New asymptotic and preasymptotic results on rainfall maxima from multifractal theory

Abstract

Contrary to common belief, Fisher-Tippett's extreme value (EV) theory does not typically apply to annual rainfall maxima. Similarly, Pickands' extreme excess (EE) theory does not typically apply to rainfall excesses above thresholds on the order of the annual maximum. This is true not just for long-averaging durations d but also for short d and in the high-resolution limit as d → 0. We reach these conclusions by applying large-deviation theory to multiplicative rainfall models with scale-invariant structure. We derive several asymptotic results. One is that, as d → 0, the annual maximum rainfall intensity in d, I[subscript yr,d,] has generalized extreme value (GEV) distribution with a shape parameter k that is significantly higher than that predicted by EV theory and is always in the EV2 range. The value of k depends not on the upper tail of the marginal distribution but on regions closer to the body. Under the same conditions, the excesses above levels close to the annual maximum have generalized Pareto distribution with parameter k that is always higher than that predicted by Pickands' EE theory. For finite d, the distribution of I[subscript yr,d] is not GEV, but in accordance with empirical evidence is well approximated by a GEV distribution with shape parameter k that increases as d decreases. We propose a way to estimate k under preasymptotic conditions from the scaling properties of rainfall and suggest a near-universal k(d) relationship. The new estimator promises to be more accurate and robust than conventional estimators. These developments represent a significant conceptual change in the way rainfall extremes are viewed and evaluated.