Imperfect scaling of time and space–time rainfall

Abstract

Scale invariance is the most fertile concept to be introduced in stochastic rainfall modeling in 15 years. In particular, a form of scale invariance called multifractality has been exploited to construct parsimonious representations of rainfall in time and space and address fundamental problems of hydrology such as rainfall extremes, downscaling, and forecasting. However, several authors have observed that rainfall is scale invariant only in approximation and within limited ranges. Here, we make a systematic analysis of the deviations of time and space–time rainfall from multifractality. We use a flexible multiplicative cascade model, which produces multifractality as a special case while allowing deviations from scale invariance to occur. By fitting the model to rainfall records from different climates and over land or ocean, we find significant and consistent departures from multifractality in both the alternation of wet and dry conditions and the fluctuations of precipitation intensity when it rains. The fractal dimension of the rain support increases with increasing rain rate and the (multiplicative) fluctuations are larger at smaller scales and for lighter rainfall. A plausible explanation of these departures from scaling is that the rate of water vapor condensation in the atmosphere is a multifractal process in three space dimensions plus time, but multifractality is destroyed when the condensation rate is integrated to produce rainfall intensity at fixed altitudes.