Two‐distribution method for modeling and sequential generation of monthly streamflows

Abstract

Distributions of observed monthly streamflows at many gaging stations exhibit slight, moderate, or sharp reverse curvatures that cannot be accommodated by the commonly used classical distributions such as the normal and gamma (Pearson type 3) distributions applied to flows or their logarithms. The two distribution, essentially a bimodal distribution, fits most of these observed distributions very well. Herein is presented a unique and versatile method for modeling and sequential generation of monthly streamflows. In order to assure the normality of the multivariate distribution of variables representing monthly flows the flows or their logarithms are converted to standardized normal deviates by suitable transforms. The distributions of monthly streamflows generated by the two-distribution method fit best the observed flow distributions. Annual flows obtained by summing generated monthly flows fit the observed annual flows very well. The use of the two-distribution method can obviate arbitrary adjustments in generated monthly flows to bring the annual flows obtained from them in line with observed annual flows.A historical hydrologic sequence or data series is a chance event that may never occur again in the same way. The single response obtained with this sequence is unlike!y*to be a satisfactory representation of future response of a system under consideration. A set of system responses can be obtained from simulated hydrologic sequences called synthetic sequences [Fiering, !966]. A number of these sequences of any desired length can be sequentially generated to aid in decision making with respect to system design, operation, and performance. These synthetic sequences must resemble the historical sequence in terms of certain properties that characterize the historical sequence [Matalas, 1967].An inspection of monthly flows of various streams and rivers indicates a tendency for high flows to follow high flows and low flows to follow low flows. This tendency is designated as hydrologic persistence and is attributed to storage processes in the atmosphere and/or in the drainage basin. Persistence is described by the correlations between various months of the year. Usually, the correlation between two consecutive monthly series is taken to be a measure of persistence. This assumption implies that the sequence is generated by a first-order Markov process.Stochastic models of streamflow generation are becoming more popular as a tool in hydrologic planning and design. A stochastic model is predicated to preserve meaningful statistical properties of a historical record in equally likely generated sequences that can be used to give a broad spectrum of response of the system under consideration. Thomas and Fiering [1962] presented a mathematical model for the sequential generation of nonhistorical monthly streamflows x. This model was formulated as a recursion relation of the type x,.i = •', q-b,(x•_•.i --.•_•) q-t•.is,(1 --ri2) ø'• (1) i= 2,3,-.-,12 and x,.• = •.• + b,(x•.•_• -5:•) + t•,•s• (1 -r,•) ø'• (2) in which subsc...

PREDICTION OF MEAN ANNUAL FLOWS IN NORTH AND CENTRAL FLORIDA1

“…As data following a lognormal distribution should plot as a straight line on this graph paper, these systematic errors, indicated by marked breaks of slope in the observed data, suggest that it may be the choice of the model type, rather than the spatial interpolation and prediction of parameters that may be in error. The break of slope occurs at cumulative probabilities of about 0.30, and although none of the fits could be considered statistically different at the control stations, Lonquist, 1974). It has been suggested that the El Niño/Southern Oscillation affects mean monthly flows across north central Florida during winter and spring (Kahya and Dracup, 1993) as well as the variability in these flows during summer and fall (Zorn and Waylen, 1997).…”

Seasonal Response of Mean Monthly Streamflow to El Niño/Southern Oscillation in North Central Florida

“…This analysis supports the use of the twoparameter lognormal distribution to model mean monthly streamºows of the Santa Fe river, for populations subdivided on the basis of ENSO conditions. The model has a strong theoretical basis in that standard t-and F-tests can be used to investigate differences between means and variances of lognormally distributed data, and it has been applied to monthly ºows in previous analyses (Harms and Campbell 1967;Singh and Lonquist 1974;Stedinger 1980). Furthermore, the parameters (mean and standard deviation) can be given a physical interpretation by linking them to documented changes in precipitation-generating processes.…”

“…Hence, an appropriate probability distribution must be sufªciently ºexible to accommodate a wide range in values of its estimated parameters and shapes of its cumulative distribution function. Previous research suggests that monthly and seasonal streamºow and precipitation generally follow a lognormal distribution (Harms and Campbell 1967;Moreau and Pyatt 1970;Singh and Lonquist 1974;Stedinger 1980;Richards 1995), which is characterized by two easily interpretable parameters, mean and standard deviation of the natural logarithms of the raw data (Haan 1977).…”

Comment on ‘Some comparisons of flood statistics in real and log space’ by J. Maciunas Landwehr, N. C. Matalas, and J. R. Wallis

“…The writer has been experimenting with the concept of mixed distributions [Singh, 1968;Singh and Sinclair, 1972;Singh, 1974;Singh and Lonnquist, 1974] for fitting the historical flood and other hydrologic sequences. The two-distribution method developed from these studies is based on the following equations: 1908-1976 with 1914 missing.…”