Abstract
There is a multitude of methods for estimating parameters of hydrologic frequency models. Some of the popular methods used in hydrology include (1) method of moments (Nash, 1959; Dooge, 1973; Harley, 1967; O’Meara, 1968; Van de Nes and Hendriks, 1971; Singh, 1988); (2) method of probability weighted moments (Greenwood, et al., 1979); (3) method of mixed moments (Rao, 1980, 1983; Shrader, et al., 1981); (4) L-moments (Hosking, 1986, 1990, 1992); (5) maximum likelihood estimation (Douglas, et al., 1976; Sorooshian, et al., 1983; Phien and Jivajirajah, 1984); and (6) least squares method (Jones, 1971; Snyder, 1972; Bree, 1978a, 1978b). A brief review of these methods is given here.
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References
Ashkar, F., Bobee, B., Leroux, D. and Morisette, D., 1988. The generalized method of moments as applied to the generalized gamma distribution. Stochastic Hydrology and Hydraulics, Vol. 2, pp. 161–174.
Bobee, B., Perreault, L. and Ashkar, F., 1993. Two kinds of moment diagrams and their applications in hydrology. Stochastic Hydrology and Hydraulics, Vol. 7, pp. 41–65.
Bree, T., 1978a. The stability of parameter estimation in the linear model. Journal of Hydrology, Vol. 37, pp. 47–66.
Bree, T., 1978b. The general linear model with prior information. Journal of Hydrology, Vol. 39, pp. 113–127.
Clarke, R.T., 1996. Residual maximum likelihood (REML) methods for analysing hydrological data series. Journal of Hydrology, Vol. 182, pp. 277–295.
Cohen, A.C. and Whitten, B., 1982. Modified maximum likelihood and modified moment estimators for the three-parameter Weibull distribution. Communications in Statistics-Theoretical Methods, Vol. 11, o. 23, pp. 2631–2656.
Cunnane, C., 1989. Statistical distribution for flood frequency analysis. WMO Operational Hydrology Report No-33, WMO-No 718, Geneva, Switzerland.
Diskin, M.H., 1967. A Laplace transform proof of the theorem of moments for the instantaneous unit hydrograph. Water Resources Research, Vol. 3, No. 2, pp. 385–388.
Diskin, M.H. and Boneh, A., 1968. Moments of the input, output, and impulse response functions of linear systems about arbitrary points. Water Resources Research, Vol. 4, No. 4, pp. 727–735.
Dooge, J.C.I., 1973. Linear theory of hydrologic systems. Technical Bulletin No. 1468, 327 pp., Agricultural Research Service, U.S. department of Agriculture, Washington, D.C.
Douglas, J.R., Clarke and Newton, S.G., 1976. The use of likelihood functions to fit conceptual models with more than one dependent variable. Journal of Hydrology, Vol. 29, pp. 181198.
Duan, Q., Sorooshian, S. And Gupta, V.K., 1988. A maximum likelihood criterion for use with data collected at unequal time intervals. Water Resources Research, Vol. 24, No. 7, pp. 1163–1173.
Dubey, S.D., 1967. Some percentile estimators for Weibull parameters. Technometrics, Vol. 9, pp. 119–129.
Fiering, M.B. 1982b. Alternative indices of resilience. Water Resources Research, Vol. 18, No. 1, pp. 33–39.
Field, C., 1985. Concepts of robustness. Chapter 15 in A Celebration of Statistics, edited by A.C. Atkinson and S.E. Fienberg, pp. 369–375, Springer-Verlag, Heidelberg, Germany
Fiering, M.B. 1982 a. A screening model to quantify resilience. Water Resources Research, Vol. 18, No. 1, pp. 27–32.
Fiering, M.B. 1982c. Estimates of resilience indices by simulation. Water Resources Research, Vol. 18, No. 1, pp. 41–50.
Fiering, M.B. 1982d. Estimating resilience by canonical analysis. Water Resources Research, Vol. 18, No. 1, pp. 51–57.
Greenwood, J.A., Landwehr, J.M., Matalas, N.C. and Wallis, J.R., 1979. Probability-weighted moments: definition and relation to parameters of several distributions expressible in inverse form. Water Resources Research,Vol. 15, pp. 1049–1054.
Gupta, V.K. and Sorooshian, S., 1983. Uniqueness and observability of conceptual rainfall-runoff model parameters: The percolation process examined. Water Resources Research, Vol. 19, No. 1, pp. 29–276.
Gupta, V.K. and Sorooshian, S., 1985. The rel;ationship between data and the precision of parameter estimates of hydrologic models. Journal of Hydrology, Vol. 81, pp. 57–77.
Haktanir, T., 1996. Probability-weighted moments without plotting position formula. Journal of Hydrologic Engineering, Vol. 1, No. 2, pp. 89–91.
