Spectral multiscaling of spatially averaged rain rate: a hint for spatio-temporal modelling

Abstract

Recent research has shown that, conditionally on rain, probability moments of instantaneous spatial averages of rain rate, over differently scaled subregions of a rain field, possess an interesting scaling property called wide sense multiscaling, with respect to magnifying spatial scales. This empirical fact, along with some theoretical considerations, have led to implementation of the so called multiplicative random cascades in order to model the instantaneous spatial variability of rain rate, given that it rains. However, random cascades do not account for the evolution of spatial variability through time. In this article an alternative notion of multiscaling is introduced, providing new theoretical basis for further consideration of multiplicative random cascades, as a class of models which potentially capture not only the spatial variability of spatially averaged rain rate fields, but also accounting for its evolution in the course of time. The new notion of multiscaling introduced here is referred to as spectral multiscaling, is defined for second order stationary processes of spatially averaged rain rate, and (similar to probabilistic multiscaling) is discerned to that of strict and of wide sense, with regard to the normalized spectral distribution and its spectral moments, respectively. In fact, the validity of spectral multiscaling properties is verified statistically, using time series of spatially averaged rain rate data from a regularly observed (by radar) tropical rain field known with the acronym TOGA-COARE. The results of this empirical non-model-based analysis, point to the validity of multiscaling of the normalized spectral distribution across the entire spectrum of frequencies, and also of multiscaling of the corresponding spectral moments. Copyright © 1998 John Wiley & Sons, Ltd.