Application Of Asymmetric ARCHIMEDEAN COPULAS Functions In Drought Analysis

Based on copula theory, monthly precipitation data of 90 Gauging stations in Wei River Basin were selected for drought identification and to obtain drought variables, namely drought duration, severity and peak. Then, four symmetric Archimedean copulas and five asymmetric Archimedean copulas were applied to construct the joint distribution of drought duration, severity and peak. Subsequently, the characteristics of drought occurrences in the basin were analyzed, which has great significance and guidance to water resources rational allocation; the planning, design and management of water conservancy project, as well as to drought and disaster reductions. The main conclusions drawn from this paper are as follows:(1) Based on the run theory, the drought duration, severity and peak were chosen to describe the drought events. By means of the goodness-of-fit tests, the type of marginal distribution of drought variables in Wei River basin are given as follows: most of the drought durations are followed exponential distribution; most of the drought severities are followed Weibull distribution, while most of the severity peaks are followed generalized Pareto distribution.(2) Using dependency measurement, it was easily found that prominent positive correlation existed between drought duration, severity and peak at any gauge, furthermore, the correlation between severity and peak was higher than the correlation between drought duration and severity, and correlation between duration and peak was lowest.(3) RMSE, AIC and Bias were used to describe the goodness-of-fit test for the joint distribution between drought duration and severity, severity and peak, duration and peak. On this basis, bivariate M12 copula was revealed to have the best fit result for the whole basin.(4) It was feasible and effective to estimate the parameters of trivariate Archimedean copulas using maximum likelihood method. Considered the parameter scopes, the parameter estimations of M3, M4 and M5 copula for most gauging stations were effective apart from the estimation of M3 copula for Baoji gauging station was invalid, along with this method failed to estimate the parameters of M4 copula for the Cheng Cheng, Da Li, Fu ping, Hua Xian and Bin Xian gauging station.Taking RMSE, AIC and Bias as goodness-of-fit assessment index to choose best fit copulas, the less of the index, the better fit of the copula. On this basis, among the symmetric Archimedean copulas, Clayton copula fit better than AMH, Frank and Gumbel copula. Among the asymmetric Archimedean copulas, M12 copula fit best, M3 copula was inferior to M12 copula but better than M4, M5 and M6 copula. Overall, M12 copula fit better than Clayton copula. Therefore, M12 copula can be used as the best-fit copula function for construction of multivariate joint probability distribution of drought duration, severity and peak at all the gauges in Weihe river basin.(5) An approach and procedures of a bootstrap version based on Rosenblatt's transformation was used to generate simulation sequence, therefore, combined with Anderson-Darling (A-D) test, the goodness-of-fit testing for trivariate copulas was achieved. Results of goodness-of-fit test of copulas showed that M12 copula could be accepted to model trivariate joint probability distributions of drought events for most of the gauging stations, except GanGu, He Shui, Hua Ting, Hui Ning, Jing Ning Long De, Peng Xian, and Chang'an gauging station. Therefore, M12 copula was used to compute the joint probability, joint return periods, conditional probability, conditional return periods, and corresponding figures were plotted as well.(6) Numerically computed results found that the co-occurrence return period was much greater than the joint return period under the same condition for bivariate copula. For trivariate copula, the co-occurrence return period was much greater than the joint return period when the drought variables take the same value; the joint return period of two drought variables under the condition that other drought variable was greater than a certain value was longer than that under the condition of the other drought variable being smaller than the same certain value. Moreover, the conditional probability and conditional return period under the condition of( D≤3, S≤100 / P≥50) were also computed for the whole basin.