Causal space‐time multifractal processes: Predictability and forecasting of rain fields

Abstract

Building on earlier cascade models of rainfall, we propose a model of space‐time rain fields based on scaling dynamics. These dynamics are indeed related to the space‐time symmetries of the turbulent medium within which rainfall occurs: the underlying phenomenology corresponds to a cascade of structures with lifetimes depending only on the scale of the structures. In this paper we clarify two major issues: the scaling anisotropy between space and time, and the need to respect causality, i.e., a fundamental asymmetry between past and future. We detail how this ”arrow of time” breaks the mirror symmetry with respect to the spatial hyperplane, and how it can be introduced in continuous multiplicative cascade models so as to remove the artificial temporal mirror symmetry of earlier models. We show that such a causal multifractal field can be understood as the result of an anomalous diffusion acting on the singularities of the field. Finally we will exploit and test these models through (1) a succinct analysis of rainfall data, (2) numerical simulations of the temporal decorrelation of two initially similar fields (accounting for the loss of predictability of the process), and (3) a forecasting method for multifractal rain fields.