Estimating Joint Probabilities for Confluences across CONUS Using Copula

Abstract

Hydraulic structure design near stream confluences requires meticulous consideration of combined flows from both tributaries, especially when the structure is located on small tributaries that drain into larger downstream rivers. The recently revised design approach for confluent structures, as presented in Hydrologic Engineering Circular No. 19 (HEC-19) [FHWA (Federal Highway Administration). 2023. Highway hydrology: Evolving methods, tools, and data. Hydraulic Engineering Circular No. 19, FHWA-HIF-23-050. Washington, DC: FHWA], utilizes copula-based joint probability estimates. However, the applicability of the revised approach is limited due to restrictions on maximum cumulative drainage area and an implicit maximum drainage area ratio (DAR) limit. This study aims to address these limitations and furthermore also provide regionally suitable recommendations and account for variations in streamflow regimes in climatically distinct regions, which HEC-19 does not adequately address. To that effect, this study systematically investigates the interdependence of flows in mainstream and tributary channels using data from 145,000 confluences. A Gumbel–Hougaard copula-based methodology is developed for estimating joint probabilities at confluences across the contiguous United States (CONUS). Additionally, the study provides a method for generating regionally suitable design criteria. Results of the study find significant correlations emerge between mainstream and tributary streamflow, denoted by Kendall’s tau (τ), ranging generally from 0.5 to 0.8. Importantly, τ, a pivotal parameter in the Gumbel–Hougard copula framework, exhibits a substantial association with the DAR. The correlation between DAR and τ is also found to vary considerably across regions, with Midwestern Plains showing good relation while the mountainous and relatively dry Western US exhibits poor relation. Utilizing regional and CONUS-wide regression equations linking DAR and τ, the study establishes a robust foundation for correlating DAR with joint probabilities at confluences. The results from this study provide a more precise approach for hydraulic design at stream confluences and advances our understanding of mainstream and tributary flow interdependencies at confluences. Importantly, the findings of the study demonstrate a more generalized methodology with wider applicability, not constrained by the upper limits on maximum drainage area ratio or cumulative drainage area imposed by HEC-19.