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Geometric Modeling of Rainfall Fields
Carlos E. Puente, University of California, Davis
UC Water Resources Center Technical Completion Report W-804
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ABSTRACT: Accurate rainfall modeling is of vital importance for the proper management of our
environment. Rainfall descriptions are required, among others, to model pollution migration,
to address issues related to climate change (i.e. global circulation), to estimate
extreme weather events, and to manage our watersheds. Adequate environmental planning
can only be accomplished with reliable rainfall quantification. Even though several
sophisticated (stochastic) rainfall models exist, they do not capture all the variability
observed at a fixed location when a storm passes by. Typical models approximate the
irregular and intermittent rain patterns (of a fractal and/or multifractal nature) by
superimposing randomly arriving smooth Euclidean objects (i.e. rectangular pulses),
and consequently preserve only some statistical features of the rainfall series (fields)
(e.g. mean, variance, spatial correlations, etc.). Since these representations are typically
limited by their analytical tractability and because there has been a recognition
of chaotic effects in rainfall, a new approach for rainfall modeling based on multinomial
multifractal measures and fractal interpolating functions has been developed by
Puente (1992, 1995). The basis for these fractal-multifractal models is the fact that
predictability could only be improved when the observed (intermittent) details present
in rainfall events are considered explicitly. One advantage of the fractal geometric
procedure is that its outcomes are entirely deterministic. This follows because the two
components that make up the approach are deterministic.
This work reports on the use of the new models to represent: (i) high resolution
rainfall time series, (Ii) natural processes of two kinds, namely those termed chaotic
or stochastic, and (iii) spatial rainfall (geophysical) patterns. In relation to the high
resolution rainfall, it is shown that the intrinsic shape and variability of three storms
gathered every few seconds (5 to 15) may be captured employing the fractal geometric
methodology. It is illustrated how these data sets are parsimoniously encoded wholistically,
resulting in very faithful descriptions of both major trends and small (noisy)
fluctuations. These results suggest that a stochastic framework for rainfall may not
be required. In regards to the geometric description of general natural time series,
it is exhibited via simulations that the fractal-multifractal approach does provide a
convenient framework to describe a large class of records that would pass, according
to typical statistical and chaotic criteria, as low-dimensional and chaotic or as highdimensional
and stochastic. In order to handle rainfall (radar reflectivities) patterns
in space, extensions of the fractal-multifractal procedure to higher dimensions are also
included. The potential for the development of a new approach to rainfall dynamics in
space, based on the geometric features of spatial patterns, is discussed.
SUGGESTED CITATION:
Carlos E. Puente,
"Geometric Modeling of Rainfall Fields"
(August 1, 1995).
University of California Water Resources Center.
Technical Completion Reports.
Paper 804.
http://repositories.cdlib.org/wrc/tcr/804
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