Nonstationary Hydrological Distribution Estimation Using Hierarchical Model with Stochastic Covariates

In nonstationary hydrological frequency analysis, the nonstationarity of hydrological series is often characterized by expressing the distribution parameters as functions of some covariates. The distribution parameters with the stochastic covariates such as climatic indices could result in time variations in forms of wide random fluctuations, which would fail to give a clear-cut description and explanation for the nonstationarity. In this paper, a hierarchical model is proposed to derive the nonstationary hydrological distribution with smooth changes by characterizing stochastic covariates as random processes. Under this model, the conditional distribution of the interested hydrological variable given its covariates is first constructed, and then the distribution of the interested hydrological variable is derived by combining its conditional distribution with the probability distribution of the stochastic covariates. The results of a simulation study indicate satisfactory performance of the proposed hierarchical model in fitting the generated series. The annual runoff series of the Weihe River is employed to perform a case study. The results reveal that the hierarchical model is able to give an intuitive description of the nonstationarity of the annual runoff. On the basis of the smoothly changing hydrological distribution derived by the hierarchical model, the nonstationarity of annual runoff series can be clearly described and explicitly associated with the changes of the covariates. It is found that the decline in the mean of annual runoff of the Weihe River is attributed to precipitation decline, climate warming, and agricultural irrigation. The hierarchical model with stochastic covariates is more applicable in engineering practice than the theoretical distribution with time-varying parameters.