# Abstract— methods. The drawback of using Fourier series is

Abstract— In this paper an attempt is made to generate a synthetic
seismic signal by using wavelets. There are many wavelets to generate
synthetic seismic signals; some of them are klauder, ormsby, ricker wavelets,
etc. Each wavelet varies in their frequency of operation. Seismic reflection is
well-known geophysical technique which can give the information about earth and
its inner core, the earthquakes. The information given by seismic reflection is
used to monitor the earthquakes, petroleum exploration, determination of the
earth’s core structure, etc. Earth quake has become a serious problem in the
world due to its damage. There are many methods to analyze the signals due to
earthquakes. Among those, we are using a sophisticated method called wavelet
transformation. The advantage of this technique is that, it suppresses the
noise and enhances the signal strength there by seismic information due to the
earthquake. Seismic signal is synthesized by convoluting a klauder wavelet with
a seismic signal.

Keywords—
Seismic signal, Klauderwavelet,
Convolution, Synthetic signal.

We Will Write a Custom Essay Specifically
For You For Only \$13.90/page!

order now

I.
Introduction

Wavelet
transforms method having its basis in applied mathematics. This method having
high resolution when compared to other frequency based methods using Fourier
transforms. The specific advantage of this method is to drive low frequency
components with considerably good resolution when compared to other methods.
The drawback of using Fourier series is overcome by wavelet transform method, particularly
for short time analysis of seismic signals (or) earthquake signals.

The wavelet is nothing but a wave
like structure, which initially oscillates at zero amplitude and increases
thereby decreasing the initial oscillated amplitude to zero. To get information
from an unknown signal a wavelet can be convoluted with portion of a known

Signal.
Convolution is a mathematical way of combining two signals in order to form a
third signal. The technique is popularly used for processing the signals
digitally (DSP). The method involves in finding the zero phase of a given
signal.  The wavelet transform is a
method, which decomposes the signal giving details as a function of time.

This involves
Scattering and shifting properties to generate a wavelet and it is limited to a specified length.
. The wavelets has two major characteristics namely 1) wavelets are compact support in time domain, and 2) The wavelet alternates from positive to negative. The process includes two steps 1) shifting the
function of
the wavelet and 2) making product
with an inner scale function.

For best resolution wavelet is a method giving
complete picture for the signals such as of seismic origination. Here the resolution of the signal is controlled by
the bandwidth of the signal. Moreover seismic
signals are transient in nature radiating
natural (or) manmade noise along with the seismic sources. This finally locates the, source mechanisms and the structure of the propagation medium through which they travel. The data taken from the seismograms, where
the amplitude of the seismic waves are
recorded as a function of distance from the epicenter, where earthquake occur the intensity
of this earthquake is measured in terms of
the magnitude taken
over a scale called
Richter scale.

The output of a monitoring
instrument namely
seismograph gives a signal corresponding
to the earthquake, by
analyzing these seismic signals one can design an earth warning system for predicting
the earth quakes in future avoid natural
disasters.

II.
Methodology

A Klauder
wavelet represents the autocorrelation of a linearly swept frequency-modulated
sinusoidal signal. It is defined by its terminal low frequency, “f1”;
its terminal high frequency, “f2”; and the duration of the input
signal “T”, often 6 or 7 seconds. The real part of the following
formula will generate a Klauder wavelet.

Klauder(t)
=real sin(tkt(T-t))/(tkt) exp (2ifot)

where k = (f2 – fl)/T (rate of change of frequency with
time)

fo=
(f2 + fl)/2 (midfrequency of bandwidth)

i = 1
(an imaginary number)

Figure-1

a Klauder wavelet (fig 1) is
symmetrical about a vertical line through its central peak at time zero.

Figure-2

The frequency
spectrum of a Klauder wavelet (fig 2) shows the substantial
similarities between a Klauder and an Ormsby wavelet.

wavelet is taken as an input signal and is represented as x(t) 2Impulse signal
h(t) is taken as an seismic signal(earthquake signal) 3Y(t) is the output of
the which is obtained by convoluting input signal and the impulse signal 4Noise is added to the y(t) in
order to obtain an noisy seismic signal 5 Due to  unwanted waves such as surface waves ,
corrupt the  seismic signal by adding
noise.6So on comparing the y(t) the output signal without noise and with
noise  we can easily understand about the
disturbance caused to our original signal.

