Geology topics
Publications and source records attributed to N.C. Matalas.
Distributions whose inverse forms are explicitly defined, such as Tukey's lambda, may present problems in deriving their parameters by more conventional means. Probability weighted moments are introduced and shown to be potentially useful in expressing the parameters of these distributions.
Estimates of the parameters and quantiles of the Gumbel distribution by the methods of probability weighted moments, (conventional) moments, and maximum likelihood were compared. Results were derived from Monte Carlo experiments by using both independent and serially correlated Gumbel numbers. The method of probability weighted moments was seen to compare favorably with the other two techniques.
The applicability of the James-Stein estimator in regional hydrologic studies which entail the estimation of an N -dimensional location parameter is discussed. Regional studies are frequently characterized by relatively short, generally correlated, samples drawn from nonsymmetric and bounded, i.e., nonnormal, populations. By means of computer simulation studies the James-Stein estimator, subject to the Lindley modification and adoption of the positive part rule suggested by Efron and Morris and conditioned on the assumption of independence, was shown to be robust in the case of the hydrologically plausible distribution considered here, namely, Weibull distributions with coefficient of skewness ranging from 0 to 10. However, in contrast to traditional methods of regionalization the effect of cross correlation is a diminishment of the relative risk advantage of the James-Stein estimator, even in the best case of normal variables : this is discussed and illustrated.
Hydrologic processes are generally held to be persistent and not secularly independent. Impetus for this view was given by Hurst in his work which dealt with properties of the rescaled range of many types of long geophysical records, in particular dendrochronologic records, in addition to hydrologic records. Mandelbrot introduced an infinite memory stationary process, the fractional Gaussian noise process (F), as an explanation for Hurst's observations. This is in contrast to other explanations which have been predicated on the implicit non-stationarity of the process underlying the construction of the records. In this work, we introduce a stationary finite memory process which arises naturally from a physical concept and show that it can accommodate the persistence structures observed for dendrochronological records more successfully than an F or any other of a family of related processes examined herein. Further, some question arises as to the empirical plausibility of an F process. Dendrochronologic records are used because they are widely held to be surrogates for records of average hydrologic phenomena and the length of these records allows one to explore questions of stochastic process structure which cannot be explored with great validity in the case of generally much shorter hydrologic records.
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Hydrologic time series of annual minimum mean monthly rainfall and annual minimum 1-day and 7-day discharge, considered as drought indices, were used to study the distribution of droughts with respect to time. The rainfall data were found to be nearly random. The discharge data, however, were found to be nonrandomly distributed in time and generated by a first-order Markov process. The expected value of the variance for a time series generated by a first-order Markov process was compared with the expected value of the variance for a random time series. This comparison showed that the expected value of the variance for a nonrandom time series converged to the population variance with an increase in sample size at a slower rate than for a random time series.
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A statistical analysis is made of the sequences of annual tree ring widths and indices. The expected value of growth during any year is shown to be proportional to the standard deviation of the growth, so that the coefficient of variation is a measure of the sensitivity of the growth of a tree . Tree ring data were found to be non-randomly distributed in time. The large values of serial correlation indicated that the non-randomness cannot be attributed entirely to climatic factors. Correlogram and power spectrum analyses applied to a sequence of tree ring indices for a pinyon pine showed that the data were generated by an autoregressive process. The statistical parameter measuring the sensitivity or complacency of growth is used to derive a growth function.