Abstract
Recognizing that simple watershed conceptual models such as the Nash cascade ofn equal linear reservoirs continue to be reasonable means to approximate the Instantaneous Unit Hydrograph (IUH), it is natural to accept that random errors generated by climatological variability of data used in fitting an imprecise conceptual model will produce an IUH which is random itself. It is desirable to define the random properties of the IUH in a watershed in order to have a more realistic hydrologic application of this important function. Since in this case the IUH results from a series of differential equations where one or more of the uncertain parameters is treated in stochastic terms, then the statistical properties of the IUH are best described by the solution of the corresponding Stochastic Differential Equations (SDE's). This article attempts to present a methodology to derive the IUH in a small watershed by combining a classical conceptual model with the theory of SDE's. The procedure is illustrated with the application to the Middle Thames River, Ontario, Canada, and the model is verified by the comparison of the simulated statistical measures of the IUH with the corresponding observed ones with good agreement.
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References
Bernier, J. 1970: Invaentaire des modeles de processus stochastiques a la description des debits journaliers des rivieres. Revue de l'Institut International de Statistique 38, 49–61
Bodo, B.A.; Unny, T.E. 1987: On the outputs of dtochastisized Nash-Dooge linear reservoir cascades. In: MacNeill, I.B.; Umphrey, G.I. (eds.) Stochastic hydrology, pp. 131–137, Reidel Publ. Co.
Bras, R.; Rodriguez-Iturbe, I. 1985: Random functions in hydrology. Addison Wesley
Chow, V.T.; Maidment, D.R.; Mays, L.W. 1988: Applied hydrology. New York: McGraw-Hill Book Co.
Dooge, J.C.I. 1959: A general theory of the unit hydrograph. J. Geophys. Res. 64, 241–256
Dooge, J.C.I. 1977: Problems and methods of rainfall-runoff modeling. In: Cirani, T.T.; Maione, U.; Wallis, J.R. (eds.) Mathematical models for surface water hydrology, pp. 70–101, John Wiley & Sons
Haan, C.T. 1977: Statistical methods in hydrology. Ames, Iowa: The Iowa State University Press
Huggins, L.F.; Burney, J.R. 1982: Surface runoff, storage and routing. In: Haan, C.T.; Johnson, H.P.; Brakensiek, D.L. (eds.) Hydrologic modeling of small watersheds, Am. Soc. Agr. Eng. Publ. No. 5, pp. 169–224
Karlsson, M.; Yakowitz, S. 1987: Rainfall-runoff forecasting methods: Old and new. Stoch. Hydrol. Hydraul. 1, 304–318
Koch, R.W. 1985: A stochastic streamflow model based on physical principles. Water Res. Res. 21, 545–553
Linsley, R.K.; Kohler, M.A.; Paulhus, J.L.H. 1982: Hydrology of engineers. New York: McGraw-Hill Book Co., 2nd ed.
Moran, P.A.P. 1971: Dams in series with a continuous release. J. Appl. Prob. 4, 380–388
Nash, J.E. 1957: The form of the instantaneous unit hydrograph. Int. Assoc. Hydrol. Sci., General Assembly of Toronto., Publ. 45, 114–119
Quimpo, R.G. 1971: Structural relation between parametric and stochastic hydrology. In: Mathematical models in hydrology, vol. 1, Int. Assoc. Hydrol. Sci., Publ. 100; 151–157
Quimpo, R.G. 1973: Link between stochastic and parametric hydrology. J. Hydraul. Div. Am. Soc. Civil Eng. 99, 461–470
Serrano, S.E. 1988: General solution to random advective-dispersive equation in porous media. Part 2: stochasticity in the parameters. Stoch. Hydrol. Hydraul. 2, 99–112
Serrano, S.E.; Unny, T.E. 1987: Predicting groundwater flow in a phreatic aquifer. J. Hydrol. 95, 241–268
Serrano, S.E.; Whitley, H.R.; Irwin, R.W. 1985: Effects of agricultural drainage on streamflow in the Middle Thames River, Ontario, 1949–1980. Can. J. Civil Eng. 12, 875–885
Singh, V.P. 1988: Hydrologic systems: Rainfall-runoff modeling. Vol. 1. New Jersey: Prentice Hall
Soong, T.T. 1973: Random differential equations in science and engineering. New York: Academic Press
Sorooshian, S. 1983: Surface water hydrology: On-line estimation. Rev. Geophys. Space Phys. 21, 706–721
Unny, T.E.; Karmeshu, 1984: Stochastic nature of outputs from conceptual reservoir model cascades. J. Hydrol. 68, 161–180
Unny, T.E. 1984: Numerical integration of stochastic differential equations in cathment modeling. Water Res. Res. 20, 360–368
Unny, T.E. 1987: Solutions to non-linear stochastic differential equations in catchment modeling. In: Mac-Neill, I.B.; Umphrey, G.J. (eds) Stochastic Hydrology, pp. 87–110, Kluwer Academic Publisher
Weiss, G. 1973: Shot noise models for synthetic genration of multisite daily streamflow data. In: Water resources projects with inadequate data. Int. Assoc. Hydrol. Sci. Publ. 108, 457–467
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Sarino, Serrano, S.E. Development of the instantaneous unit hydrograph using stochastic differential equations. Stochastic Hydrol Hydraul 4, 151–160 (1990). https://doi.org/10.1007/BF01543288
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DOI: https://doi.org/10.1007/BF01543288