ORIGINAL PAPER Bayesian power-law regression with a location parameter, with applications for construction of discharge rating curves Trond Reitan Æ Asgeir Petersen-Øverleir Published online: 30 March 2007 Ó Springer-Verlag 2007 Abstract This paper presents a Bayesian approach for fitting the standard power-law rating curve model to a set of stage-discharge measurements. Methods for eliciting both regional and at-site prior information, and issues concerning the determination of prior forms, are discussed. An efficient MCMC algorithm for the specific problem is derived. The appropriateness of the proposed method is demonstrated by applying the model to both simulated and real-life data. However, some problems came to light in the applications, and these are discussed. Keywords Power-law  Location parameter  Non-linear regression  Bayesian statistics  Stage-discharge  Rating curve 1 Introduction Power-law regression with multiplicative measurement error and a location parameter representing the unknown zero plane displacement has many applications in geo- physics, e.g. stage-discharge rating curve estimation (Lambie 1978; Rantz et al. 1982; Mosley and McKerchar 1993; Hershey 1995; ISO 1998), modelling valley profiles (Greenwood and Humphrey 2002), height–velocity rela- tionships in various settings such as wind engineering problems (Balendra et al. 2002) and fluid mechanics (Stephan and Gutknecht 2002). This study will focus on standard stage-discharge rating curve estimation, with the following assumption as starting point: Q i ¼ Cðh i  h 0 Þ b E i if h i [h 0 0 otherwise;  ð1Þ where i is an index running from 1 to the number of measurements, n. Here, the discharge, Q, is explained using a nonlinear function of the stage, h, times independent and identically distributed noise. The noise, E, needs to be positive in order to ensure a positive discharge, implying that a log–normal distribution could be an appropriate distribution for this quantity, thus E ~ log N(0,r 2 ), where the log–normal distribution is parametrised so that x ~ log N(l,r 2 ) implies log(x) ~ N(l, r 2 ). The positive constant, C, the exponent, b, the cease-to-flow point, h 0 , and the noise magnitude, r, are parameters. Clearly, the applicability of Eq. 1 depends on the stage-discharge relationship being only negligibly affected by unsteadiness or backwater effects, and that the channel friction factors and geometry are virtually unchanged within the time period under consideration. These assumptions are accepted for this study. It is easier to study Eq. 1 for stage values, h > h 0 , after doing a logarithmic transformation: q i ¼ a þ b logðh i  h 0 Þþ  i ; ð2Þ where e i ~ N(0,r 2 ) is independent noise, q i = log(Q i ) and a = log(C). Traditionally, this classical rating curve model was fitted to measurement using manual methods. Venetis (1970), was the first to provide a thorough statistical treatment for this regression model, basing his study on T. Reitan (&) Department of Mathematics, University of Oslo, P. O. Box 1053, Blindern, 0316 Oslo, Norway e-mail: trondr@math.uio.no A. Petersen-Øverleir Norwegian Water Resources and Energy Directorate, P. O. Box 5091, Majorstua, 0301 Oslo, Norway e-mail: apoe@nve.no 123 Stoch Environ Res Risk Assess (2008) 22:351–365 DOI 10.1007/s00477-007-0119-0