Fractal properties of rain, and a fractal model

Abstract

This paper amplifies upon earlier theories and observations by the authors, advances a probabilistic model of rain fields, and exhibits realistic simulations, which the reader is advised to scan before continuing with the text. The model is also compared with other approaches. The theory of fractals has been in part motivated by the Hurst effect, which is an empirical observation in hydrology and climatology. A fractal is an unsmooth shape that is scaling, that is, where shape appears unchanged when examined by varying magnifications. The study of fractals is characterized by the prevalence of hyperbolically distributed random variables, for which Pr( U > u ) ∝ u a, where Pr is the probability that the value of the variable exceeds u , and a is a positive exponent. Lovejoy established the applicability of fractals in meteorology, by showing that cloud and rain areas project on the Earth along shapes whose boundaries are fractal curves, and that the temporal and spatial structure of rain is rife with hyperbolically distributed features. These observations, as amplified in this paper, set up the challenge of constructing fractal models with the observed properties. The models presented here belong to a very versatile family of random processes devised in Mandelbrot: fractal sums of (simple) pulses, or FSP processes. The simulations to be presented reveal that these processes involve a scaling heirarchy of “bands”, “fronts”, and “clusters”, as well as other complex shapes, none of which had been deliberately incorporated in the process. This very rich morphology and the related statistical distributions exemplify the power of simple fractal models to generate complex structures, and are in accord with the wide diversity of actual rainfall shapes. It is argued that this model already provides a useful context in which the basic statistical properties of the rainfield, including the relationship between the temporal and spatial structure, may be studied. DOI: 10.1111/j.1600-0870.1985.tb00423.x