A Bayesian hierarchical model for estimating the statistical parameters in a three-parameter log-normal distribution for monthly average streamflows

Abstract

We develop a Bayesian hierarchical model (BHM) for estimating the statistical parameters for monthly average streamflows. We assume monthly average streamflow can be characterized by a three-parameter log-normal distribution (LN3). The three underlying statistical parameters are shift, shape, and location. When estimating a parameter, such as the shape, of a given month the BHM utilizes historical observations not only from the month under consideration but also from all other months. This is different from traditional statistical parameter estimation methods that only use historical observations for the month under consideration. We apply the proposed BHM for parameter estimation to eight watersheds in the United States, where historical unimpaired streamflows have been collected. We also carry out parameter estimation using traditional methods, such as the maximum likelihood estimation, the method of moments, and the L-moment method. Using cross-validation with test data log-likelihood as the measure of performance, the results show that BHM outperforms traditional estimation methods. In addition, we show that as available observation data decreases, the more the proposed method improves relative to traditional methods. Since BHM utilizes information contained in the entire data set, it is especially suited for parameter estimation where historical observations are limited. Furthermore, we conduct a comparative analysis between BHM and an autoregressive model to demonstrate the advantage of BHM.