The use of Bayesian methods for fitting rating curves, with case studies

Using data from two very large watersheds and five smaller, this paper explores the use of Bayesian methods for fitting rating curves. Posterior distribution of rating-curve parameters were calculated using Markov Chain Monte Carlo (MCMC) methods, and 95% credible intervals were calculated for predicted discharges, given stage. Expected discharge was related to stage using a link function. For the five smaller watersheds, the assumptions were (a) that the distribution of discharge Q, given stage h, is Normal, with variance proportional to h; (b) that a log link function relates μQ, the mean of Q∣h, to a function of stage, of the form μQ = β(h + α)λ. For the two large watersheds, however, a better fit was obtained by taking the distribution of Q to be log-Normal, and the link function as ln μQ = β0 + β1h. For the two large watersheds, priors for all three parameters were taken as uninformative; for the five smaller, the prior for parameter λ was taken as Normally distributed, N(2, 0.5). Acceptable ratings were obtained for all seven sites. It is argued that distributions of derived variables (such as annual maximum discharge) can be derived directly from (a) the posterior distribution of rating-curve parameters, and (b) the stage record, without recourse to additional assumptions. Estimates thus obtained for the T-year event will incorporate rating-curve uncertainty. It is argued that Bayesian methods are appropriate for rating-curve calculation because their inherent flexibility (a) allows the incorporation of prior information about the nature of a rating curve; (b) yields credible intervals for predicted discharges and quantities derived from them; (c) can be extended to allow for uncertainty in stage measurements.