A Geometric Platonic Approach to Multifractality and Turbulence

Stoch Environ Res Risk Assess

2008

Self Cite

We present the extension of a deterministic fractal geometric procedure aimed at representing the complexity of the spatio-temporal patterns encountered in environmental applications. The original procedure, which is based on transformations of multifractal distributions via fractal functions, is extended through the introduction of nonlinear perturbations to the underlying iterated linear maps. We demonstrate how the nonlinear perturbations generate yet a richer collection of patterns by means of various simulations that include evolutions of patterns based on changes in their parameters and in their statistical and multifractal properties. It is shown that the nonlinear extensions yield structures that closely resemble complex hydrologic temporal data sets, such as rainfall and runoff time series, and width-functions of river networks as a function of distance from the basin outlet. The implications of this nonlinear approach for environmental modeling and prediction are discussed.

Section: Sample Patterns Via the Nonlinear Fractal-multifractal Approachsupporting

confidence: 77%

Section: Nonlinear Extensions Of the Fractal Multifractal Approachmentioning

confidence: 99%

Nonlinear extensions of a fractal–multifractal approach for environmental modeling

Cortis

Stoch Environ Res Risk Assess

2008

Self Cite

“…The purpose of this article is to illustrate by means of few examples that a deterministic fractal-multifractal (FM) repCorrespondence to: C. E. Puente (cepuente@ucdavis.edu) resentation (Puente, 1992(Puente, , 1994, successfully employed to model data sets corresponding to a host of geophysical processes, such as rainfall (Puente and Obregón, 1996;Obregón et al, 2002a, b), turbulence (Puente and Obregón, 1999), and groundwater contamination transport (Puente et al, 2001a, b), may also be used to represent the overall structure of the width function of natural catchments. Figure 1 illustrates the construction of a derived distribution via the FM approach (Puente, 1992(Puente, , 1994.…”

Section: Introductionmentioning

confidence: 99%

A deterministic width function model

Nonlin. Processes Geophys.

2003

Abstract.Use of a deterministic fractal-multifractal (FM) geometric method to model width functions of natural river networks, as derived distributions of simple multifractal measures via fractal interpolating functions, is reported. It is first demonstrated that the FM procedure may be used to simulate natural width functions, preserving their most relevant features like their overall shape and texture and their observed power-law scaling on their power spectra. It is then shown, via two natural river networks (Racoon and Brushy creeks in the United States), that the FM approach may also be used to closely approximate existing width functions.

“…2)} whose graph has a fractal dimension of 1.485, and the invariant (and hence deterministic) measure dx, as identified in turbulence studies (Puente and Obregón, 1999). As a way of clarification, it should be emphasized that although it may appear to a casual reader that the obtained patterns may depend on the "coin" mentioned above, such is not the case, since the successive iterations are just a suitable Monte Carlo approach that always converges to the same deterministic pattern (Barnsley, 1988).…”

Section: Figmentioning

confidence: 99%

Modeling geophysical complexity: a case for geometric determinism

Hydrol. Earth Syst. Sci.

2007

Abstract. It has been customary in the last few decades to employ stochastic models to represent complex data sets encountered in geophysics, particularly in hydrology. This article reviews a deterministic geometric procedure to data modeling, one that represents whole data sets as derived distributions of simple multifractal measures via fractal functions. It is shown how such a procedure may lead to faithful holistic representations of existing geophysical data sets that, while complementing existing representations via stochastic methods, may also provide a compact language for geophysical complexity. The implications of these ideas, both scientific and philosophical, are stressed.