Event-specific multiplicative cascade models and an application to rainfall

Multiplicative cascades offer parsimonious models capable of capturing the scale-invariant (multifractal scaling) behavior of some geophysical phenomena, such as rainfall, over a large range of scales. While these models achieve a remarkable degree of universality, it is still unclear how to characterize individual events within this framework. The present work offers an event description based on a few most important (amplitude-wise) branchings of the event's multiplicative cascade generator. The proposed method is based on the modulus extrema of wavelet transforms and indexes the branches (or generator weights in the multiplicative cascade model) such that their number at each branching, magnitude, and the relative scales at which they occur can be extracted and memorized. In this way, a particular event can be characterized in a multiplicative cascade framework by only a few significant weights and their respective positioning within the cascade. The application of the present model to rainfall is supported by the evidence of branching of the wavelet modulus extrema as well as by the findings [Venugopal and Foufoula-Georgiou, 1996; Cârsteanu et al., 1997] that an important part of the signal energy of temporal rainfall events can be recovered from a few wavelet-packet components.