Bayesian power-law regression with a location parameter, with applications for construction of discharge rating curves

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.

Appendix: Case study data and code

The stage-discharge measurement data used in the case studies as well as the code used in this paper can be accessed at http://www.folk.uio.no/trondr/hydrasub/ratingcurve.html.

1.1 Finiteness of the expected discharge for the semi-conjugate prior

In order to calculate the posterior expectation of the discharge, Q, given h ₀, one needs to integrate the product of π(a, b), π(σ²), f(D|a, b, h ₀, σ²) and Q(h, a, b, h ₀) over the parameters a, b and σ². Only π(σ²) and f(D|a, b, h ₀, σ²) contains σ². The integral

$$\begin{aligned} &\,\int_0^{\infty}\pi(\sigma^2) f(D|a,b,h_0,\sigma^2) {\rm d}\sigma^2\\ &\quad\propto \int_0^{\infty} (\sigma^2)^{-(\alpha_0+n/2+1)}\exp\left(-{\frac{1}{\sigma^2}} \left[ \beta_0 + (Y-X\gamma)^t(Y-X \gamma)/2 \right] \right) {\rm d}\sigma^2 \\ &\quad= \Gamma(\alpha_0+n/2)(\beta_0+(Y-X \gamma)^t(Y-X\gamma)/2)^{-(\alpha_0+n/2)}\\ &\quad \leq \Gamma(\alpha_0+n/2)\beta_0^{-(\alpha_0+n/2)} \equiv K \quad \forall \gamma \in \Re^2\end{aligned}$$

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is obtained by identifying the integral as the normalization of an inverse-gamma distribution, see also Eq. 9.

Defining the prediction design vector x ₀(h) = (1 log(h − h ₀))^t, an upper limit for the expected discharge can be found:

$$ \begin{aligned} E(Q(h) | h_0))&=E(\exp(x_0(h)^t \gamma) | h_0)\\ & \propto \int \exp(x_0(h)^t \gamma) \pi(a,b) \pi(\sigma^2) f(D|a,b,h_0,\sigma^2) {\rm d}\gamma\,{\rm d}\sigma^2\\ &\leq K \int \exp(x_0(h)^t \gamma - {\frac{1}{2}}(\gamma-\gamma_0)V^{-1}(\gamma-\gamma_0)) {\rm d}\gamma \end{aligned} $$

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This is a two-dimensional quadratic term integral, which has a finite value since the quadratic term in the exponent is negative. Thus given h ₀, the expected discharge is finite.