Relating Autocorrelations and Crossing Rates of Continuous- and Discrete-Valued Hydrologic Processes

The return period and risk of extreme droughts can be derived from hydrologic series of wet and dry years. If \IZ\dt\N denotes a continuous-valued hydrologic series such as annual streamflows, a series of wet and dry years, \IX\dt\N, can be obtained by clipping \IZ\dt\N by \Iz\dO\N such that \IX\dt\N=1 if \IZ\dt\N≥\Iz\dO\N, and \IX\dt\N=0 if \IZ\dt\N<\Iz\dO\N. A method is presented for relating the autocorrelation functions \Iρ\dk(Z)\N and \Iρ\dk(X)\N. In addition, the relationships between the crossing rate \iγ and \Iρ\d1(Z)\N and \Iρ\d1(X)\N are derived. The method assumes that the underlying hydrologic series is stationary and normally distributed. The applicability of the methods and derived relationships has been examined and tested by using annual streamflow series at several sites and by simulation experiments based on low-order ARMA and DARMA models. The analysis of 23 series of annual flows reveals that the derived relationship between \Iρ\dk(X)\N and \Iρ\dk(Z)\N are applicable and reliable. The same conclusion is reached when simulated samples from the ARMA(1,1) model are utilized. In addition, it has been shown that the autocorrelation function \Iρ∼\dk(X)\N obtained (by using the derived relationship) from \Iρ\dk(Z)\N of a low-order ARMA model, can be fitted by a low-order DARMA model. The significance of the relationships between the referred autocorrelation functions has been documented in terms of estimating certain drought properties. It has been shown that significant differences can be obtained for estimating the return periods and risks of certain drought events if the sample autocorrelations \Iρˆ\dk(X)\N are used instead of the derived autocorrelations \Iρ∼\dk(X)\N. Furthermore, it has been shown that the derived relationships between \Iγ\N and \Iρ\d1(Z)\N and \Iγ\N and \Iρ\d1(X)\N apply quite well for annual streamflows.

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