Geometric Modeling of Rainfall Fields
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.