Estimating Joint Flow Probabilities at Stream Confluences by Using Copulas

Highway drainage structures are often located near the confluence of two streams, where the structures may be subjected to inundation by high flows from either stream. These structures must be designed to meet specified performance objectives for floods. Although many tools estimate the probability of flooding on a single stream, there is little guidance for quantifying the probability of flooding on two streams simultaneously, which is a problem of estimating the joint probability of coincident flows. The objective of the research summarized in this paper was to develop practical procedures for estimating joint probabilities of design coincident flows at stream confluences and guidelines for applying these procedures. The scope was limited to riverine areas and did not include coastal areas. Two practical strategies emerged from the analyses: ( a ) a strategy for determining a set of combinations of exceedance probabilities associated with the desired joint probability for design based on copulas and ( b ) a series of conditional probability matrices for use in the total probability method. Only the copula strategy is addressed in this paper. The use of a series of design tables makes the procedure straightforward for the designer to apply without the need for a background in statistics. An example application of the method that uses a pair of confluent streams in Washington State is provided. The application illustrates the steps needed for a designer to implement the method. The procedure advances the state of the art by providing a tool applicable to confluent watersheds in the coterminous United States.