# Hydrologic Series Generation from the Spectral Density Function

Report

January 1982

## Abstract

Computer simulation of hydrologic processes has become an effective analytical tool for analysis of complicated water resource systems. Present methods of generating sequences place practical limits on the ability of the methods to preserve an observed autocorrelation structure. This report explores the use of spectral characteristics in the generation of time series. Because of the correspondence between the spectral density function and the autocorrelation function of a time series, the spectrum can be used to fit any of the empirical autocorrelation structures observed in hydrologic sequences. The problem of maintaining an observed correlation structure in a generated sequence is reduced to the much easier problem of maintaining the observed spectral characteristics. The method requires that the series be generated in the frequency domain. The fast Fourier transform is then used to transform the frequency domain into the sequential domain of the process. A problem similar to aliasing (encountered when estimating the spectral density function) is discussed. When generating sequences from the spectrum it is shown that aliasing causes a distortion of the true correlation function of the series. Direction in choice of design parameters of the Monte Carlo mode to effectively eliminate the problem is given. A FORTRAN language program is provided which can be used to generate a normally distributed sequence with a given spectral density function (or equivalently, a given autocorrelation function). Bias in the standard estimate of autocorrelation can be a particularly dangerous problem in the generation of sequences. The Monte Carlo model is usually designed to retain the observed autocorrelation of a natural sequence. If the observed autocorrelation has been estimated with considerable bias the associated Monte Carlo model can lead to erroneous conclusions. A method of unbiased estimation of the autocorrelation function is developed and tested. In particular it is shown that the expected value of the vector of lagged autocorrelation estimates r is the form: E® = Ao Where A is a square matrix of known constants and o is the vector of true population lagged autocorrelations. Thus A-r is an unbiased estimate of o.

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