Z-Transform Based Instantaneous Unit Hydrograph for Hilly Watersheds
DOI: 10.4236/jwarp.2009.16046   PDF    HTML     6,180 Downloads   11,022 Views   Citations


Present study emphasizes the applicability of linear theory concept onto hilly watersheds. For this purpose, Z-transform technique was used to derive the instantaneous unit hydrograph (IUH) from the transfer function of autoregressive and moving average (ARMA) type linear difference equation. Parameters of the ARMA type rainfall-runoff process were estimated by least-squares method. The derived IUH from Z-transform (i.e. ARMA-IUH) has been used to compute the hydrologic response i.e. direct runoff hydrograph (DRH). Fur-ther, the superiority of the proposed approach has been tested by comparing the results through the results obtained from the Nash-IUH. Analyzing the results obtained from ARMA-IUH and Nash-IUH for the two hilly watersheds of North Western Himalayas shows the applicability of the linear theory concept even in turbulent flow conditions which are frequently encountered in hilly terrains under similar conditions of flow.

Share and Cite:

RAI, R. , OJHA, C. and UPADHYAY, A. (2009) Z-Transform Based Instantaneous Unit Hydrograph for Hilly Watersheds. Journal of Water Resource and Protection, 1, 381-390. doi: 10.4236/jwarp.2009.16046.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] S. K. Spolia and S. Chander, “Modeling of surface runoff systems by an ARMA model,” Journal of Hydrology, Vol. 22, pp. 317–332, 1974.
[2] J. E. Nash, “The forms of the instantaneous unit hydro-graph,” International Association of Science and Hydrau-lics Division, Proceedings of the American Society of Civil Engineers (ASCE), Vol. 104 (HY 2), pp. 262–276, 1957.
[3] K. M. O’Connor, “A discrete linear cascade model for hy-drology,” Journal of Hydrology, Vol. 29, pp. 203–242, 1976.
[4] K. M. O’Connor, “Derivation of discretely coincident forms of continuous linear time-invariant models using the transfer function on approach,” Journal of Hydrology, Vol. 59, pp. 1–48, 1982.
[5] G. T. Wang and K. Wu, “The unit-step function response for several hydrological conceptual models,” Journal of Hydrology, Vol. 62, pp. 119–128, 1983.
[6] G. T. Wang, V. P. Singh, and F. X. Yu, “A rainfall-runoff model for small watersheds,” Journal of Hydrology, Vol. 138, pp. 97–117, 1992.
[7] L. K. Sherman, “Stream flow from rainfall by the unit-graph method,” Eng. News Rec., Vol. 108, pp. 501–505, 1932.
[8] J. E. Nash, “Systematic determination of unit hydrograph parameters,” Journal of Geophysical Research, Vol. 64, No. 1, pp. 111–115, 1959.
[9] J. C. Dooge, “A general theory of the unit hydrograph,” Jour-nal of Geophysical Research, Vol. 64, No. 2, pp. 241–256, 1959.
[10] J. C. Dooge, “Linear theory of hydrologic Systems,” Tech-nical Bulletin No. 1468, U.S. Department of Agriculture, Agricultural Research Service, Washington, D.C., 1973.
[11] V. P. Singh, “Hydrologic Systems: Rainfall-Runoff Mod-elling,” Vol. I, Prentice Hall, Englewood Cliffs, New Jersey, 1988.
[12] J. E. Turner, J. C. I. Dooge, and T. Bree, “Deriving the unit hydrograph by root selection,” Journal of Hydrology, Vol. 110, pp. 137–152, 1989.
[13] K. K. Singh, “Flood estimation for selected Indian river basins,” An unpublished Ph.D. Thesis, Kurukshetra Uni-versity, Kurukshetra, India, 1997.
[14] C. S. P. Ojha, K. K. Singh, and D. V. S. Verma, “Sin-gle-storm runoff analysis using Z-transform,” Journal of Hydrologic Engineering, American Society of Civil En-gineers (ASCE), Water Resources Engineering Division, Vol. 4, No. 1, pp. 80–82, 1999.
[15] R. K. Rai, M. K. Jain, S. K. Mishra, C. S. P. Ojha, and V. P. Singh, “Another look at Z-transform for deriving the unit impulse response function,” Water Resources Man-agement, Vol. 21, No. 11, pp. 1829–1848, 2007.
[16] G. E. P. Box and G. M. Jenkins, “Time series analysis: Forecasting and control,” Revised Edition, Holden Day, San Francisco, California, 1976.
[17] V. T. Chow, “Hydrologic modeling – The seventh John R. Freeman memorial lecture,” Proceeding of Boston Soci-ety of Civil Engineers, Vol. 60, No. 5, pp. 1–27, 1972.
[18] E. J. Muth, “Transform Methods, with applications to engineering and operations research,” Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1977.
[19] A. Gelb, Applied Optimal Estimation, MIT Press, Cam-bridge, Mass, 1974.
[20] G. T. Wang and Y. S. Yu, “Estimation of parameters of the discrete, linear, input-output model,” Journal of Hy-drology, Vol. 85, pp. 15–30, 1986.
[21] I. P. Wu, J. W. Delleur, and M. H. Diskin, “Determina-tion of peak discharge and design hydrographs for small watersheds in Indiana,” Bulletin Indiana Flood Control Resources Commission at Purde University, 1964.
[22] L. N. Singh and R. S. Rana, “Indigenous flora and soil-water conservation practices in Kandi region of Hi-machal Pradesh,” Directorate of Research, Himachal Pradesh Krishi Vishvavidyalaya, H. P. Palampur, 1998.
[23] V. T. Chow, Handbook of Applied Hydrology, McGraw Hill, New York, U.S.A., 1964.
[24] V. T. Chow, D. R. Maidment, and L. W. Mays, “Applied Hydrology,” McGraw-Hill Book Company, New York, 1988.
[25] J. E. Nash and J. V. Sutcliffe, “River flow forecasting through conceptual model. Part-3 A discussion of the principle,” Journal of Hydrology, Vol. 10, pp. 282–290, 1970.

Copyright © 2024 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.