ABSTRACT
Wavelets and fractals are relatively new tools in geophysical analysis; however, in the last two decades, they have gained substantial popularity due to their applications in nonlinear signal analysis (Dimri 2000; Dimri et al. 2012). The essence of fractal analysis lies in fractal dimension analysis. Furthermore, the concept of fractal analysis is extended to time series analysis using the concept of fractional Brownian motion (fBm) and fractional Gaussian noise (fGn). A widely used concept called the box counting method for fractal dimension analysis is explained with examples. Wavelet analysis can be seen as an advanced substitute for the Fourier analysis, which has been widely used in geophysics for noise attenuation, power spectrum analysis (Maus and Dimri 1995a,b), characterization of fractal/scaling behavior of the time/space series
CONTENTS
Introduction ......................................................................................................... 155 Fractal Time Series Characterization ................................................................ 156
Concept of fBm and fGn ................................................................................ 156 R/S Analysis ........................................................................................................ 157 Determination of Fractal Dimension: Box Counting Method ...................... 159
Box Counting Method ................................................................................... 159 Design of Survey Network ................................................................................ 162
Spectral Approach .......................................................................................... 162 Fractal Dimension Approach ........................................................................ 163 Optimum Grid ................................................................................................ 164 Detectability Limit of the Survey ................................................................. 166
Fast Multiplication of Large Matrices: A Useful Application for Gradient-Based Inversion .................................................................................. 167 Wavelet Transform of Gravity Data for Source Depth Estimation .............. 170 Acknowledgments .............................................................................................. 173 References ............................................................................................................. 173
using power spectrum approach (Dimri et al. 2011; Srivastava et al. 2007; Dimri and Vedanti 2005) and frequency analysis of the seismic time series (Dimri 1992; Srivastava and Sen 2009, 2010), and many more similar applications. An intriguing application of wavelet transform is given in Moreau et al. (1999), in which they present the estimation of source parameters, that is, horizontal location, depth, multipolar nature, and strength of the homogeneous potential eld sources based on a special class of wavelets that are invariant while upward continuation. This makes source parameter analysis very easy and intuitive simply by looking at the wavelet transform, whereas quantitative estimates can be drawn with the help of lines of extrema of the wavelet transform. Other applications of wavelet-based analysis in geophysics include inversion of geophysical data using wavelets (Li et al. 1996) and fast computation of the inverse of large matrices. In this article, some examples of wavelet transform for fast computation of large matrix inversion and source parameter estimation from gravity data are presented.
