Abstract. We prove the existence of the total length process for the genealogical tree of a population model with random size given by a quadratic stationary continuous-state branching processes. We also give, for the one-dimensional marginal, its Laplace transform as well as the fluctuation of the corresponding convergence. This result is to be compared with the one obtained by Pfaffelhuber and Wakolbinger for constant size population associated to the Kingman coalescent. We also give a time reversal property of the number of ancestors process at all time, and give a description of the so-called lineage tree in this model.

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Recent evidence of the impact of watershed underlying conditions on hydrological processes have made the assumption of stationarity widely questioned. In this study, the temporal variations of frequency distributions of the annual maximum flood were investigated by continuous hydrological simulation considering nonstationarity for Weihe River Basin (WRB) in northwestern China. To this end, two nonstationary versions of the GR4J model were introduced, where the production storage capacity parameter was regarded as a function of time and watershed conditions (e.g., reservoir storage and soil-water conservation land area), respectively. Then the models were used to generate long-term runoff series to derive flood frequency distributions, with synthetic rainfall series generated by a stochastic rainfall model as input. The results show a better performance of the nonstationary GR4J model in runoff simulation than the stationary version, especially for the annual maximum flow series, with the corresponding NSE metric increasing from 0.721 to 0.808. The application of the nonstationary flood frequency analysis indicates the presence of significant nonstationarity in the flood quantiles and magnitudes, where the flood quantiles for an annual exceedance probability of 0.01 range from 4187 m3/s to 8335 m3/s for the past decades. This study can serve as a reference for flood risk management in WRB and possibly for other basins undergoing drastic changes caused by intense human activities.

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