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In this study, a bivariate hydrologic risk framework is proposed through coupling Gaussian mixtures into copulas, leading to a coupled GMM-copula method. In the coupled GMM-Copula method, the marginal distributions of flood peak, volume and duration are quantified through Gaussian mixture models and the joint probability distributions of flood peak-volume, peak-duration and volume duration are established through copulas. The bivariate hydrologic risk is then derived based on the joint return period of flood variable pairs. The proposed method is applied to the risk analysis for the Yichang station on the main stream of the Yangtze River, China. The results indicate that (i) the bivariate risk for flood peak-volume would keep constant for the flood volume less than 1.0 × 10 5 m 3 /s day, but present a significant decreasing trend for the flood volume larger than 1.7 × 10 5 m 3 /s day; (ii) the bivariate risk for flood peak-duration would not change significantly for the flood duration less than 8 days, and then decrease significantly as duration value become larger. The probability density functions (pdfs) of the flood volume and duration conditional on flood peak can also be generated through the fitted copulas. The results indicate that the conditional pdfs of flood volume and duration follow bimodal distributions, with the occurrence frequency of the first vertex decreasing and the latter one increasing as the increase of flood peak. The obtained conclusions from the bivariate hydrologic analysis can provide decision support for flood control and mitigation.

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In this study, a recourse‐based interval fuzzy programming (RIFP) model is developed for tackling uncertainties expressed as fuzzy, interval, and/or probabilistic forms in an effluent trading program. It can incorporate preregulated water‐pollution control policies directly into its optimization process, such that an effective linkage between environmental regulations and economic implications (i.e., penalties) caused by improper policies due to uncertainty existence can be provided. The RIFP model is applied to point‐nonpoint source effluent trading of the Xiangxi River in China. The efficiency of trading efforts between water quality improvement and net system benefit under different degrees of satisfying discharge limits is analyzed. The results are able to help support (1) formulation of water‐pollution control strategies under various economic objectives and system‐reliability constraints, (2) selection of the desired effluent trading pattern for point and nonpoint sources, and (3) generation of tradeoffs among system benefit, satisfaction degree, and pollutant mitigation under multiple uncertainties. Compared with the traditional regulatory approaches, the results demonstrate that the water‐pollution control program can be performed more cost‐effectively through trading than nontrading.

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