Probabilistic Solution to Stochastic Overland Flow Equation
This article is a reply.
VIEW THE ORIGINAL ARTICLEPublication: Journal of Hydrologic Engineering
Volume 8, Issue 2
Abstract
In this paper, the second in a series of two, the theory developed in the companion paper is applied to the case of the stochastic overland flow equation, and a numerical solution method is presented for the resulting Fokker-Planck equation (FPE), which describes the evolution of the probability-density function (PDF) of overland flow depth at the downstream section of a hillslope. The derived FPE is evaluated for two different approximations to the diffusion coefficient of the FPE. The Monte Carlo analysis of stochastic overland flow equation is then performed using the random rainfall sequences, generated by a compound filtered Poisson process model for the stochastic rainfall, in order to provide a benchmark for the results obtained from the FPEs. When compared to the Monte Carlo simulation based PDFs and their ensemble average, the second approximation to the diffusion coefficient gives a good fit in terms of the shape of the PDF and the ensemble average of the overland flow depth. Therefore, the theory proposed here is quite promising for obtaining the ensemble averages of nonlinear hydrological processes.
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References
Barfield, B. J., Barnhisel, R. I., Powell, J. L., Hirschi, M. C., and Moore, I. D. (1983). “Erodibilities and eroded size distribution of Western Kentucky mine spoil and reconstructed topsoil.” Rep., Institute for Mining and Minerals Research, Univ. of Kentucky, Lexington, Ky.
Bierkens, M. F. P., and Puente, C. E.(1990). “Analytically derived runoff models based on rainfall point processes.” Water Resour. Res., 26(11), 2653–2659.
Binley, A., Beven, K., and Elgy, J.(1989a). “A physically based model of heterogeneous hillslopes, 2. Effective hydraulic conductivities.” Water Resour. Res., 25(6), 1227–1233.
Binley, A., Elgy, J., and Beven, K.(1989b). “A physically based model of heterogeneous hillslopes, 1. Runoff production.” Water Resour. Res., 25(6), 1219–1226.
Cadavid, L., Obeysekera, J. T. B., and Shen, H. W.(1991). “Flood-frequency derivation from kinematic wave.” J. Hydraul. Eng., 117(4), 489–510.
Chang, J. S., and Cooper, G.(1970). “A practical difference scheme for Fokker-Planck equations.” J. Comput. Phys., 6, 1–16.
Diaz-Granados, M. A., Valdes, J. B., and Bras, R. L.(1984). “A physically based flood frequency distribution.” Water Resour. Res., 20(7), 995–1002.
Eagleson, P. S.(1972). “Dynamics of flood frequency.” Water Resour. Res., 8(4), 878–898.
Freeze, R. A.(1980). “A stochastic conceptual analysis of rainfall-runoff processes on a hillslope.” Water Resour. Res., 16(2), 391–408.
Gelhar, L. W., and Axness, C. L.(1983). “Three-dimensional stochastic analysis of macrodispersion in aquifers.” Water Resour. Res., 9, 161–180.
Gerald, C. F., and Wheatley, P. O. (1994). Applied numerical analysis, Addison-Wesley, Reading, Mass.
Goel, N. K., Kurothe, R. S., Mathur, B. S., and Vogel, R. M.(2000). “A derived flood frequency distribution for correlated rainfall intensity and duration.” J. Hydrol., 228, 56–67.
Govindaraju, R. S., Jones, S. E., and Kavvas, M. L.(1988). “On the diffusion wave modeling for overland flow, 1. Solution for steep slopes.” Water Resour. Res., 25(5), 734–744.
Govindaraju, R. S., and Kavvas, M. L.(1991). “Stochastic overland flows. Part 2: Numerical solutions evolutionary probability density functions.” Stochastic Hydrol. Hydraul., 5, 105–124.
Hebson, C., and Wood, E. F.(1982). “A derived flood frequency distribution using Horton order ratios.” Water Resour. Res., 18(5), 1509–1518.
Kavvas, M. L.(2003). “Nonlinear hydrologic processes: Conservation equations for determining their means and probability distributions.” J. Hydrologic Eng., 8(2), 44–53.
Kavvas, M. L., and Govindaraju, R. S.(1991). “Stochastic overland flows. Part 1: Physics-based evolutionary probability distributions.” Stochastic Hydrol. Hydraul., 5, 89–104.
Klemes, V.(1978). “Physically based stochastic hydrologic analysis.” Adv. Hydrosci., 11, 285–356.
Koch, R.(1985). “A stochastic streamflow model based on physical principles.” Water Resour. Res., 21(4), 545–553.
Merz, B., and Plate, E.(1997). “An analysis of the effects of spatial variability of soil and soil moisture on runoff.” Water Resour. Res., 33(12), 2909–2922.
Onof, C., Chandler, R. E., Kakou, A., Northrop, P., Wheater, H. S., and Isham, V.(2000). “Rainfall modelling using Poisson-cluster processes: A review of developments.” Stochastic Environmental Research and Risk Assessment, 14, 384–411.
Parzen, E. (1962). Stochastic processes, Holden-Day, Oakland, Calif.
Salas, J. D., Delleur, J. W., Yevjevich, V., and Lane, W. L. (1980). Applied modeling of hydrologic time series, Water Resources Publications, Littleton, Colo.
Shen, H. W., Koch, G. J., and Obeysekera, J. T. B.(1990). “Physically based flood features and frequencies.” J. Hydraul. Eng., 116(4), 494–514.
Tayfur, G., and Kavvas, M. L.(1998). “Areally-averaged overland flow equations at hillslope scale.” Hydrol. Sci. J., 43(3), 361–378.
Yue, S., Hashino, M., Bobee, B., Rasmussen, P. F., and Ouarda, T. B. M. J.(1999). “Derivation of streamflow statistics based on a filtered point process.” Stochastic Environmental Research and Risk Assessment, 13, 317–326.
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Copyright © 2003 American Society of Civil Engineers.
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Received: Feb 5, 2002
Accepted: Aug 6, 2002
Published online: Feb 14, 2003
Published in print: Mar 2003
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