Institute of Applied Mathematics,

Odesa University OSEU,

P.O.Box 24A,

65009 Odesa,

UKRAINE

E-mail: loboda@paco.net

**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 *F*2(*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.