Application Of Archimedean Copulas Functions In Drought Analysis

Based on Copula theory, monthly precipitation data of 90 Gauging stations in Wei River Basin were selected. Archimedean Copulas were applied to construct the joint distribution of drought variables and analyzed the characteristics of drought occurrences in the basin, 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. Main results in this study are as followed:(1) Through the optimal selection of marginal distributions and Kolmogorov-Smirnov(K-S) test, the type of marginal distribution of drought variables in Wei River basin are followed: most of the drought durations are subjected to exponential distribution; most of the drought severities are subjected to Gamma distribution, while most of the severity peaks are subjected to generalized Pareto Distribution.(2) Though correlation measurement, the correlation between drought duration and drought severity and that between drought severity and severity peak in Wei River Basin are high, while the correlation between drought duration and severity peak is low.(3) Maximum likelihood method and fitting curve method were applied to the parametesr estimation to 20 kinds of two dimensional Archimedean Copulas and 4 kinds of three dimensional Archimedean Copulas respectively. Take RMSE, AIC and Bias as goodness-of-fit assessment index to make comparison, the results show that fitting curve method is better than maximum likelihood method.(4) The most appropriate Archimedean Copulas for Wei River Basin were optimaly selected: for the joint distribution of drought duration and drought severity, Gumbel Copula shows the best result to TongChuan, WeiNan, Xi'an, XianYang and etc.,while Frank Copula shows the best result to BaoJi area, however, for the whole Wei River basin, Gumbel Copula gives the best applicability and shows the best result; for the joint distribution of drought duration and drought severity, Nelsen No 20 Copula shows the best result of goodness-of-fit to the whole basin, next to Nelsen No 20 Copula which show better results are Nelsen No 13 Copula and Frank Copula; for the joint distribution of drought severity and severity peak, Nelsen No 20 Copula shows best result to BaoJi and XianYang regions, but to the whole basin, Clayton copula shows the best result; for the three dimensional joint distribution of drought duration, drought severity and severity peak, Clayton Copula also shows the best result.(5) The joint distribution model for drought variables were established by the optimal selected Archimedean Copulas, joint probability, joint return period, co-occurrence return period and conditional return period were calculated, corresponding figures were drawn out as well. From those figures, different probabilities and return periods can be given, which can provide basis for systemic analyzing of drought characteristics in Wei River Basin.(6) For two dimensional distribution, co-occurrence return period is much greater than the joint return period under the same condition. For three dimensional distribution, the co-occurrence return period is 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 is greater than a certain value is longer than that under the condition of the other drought variable being smaller than the same certain value.(7) Taking the monthly average rainfall in years as truncation level, draw the conditional probabilities of drought variables in Wei River Basin, which shows the spatial distribution characteristics of the probability under typical combinational conditions. The general spatial distribution characteristic of conditional probability P(S≥100/D≥3),P(M≥50/D≥3) is that it is increasing from southeast to northwest; for the conditional probability P(S≥100/M≥50), it takes HeShui, QingYang, XiFeng and ZhenYuan which lies in the upper stream of Jing River (which is a branch of Wei River) and HuaTing, ZhuangLang, ZhangJiaChuan in GanSu and JingYuan in NingXia as center, and it increases to the around, there into, the increasing tend in northwest area is much more significant than that in southeast area.