Final Report: Modeling Temporal Rainfall via a Fractal Geometric Approach

EPA Grant Number: R824780
Title: Modeling Temporal Rainfall via a Fractal Geometric Approach
Investigators: Puente, Carlos E.
Institution: University of California - Davis
EPA Project Officer: Packard, Benjamin H
Project Period: November 1, 1995 through October 1, 1998
Project Amount: $198,000
RFA: Water and Watersheds (1995) Recipients Lists
Research Category: Water and Watersheds , Water

Objective:

The analysis and synthesis of complex one-dimensional geophysical data sets (i.e., rainfall time series) is one of the most important practical problems facing scientists today. As these sets often may be described as rugged, intermittent, noisy, and in short "random," it has become natural to try to employ stochastic (fractal) theories to model them. This has resulted in a host of approaches that, even though they give realizations which preserve relevant statistical and physical information (e.g., clustering of arrivals, energy cascades, power spectrum, etc.), often are unable to capture the detailed information as found in individual data sets.

Although usage of stochastic methods is widespread, the outcomes they produce typically miss the actual "textures" present in many natural records. Due to this limitation and guided by the success in defining certain classical deterministic fractal sets (e.g., Mandelbrot, 1982; Barnsley, 1988), this work studied the plausibility of defining a new approach to the modeling of natural sets, one aimed at preserving not only statistical information, but also geometric details present in the sets.

Summary/Accomplishments (Outputs/Outcomes):

This work reported usage of a new vision aimed at addressing the complexity of some of nature's patterns as projections off fractal functions, interpreting observed records as deterministic derived measures constructed transforming simple multifractals via fractal interpolating functions (Puente, 1992 and 1994). It was illustrated how this framework leads to whole sets, fully characterized in terms of few "surrogate" parameters (i.e., the quantities that define the fractal function and the simple multifractal), which closely resemble several geophysical patterns and which may be used to: (1) model rainfall series, both at high and low resolution scales; (2) reproduce the intermittency (multifractality) of fully developed turbulent flows; and (3) simulate time series that encompass both "chaotic" and "stochastic" dynamical systems.

These applications illustrate that the projection's framework may be used to simulate data sets preserving the records' overall shapes and inherent texture, in a fashion that is consistent with typically used statistical characteristics as the data's power and multifractal spectra, and even in cases that would be classified as "random."

Although it is shown that the ideas hold promise to study the precise details in individual rainfall sets, as illustrated for storms in Boston, Iowa City, and Sacramento, it is explained that progress needs to be made on the solution of an inverse problem, which finds the appropriate surrogate parameters for a data set. This is the case because such a problem typically was found to be plagued by multiple local minima that require non-trivial search algorithms, which unfortunately do not converge to the desired optimum. Because of these constraints, it was not possible to establish if the parameters of the new geometric approach could discriminate rainfall records under a variety of climatic conditions or if they could serve as a suitable basis for understanding the dynamics of rainfall.

Even though the results of the work are not fully conclusive due to the non-trivial inverse problem, it is believed that generalization of the ideas to higher dimensions and/or usage of other suitably parameterized fractal functions could provide a vast framework for representing natural complexity in one, two, and three dimensions, which is vastly unexplored and may indeed hold the key for understanding rainfall dynamics in the near future.

References:

Barnsley MF. Fractals everywhere. Academic Press, 1988.

Mandelbrot BB. The fractal geometry of nature. Freeman, 1982.

Puente CE. Multinomial mintifractals, fractal interpolators, and the Gaussian distribution. Physics Letters A 1992;161:441-447.

Puente CE. Deterministic fractal geometry and probability. Bifurcation and Chaos 1994;4(6):1613-1629.


Journal Articles on this Report : 9 Displayed | Download in RIS Format

Other project views: All 19 publications 9 publications in selected types All 9 journal articles
Type Citation Project Document Sources
Journal Article Obregon N, Sivakumar B, Puente CE. A deterministic geometric representation of temporal rainfall: sensitivity analysis for a storm in Boston. Journal of Hydrology 2002;269(3-4):224-235. R824780 (Final)
  • Full-text: Science Direct
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  • Abstract: Science Direct
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  • Other: Science Direct PDF
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  • Journal Article Obregon N, Puente CE, Sivakumar B. Modeling high-resolution rain rates via a deterministic fractal-multifractal approach. Fractals 2002;10(3):387-394. R824780 (Final)
  • Abstract: World Scientific Net
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  • Journal Article Puente CE. A new approach to hydrologic modeling: derived distributions revisited. Journal of Hydrology 1996;187(1-2):65-80. R824780 (Final)
  • Full-text: Science Direct PDF
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  • Abstract: Science Direct abstract
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  • Journal Article Puente CE, Obregon N. A deterministic geometric representation of temporal rainfall: results for a storm in Boston. Water Resources Research 1996;32(9):2825-2839. R824780 (Final)
  • Full-text: UC Davis full text
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  • Abstract: AGU abstract
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  • Journal Article Puente CE, Obregon N. A geometric platonic approach to multifractality and turbulence. Fractals 1999;7(4):403-420. R824780 (Final)
  • Abstract: World Scientific Net
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  • Journal Article Puente CE, Obregon N, Robayo O, Puente MG, Simsek D. Projections off fractal functions: a new vision of nature's complexity. Fractals 1999;7(4):387-401. R824780 (Final)
  • Abstract: World Scientific Net
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  • Journal Article Puente CE, Robayo O, Diaz MC, Sivakumar B. A fractal-multifractal approach to groundwater contamination. 1. Modeling conservative tracers at the Borden site. Stochastic Environmental Research and Risk Assessment 2001;15(5):357-371. R824780 (Final)
  • Abstract: SpringerLink
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  • Journal Article Puente CE, Robayo O, Sivakumar B. A fractal-multifractal approach to groundwater contamination. 2. Predicting conservative tracers at the Borden site. Stochastic Environmental Research and Risk Assessment 2001;15(5):372-383. R824780 (Final)
  • Abstract: SpringerLink
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  • Journal Article Puente CE, Obregon N, Sivakumar B. Chaos and stochasticity in deterministically generated multifractal measures. Fractals 2002;10(1):91-102. R824780 (Final)
  • Full-text: UC Davis full text
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  • Abstract: Worldscinet
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  • Supplemental Keywords:

    precipitation, modeling, hydrology, scaling,, RFA, Scientific Discipline, Air, Geographic Area, Water, Hydrology, Water & Watershed, Environmental Chemistry, climate change, State, Air Pollution Effects, Wet Weather Flows, Geology, Atmosphere, EPA Region, Watersheds, environmental monitoring, Region 9, hydrologic dynamics, precipitation monitoring, temporal rainfall, alternative climate conditions, fractal geometric approach, rainfall patterns, aquatic ecosystems, California (CA), Cantorian parent measures, climate variability

    Progress and Final Reports:

    Original Abstract
  • 1996
  • 1997