Bayesian estimation of intensity–duration–frequency curves and of the return period associated to a given rainfall event

Abstract

Intensity–duration–frequency (IDF) curves are used extensively in engineering to assess the return periods of rainfall events and often steer decisions in urban water structures such as sewers, pipes and retention basins. In the province of Québec, precipitation time series are often short, leading to a considerable uncertainty on the parameters of the probabilistic distributions describing rainfall intensity. In this paper, we apply Bayesian analysis to the estimation of IDF curves. The results show the extent of uncertainties in IDF curves and the ensuing risk of their misinterpretation. This uncertainty is even more problematic when IDF curves are used to estimate the return period of a given event. Indeed, standard methods provide overly large return period estimates, leading to a false sense of security. Comparison of the Bayesian and classical approaches is made using different prior assumptions for the return period and different estimation methods. A new prior distribution is also proposed based on subjective appraisal by witnesses of the extreme character of the event.

Appendices

Appendix 1: Generalized extreme value distribution

The distribution of annual maximal rainfall intensity is represented in the paper by the GEV distribution. The GEV is the limiting distribution of the maximum of a sequence of independent and identically distributed random variables. The parametrization for the cdf and pdf follows Hosking et al. (1985):

$$ {F_{\rm GEV}}(x|\kappa, \alpha, \xi) = \exp\{-\exp\{-y\}\} $$

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$$ {f_{\rm GEV}}(x|\kappa, \alpha, \xi) = \alpha^{-1} \exp\{-(1-\kappa)y-e^{-y}\} $$

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where

$$ y =\left\{\begin{array}{ll} -\kappa^{-1}\log\{1-\kappa(x-\xi)/\alpha\}, & \kappa\neq 0 \\ (x-\xi)/\alpha, &\kappa=0.\\ \end{array}\right. $$

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The inverse cdf, also called the quantile function, has a simple analytical form:

$$ F_{{{\text{GEV}}}}^{{ - 1}} (q) = \left\{\begin{array}{lll} \xi + \alpha\{1-(-\log q)^\kappa\}/\kappa,& \kappa\neq 0 \\ \xi - \alpha\log(-\log q), & \kappa = 0. \end{array}\right. \quad \hbox{for}\; q\in[0,1] $$

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There are no constraints on the three parameters κ (shape), ξ (location) and α (scale), but the range of x is defined as:

$$ x \left\{\begin{array}{ll} -\infty < x \leq \xi+\alpha/\kappa, & \hbox{if}\; \kappa >0 \\ -\infty < x < \infty, & \hbox{if}\; \kappa=0\\ \xi+\alpha/\kappa \leq x < \infty, & \hbox{if}\; \kappa<0.\\ \end{array}\right. $$

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In hydrological practice however, the parameters are usually confined to −1/2 < κ < 1/2, ξ > 0 and α > 0.

Appendix 2: Classical estimators

2.1 Maximum likelihood

The maximum likelihood method seeks the parameters of a distribution maximizing the likelihood:

$$ \hat{\kappa}, \hat{\xi}, \hat{\alpha} = \mathop{\hbox{argmax}}\limits_{(\kappa, \xi, \alpha)}{\prod_i {f_{\rm GEV}}(x_i|\kappa, \xi, \alpha)}, $$

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where f _GEV is the GEV likelihood (Eq. 16 of Appendix 1) and x _i the annual maxima. No explicit solution exists for the AMS/GEV model and ML estimators have to be determined numerically. Although the method is straightforward to apply, it may lead to poor estimates for short series (Madsen et al. 1997).

2.2 Probability weighted moments

The method of probability weighted moments, also denoted as the L-moments method, leads to the following parameter estimators (Hosking et al. 1985):

$$ \hat{\xi} = \hat{\lambda}_1 + {\frac{\hat{\alpha}} {\hat{\kappa}}}(\Gamma(1+ \hat{\kappa})-1) $$

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$$ \hat{\alpha} = {\frac{\hat{\lambda_{2}} \hat{\kappa}} {(1-2^{-\hat{\kappa}}) \Upgamma(1+\hat{\kappa})}} $$

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$$ \hat{\kappa} = 7.8590 \; c + 2.9554 \; c^{2} $$

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$$ c = {\frac{2}{\hat{\tau}_3+3}}- {\frac{\ln(2)}{\ln(3)}} $$

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The L-moments estimators $\hat{\lambda}_1,$ $\hat{\lambda}_2$ and $\hat{\tau}_3 = \hat{\lambda}_3/\hat{\lambda}_2$ are obtained using the first three PWM estimators (Landwehr et al. 1979). The numerical routines for the GEV PWM parameter estimation are taken from Hosking (1996).

2.3 Method of moments

Given the first three moments $\hat{\mu},$ $\hat{\sigma},$ $\hat{\gamma},$ the estimators of the GEV parameters are given by the following expressions (Stedinger et al. 1993):

$$ \hat{\xi} = \hat{\mu} + {\frac{\hat{\alpha}} {\hat{\kappa}}}(\Upgamma(1 + \hat{\kappa}) - 1) $$

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$$ \hat{\alpha} = \hbox{sign}{\{\hat{\kappa}\}} {\frac{\hat{\sigma}\hat{\kappa}}{\{\Upgamma(1 + 2 \hat{\kappa})- [\Upgamma(1 + \hat{\kappa})]^{2}\}^{1/2}}} $$

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$$ \hat{\gamma} = \hbox{sign}{\{\hat{\kappa}\}} {\frac{-\Upgamma(1+3 \hat{\kappa}) + 3 \Upgamma(1+ \hat{\kappa}) \Upgamma(1+2 \hat{\kappa}) - 2 \Upgamma^{3}(1 + \hat{\kappa})}{\{\Upgamma(1 + 2 \hat{\kappa})- [\Upgamma(1 + \hat{\kappa})]^{2}\}^{1/2}}} $$

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The last equation for $\hat{\kappa}$ must be solved iteratively. It should be noted that the MOM estimates are limited to $\hat{\kappa} > -1/3,$ a condition generally satisfied in hydrological applications.