SEISMIC SIGNAL ANALYSIS USING CORRELATION DIMENSION
Soundararajan Ezekiel Matthew J. Barrick Matthew Lang
Computer Science Department
Indiana University of Pennsylvania,
Indiana, PA 15705
USA
ABSTRACT
Multifractal analysis is a new and promising approach to
analyze and characterize non-stationary signals such as
seismic signals, electrocardiograms, heart rate variability
signals, stock prices, etc. Traditional Fourier analysis
imposes the assumption that signals are stationary over
temporal segments. Such an assumption is inappropriate
for seismic signals because conditions in mines undergo
constant change. Multifractal analysis does not impose
this assumption, and is therefore useful for processing
seismic signals. In this paper we propose a novel
correlation dimension based method to analyze seismic
signals, which is superior to many conventional used
methods. The correlation fractal dimension is calculated
for overlapping windows for the entire signal producing a
multifractal spectrum. Using this strategy our method
pinpoints seismic events in the seismic signal. Our results
show that a small correlation dimension corresponds to a
region of seismic activity in the signal. The approach
applied here may be useful for analyze of other nonstationary signals.
KEY WORDS
Seismic Signal, Fractal dimension, Correlation dimension,
Multifractal Spectrum
1. INTRODUCTION
One of the most important tasks in seismic signal
processing is to be able to both detect and identify the
seismic signals. Detection consists of recognizing that a
seismic event has occurred and locating the source of the
seismic signals. Once a seismic event has been detected,
the next task is to determine if an underground nuclear
explosion created it or if it was created by other seismic
events including natural earthquakes, rock bursts in
mines, and chemical explosions conducted for mining,
quarry blasting, and construction. Rock bursts are
experienced in underground mining at various localities in
the world, causing death and injury to underground
miners and damaging mine structures. To date, many
studies have been conducted to understand the cause of
rock bursts and outbursts to predict their occurrence. The
approaches made to detect rock bursts include the
traditional time-frequency analysis such as the Windowed
Fourier Transform (WFT), or any standard statistical
methods like the microgravity method, rheological
method, rebound method, drilling-yield method,
microseismic method and so on. Although all of these
methods have been used, none is completely reliable and
few are useful in the rapidly advancing mining
environment. The U.S. Bureau of mines recognized
microsesismic technology as a potential tool for rock
burst prediction as early as 1939. However, to date only a
few successful predictions have been achieved. In this
paper, we introduce a different and efficient detection
method, which is based on multifractal spectrum by using
correlation fractal dimension. In section 2 we recall some
fundamental definitions of fractal measure and
multifractal spectrum. Section 3 describes the algorithm
proposed in this paper, which is tested on seismic data the
results of which are given in section 4. Section 5
concludes the paper.
2. FRACTAL DIMENSION
Multifractal theory was described for the first time by B.
B. Mandelbrot [1,3,4,5] in the context of fully developed
turbulence. Since then, mathematicians have increasingly
studied multifractals. Multifractal analysis has recently
drawn much attention as a tool for studying singular
measures and functions in both theory and applications.
Multifractal analysis has been extended as an application
of correlation dimensions in [2]. In the multifractal
scheme, the pointwise structure of a singular measure is
analyzed through the so called ''multifractal spectrum'',
which gives either geometrical or probabilistic
information about the distribution of points having the
same degree of singularity. Several definitions of a
multifractal spectrum exist. In this section, we take an
approach to multifractal analysis pioneered by Levy
Vehel et al [6][7]. However, instead of computing the
local singularity exponent by using Choquet capacities,
we use a correlation dimension. Using this method, we
capture the dynamics of seismic signals.
We can visualize the seismic data by constructing a phase
space of the time series. Packard et al. [8] outlined a
simple method for reconstructing a phase space for a time
series.
The fractal dimension gives us important
information about the time series. It can be approximated
by covering the data points in the phase space with circles
and taking the following measure:
log N
D
log(1/ R)
where N = number of circles of diameter R
R = diameter
This method works for fractal embedded in a twodimensional space, for higher dimensional space we to
use hyperspheres of dimensionality of 3 or higher. A
similar more practical method developed by Grassberger
and Procaccia [2] is the correlation dimension, an
approximation of the fractal dimension [9] that uses the
correlation integral, which is explained in the following
section.
3. METHODOLOGY
The correlation integral is the probability that a pair of
points in the phase space is within a distance R of one
another. We count the number of pairs of points in the
following manner. First, we construct our seismic signal
as phase space with an embedding dimension of two.
Then starting with a small R, we calculate the correlation
integral Cm ( R) for this distance, according to the
following equation:
1 N

N  N 2
j 1
Cm ( R)  lim
where

N
  ( R | X
i  j 1
i
x  X i , y  X i  where  is the
time lag. Second, calculating the probability C m that any
space by selecting
given pair of data points in the phase space is within
distance R of one another for various distances R. Finally,
perform a linear regression of log(Cm ) vs. log( R ) . The
slope of the resulting line is the correlation dimension.
Then plots the correlation dimensions against the number
of windows. The resulting graph is the multifractal
spectrum.
4. RESULTS
To illustrate the proposed method, we applied it to 30
different seismic signals, which were collected over a 24hour period on different days. We divided the signal
overlapping segments of 500 points with an overlap of
100 points. For each segment, we calculated the
correlation dimension as explained in the previous
section. The multifractal spectrums were then plotted for
each signal. We observed that dip in the spectrum
correlated to bursts of seismic activity in the signal.
Figures 1, 2, 3, 4, 5, 6, and 7 are each a seismic signal
along with the corresponding multifractal spectrum.
Figures 1 and 2 are the first and second halves of the
small signal, presented in different scales for clarity.
Figures 3 and 4 are split similarly. Figures 5, 6, and 7 are
full sized signal.
 X j |)
is the Heaviside step function described as,
1 0  ( R | X i  X j |)
0 1>( R | X i  X j |)
 ( R | X i  X j |)  
The correlation integral is the probability that two points
chosen at random are less than R units apart. If we
increase the value of R, C m should increase as the rate of
kR D , where k is a constant and D is the correlation
dimension This gives the following relation:
Cm  R D
or
log(Cm )  D *log( R)  constant
Cm for increasing values of R. By
finding the slope of a graph of the log(Cm ) versus the
We can calculate
log(R), through a linear regression, we can estimate the
correlation dimension for the embedded dimension.
A summary of the algorithm is as follows:
Divide the given single in a number of overlapping
windows. For each window calculate the correlation
dimension. This is done by first, reconstructing the phase
Figure 1
Figure 2
Figure 5
Figure 3
Figure 6
Figure 4
Figure 7
5. CONCLUSION
A framework using correlation fractal dimension and
multifractal spectrum was presented. This framework is
suitable to analysis non-stationary signals, especially
seismic signals. Although in principal the techniques of
correlation dimension based multifractal analysis have
been shown to be powerful tools for characterization of
various non-stationary signals no major break-through has
yet been achieved by their application to seismic signal
analysis. However, further experimental analysis needs to
be carried out to fine-tune the various fractal and
statistical parameters.
6. ACKNOWLEDGEMENT
The authors would like to thank Dr. Lind S. Gee of the
Seismological Laboratory at University of California at
Berkley for providing the seismic data used in this study.
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