Multiscaling properties of rainfall and bounded random cascades

Abstract

Abstract. The connection among different types of exponents characterizing multiscaling properties of rainfall and a criterion for stationarity of random fields are discussed, and a new phenomenological model for rainfall time series simulation is proposed. The bounded random cascade model presented here is a generalization of the well-known a model with the multiplicative weights of the generator converging to unity as the cascade proceeds to smaller scales. This allows one to directly simulate a multiscaling random field with an energy spectrum exponent, •3 > 1, which is typical for rainfall time series data but which cannot be produced by a standard a model. A procedure is proposed for estimating the cascade parameters of this new bounded a model from observed data. Parameters are estimated from two data sets with different degrees of intermittency and different spectral exponents. The bounded a model simulations using these parameters produced realistic rainfall time series with spectral exponents similar to their observed counterparts. We expect that the occurrence of rainfall and turbulence have something in common since rain is a tracer of turbulence at small scales and the larger eddies may be involved in the processes which produce rainfall. This suggests the two phenomena might be described by the same models. However, we believe it is clear that the physical processes which produce large areas of uniform rain and small convective showers, for example, are radically different and are extremely unlikely to be described as different realizations of a random multiplicative cascade with the same parameters. In our view the best justification for the use of the multiplicative cascade models in rainfall pattern analysis is the remarkable ability of the technique to reproduce the observed structure of rainfall patterns and its statistical properties. We therefore believe that the careful analysis of the goodness of fit of the statistical properties of the simulated pattern with the observed one is actually crucial, especially taking into account the phenomenological character of the cascade models. As already mentioned, /3 is an important exponent in classifying whether an observed field can be modeled by a bounded or unbounded cascade depending on whether/3 > 1 or/3 < 1, respectively. This condition on/3 has often been related to the stationarity of a field. However, we show that this is not the only possible interpretation and illustrate the point with a counterexample of a stationary random process with scaling energy spectrum with /3 > 1. Another important result that follows from this counterexample is that the condition/3 > 1 implies that the moments of the raw field do not scale, and thus K(q) function is not defined. When using discrete cascades to model rainfall (and many other geophysical fields) one runs into the familiar problemSection 3 presents a new type of bounded cascade along with a parameter estimation method for the model from real data.As an applicatioh, the bounded a model parameters are...

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