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A Statistical Analysis of Mesoscale Rainfall as a Random Cascade

Vijay K. GuptaCenter for the Study of Earth from Space/CIRES and Department of Geological Sciences, University of Colorado, Boulder, Colorado

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Edward C. WaymireDepartments of Mathematics and Statistics, Oregon State University, Corvallis, Oregon

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Abstract

Theoretical attempts to model space–time rainfall during the last 15 years have evolved along two separate lines. The first group of approaches is based on an assumed hierarchy of scales in spatial rainfall as noted by numerous empirical interpretations of remotely sensed observations. The second group of approaches rests on the assumption of an invariance property in statistical distributions of spatial rainfall called self-similarity, or simple scaling. The current theoretical developments involve a common modification of each of these assumptions. It is based on the notion of spatial random mass distribution or random measures generated by random cascades. The current mathematical foundations of the theory of random cascades involve the study of their ensemble properties and their sample-average properties in space. Recent results have shown that these two are not the same because the spatial law of large numbers does not hold for random cascades due to strong spatial correlations. Two key results from the current literature on the scaling of the statistical moments and the tail probability of the random cascade measures are derived. These ensemble computations only involve the marginal probability distributions of the random measures. By contrast, the sample-average computations involve their spatial-dependence structure and so far have been carded out only under certain boundedness assumptions on the generators of the random measures. Applications of the sample-average results are given to a space–time analysis of fractional wetted area of GATE I rainfall. This quantity is of basic significance in a variety of hydrometeorologic and hydroclimatologic studies.

Abstract

Theoretical attempts to model space–time rainfall during the last 15 years have evolved along two separate lines. The first group of approaches is based on an assumed hierarchy of scales in spatial rainfall as noted by numerous empirical interpretations of remotely sensed observations. The second group of approaches rests on the assumption of an invariance property in statistical distributions of spatial rainfall called self-similarity, or simple scaling. The current theoretical developments involve a common modification of each of these assumptions. It is based on the notion of spatial random mass distribution or random measures generated by random cascades. The current mathematical foundations of the theory of random cascades involve the study of their ensemble properties and their sample-average properties in space. Recent results have shown that these two are not the same because the spatial law of large numbers does not hold for random cascades due to strong spatial correlations. Two key results from the current literature on the scaling of the statistical moments and the tail probability of the random cascade measures are derived. These ensemble computations only involve the marginal probability distributions of the random measures. By contrast, the sample-average computations involve their spatial-dependence structure and so far have been carded out only under certain boundedness assumptions on the generators of the random measures. Applications of the sample-average results are given to a space–time analysis of fractional wetted area of GATE I rainfall. This quantity is of basic significance in a variety of hydrometeorologic and hydroclimatologic studies.

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