A space‐time theory of mesoscale rainfall using random cascades

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.

[1]  Shaun Lovejoy,et al.  Causal space‐time multifractal processes: Predictability and forecasting of rain fields , 1996 .

[2]  Shaun Lovejoy,et al.  Multifractals and extreme rainfall events , 1993 .

[3]  Shaun Lovejoy,et al.  Universal Multifractals: Theory and Observations for Rain and Clouds , 1993 .

[4]  G. Taylor.  The Spectrum of Turbulence , 1938 .

[5]  A. M. Oboukhov.  Some specific features of atmospheric tubulence , 1962, Journal of Fluid Mechanics.

[6]  Thomas L. Bell,et al.  A space‐time stochastic model of rainfall for satellite remote‐sensing studies , 1987 .

[7]  E. Foufoula‐Georgiou,et al.  Linkage of scaling and thermodynamic parameters of rainfall: Results from midlatitude mesoscale convective systems , 1996 .

[8]  Thomas M. Over,et al.  On scaling exponents of spatial peak flows from rainfall and river network geometry , 1996 .

[9]  Shaun Lovejoy,et al.  Generalized Scale Invariance in the Atmosphere and Fractal Models of Rain , 1985 .

[10]  Edward C. Waymire,et al.  A statistical analysis of mesoscale rainfall as a random cascade , 1993 .

[11]  Lewis F. Richardson,et al.  Weather Prediction by Numerical Process , 1922 .

[12]  B. Mandelbrot,et al.  Fractal properties of rain, and a fractal model , 1985 .

[13]  Shaun Lovejoy,et al.  Multifractal Analysis Techniques and the Rain and Cloud Fields from 10−3 to 106m , 1991 .

[14]  Vijay K. Gupta,et al.  On Taylor's Hypothesis and Dissipation in Rainfall , 1987 .

[15]  Thomas M. Over,et al.  Statistical Analysis of Mesoscale Rainfall: Dependence of a Random Cascade Generator on Large-Scale Forcing , 1994 .

[16]  J. Kahane,et al.  Sur certaines martingales de Benoit Mandelbrot , 1976 .

[17]  René Laprise,et al.  Predictability of Precipitation Patterns: An Operational Approach , 1994 .

[18]  David R. Dawdy,et al.  Physical interpretations of regional variations in the scaling exponents of flood quantiles , 1995 .

[19]  Shaun Lovejoy,et al.  Fractals, Raindrops and Resolution Dependence of Rain Measurements , 1990 .

[20]  J. Olsson,et al.  Limits and characteristics of the multifractal behaviour of a high-resolution rainfall time series , 1995 .

[21]  Robert K. Crane,et al.  Space‐time structure of rain rate fields , 1990 .

[22]  D. Schertzer,et al.  Physical modeling and analysis of rain and clouds by anisotropic scaling multiplicative processes , 1987 .

[23]  Pierre Y. Julien,et al.  Runoff model sensitivity to radar rainfall resolution , 1994 .

[24]  A. Kolmogorov.  A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number , 1962, Journal of Fluid Mechanics.

[25]  Efi Foufoula-Georgiou,et al.  Assessing dependence among weights in a multiplicative cascade model of temporal rainfall , 1996 .

[26]  Edward C. Waymire,et al.  A Spectral Theory of Rainfall Intensity at the Meso‐β Scale , 1984 .

[27]  Praveen Kumar,et al.  A multicomponent decomposition of spatial rainfall fields: 2. Self‐similarity in fluctuations , 1993 .

[28]  I. Zawadzki.  Statistical Properties of Precipitation Patterns , 1973 .

[29]  Edward C. Waymire,et al.  Multifractal Dimensions and Scaling Exponents for Strongly Bounded Random Cascades , 1992 .

[30]  Keith Beven,et al.  Linking parameters across scales: Subgrid parameterizations and scale dependent hydrological models. , 1995 .

[31]  A. Bellon,et al.  The use of digital weather radar records for short‐term precipitation forecasting , 1974 .

[32]  C. R. Laughlin,et al.  On the effect of temporal sampling on the observation of mean rainfall , 1981 .

[33]  B. Mandelbrot.  Intermittent turbulence in self-similar cascades : divergence of high moments and dimension of the carrier , 2004 .