Drought Characteristics Analysis Using Meta-Elliptical Copulas

As a complex natural phenomenon, drought is stochastic and multi-attributed. This study aims to investigate the multivariate characteristics of droughts in Weihe river basin, China, by combining stochastic theory and copula functions theory. First, through analysis of the runs, monthly rainfall data of 90 meteorological gauges in Weihe river basin was used for drought identification and to obtain drought variables, i.e. drought duration, severity and peak. Then, two meta-elliptical copulas, i.e. Gaussian copula and Student t copula were applied to model association and dependency structures among drought variables and to construct the multivariate joint probability distribution of drought duration, severity and peak. Subsequently, statistical properties and occurrence patterns of droughts in Weihe river basin were analyzed to provide reference and data support for regional water resource planning and management, water conservancy project design and operation, protection against drought and disaster-reduction, and so on in future. The main conclusions drawn from this paper are as follows:(1) By means of goodness-of-fit tests, it was found that exponential distribution, Weibull distribution, and generalized Pareto distribution could be used as theoretical marginal probability distributions of drought duration, severity and peak, respectively, for all the gauges in Weihe river basin. Furthermore, prominent positive correlation was revealed between drought duration, severity and peak at any gauge, with the dependent degree of drought variables roughly being SP > DS>DP.(2) It was feasible and effective to estimate the parameters of bivariate as well as trivariate Gaussian copula and Student t copula using maximum likelihood method. And fitting of bivariate and trivariate joint probability distributions of drought duration, severity and peak indicates that both Gaussian copula and Student t copula could reflect the empirical probability distribution of observed points satisfactorily.(3) An approach and procedures of a bootstrap version based on Rosenblatt's transformation for goodness-of-fit test of bivariate copulas was reviewed and summarized, and on this basis, a modified version was proposed to realize the goodness-of-fit testing for trivariate copulas. Results of goodness-of-fit test of copulas showed that both Gaussian copula and Student t copula could be accepted to model bivariate and trivariate joint probability distributions of drought events at the significance levelα=0.05.(4) Through fitting evaluation, it was found that trivariate Gaussian copula and Student t copula gave a better fit result than Gumbel-Hougaard, Frank, Clayton, M5, M6 and M12 copulas. Although fitting results of bivariate Student t copula were much closed to Gaussian copula under large enough degrees of freedom, fitting results of trivariate Gaussian copula were better than Student t copula. Therefore, Gaussian 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) Drought return periods of multivariate joint probability distribution were numerically computed, which indicated (i) drought return period of several drought variables simultaneously exceeding specific values was greater than that of only one of them exceeding specific value under the same condition; (ii) given two of drought variables simultaneously exceeding or not exceeding specific values, then conditional period of the third drought variable was greater under the former situation; (iii) given one of drought variables exceeding or not exceeding specific value, then conditional period of the other two drought variables was also greater under the former situation; (iv) drought return period of univariate marginal probability distribution was always between the corresponding two kinds of return periods (i.e. all of the drought variables simultaneously exceeding/not exceeding specific values or only one of them exceeding/not exceeding specific value) of multivariate joint probability distribution.(6) Taking average monthly precipitation series as the threshold level, twenty kinds of multivariate joint and conditional return periods of droughts considering various combinations of D = 3 month, S = 70 mm and P = 40 mm at all the gauges in Weihe river basin were calculated and plotted, and spatial distributions of them might provide some references drought forecast and management in practice.(7) A set of general computing software was developed in MATLAB R2008a, which includes three calculation modules, i.e. drought identification, marginal probability distribution fitting and multivariate joint probability distribution fitting using copulas. This system provides a visual platform for the process of drought frequency analysis based on precipitation data and facilitates the subsequent research greatly.The stochastic theory and method is an effective tool to study droughts, and application of copulas remedies the deficiency of univariate analysis and overcomes the limitations of conventional bivariate models. Multivariate joint probability distribution models of droughts constructed by Gaussian copula and Student t copula are capable to reflect the real features of drought events comprehensively and objectively. At the same time, to investigate the characteristics of droughts using meta-elliptical copulas also expands the extent of dependence described by copulas function between hydrological variables.