Estimating storm surge intensity with Poisson bivariate maximum entropy distributions based on copulas

Abstract

This paper introduces four kinds of novel bivariate maximum entropy distributions based on bivariate normal copula, Gumbel–Hougaard copula, Clayton copula and Frank copula. These joint distributions consist of two marginal univariate maximum entropy distributions. Four types of Poisson bivariate compound maximum entropy distributions are developed, based on the occurrence frequency of typhoons, on these novel bivariate maximum entropy distributions and on bivariate compound extreme value theory. Groups of disaster-induced typhoon processes since 1949–2001 in Qingdao area are selected, and the joint distribution of extreme water level and corresponding significant wave height in the same typhoon processes are established using the above Poisson bivariate compound maximum entropy distributions. The results show that all these four distributions are good enough to fit the original data. A novel grade of disaster-induced typhoon surges intensity is established based on the joint return period of extreme water level and corresponding significant wave height, and the disaster-induced typhoons in Qingdao verify this grade criterion.

Appendix

The distribution function of Gumbel distribution is

$$ F(x) = \exp \left[ { - \exp \left( { - \frac{{x - \mu }}{\sigma }} \right)} \right] $$

(20)

in which μ and σ are the location parameter and scale parameter, respectively.

The distribution function of Weibull distribution is

$$ F\left( x \right) = \left\{ {\begin{array}{*{20}c} {1 - \exp \left[ { - \frac{{\left( {x - \mu } \right)^{\gamma } }}{\sigma }} \right],} \hfill & {x \ge \mu } \hfill \\ {0,} \hfill & { \, x < \mu } \hfill \\ \end{array} } \right. $$

(21)

in which μ > 0 is the location parameter, σ > 0 is the shape parameter, γ > 0 is the scale parameter.

The distribution function of lognormal distribution is

$$ F(x) = \int\limits_{ - \infty }^{x} {\frac{1}{{t\sigma \sqrt {2\pi } }}\exp \left[ { - \frac{1}{{2\sigma^{2} }}\left( {\ln t - \mu } \right)^{2} } \right]{\text{d}}t},\quad x \ge a $$

(22)

in which μ and σ are the location parameter and scale parameter, respectively.