Rainfall extremes, excesses, and intensity‐duration‐frequency curves: A unified asymptotic framework and new nonasymptotic results based on multifractal measures

Abstract

We present a unified asymptotic theory of rainfall extremes including annual maxima, excesses above high thresholds, and intensity‐duration‐frequency (IDF) curves that builds on previous findings and derive new nonasymptotic results. The analysis is based on stationary multifractal representations of rainfall and produces extensions of the familiar results from extreme value (EV) and extreme excess (EE) theories. The latter results apply to the T‐yr maximum as T→∞ and the excess above z as z→∞. By exploiting the scaling relationship among the distributions of rainfall intensity for different averaging durations d, the multifractal asymptotics include, in addition, results in the small‐scale limits d→0 and with α > 0. In all cases, the maximum distributions are of the generalized extreme value (GEV) type, but the index k depends on the limit considered. Multifractal models also produce asymptotic scaling results for the IDF curves. For the nonasymptotic case (d and T finite), we obtain accurate approximations of the IDF curves and derive a semitheoretical formula for the index k of the GEV model that best approximates the distribution of the annual maximum over a finite range of return‐period intensities. The nonasymptotic analysis explains several observed deviations of rainfall extremes from the asymptotic predictions, such as the tendency of k to decrease as the averaging duration d increases and the tendency of the IDF curves to converge as d or the return period T increase.