Relating autocorrelations and crossing rates of continuous- and discrete-valued hydrologic processes

The return period and risk of extreme droughts can be derived from hydrolog ic series of wet and dry years. If Z(t) denotes a continuous-valued hydrolo gic series such as annual streamflows, a series of wet and dry years, X-t, can be obtained by clipping Z(t) by z(0) such that X-t = 1 if Z(t) greater than or equal to z(0), and X-t = 0 if Z(t) < z(0). A method is presented fo r relating the autocorrelation functions <rho>(k)(Z) and rho (k)(X). In add ition, the relationships between the crossing rate gamma and rho (j)(Z) and rho (j)(X) are derived. The method assumes that the underlying hydrologic series is stationary and normally distributed. The applicability of the met hods 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 rho (k)(X) and rho (k)(Z) are applicable and reliable. The same conclusion is reached when simulated sam ples from the ARMA model are utilized. In addition, it has been shown that the autocorrelation function,(X) obtained (by using the derived relationshi p) from rho (k)(Z) of a low-order ARMA model, can be fitted by a low-order DARMA model. The significance of the relationships between the referred aut ocorrelation functions has been documented in terms of estimating certain d rought properties. It has been shown that significant differences can be ob tained for estimating the return periods and risks of certain drought event s if the sample autocorrelations rho (k)(X) are used instead of the derived autocorrelations <(<rho>)over tilde>X-k). Furthermore, it has been shown t hat the derived relationships between gamma and rho (t)(Z) and gamma and rh o (i)(X) apply quite well for annual streamflows.