Assessing dependence among weights in a multiplicative cascade model of temporal rainfall

"Classical" multifractal analysis shows, with a good degree of confidence, that the fit of multiplicative cascades to rainfall time series is appropriate, at least from the point of view of preserving the f(α) spectrum of scaling exponents. However, basing the analysis only on the f(α) curve allows only limited discrimination between different types of cascade models; thus other descriptors with more discriminating power are needed. Also, the question of whether cascades with independent weights are appropriate for rainfall remains unanswered and needs to be addressed. In the present work we address this question and provide an assessment of the dependence structure among weights in a multiplicative cascade model of temporal rainfall. We introduce a quantity based on oscillation coefficients (describing how many of the total n-tuples of the series are obeying a certain pattern up-down-up etc.), and find that this quantity is invariant under aggregation for a multiplicative cascade model and has the ability to depict the presence and type of correlation in the weights of the cascade generator. Application of this development to high-resolution temporal rainfall series consistently suggests the need for negative correlation in weights of a binary multiplicative cascade in order to match the oscillation coefficient structure of rainfall. This is interpreted as an indication of dependence in the splitting mechanisms of intensities cascading over successive scales and might have important implications for rainfall modeling and process understanding.