This study aims to model the joint drought duration and severity distribution using twodimensional copulas. The method of inference function for margins (IFM method) is employed to construct copulas. Two separate maximum likelihood estimations of univariate marginal distributions are performed first, then followed by a maximization of the bivariate likelihood as a function of the dependence parameters. The drought duration and severity are assumed to be exponential and gamma distributions, respectively. Several copulas are tested to determine the best data fitted copula. Droughts, defined using the Standardized Precipitation Index (SPI), of Wushantou (Taiwan) are employed as an example to illustrate the proposed methodology. The copula fitting results for drought duration and severity are quite satisfactory. The bivariate drought analyses, including the joint probabilities and bivariate return periods, based on the derived copula-based joint distribution are also investigated to demonstrate the advantages of bivariate modeling of droughts.

Extreme hydrological events are inevitable and stochastic in nature. Characterized by multiple properties, the multivariate distribution is a better approach to represent this complex phenomenon than the univariate frequency analysis. However, it requires considerably more data and more sophisticated mathematical analysis. Therefore, a bivariate distribution is the most common method for modeling these extreme events. The return periods for a bivariate distribution can be defined using either separate single random variables or two joint random variables. In the latter case, the return periods can be defined using one random variable equaling or exceeding a certain magnitude and/or another random variable equaling or exceeding another magnitude or the conditional return periods of one random variable given another random variable equaling or exceeding a certain magnitude. In this study, the bivariate extreme value distribution with the Gumbel marginal distributions is used to model extreme flood events characterized by flood volume and flood peak. The proposed methodology is applied to the recorded daily streamflow from Ichu of the Pachang River located in Southern Taiwan. The results show a good agreement between the theoretical models and observed flood data.

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Drought is a complex and multi-attribute natural hazard that has worldwide effects. Defined by a commonly used standardized precipitation index (SPI), each drought event is characterized by three correlated attributes: severity, duration and frequency. A probabilistic approach is developed to establish a drought severity-duration-frequency (SDF) relationship. Copulas are employed to construct the joint distribution function of drought severity and duration. Drought frequency, in terms of recurrence interval of drought events, is then related to the copula-based distribution function via a conditional distribution function. The derived analytic drought SDF thus becomes a function of univariate distribution functions of drought severity and duration, a copula function which links the fitted univariate models, and the arrival rate of drought events. In this study, rainfall data for the period of 1954-2003 from two gauge stations in Iran, Abadan in the southwestern semi-arid region and Anzali in the north humid region, are employed as an example to illustrate the proposed approach. From the derived drought SDF, drought severity in Anzali station is greater than those in Abadan station for given drought duration and recurrence interval. The results imply that the drought severity in humid region might be more severe if high rainfall fluctuations exist in that region.