Ascertaining symmetry and fractal dimensions for hydrological variables using the empirical orthogonal function decomposition

Abstract:
Many patterns have the fundamental property of geometric regularity known as an invariance with respect to scale or as "self-similarity". In other words, if we consider the objects in various geometrical scales then same fundamental elements (fractals) are revealed. It is important to emphasize that the fractals were primarily embedded as a language of geometry. But they are not accessible for direct observation. In this respect, the fractals differ in principle from usual objects of Euclidean geometry such as a straight line or a circle. Fractals not appear as a basic geometric shape but as algorithms or as a set of mathematical procedures. Namely the determination and ground of these algorithms are central problem for the modern theory of fractals. In spite of the fact that many natural processes conform to certain deterministic laws these processes are in principle (for sufficiently large temporal scales) unpredicted ones and show similar patterns in variations for various temporal scales just as objects possessing scale invariance reveal similar structural patterns for various spatial scale. This means that the conformity  between fractals and chaos is not incidental but this is indicator of their fundamental relationship: the fractal geometry is the geometry of chaos. Regardless of the nature or building technique all fractals has substantial common property: the degree of irregularity or complexity for their structure can be measured by some eigen-value namely by the fractal dimension. Therefore a fractal represents the mathematical object with fractional dimension in contrast to the conventional mathematical figures with integral dimensions. We consider methods for an ascertaining symmetry and fractal dimensions of some hydrological variables. Specifically, we investigate the annual runoff for the Ukrainian rivers and reveal scale invariance for distribution of this variable by using statistical parameters such as arithmetic average, coefficients of variation, skewness, and auto-correlation. It is shown that the fractal dimensions for the arithmetic average and coefficient of variations amount to 1.72 and 1.63 respectively. The coefficients of skewness and auto-correlation are related to the spatially un-correlated variables. Temporal components of empirical orthogonal function decomposition for the annual runoff  are used to reveal properties of time invariance for the annual runoff. The first components of decomposition are analyzed and its connection with factors of creation for annual runoff is investigated. It is shown that first and second components represent the large-scale atmospheric forcing of annual runoff creation. The time part of first component describes most general patterns for the annual runoff fluctuations of Ukrainian rivers.Namely this variable is subject to the fractal analysis. Here the variational function F2(s) ~ s^H is used as a property of spatial-time variation for the annual runoff  (H is the exponent of scaling identical with the fractal dimension). It is determined that H = 0.77 and this agrees to the hypothesis of Hurst's universal exponent.