A stochastic fractional dynamics model of space‐time variability of rain

Abstract

[1] Rainfall varies in space and time in a highly irregular manner and is described naturally in terms of a stochastic process. A characteristic feature of rainfall statistics is that they depend strongly on the space-time scales over which rain data are averaged. A spectral model of precipitation has been developed based on a stochastic differential equation of fractional order for the point rain rate, which allows a concise description of the second moment statistics of rain at any prescribed space-time averaging scale. The model is thus capable of providing a unified description of the statistics of both radar and rain gauge data. The underlying dynamical equation can be expressed in terms of space-time derivatives of fractional orders that are adjusted together with other model parameters to fit the data. The form of the resulting spectrum gives the model adequate flexibility to capture the subtle interplay between the spatial and temporal scales of variability of rain but strongly constrains the predicted statistical behavior as a function of the averaging length and time scales. We test the model with radar and gauge data collected contemporaneously at the NASA TRMM ground validation sites located near Melbourne, Florida and on the Kwajalein Atoll, Marshall Islands in the tropical Pacific. We estimate the parameters by tuning them to fit the second moment statistics of radar data at the smaller spatiotemporal scales. The model predictions are then found to fit the second moment statistics of the gauge data reasonably well at these scales without any further adjustment.Citation: Kundu, P. K., and J. E. Travis (2013), A stochastic fractional dynamics model of space-time variability of rain,

Disdrometer Network Observations of Finescale Spatial–Temporal Clustering in Rain

“…That rain is spatially variable is well recognized (e.g., Krajewski et al 2003;Koutsoyiannis 2006;Molini et al 2009;Smith et al 2009;Jaffrain et al 2011;Jaffrain and Berne 2012). While there have been important strides in the development of a broad spectral statistical framework for treating the spatial-temporal statistical characterization of some bulk parameters such as the rainfall rate (Kundu and Travis 2013), no such framework is possible at the level of individual raindrops without vastly improved observations. The humble purpose of this study is simply to present some observations that we hope will contribute toward reaching that goal one day.…”

Statistical intercomparison of idealized rainfall measurements using a stochastic fractional dynamics model

“…The physical picture that we have in mind is somewhat similar to hydrodynamics or kinetic theory of gases with raindrops playing the role of molecules. A further extension of the model was formulated in a recent paper by Kundu and Travis [] (hereinafter KT13), which was shown to describe both radar and gauge statistics derived from space‐time colocated data in a remarkably accurate manner. It introduced two further refinements of the original version: (i) a fractional order time derivative leading to a new temporal exponent in addition to the spatial exponent and (ii) a spectral cutoff in the wave number space of the spatial Fourier components of the precipitation field.…”

Stochastic modeling of time series with intermittency, persistence and extreme variability, with application to spatio-temporal averages of rainfall fields

“…Much like RPM's, MCM's may also be discerned as temporal, spatial, spatio-temporal, depending on the type of disaggregation (spatial or temporal) to which they apply. The appeal of MCM's as a modelling framework suitable for multi-fractal mathematical representations of rainfall fields, has emerged on the basis of several studies of rainfall data, revealing multiscaling properties of statistical characteristics of rain rate fields, with respect to the scale of aggregation over space and/or time [Lovejoy and Mandelbrot (1985), Schertzer and Lovejoy (1987), Lovejoy and Schertzer (1985, Waymire (1990,1993), Olsson et al (1993), Marshak et al (1994), Olsson (1995Olsson ( , 1996, Over (1995), Over andGupta (1994, 1996), Carsteanu and Foufoula-Georgiou (1996), Harris (1996), Marsan et al (1996), Olsson and Niemczynowicz (1996) Another modelling effort is based on a stochastic fractional diffusion equation driven by white noise, for the spatial Fourier components of the point rain rate field {r(t, a); (t, a) ∈ T × A}, leading to an explicit form of its space-time covariance function [Bell (1987), Bell and Kundu (1996), Bell (2003, 2006), Kundu and Travis (2013)]. That model is consistent with dynamic scaling of probability distributions of temporal increments of logarithmically transformed SARR measurements.…”