Abstract
The three-parameter lognormal (TPLN)distribution is frequently used in hydrologic analysis of extreme floods, seasonal flow volumes, duration curves for daily streamflow, rainfall intensity-duration, soil water retention, etc. It is also popular in synthetic streamflow generation. Properties of this distribution are discussed by Aitchison and Brown (1957), and Johnson and Kotz (1970). Its applications are discussed by Slade (1936), Chow (1954), Matalas (1967), Sangal and Biswas (1970), Fiering and Jackson (1971), Snyder and Wallace (1974), Burges et al. (1975), Burges and hoshi (1978), Charbeneau (1978), Stedinger (1980), Singh and Singh (1987), Kosugi (1994), among others. Burges et al. (1975) discussed properties of the three-parameter lognormal distribution and compared two methods of estimation of the third parameter “a”. Kosugi (1994) applied the three-parameter lognormal distribution to the pore radius distribution function and to the water capacity function which was taken to be the pore capillary distribution function. He found that three parameters were closely related to the statistics of the pore capillary pressure distribution function, including the bubbling pressure, the mode of capillary pressure, and the standard deviation of transformed capillary distribution function. Burges and Hoshi (1978) proposed approximating the normal populations with 3-parameter lognormal distributions to facilitate multivariate hydrologic disaggregation or generation schemes in cases where mixed normal and lognormal populations existed.
This is a preview of subscription content, log in via an institution to check access.
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
Aitchison, J. And Brown, J.A.C., 1957. The lognormal Distribution with Special Reference to its Uses in Economics. Cambridge University Press, London.
Bates, C.L., Lettenmaier, D.P. and Burges, S.J., 1974. Tables of parameters for the three parameter log normal probability distribution. Technical Report No. 41, C.W. Harris Hydraulics Laboratory, Department of Civil Engineering, University of Washington, Seattle, Washington.
Burges, S.J. and Hoshi, K., 1978. Approximation of a normal distribution by a three-parameter lognormal distribution. Water Resources Research, Vol. 14, No. 4, pp. 620–622.
Burges, S.J., Lettenmaier, D.P. and Bates, C.L., 1975. Properties of the three-parameter log normal probability distribution. Water Resources Research, Vol. 11, No. 2, pp. 229–235.
Charbeneau, R.J., 1978. Comparison of the two-and three-parameetr log-normal distributions used in streamflow synthesis. Water Resources Research, Vol. 14, No. 1, pp. 149–150.
Chow, V.T., 1954. The log probability law an dits engineering applications. Proceedings, American Society of Civil Engineers, Vol. 80, No. 5, pp. 1–25.
Cohen, A.C. and Whitten, B.J., 1980. Estimation in the three parameter lognormal distribution. Journal of American Statistical Association, Vol. 70, pp. 370–399.
Fiering, M.B. anf Jackson, B.B., 1971. Synthetic Streamflows. Water Resources Monograph No. 1, American Geophysical Union, Washington, D.C.
Hoshi, K., Stedinger, J.R., and Burges, S.J., 1984. Estimation of lo g-normal quantiles: Monte Carlo results and first-order approximations. Journal of Hydrology, Vol. 71, pp. 1–30.
Hosking, J.R.M., 1990. L-moments: Analysis and estimation of distributions using linear combinations of order statistics. Journal of Royal Statistical Society, Series B, Vol. 52, No. 1, pp. 105–124.
Johnson, W.L. and Kotz, S., 1970. Distributions in Statistics: Continuous Univariate Distributions 1. Houghton-Mifflin, Boston, Massachsetts.
Kite, G.W., 1978. Frequency and Risk Analyses in Hydrology. Water Resources Publications, Fort Collins, Colorado.
Kosugi, K., 1994. Three-parameter lognormal distribution model for soil water retention. Water Resources Research, Vol. 30, No. 4, pp. 891–901.
Kuczera, G., 1982a. Combining site-specific and regional information: An empirical Bayes approach. Water Resources Research, Vol. 18, pp. 306–314.
Kuczera, G., 1982b. Robust flood frequency models. Water Resources Research, Vol. 18, pp. 315–324.
Matalas, N.C., 1967. Mathematical assessment of synthetic hydrology. Water Resources Research, Vol. 3, No. 4, pp. 937–946.
Sangal, B.P. and Biswas, A.K., 1970. Three parameter lognormal distribution and its applications in hydrology. Water Resources Research, Vol. 6, No. 2, pp. 505–515.
Singh, V.P. and Singh, K., 1987. Parameter estimation for TPLN distribution for flood frequency analysis. Water Resources Bulletin, Vol. 23, No. 6, pp. 1185–1191.
Singh, V.P., Cruise, J.F. and Ma, M., 1990. A comparative evaluation of the estimators of the three-parameter lognormal distribution by Monte Carlo simulation. Computational Statistics and data Analysis, Vol. 10, pp. 71–85.
Singh, V.P. and Rajagopal, A.K., 1986. A new method of parameter estimation for hydrologic frequency analysis. Hydrological Science and Technology, Vol. 2, No. 3, pp. 33–40.
Singh, V.P., Rajagopal, A.K. and Singh, K., 1986. Derivation of some frequency distributions using the principle of maximum entropy. Advances in Water Resources, Vol. 9, pp. 91106.
Singh, V.P., Singh, K. and Rajagopal, A.K., 1985. Application of the principle of maximum entropy (POME) to hydrologic frequency analysis. Completion Report 06, 144 p., Louisiana Water Resources Research Institute, Louisiana State University, Baton Rouge, Baton Rouge, Louisiana.
Slade, J.J., 1936. An asymmetric probability function. Transactions, American Society of Civil Engineers, Vol. 101, pp. 481–490.
Snyder, W.M. and Wallace, J.R., 1974. Fitting a three-parameter lognormal distribution by least squares. Nordic Hydrology, Vol. 5, pp. 129–145.
Stedinger, J.R., 1980. Fitting log nomal distributions to hydrologic data. Water Resources Research, Vol. 16, No. 3, pp. 481–490.
Stevens, E.W., 1992. Uncertainty of extreme flood estimates incorporating historical flood data Water Resources Research, Vol. 28, No. 6, pp. 1057–1068.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Singh, V.P. (1998). Three-Parameter Lognormal Distribution. 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_7
Download citation
DOI: https://doi.org/10.1007/978-94-017-1431-0_7
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-5089-2
Online ISBN: 978-94-017-1431-0
eBook Packages: Springer Book Archive