Harley, B.M., 1967. Linear routing in uniform open channels. M. Eng. Sc. Thesis, University College, Cork, Ireland.
Hosking, J.R.M., 1985. A correction for the bias of maximum likelihood estimators of Gumbel parameters-comment. Journal of Hydrology, Vol. 78, pp. 393–396.
Hosking, J.R.M., 1986. The theory of probability-weighted moments. Technical Report RC 12210, Mathematics, 160 pp., IBM Thomas J. Watson Research Center, Yorktown Heights, New York.
Hosking, J.R.M., 1992. Moments or L-moments. An example comparing two measures of distributional shape. American Statistician, Vol, 46, No. 3, pp. 186–189.
Hosking, J.R.M., 1990. L-Moments: Analysis and estimation of distribution using linear combination of order statistics. Journal of Royal Statistical Society, Series B, Vol. 52, No. 1, pp. 105–124.
Hosking, J.R.M. and Wallis, J.R., 1991. Some statistics useful in regional frequency analysis. Research Report RC 1709623 AP, IBM Research Division, T.J. Watson Research Center, Yorkstown Heights, N.Y.
Hosking, J.R.M. and Wallis, J.R., 1987. An “index-flood” procedure for regional rainfall frequency analysis. EOS Transactions, AGU, Vol. 68, pp. 312.
Hosking, J.R.M. and Wallis, J.R., 1995. A comparison of unbiased and plotting position estimators of L moments. Water Resources Research, Vol. 31, No. 8, pp. 2019–2025.
Jones, L.E., 1971. Linearizing weight factors for least quares fitting. Journal of the Hydraulics Division, ASCE, Vol. 97, No. HY5, pp. 665–675.
Johnson, N.L. and Kotz, S., 1985. Moment ratios. Encyclopedia of Statistical Sciences, Vol. 1, pp. 603–604.
Kappenman, R.F., 1985. Estimation for the three-parameter Weibull, lognormal, and gamma distributions. Computational Statistics and Data Analysis, Vol. 3, No. 1, pp. 11–23.
Kitanidis, P. K., 1986. Parameter uncertainty in estimation of spatial functions: Bayesian analysis. Water Resources Research, Vol. 22, No. 4, pp. 499–507.
Kitanidis, P.K. and Lane, R.W., 1985. Maximum likelihood parameter estimation of hydrologic spatial processes by the Gauss-Newton method. Journal of Hydrology, Vol. 79, pp. 53–71.
Kline, D.E. and Bender, D.A., 1990. Maximum likelihood estimation for shifted Weull and lognormal distributions. Transactions in Agriculture, Vol. 33, No. 1, pp. 330–335.
Koch, S.P., 1991. Bias error in maximum likelihood estimation. Journal of Hydrology, Vol. 122, pp. 289–300.
Kroll, C.N. and Stedinger, J.R., 1996. Estimation of moments and quantiles using censored data Water Resources Research, Vol. 32, No. 4, pp. 1005–1012.
Kuczera, G., 198a. Combining site-specific and regional information: An empirical Bayes approach. Water Resources Research, Vol. 18, No. 2, pp. 306–314.
Kucczera, G., 1982b. Robust flood frequency models. Water Resources Research, Vol. 18, No. 2, pp. 315–324.
Kuczera, G., 1982c. On the relationship between the reliability of parameter estimates and hydrologic time series data used in calibration. Water Resources Research, Vol. 18, No. 1, pp. 146–154.
Kuczera, G., 1983a. Improved parameter inference in catchment models: 1. Evaluating parameter uncertainty. Water Resources Research, Vol. 19, No. 5, pp. 1151–1162.
Kuczera, G., 1983b. Improved parameter inference in catchment models: 2.Combining different kinds of hydrologic data and testing their compatibity. Water Resources Research, Vol. 19, No. 5, pp. 1163–1172.
Landwehr, J.M., Matalas, N.C. and Wallis, J.R., 1979a. Probability-weighted moments compared with some traditional techniques in estimating Gumbel parameters and quantiles. Water Resources Research, Vol. 15, pp. 1055–1064.
Landwehr, J.M., Matalas, N.C. and Wallis, J.R., 1979b. Estimation of parameters and quantiles of Wakeby distributions. Water Resources Research, Vol. 15, pp. 1361–1379; correction, p. 1672.
Natale, L. and Todini, E., 1974. A constrained parameter estimation technique for linear models in hydrology. Publication No. 13, Institute of Hydraulics, University of Pavia, Pavia, Italy.
Nash, J.E., 1959. A systematic determination of unit hydrograph parameters. Journal of Geophysical Research, Vol. 64, No. 1, pp. 111–115.
