Table Of ContentFRACTAL ANALYSIS AND
CHAOS IN GEOSCIENCES
Edited by Sid-Ali Ouadfeul
FRACTAL ANALYSIS AND
CHAOS IN GEOSCIENCES
Edited by Sid-Ali Ouadfeul
Fractal Analysis and Chaos in Geosciences
http://dx.doi.org/10.5772/3309
Edited by Sid-Ali Ouadfeul
Contributors
Reik V. Donner, Manuel Tijera, Gregorio Maqueda, Carlos Yagüe, José L. Cano, Noboru
Tanizuka, Sid-Ali Ouadfeul, Mohamed Hamoudi, O.A. Khachay, A.Yu. Khachay, O.Yu.
Khachay, Asim Biswas, Hamish P. Cresswell, C. Si. Bing, Leila Aliouane, Amar Boudella
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Contents
Preface VII
Chapter 1 Complexity Concepts and Non-Integer Dimensions
in Climate and Paleoclimate Research 1
Reik V. Donner
Chapter 2 Analysis of Fractal Dimension of the Wind Speed and Its
Relationships with Turbulent and Stability Parameters 29
Manuel Tijera, Gregorio Maqueda, Carlos Yagüe and José L. Cano
Chapter 3 Evolution of Cosmic System 47
Noboru Tanizuka
Chapter 4 Fractal Analysis of InterMagnet Observatories Data 65
Sid-Ali Ouadfeul and Mohamed Hamoudi
Chapter 5 Dynamical Model for Evolution of Rock Massive State as
a Response on a Changing of Stress-Deformed State 87
O.A. Khachay, A.Yu. Khachay and O.Yu. Khachay
Chapter 6 Application of Multifractal and Joint Multifractal Analysis
in Examining Soil Spatial Variation: A Review 109
Asim Biswas, Hamish P. Cresswell and C. Si. Bing
Chapter 7 Well-Logs Data Processing Using
the Fractal Analysis and Neural Network 139
Leila Aliouane, Sid-Ali Ouadfeul and Amar Boudella
Chapter 8 Fractal and Chaos in Exploration Geophysics 155
Sid-Ali Ouadfeul, Leila Aliouane and Amar Boudella
Preface
The fractal analysis has becoming a very useful tool to process obtained data from
chaotic systems in geosciences, it can be used to resolve many ambiguities in this
domain. This book contains eight chapters showing the recent applications of the
Fractal/Multifractal analysis in geosciences.
The first chapter explains how the fractal formalism can be used for analysis of
complexity concepts and non-integer dimensions in climate and paleoclimate research.
The second chapter develops the analysis of fractal dimension of the wind speed and
its relationships with turbulent and stability parameters.
Another chapter studies the evolution of the cosmic system, while the fourth shows
the fractal analysis of InterMagnet observatories data.
One of the chapters suggests a dynamical model for evolution of rock massive state as
a response on a changing of stress-deformed state.
A review paper of the application of multifractal analysis in examining soil spatial
variation is detailed in the next chapter.
The two last chapters are devoted to the application of the fractal analysis in
exploration geophysics.
The current book shows a range of applications of the fractal analysis in earth sciences.
I believe that this last will be an important source for researchers and students from
universities.
Dr Sid-Ali Ouadfeul
Algerian Petroleum Institute, IAP Corporate University, Boumerdès,
Algeria
Chapter 1
Complexity Concepts and
Non-Integer Dimensions
in Climate and Paleoclimate Research
Reik V. Donner
Additional information is available at the end of the chapter
http://dx.doi.org/10.5772/53559
1. Introduction
The ongoing global climate change has severe effects on the entire biosphere of the Earth.
According to the most recent IPCC report [1], it is very likely that anthropogenic influences
(like the increased discharge of greenhouse gases and a gradually intensifying land-use) are
important driving factors of the observed changes in both the mean state and variability of
the climate system. However, anthropogenic climate change competes with the natural
variability on very different time-scales, ranging from decades up to millions of years, which
is known from paleoclimate reconstructions. Consequently, in order to understand the
crucial role of man-made influences on the climate system, an overall understanding of the
recent system-internal variations is necessary.
The climate during the Anthropocene, i.e. the most recent period of time in climate history
that is characterized by industrialization and mechanization of the human society, is well
recorded in direct instrumental measurements from numerous meteorological stations. In
contrast to this, there is no such direct information available on the climate variability before
this epoch. Besides enormous efforts regarding climate modeling, conclusions about climate
dynamics during time intervals before the age of industrial revolution can only be derived
from suitable secondary archives like tree rings, sedimentary sequences, or ice cores. The
corresponding paleoclimate proxy data are given in terms of variations of physical,
chemical, biological, or sedimentological observables that can be measured in these archives.
While classical climate research mainly deals with understanding the functioning of the
climate system based on statistical analyses of observational data and sophisticated climate
models, paleoclimate studies aim to relate variations of such proxies to those of observables
with a direct climatological meaning.
2 Fractal Analysis and Chaos in Geosciences
Classical methods of time series analysis used for characterizing climate dynamics often
neglect the associated multiplicity of processes and spatio-temporal scales, which result in a
very high number of relevant, nonlinearly interacting variables that are necessary for fully
describing the past, current, or future state of the climate system. As an alternative, during
the last decades concepts for the analysis of complex data have been developed, which are
mainly motivated by findings originated within the theory of nonlinear deterministic
dynamical systems. Nowadays, a large variety of methods is available for the quantification
of the nonlinear dynamics recorded in time series [2,3,4,5,6,7,8,9], including measures of
predictability, dynamical complexity, or short- as well as long-term scaling properties,
which characterize the dynamical properties of the underlying deterministic attractor.
Among others, fractal dimensions and associated measures of structural as well as
dynamical complexity are some of the most prominent nonlinear characteristics that have
already found wide use for time series analysis in various fields of research.
This chapter reviews and discusses the potentials and problems of fractal dimensions and
related concepts when applied to climate and paleoclimate data. Available approaches
based on the general idea of characterizing the complexity of nonlinear dynamical systems
in terms of dimensionality concepts can be classified according to various criteria. Firstly,
one can distinguish between methods based on dynamical characteristics estimated directly
from a given univariate record and those based on a (low-dimensional) multivariate
projection of the system reconstructed from the univariate signal. Secondly, one can classify
existing concepts related to non-integer or fractal dimensions into self-similarity approaches,
complexity measures based on the auto-covariance structure of time series, and complex
network approaches. Finally, an alternative classification takes into account whether or not the
respective approach utilizes information on the temporal order of observations or just their
mutual similarity or proximity. In the latter case, one can differentiate between correlative and
geometric dimension or complexity measures [10]. Table 1 provides a tentative assignment of
the specific approaches that will be further discussed in the following. It shall be noted that
this chapter neither gives an exhaustive classification, nor provides a discussion of all existing
or possible approaches. In turn, the development of new concepts for complexity and
dimensionality analysis of observational data is still an active field of research.
Methods based on univariate Methods based on multivariate
time series reconstruction
Self-similarity / scaling Correlative:Higuchi estimator Geometric:fractal dimensions
approaches for D0, estimators of the Hurst based on box-counting and
exponent (R/S analysis, box-probability, Grassberger-
detrended fluctuation analysis, Procaccia estimator for D2
and others)
Approaches based on auto- Correlative:LVD dimension
covariance structure density
Complex network approaches Correlative:visibility graph Geometric:recurrence network
analysis analysis
Table 1. Classification of some of the most common dimensionality and complexity concepts mainly
discussed in this chapter.
