Space–time scaling behavior of rain statistics in a stochastic fractional diffusion model

Abstract

Statistical properties of rain exhibit the interesting feature that they depend in a nontrivial way on the length and time scales over which rain rate is averaged. A quantitative understanding of this dependence can be utilized to relate statistics at different scales and is important for inter-comparison of rainfall data obtained from measuring devices with differing space–time resolutions. A stochastic dynamical model of rainfall based on a fractional diffusion type kinetic equation introduced earlier by the authors describes fairly well how the second moment statistics of area-averaged rain rate depend on the averaging length L and predicts a power law scaling behavior as L→0. The model pictures the correlation of the precipitation field as arising from two-dimensional Lévy flights. The present paper extends the investigation to the full space–time covariance function of the precipitation field. In particular, a scaling regime is identified in which the various second moment statistics of area- and/or time-averaged rain field exhibit invariance under a combined rescaling of the space and time variables—a property known as dynamic scaling, the scaling exponent being identified with the Lévy index. Although the space and time scales resolved in the radar data used to establish the model turn out to be too coarse for the dynamic scaling behavior to be experimentally demonstrated, we predict that it should be observable in high frequency rain gauge data from dense gauge networks.