716
August 2014
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
BOOK
REVIEW
Wavelets and Fractals in Earth System
Sciences
Editors: E. Chandrasekhar, V.P. Dimri and V.M.
Gadre
CRC Press, Taylor & Francis Group, 2014, 300 pp., list of
index
ISBN 13: 978-1-4665-5359-0 (Hardback), $99.95
Reviewed by:
Fei Ma, Ph.D., CP,
Photogrammetric consulting, Renton, WA
Both wavelets and fractals are newly developed mathematical
functional analysis tools developed about 30 years ago. Both
of these two independent but related methodologies and their
fundamental theories have been raising enormous attention since
their emergence across many disciplines including earth system
science and its branches. A combined study on how wavelets
and fractals have been applied on specific analysis within
earth system science as well as other scientific applications is
the preliminary motivation of the book
Wavelets and Fractals
in Earth System Sciences
. This book “attempts to highlight the
role of such advanced data processing techniques in present-day
research in various fields of earth system sciences”.
The book consists of 10 chapters. The first half of the
book (chapters 1-4), focuses on the rudimentary concept
and fundamental theory of both wavelet transform and
fractal analysis. These chapters outline the basic theoretical
framework of wavelets and fractals and how these newly
developed techniques change the way of traditional analysis
on scaling and self-similarity. Here, the author provides the
reader with the necessary foundation and preparation from
theory to practice. Applications on earth system sciences are
indispensably analyzed on the case study basis in the second
half of the book (chapters 5-10).
The first chapter makes a succinct introduction of wavelets
and fractals from analytical function and its historical point of
view. While the introduction identifies essential mathematical
characters of these two frameworks, the rudimentary definition
of wavelets and fractals are also introduced and presented. The
correlation of these two mathematical frameworks is lightly
discussed in the conclusion.
Chapter 2 describes wavelet principles from construction
to analysis. The essence of wavelet transforms is introduced
through the adequate explanation (with examples) of various
filter bank constitutions and their effects and differentiation.
The well-known Cascade algorithm for multi-resolution wavelet
analysis is then analyzed. In the remainder of the chapter the
author examines the methodology of designing an appropriate
wavelet and filter bank in terms of uncertainty principles and
time-frequency localization properties. Then the suggested
design method is illustrated with an image compression example.
Chapter 3 is a supplement of chapter 1 on wavelet
development from the history of integral transforms to the so-
called second-generation wavelets. While this chapter focuses
on the genesis of wavelet transform and its general application,
it also briefly discusses the limitation of wavelet transform
and the form of two-dimensional wavelet transforms. These
are the only discussions on these two elemental but critical
components in the entire book.
The beginning of chapter 4 (i.e. multiscale analysis) abridges
the axioms of dyadic multi-resolution analysis (MRA). Then,
based on the theory of MRA, both fractal and wavelet-based
self-similar functions and their applications on self-similarity
descriptions are analyzed. Two case studies are also presented
using wavelets in singularity detection. The contents of this
chapter serve as the direct link from theoretical foundation
(chapters 1-4) to the following practical applications (chapters
5-10) in this book.
Chapter 5 introduces the basic application of both fractals
and wavelets in geophysics measurements (e.g., gravity/
magnetic data). The authors address the advantages of fractal
based time series characterization and fractal dimension
determination. The application of fractals in geophysics is
further explored in chapter 6 where earthquake prediction
using the extended multifractal concept and generalized fractal
dimensions, is highlighted based on the analysis of seismicity
across a larger data set observed around Himalaya region.
Chapters 7-10 focus on wavelet transform applications
from multiple earth system science perspectives. Chapter 7
describes the complex wavelet useful to geomagnetic jerks
in terms of time-frequency localization. The complex wavelet
(part real, part imaginary) is examined and identified as a very
powerful tool in region-wise geomagnetic jerk study according
