A space‐time theory of mesoscale rainfall using random cascades

Abstract

Following a brief review of relevant theoretical and empirical spatial results, a theory of space‐time rainfall, applicable to fields advecting without deformation of the coordinates, is presented and tested. In this theory, spatial rainfall fields are constructed from discrete multiplicative cascades of independent and identically distributed (iid) random variables called generators. An extension to space‐time assumes that these generators are iid stochastic processes indexed by time. This construction preserves the spatial structure of the cascades, while enabling it to evolve in response to a nonstationary large‐scale forcing, which is specified externally. The construction causes the time and space dimensions to have fundamentally different stochastic structures. The time dimension of the process has an evolutionary behavior that distinguishes between past and future, while the spatial dimensions have an isotropic stochastic structure. This anisotropy between time and space leads to the prediction of the breakdown of G. I. Taylor's hypothesis of fluid turbulence after a short time, as is observed empirically. General, nonparametric, predictions of the theory regarding the spatial scaling properties of two‐point temporal cross moments are developed and applied to a tracked rainfall field in a case study. These include the prediction of the empirically observed increase of correlation times as resolution decreases and the scaling of temporal cross moments, a new finding suggested by this theory.

Scaling statistics in a critical, nonlinear physical model of tropical oceanic rainfall

“…Near the critical value of this parameter, the model rainfall field exhibits multifractal scaling determined from a fractional wetted area analysis and a moment scaling analysis. It therefore must exhibit long-range spatial correlations at this point, a situation qualitatively similar to that shown by multiplicative random cascade models and GATE rainfall data sets analyzed previously (Over and Gupta, 1994;Over, 1995). However, the scaling exponents associated with the model data are different from those estimated with real data.…”

“…A key feature of random cascades is that they have longrange spatial correlations. This means that the correlation length scale is infinite; see Over (1995) for a careful derivation of this result. What this means is that the existence of scaling relations in the spatial moments of a random field, as depicted by Eq.…”

“…However, applications of spatial random cascade theory to real data have required two sets of generalizations, both of which greatly complicate the underlying mathematical structures. The first is the extension of cascade models from only space to space-time rainfall (Over and Gupta, 1996;Marsan. et al, 1996).…”

Modulation of homogeneous space-time rainfall cascades to account for orographic influences

“…Mainly due to empirical evidences, the application of multifractal models for rainfall downscaling, which allows to reproduce hierarchical structures of rainfall fields shown by Austin and Houze (1972), is widespread in literature (Lovejoy and Schertzer and Lovejoy, 1987;Gupta and Waymire, 1993;Over and Gupta, 1996;Perica and Foufoula-Georgiou, 1996;Deidda, 2000, among the others).…”

Multifractal features of sea clutter

“…In this paper, we work with a specific type of multifractal, called random multiplicative process model, to analyze sea clutter. Such models have been used to describe the energy dissipation of turbulence [5], rain fall [16], amount of radiation the earth received from the Sun [4], stock variation [13], and network traffic [8], [9].…”