Nonlinear extensions of a fractal–multifractal approach for environmental modeling

Abstract

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.

Encoding hydrologic information via a fractal geometric approach and its extensions

“…A parent multifractal distribution dx (bottom left) is transformed by a fractal interpolating function f, from x to y, (upper left) to generate the derived distribution dy (upper right). A fractal interpolating function is constructed by iterating at least two contractive affine maps of the form (Barnsley 1988):…”

“…The distribution dy is similarly obtained as a histogram of projected values over the y-axis, that is hence defined as a (fractional) integration of the measure in x by adding contributions associated with the crossings of the function f, or in other words, dy is the derived distribution of dx as defined via the function f, which itself is obtained just by plotting the 2 18 (x, y) points. The set of points in the x-y plane define indeed a nontrivial function that is typically non-monotonic and not oneto-one (Barnsley 1988). However, such a set is a stable attractor for the affine maps, irrespective of the random order for the iterations; and, hence, the geometric construction just illustrated turns out to be fully deterministic.…”

“…However, such a set is a stable attractor for the affine maps, irrespective of the random order for the iterations; and, hence, the geometric construction just illustrated turns out to be fully deterministic. The parameters a n and d n determine the fractal dimension of the graph of the function f (Barnsley 1988), which for the example has a value of 1.48. As may be appreciated, the parameters for Fig.…”

“…Another way to extend the FMFP is to use affine maps defined over more than two dimensions such that they generate fractal interpolating functions in higher dimensions (e.g. Barnsley 1988;Puente and Klebanoff 1994). As illustrated in Fig.…”

Multifractal characteristics of the jet turbulent intensity depending on the outfall nozzle geometry

“…These measures are useful for describing the experimental observations, as reported by Frisch and Parisi (1985), Meneveau and Sreenivasan (1987), and Puthenveettil et al (2005). Multifractal is also used to characterize the wetland topography (Tchiguirinskaia et al 2000), airborne geophysical data (Tennekoon et al 2005), and to generate complex hydrologic spatiotemporal datasets (Cortis et al 2009(Cortis et al , 2010. For river networks, Seo et al (2014) examined the peak flow distribution on stochastic network models.…”