Multivariate Frequency Analysis Of Drought Events Using Drought Indices And Copula Functions

Drought has affected and restricted the development of economy and society since ancient times.China is more deeply affected with drought occurrence due to its geography,climate and hydrological features.Therefore,the proper understanding of the spatiotemporal characteristics of multi-year droughts and return periods is important for drought risk assessment.This study evaluated the regional and spatial comparability of drought probabilities and return periods within mainland China and seven sub-regions between 1961 and 2013 using meteorological data from 552 stations.In order to provide some theoretical support for regional and national drought prevention and control,allocation of water consumption and sustainable development of water resources,the Standardized Precipitation Index(SPI),Standardized Precipitation Evapotranspiration Index(SPEI),Composite Index(CI)and Effective Drought Index(EDI)were calculated at multiple timescales at each stations and sub-regions over the entire study period.The run theory was used for the objective identification and characterization of drought events while Classical,Spearman,and Kendall's method were used to analyze their dependencies.Within the one-variate framework,the drought duration(Dd),severity(Ds),peak(Dp),and inter-arrival time(Di)were fitted with the best marginal distributions in each sub-region and the entire mainland of China.As these drought variables showed significant dependence properties and followed different marginal distributions,copula functions were used to study their joint distributions.Appropriate copula functions for Ddvs.Ds,Ddvs.Dp and Dsvs.Dp from nineteen two-variate Archimedean copulaswere selected for bivariate analysis,and the copula parameters were estimated using the maximum likelihood estimation(MLE)and curve fitting method(CFM).Within the three-variate framework,thejoint distribution of Dd,Ds and Dp were modeled using nine trivariate copula functions,namely: Clayton,Ali-Mikhail-Haq(AMH),Gumbel–Hougaard,Frank,M3,M4,M5,M6 and M12.In order to model Dd,Ds,Dp,and Di,six four-variate symmetric and asymmetric Archimedean copulas(i.e.,Frank,Clayton,Gumbel–Hougaard)were also evaluated and compared.The performances of the different copula functions were evaluated using the Root mean square error(RMSE),Akaike information criterion(AIC),bias and graphical test.Based on the appropriate copula functions,the joint behaviors of drought variables were studied and their probabilities as well as return periods for the occurrence of the events over the next 5-,10-,20-,50,and 100 years in each sub-region were mapped and comprehensively explained.The main results obtained in this study are as follows:(1)The comparison of drought indices showed that SPEI and CI performed better than SPI in delineating spatial patterns of drought characteristics.This might be attributed to the temperature effect on evapotranspiration and therefore on drought index.EDI showed similar behaviour with SPI and daily EDI showed absrupt changes,more variations compared to monthly EDI.Considering the increasing trend in reference evapotranspiration in the 21 st century,the importance of utilizing temperature-based drought index is imperative.Severe and extreme droughts occurred in the late 1990 s in many places in China while persistent multi-year severe droughts occurred more frequently over North China,Northeast China,Northwest China and Southwest China.The spatial patterns of drought variables showed that regions characterized by higher Ds were associated with higher Dd.North China,Northwest China,and Southwest China had much longer Dd during the 1990 s and 2000 s.(2)The goodness-of-fit tests as indicated by the CFM showed thattwo-variate copula NN19 in sub-regions III,IV,V,VI and mainland China,NN20 in sub-region I and NN13 in sub-region VII are the best for modeling drought variables.The trivariate Clayton copula performed better in modeling joint dependence structure of drought variables as compared to other copulas in sub-regions I-VII and mainland China.Therefore,the Clayton copula-based joint distributions were used for three-variate drought frequency analysis,in terms of probabilities and return periods.The empirical and theoretical probabilities of the symmetric four-variate Clayton copulas in sub-regions I,III,and entire mainland China,and symmetric Frank in regions II,IV,V,VI,VII agreed well.The four-,three-and two-variate probabilities and return periods were compared with the univariate analysis and addressed the need for multivariate drought frequency analysis.(3)Generally,the drought probability across mainland China is relatively high,and the highest drought probabilities are found mainly in the Northwestern and Southwestern China.The joint probabilities in the “ ” cases were greater than that of the “ ” cases in all the seven sub-regions and mainland China.It is very natural that the possibility of drought occurrence will be presented for “ ” cases.This showed that different sub-regions might suffer varying drought risks.Accordingly,counter measures of drought hazards have to be established for proactive responses particularly in sub-regions III,V and VI where the drought risks were found to be significantly greater than other sub-regions.(4)Higher probability of droughts of longer Dd in the sub-regions also corresponds to shorter return periods with greater Ds.As droughts normally cover large areas,regional drought return periods has been showed to be more effective in providing support for drought management than station based return periods.These might imply tremendous challenges for the water resources management in different sub-regions,particularly the Northwestern and Southwestern China.In conclusion,the copula approach is a vital tool for probabilistic interpretation of drought variations.Also,the results could be helpful in the assessment of severe and extreme drought scenarios in the different climatic regions of China.