Abstract
The logarithmic normal probability law is widely used to describe the distribution of annual maximum values of hourly or daily precipitation (Weiss, 1957), flood flows (Chow, 1951, 1954), hydraulic conductivity (Freeze and Cherry, 1979), soil properties (physical, chemical and microbiological) (Parkin and Robinson, 1993), etc. Kalinske (1946) found that many times river discharge data and sand sizes followed the normal law if they were logarithmically transformed. Chow (1951, 1954, 1959) gave a historical background of the log-probability law and discussed its wide-ranging application in engineering, and exensively worked with the lognormal distribution. Aitchison and Brown (1957) presented a comprehensive statistical treatment of the lognormal distribution. Parkin et al. (1988) evaluated statistical methods for log-normally distributed variables, including the method of moments, maximum likelihood, and Finney’s method. Parkin and Robinson (1993) evaluated soil properties using log-normal distribution. Brakensiek (1958) employed the least squares method for fitting the log-normal distribution to annual runoff. Moran (1957) fitted a log-normal distribution to fifty annual values of extreme monthly flow of the River Murray in Australia. Lewis (1979) applied log-normal distribution to maximum measured discharges of River Kafue in Africa.Weiss (1957) developed a nomogram for log-normal frequency analysis.Alexander et al. (1969) discussed statistical properties of lognormal distribution. Using mean square error of estimation as a criterion, Stedinger (1980) evaluated the efficiency of alternative methods of fitting the lognormal distribution. Charbeneau (1978) compared two- and three-parameter log-normal distributions for simulation of stream flow.
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References
Aitchison, J. and Brown, J.A.C., 1957. The Lognormal Distribution. Cambridge University Press, London, U.K.
Alexander, G.N., Karoly, A. and Susts, A.B., 1969a. Equivalent distributions with application to rainfall as an upper bound to flood distributions. Journal of Hydrology, Vol. 79, No. 4, pp. 322–344.
Alexander, G.N., Karoly, A. and Susts, A.B., 1969b. Equivalent distributions with application to rainfall as an upper bound to flood distributions. Journal of Hydrology, Vol. 9, No. 3, pp. 345–373.
Brakensiek, D.L., 1958. Fitting a generalized log-normal distribution to hydrologic data.
Transactions, American Geophysical Union,Vol. 39, No.3, pp. 469–473.
Charbeneau, R.J., 1978. Comparison of the two-and three-parameter lognormal distributions used in streamflow sysnthesis. Water Resources Research, Vol. 14, No. 1, pp. 149–150.
Chow, V.T., 1951. A general formula for hydrologic frequency analysis. Transactions, American Geophysical Union, Vol. 32, pp. 231–237.
Chow, V.T., 1954. The log-probability law and its engineering application. Proceedings
American Society of Civil Engineers,Vol. 80 pp. 536–1 to 536–14.
Chow, V.T., 1959. Determination of hydrologic frequency factor. Journal of the Hydraulics Division, ASCE, Vol. 85, pp. 93–98.
Freeze, A.R. and Cherry, J.C., 1979. Ground Water. Prentice Hall, Inc., Englewood Cliff, New Jersey.
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.
Kalinske, A.A., 1946. On the log-probability law. Transactions, American Geophysical Union, Vol. 27, No. V, pp. 709–710.
Kite, G. W., 1978. Frequency and Risk Analyses in Hydrology. Water Resources Publications, Fort Collins, Colorado.
Lewis, G., 1979. A statistical estimation of flood flows. Proceedings, Institution of Civil Engineers, Part 2, Vol. 67, pp. 841–844.
Moran, P.A.P., 1957. The statistical treatment of flood flows. Transaction, American Geophysical Union, Vol. 38, No. 4, pp. 519–523.
Parkin, T.B., Meisinger, J.J., Chester, S.T., Starr, S.T., and Robinson, J.A., 1988. Evaluation of statistical estimation methods for lognormally distributed variables. Soil Science Society of America Journal, Vol. 52, pp. 323–329.
Parkin, T.B. and Robinson, J.A., 1993. Statistical evaluation of median estimators for lognormally distributed variables. Soil Society of America Journal, Vol. 57, pp. 317–323.
Singh, V.P. and Rajagopal, A.K., 1986. A new method of parameter estimation. 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. Completion Report 06, Louisiana Water Resources Research Institute, Louisiana State University, Baton Rouge, Louisiana.
Stedinger, J.R., 1980. Fitting log normal distributions to hydrologic data Water Resources Research, Vol. 16, No. 3, pp. 481–490.
Weiss, L.L., 1957. A nomogram for log-normal frequency analysis. Transactions, American Geophysical Union, Vol. 38, No. 1, pp. 33–37.
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Singh, V.P. (1998). Two-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_6
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