Operational Bayesian GLS regression for regional hydrologic analyses

Abstract

This paper develops the quasi analytic Bayesian analysis of the generalized least squares (GLS)(BGLS) model introduced by Reis et al. (2005, https://doi.org/10.1029/2004WR003445) into an operationaland statistically comprehensive GLS regional hydrologic regression methodology to estimate oodquantiles, regional shape parameters, low ows, and other statistics with spatially correlated ow records.New GLS regression diagnostic statistics include a Bayesian plausibility value, pseudo adjusted R2,pseudo analysis of variance table, and two diagnostic error variance ratios. Traditional leverage andinuence are extended to identify rogue observations, address lack of t, and support gauge network designand regionofinuence regression. Formulas are derived for the Bayesian computation of estimators,standard errors, and diagnostic statistics. The use of BGLS and the new diagnostic statistics are illustratedwith a regional logspace skew regression analysis for the Piedmont region in the Southeastern U.S. Acomparison of ordinary, weighted, and GLS analyses documents the advantages of the Bayesian estimatorover the method ofmoment estimator of model error variance introduced by Stedinger and Tasker (1985,https://doi.org/10.1029/WR021i009p01421). Of the three types of analyses, only GLS considers thecovariance among concurrent ows. The example demonstrates that GLS regional skewness models can behighly accurate when correctly analyzed: The BGLS average variance of prediction is 0.090 for SouthCarolina (92 stations), whereas a traditional ordinary least squares analysis published by the U.S.Geological Survey yielded 0.193 (Feaster & Tasker, 2002, https://doi.org/10.3133/wri024140). BGLSprovides a statistical valid framework for the rigorous analysis of spatially correlated hydrologic data,allowing for the estimation of parameters and their actual precision and computation of several diagnosticstatistics, as well as correctly attributing variability to the three key sources: time sampling error, modelerror, and signal