Fractals and Similarity Approaches in Hydrology

Abstract

The application of fractals and similarity concepts in hydrology has given rise to a better understanding of the space‐time organization of forms and processes that are relevant to the hydrologic cycle. Indeed, a variety of literature results support the conjecture that scaling holds for most hydrological variables in time and space. This chapter concentrates on fields where fractals and similarity approaches have proved helpful in fostering advances in specific hydrological studies. In particular, precipitation and drainage network morphology are addressed. A few specific applications to the study of natural forms and patterns relevant to the hydrological sciences are presented first, while the application of scaling concepts to the study of processes themselves is later discussed. Because of its highly irregular behavior, the rainfall process is one ideal candidate to be approached by means of self‐similarity and/or (multi)fractal description models. Such models involve increasing complexity and computational burden as soon as the interest moves from the one‐dimensional time series to the three‐dimensional case, where the full space‐time pattern of rainfall is considered. Examples of recent interesting results are presented with reference to the one‐, two‐, and three‐dimensional approaches to rainfall modeling based on similarity concepts.

Scale invariance and self‐similarity in kinematic wave overland flow in space and time

“…Since the excess rainfall rate i ( x , t ) is assumed to be a known self‐similar function for the example problem, one can obtain one more condition on the scaling exponents from the self‐similar functional form of this dependent variable. In this study, the functional form of i ( x , t ) is set to ‘power‐law’ form due to its well‐known self‐similar property (Lanza and Gallant, 2006). Let i ( x , t ) = I 0 · t a · x b , where I 0 , a and b are known real numbers.…”

Rainfall Series Fractality in the Baja California State

“…Due to the application of the previous methods, it has been found that rainfall possesses fractal properties (Venugopal et al 1999;Amaro et al 2004;Beran 1994;Oñate 1997;Peters et al 2002;Schertzer et al 2010;Turcotte 1994;Lanza and Gallant 2006;Hubert and Carbonnel 1990); however the distribution of rainfall drops do not have these properties (Malinowki et al 1993;Lovejoy and Schertzer 1987Hentschel and Procaccia 1984;Lombardo et al 2012). …”