Estimating Joint Probabilities of Design Coincident Flows at Stream Confluences

The results of a research study to develop practical procedures for estimating joint probabilities of coincident flows at stream confluences and guidelines for applying the procedures to design problems are documented in this report. The study was based on a database of 85 gage pairs located throughout the coterminous United States. Two practical design methods emerged from the study. The first method for determining a set of exceedance probability combinations associated with the desired joint probability for design is based on copulas. The Gumbel- Hougaard copula was selected as the most appropriate for this application. Use of the copula requires estimation of the Kendall’s τ rank correlation for an ungaged watershed pair. Tools for estimating this parameter based on watershed characteristics are also a product of the research. For a given Kendall’s τ, the appropriate combinations of events on the main and tributary streams for a desired joint exceedance probability are provided in tabular form. The second method is the use of the total probability method. Although the total probability method, based on the total probability theorem, is a well-documented procedure, this study provided recommended conditional probability matrices needed to apply the theorem. Both methods result in the design conditions representing the joint probability desired by the designer for analysis.