Comparison of Stationary and Non-Stationary Copula Models in the Analysis of Compound Meteorological-Hydrological Drought (Case Study: Minab Dam Watershed)

Introduction
Drought is one of the most significant hydro climatic hazards affecting water resources, agriculture, and ecosystems worldwide. Traditionally, drought assessment has often been conducted using univariate indices such as the Standardized Precipitation Index (SPI) for meteorological drought or the Standardized Runoff Index (SRI) for hydrological drought. However, drought events are inherently complex phenomena resulting from the interaction of multiple hydro climatic variables. Therefore, multivariate approaches have gained increasing attention in recent years. Copula functions provide a powerful statistical framework for modeling the dependence structure between variables such as precipitation and runoff. Moreover, many traditional drought analyses assume stationarity in hydrological time series, while growing evidence suggests that climate change and anthropogenic influences have introduced non stationary behavior in hydrological processes. Ignoring non stationarity may lead to biased estimation of drought characteristics and associated risks. Accordingly, the main objective of this study is to evaluate and compare the performance of stationary and non stationary copula models in the analysis of compound meteorological–hydrological drought in the Minab Dam watershed, located in Hormozgan Province, Iran.
Materials and Methods
Monthly precipitation data from 15 rain gauge stations and runoff data from the Bernetin hydrometric station were used in this study. Meteorological and hydrological drought conditions were first quantified using the SPI and SRI indices. To analyze the joint behavior of precipitation and runoff deficits, the Joint Drought Index (JDI) was derived using copula functions. Several probability distributions including Generalized Gamma, exGaussian, Generalized Inverse Gaussian, and Pearson Type VI were tested to model the marginal distributions under both stationary and non stationary frameworks. Parameter estimation was carried out using the maximum likelihood method, and model performance was evaluated using the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC).
To characterize the dependence structure between precipitation and runoff, five copula families—Gaussian, Frank, Plackett, Gumbel, and Clayton—were examined in both stationary and time varying (non stationary) forms. In the non stationary framework, copula parameters were modeled as functions of time. Subsequently, drought event characteristics such as duration, severity, magnitude, peak intensity, and frequency were extracted using run theory. Transition probabilities between drought classes were also investigated, and bivariate return periods of compound drought events were estimated under AND and OR scenarios.
Results and Discussion
The results showed that for precipitation data, the Generalized Gamma distribution provided the best fit under the stationary framework with an AIC value of approximately 3489, whereas for runoff data the Generalized Inverse Gaussian (GIG) distribution under the non stationary framework performed best with an AIC of about 2289. Among the dependence models, the non stationary Gaussian copula showed the best overall performance with an AIC value of approximately −1336, indicating a better representation of the evolving dependence structure between precipitation and runoff.
The analysis of drought index time series revealed that the non stationary framework captured larger variability and more pronounced extreme events compared with the stationary approach. In some drought periods, the JDI values in the non stationary model decreased to around −4.5, whereas the stationary model estimated values close to −3. This suggests that the stationary assumption may underestimate the intensity of severe drought events.
The analysis of drought characteristics further indicated that the number of drought events identified by the JDI increased from 11 events in the stationary model to 16 events in the non stationary model, while the mean drought duration decreased from about 26 months to nearly 15 months. Similarly, the number of hydrological drought events (SRI) increased from 10 to 13 in the non stationary framework. In addition, the maximum drought severity for SRI increased from 1.44 in the stationary model to 2.24 in the non stationary model. These findings demonstrate that the non stationary approach can detect more frequent but shorter drought events with stronger intensity.
Transition probability analysis between drought categories also showed a higher likelihood of transitions toward more severe drought states under the non stationary framework. Furthermore, the estimation of bivariate return periods revealed that considering non stationarity leads to shorter return periods for compound drought events. For example, under a severe threshold level, the joint return period decreased from approximately 50 years in the stationary model to about 32 years in the non stationary model, indicating a higher risk of extreme compound droughts than previously estimated.
Conclusion
Overall, the results highlight the importance of incorporating non stationary copula models in multivariate drought analysis. The non stationary framework provides a more realistic representation of the evolving dependence between precipitation and runoff and improves the identification of extreme compound drought events. In contrast, stationary models may underestimate drought severity and associated risks. The findings suggest that integrating multivariate drought indices such as JDI with non stationary dependence models can significantly enhance drought risk assessment and water resource management, particularly in regions experiencing climate variability and change. Future research may extend this framework by incorporating additional hydro climatic variables and developing spatial–temporal copula models to provide a more comprehensive understanding of compound drought dynamics.