model of monthly reservoir storage. Water Resour. Res., 5, 729–34.
Venetis, C. (1970). Correction. Water Resour. Res. 6, 351
Weesakul, B. (1961). First emptiness of a finite dam. J. R. Statist. Soc. B, 23, 343–51
Weiss, L. L. (1964). Sequences of wet or dry days described by a Markov chain probability model. Mon. Weath. Rev., 92, 169–76
White, J. B. (1965). Probability of emptiness, II, a variable–season model. Proceedings of the Reservoir Yield Symposium, 21–23 September 1965 Water Research Association, Medmenham, paper 6, pp. 6–1–6–11
Author information
Authors and Affiliations
Copyright information
© 1980 N. T. Kottegoda
About this chapter
Cite this chapter
Kottegoda, N.T. (1980). Probability theory applied to reservoir storage. In: Stochastic Water Resources Technology. Palgrave Macmillan, London. https://doi.org/10.1007/978-1-349-03467-3_7
Download citation
DOI: https://doi.org/10.1007/978-1-349-03467-3_7
Publisher Name: Palgrave Macmillan, London
Print ISBN: 978-1-349-03469-7
Online ISBN: 978-1-349-03467-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)