Modeling of rainfall time series using two‐state renewal processes and multifractals

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Abstract
The high variability of rainfall fields comes from (1) the occurrence of wet and dry events and (2) from the intermittency of precipitation intensities. To model these two aspects for spatial variability,Over and Gupta[1996] have proposed a lognormal cascade model with an atom at zero, which corresponds to combine in one model two independent cascade models, the β and the lognormal multifractal models. In the present work, we test this approach for time variability, using a high‐resolution rainfall time series. We built a continuous version of the discrete β model and investigate some of its dynamical properties. We show that the β model cannot fit the probability density for the duration of the wet state. In order to model the succession of wet and dry periods we therefore use a two‐state (or alternate) renewal process based on appropriate fits of the empirical densities. The synthetic series obtained this way reproduces the scaling of the original support. The intensity of the rainfall events is then modeled using the universal multifractal model, generalizing the lognormal model. We show that the fractal support of the rainfall events must be taken into account to retrieve the parameters of this model. This combination of two different models allows to closely reproduce the high variability at all scales and long‐range correlations of precipitation time series, as well as the dynamical properties of the succession of wet and dry events. Simulations of the high‐resolution rainfall field are then performed displaying the salient features of the original time series.

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