III. Result and
discussion

Abstract— In this paper an attempt is made to generate a synthetic
seismic signal by using wavelets. There are many wavelets to generate
synthetic seismic signals; some of them are klauder, ormsby, ricker wavelets,
etc. Each wavelet varies in their frequency of operation. Seismic reflection is
well-known geophysical technique which can give the information about earth and
its inner core, the earthquakes. The information given by seismic reflection is
used to monitor the earthquakes, petroleum exploration, determination of the
earth’s core structure, etc. Earth quake has become a serious problem in the
world due to its damage. There are many methods to analyze the signals due to
earthquakes. Among those, we are using a sophisticated method called wavelet
transformation. The advantage of this technique is that, it suppresses the
noise and enhances the signal strength there by seismic information due to the
earthquake. Seismic signal is synthesized by convoluting a klauder wavelet with
a seismic signal.

Keywords—
Seismic signal, Klauderwavelet,
Convolution, Synthetic signal.

I.
Introduction

Wavelet
transforms method having its basis in applied mathematics. This method having
high resolution when compared to other frequency based methods using Fourier
transforms. The specific advantage of this method is to drive low frequency
components with considerably good resolution when compared to other methods.
The drawback of using Fourier series is overcome by wavelet transform method, particularly
for short time analysis of seismic signals (or) earthquake signals.

The wavelet is nothing but a wave
like structure, which initially oscillates at zero amplitude and increases
thereby decreasing the initial oscillated amplitude to zero. To get information
from an unknown signal a wavelet can be convoluted with portion of a known

Signal.
Convolution is a mathematical way of combining two signals in order to form a
third signal. The technique is popularly used for processing the signals
digitally (DSP). The method involves in finding the zero phase of a given
signal.  The wavelet transform is a
method, which decomposes the signal giving details as a function of time.

This involves
Scattering and shifting properties to generate a wavelet and it is limited to a specified length.
. The wavelets has two major characteristics namely 1) wavelets are compact support in time domain, and 2) The wavelet alternates from positive to negative. The process includes two steps 1) shifting the
function of
the wavelet and 2) making product
with an inner scale function.

For best resolution wavelet is a method giving
complete picture for the signals such as of seismic origination. Here the resolution of the signal is controlled by
the bandwidth of the signal. Moreover seismic
signals are transient in nature radiating
natural (or) manmade noise along with the seismic sources. This finally locates the, source mechanisms and the structure of the propagation medium through which they travel. The data taken from the seismograms, where
the amplitude of the seismic waves are
recorded as a function of distance from the epicenter, where earthquake occur the intensity
of this earthquake is measured in terms of
the magnitude taken
over a scale called
Richter scale.

The output of a monitoring
instrument namely
seismograph gives a signal corresponding
to the earthquake, by
analyzing these seismic signals one can design an earth warning system for predicting
the earth quakes in future avoid natural
disasters.

II.
Methodology

A Klauder
wavelet represents the autocorrelation of a linearly swept frequency-modulated
sinusoidal signal. It is defined by its terminal low frequency, “f1”;
its terminal high frequency, “f2”; and the duration of the input
signal “T”, often 6 or 7 seconds. The real part of the following
formula will generate a Klauder wavelet.

Klauder(t)
=real sin(tkt(T-t))/(tkt) exp (2ifot)

where k = (f2 – fl)/T (rate of change of frequency with
time)

fo=
(f2 + fl)/2 (midfrequency of bandwidth)

i = 1
(an imaginary number)

Figure-1

a Klauder wavelet (fig 1) is
symmetrical about a vertical line through its central peak at time zero.

Figure-2

The frequency
spectrum of a Klauder wavelet (fig 2) shows the substantial
similarities between a Klauder and an Ormsby wavelet.

wavelet is taken as an input signal and is represented as x(t) 2Impulse signal
h(t) is taken as an seismic signal(earthquake signal) 3Y(t) is the output of
the which is obtained by convoluting input signal and the impulse signal 4Noise is added to the y(t) in
order to obtain an noisy seismic signal 5 Due to  unwanted waves such as surface waves ,
corrupt the  seismic signal by adding
noise.6So on comparing the y(t) the output signal without noise and with
noise  we can easily understand about the
disturbance caused to our original signal.

III. Result and
discussion

Representation of a klauder wavelet

Representation of a klauder wavelet