CAUSAL SPACE-TIME MULTIFRACTAL PROCESSES - PREDICTABILITY AND FORECASTING OF RAIN FIELDS

Building on earlier cascade models of rainfall, we propose a model of space-time rain fields based on scaling dynamics, These dynamics are i ndeed related to the space-time symmetries of the turbulent medium wit hin 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 scalin g 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 til e spatial hyperplane, and how it can be introduced in continuous multi plicative cascade models so as to remove the artificial temporal mirro r 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 t hese models through (1) a succinct analysis of rainfall data, (2) nume rical simulations of the temporal decorrelation of two initially simil ar fields (accounting for the loss of predictability of the process), and (3) a forecasting method for multifractal rain fields.