The areal reduction factor: A multifractal analysis

The areal reduction factor (ARF) η is a key quantity in the design against hydrologic extremes. For a basin of area a and a duration d, η(a, d, T) is the ratio between the average rainfall intensity in a and d with return period T and the average rainfall intensity at a point for the same d and T. Empirical ARF charts often display scaling behavior. For example, for large ( a/d) ratios and given T, the ARF tends to behave like (√a/d)^(-α) for some α. Here we obtain scaling properties of the ARF under the condition that space-time rainfall has multifractal scale invariance. The scaling exponents of the ARF are related in a simple way to the multifractal properties of the parent rainfall process. We consider regular and highly elongated basins, quantify the effect of rainfall advection, and investigate the bias from estimating the ARF using sparse raingauge networks. We also study the effects of departure of rainfall from exact multifractality. The results explain many features of empirical ARF charts while suggesting dependencies on advection, basin shape, and return period that are difficult to quantify empirically. The theoretical scaling relations may be used to extrapolate the ARF beyond the empirical range of a, d and T.