O’meara, W.A., 1968. Linear routing of lateral inflow in uniform open channels. M.Eng. Sc. Thesis, University College, Cork, Ireland.
Phien, H.N. and Jivajirajah, T., 1984. Fitting the SB curve by the method of maximum likelihood. Journal of Hydrology, Vol. 67, pp. 67–75.
Rao, A.R. and Hamed, K.H., 1994. Frequency analysis of upper Cauvert flood data by L-moments. Water Resources Management, Vol. 8, pp. 183–201.
Rao, A.R. and Hamed, K.H., 1997. Regional frequency analysis of Wabash River flood data by L-moments. Journal of Hydrologic Engineering, Vol. 2, No. 4, pp. 169–179.
Rao, A.R. and Mao, L.T., 1987. An investigation of the instrumental varaible-approximate maximum likelihood method of modeling and forecasting daily flows. Water Resources Manageemnt, Vol. 1, pp. 79–106.
Rao, D.V., 1980. Log-Pearson type 3 distribution: method of mixed moments. Journal of Hydraulics Division, ASCE, Vol. 106, No. HY6, pp. 999–1019.
Rao, D.V., 1983. Estimating log Pearson parameters by mixed moments. Journal of Hydraulic Engineering, Vol. 109, No. 8, pp. 1118–1132.
Shrader, M.L., Rawls, W.J., Snyder, W.M. and McCuen, R.H., 1981. Flood peak regionalization using mixed-mode estimation of the parameters of the log-normal distribution. Journal of Hydrology, Vol. 52, pp. 229–237.
Singh, V. P., 1988. Hydrologic Systems, Vol.]: Rainfall-Runoff Modeling. Prentice hall, Englewood Cliffs, New Jersey.
Snyder, W.M., 1972. Fitting of distribution functions by nonlinear least squares. Water Resources Research, Vol. 8, No. 6, pp. 1423–1432.
Sorooshian, S. And Gupta, V.K., 1983. Automatic calibration of conceptual rainfall-runoff models: the question of parameter observability and uniqueness. Water Resources Research, Vol. 19, No. 1, pp. 260–268.
Sorooshian, S., Gupta, V.K. and Fulton, J.L., 1983. Evaluation of maximum likelihood parameter estimation techniques for conceptual rainfall-runof models: Influence of calibration data variability and length on model credibility. Water Resources Research, Vol. 19, No. 1, pp. 251–259.
Stedinger, J.R. and Tasker, G.D., 1985. Regional hydrologic analysis: 1. Ordinary, weighted and generalized least squares compared. Water Resources Research, Vol. 21, No. 9, pp. 14211432.
Troutman, B.M., 1985a. Errors and parameter estimation in precipation-runoff modeling: Theory. Water Resources Research, Vol. 21, No. 8, pp. 1195–1213.
Troutman, B.M., 1985b. Errors and parameter estimation in precipation-runoff modeling: 2. Case study. Water Resources Research, Vol. 21, No. 8, pp. 1214–1222.
Van des Nes, T.J. and Hendriks, M.H., 1971. Analysis of a linear distributed modelof surface model. Report No. 1, Laboratory of Catchment Hydraulics and Hydrology, Agricultural University of Wageningen, The Netherlands.
Vogel, R.M.S. and Fennessey, N.M., 1993. L-moment diagrams should replace product moment diagrams. Water Resources Research, Vol. 29, No. 6, pp. 1745–1752.
Vogel, R.M. and Wilson, I., 1996. Probability distribution of annual maximum, mean, and minimum streamflows in the United States. Journal of Hydrologic Engineering, Vol. 1, No. 2, pp. 69–76.
Vogel, R.M.S., Thomas W.O., and McMahon, T.A., 1993a. Flood flow frequency model selection in southwestern U.S.A. Journal of Water Resources Planning and Management, ASCE, Vol. 119, No. 3, pp. 353–366
Vogel, R.M.S., McMahon, T.A., and Chiew, F.H.S., 1993b. Flood flow model selection in Australia. Journal of Hydrology, Vol. 146, pp. 421–450.
Wang, S.X. and Adams, B.J., 1984. Parameter estimation infloodfrequency analysis. Publication 84–02, Department of Civil Engineering, University of Toronto, Toronto, Canada.
Wang, Q.J., 1997. LH moments for statistical analysis of extreme events. Water Resources Research, Vol. 33, No. 12, pp. 2841–2848.
Williams, B.J. and Yeh, W.W.-G., 1983. Parameter estimation in rainfall-runoff models. Journal of Hydrology, Vol. 63, pp. 373–393.
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Singh, V.P. (1998). Methods of Parameter Estimation. In: Entropy-Based Parameter Estimation in Hydrology. Water Science and Technology Library, vol 30. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1431-0_2
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DOI: https://doi.org/10.1007/978-94-017-1431-0_2
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