analysis of seismic waves using modified periodogram...

ANALYSIS OF SEISMIC WAVES USING

MODIFIED PERIODOGRAM ALGORITHM

M.L.S.Prameela¹, Ch.Prem Abhinav², K.S.Ramesh³, S.

Koteswara Rao4,T.Vaishnavi Chandra5

Department of ECE, K L University, Vaddeswaram, Gunutr.

[email protected], [email protected]

Abstract

Earthquakes are the most terrible events on the earth. They are caused by

sudden breaks in the Earths rocky outer shell and shaking of ground. The

earthquake effect depends on breaking of rock and the distance which it shifts. So

by using signal processing techniques, power spectral density is estimated

through which the movement of earthquakes can be predicted. In this paper, non-

parametric methods are used for spectral estimation. Modified periodogram

technique is used to calculate spectral densities.

Keywords: Stochastic signal processing, Adaptive signal processing,

Seismology, Applied Statistics and Seismic signal processing.

1. INTRODUCTION

International Journal of Pure and Applied MathematicsVolume 114 No. 10 2017, 201-209ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.euSpecial Issue ijpam.eu

201

The surface of the earth is known as the crust, and it is comprised of

plates, called structural plates, that move. Seismic tremors happen

when these plates knock, rub or drag against each other. These seismic

tremors are measured utilizing seismometers. Earthquakes frequently

happen in volcanic districts because of the stream of magma in

volcanoes and are an early cautioning of volcanic emissions.Seismology

is defined as the earthquakes study and elastic waves propagation

through earth and other planet-like bodies. This field also includes

earthquake environmental effects and seismic sources such as

Tsunamis and artificial processes such as explosions. A seismogram is

earth motion recordings as a function of time.

1.1 Detection of Seismic waves

Seismic waves are the energy waves caused due to sudden breaking

of rock in the earth.It is also defined as energy that travels through the

earth and is recorded on seismographs. There are various types of

seismic waves. Those are Body waves and Surface waves. Body waves

travel through earth’s inner layer along the planet’s surface. Surface

waves travel through the crust and have lower frequencies than body

waves. These seismic wave can be detected using seismometers.

Seismometers sense and record the earth motions propagating through

elastic waves. These are moved deep into the earth’s surface.

Seismograph is an instrument that records seismic signals. A geophone

is a device that changes ground motion (velocity) into voltage, which

might be recorded at a recording station.

1.2 Non Parametric method

For a random process signal, the Non parametric methods are used

to estimate the autocorrelation sequence. In 1898, Arthur Schuster in

his study of periodicities, first introduced periodogram which is easy to

compute. For short data records, periodogram has limited ability in

estimation of power spectrum. To improve the statistical properties of

periodogram, many modification have been made which is called

Modified Periodogram. It is the periodogram of a windowed data

International Journal of Pure and Applied Mathematics Special Issue

202

sequence. In order to smooth the edges of the signal, modified

periodogram windows the time domain signal earlier to DFT

computation.

2. MATHEMATICAL MODELLING

2.1 Spectral Analysis

Process of splitting a large data signal to simple parts is known as

Spectral Analysis. It is also known as Frequency Domain Analysis.

These graphs are plotted with respect to Amplitude or Phase on x-axis

and Frequency on y-axis. In this method Frequency is only term which

is not varying on the graphs (on y-axis) that are being plotted for

Spectrum Analysis. It is being performed by common method known as

Fourier Transformation.

2.2 Fourier Transformation

The term Fourier consists of both mathematical and frequency

domain modelling in it. In mathematical it can be represented by using

the equation.

𝑓 𝑧 = 𝑓 𝑥 ∞

−∞𝑒−2𝜋𝑧𝑥 dx (1)

It is used for both time variant and non-linear systems for

calculation of power spectral density. Now a days the software is using

DFT (Discrete Fourier Transformation) to generate frequency

estimation or spectrum.

𝐹 𝑥 =1

√𝑁 𝑓(𝑛)𝑁−1

𝑛=0 𝑒−𝑗2𝜋𝑥𝑛 /𝑁 (2)

It is time taking process for the frequency estimation. A separate

algorithm was developed to perform DFT known as FFT(Fast Fourier

Transformation).

𝑋 𝑧 = 𝑥(𝑛)𝑊𝑛𝑘𝑧𝑁−1

𝑛=0 , 0 ≤ k ≤ N-1 (3)

All these methods come under parametric methods. In addition to this

separate non parametric methods are also introduced for estimation of


203

Power Spectral Density.

2.2 Modified Periodogram

The Non Parametric method which is used to calculate the power

spectral density of the input signal by using rectangular data window

is known as Periodogram. It can be represented by using mathematical

equation.

ṙ𝑥 𝑧 =1

𝑁 𝑥𝑛 𝑛 + 𝑧 𝑥𝑛

∗∞𝑛=−∞ 𝑛 =

1

𝑁𝑥𝑛 𝑧 ∗ 𝑥𝑛

∗ −𝑧 (4)

The process of reducing the side lobes or spectral leakage to

calculate power spectral density statistics of a particular signal is

known as Modified Periodogram.

2.4 Spectral estimation by averaging modified periodogram

In this method, before computing individual periodogram windows

W(n) is directly applied to data signals. Here length of the window is

given by formulae.

𝑈 =1

𝐾 𝑤(𝑛) ²𝐾−1

𝑛=0 (5)

We consider the expected value of modified periodogram to evaluate

the bias. Here W (𝑒𝑗𝜔 ) is the Fourier transform of the data window.

𝐸 Ṕ𝑀 𝑒𝑗𝜔 =1

2𝜋𝐾𝑈𝑃𝑥 𝑒

𝑗𝜔 ∗ 𝑊 𝑒𝑗𝜔 ² (6)

By Parseval’s theorem, U is the window energy divided by K

𝑈 =1

𝐾 𝑤 𝑛 2 =

1

2𝜋𝐾𝐾−1𝑛=0 𝑊(𝑒𝑗𝜔

𝜋

−𝜋²𝑑𝜔 (7)

Modified Periodogram will be asymptotically unbiased when

𝑊(𝑒𝑗𝜔 2

𝐾𝑈

will converge to a unit area impulse.

1

2𝜋𝐾𝑈 𝑊(𝑒𝑗𝜔 ) ²

𝜋

−𝜋 (8)

The advantage is that there is a trade-off between spectral solution


204

and spectral masking. For a data window modified periodogram

resolution is 3dB bandwidth.

𝑅𝑒𝑠 Ṕ𝑀(𝑒𝑗𝜔 ) = (∆𝜔)3𝑑𝐵 (9)

3. SIMULATION AND RESULTS

Step 1:The input signal is considered as reference signal in the

modified periodogram algorithm. This input signal is called as

Synthetic signal as shown in Fig.1.

Step2: Power Spectrum analysis using normalized frequencies is

performed for the synthetic signal using modified Periodogram

algorithm as shown in Fig.2.

Step 3:Raw Seismic Signal is recorded during Dynamite Blast which is

performed within 80-100 feet below the Earth and is loaded to

MATLAB by using Book_Seismic_Data.mat as shown in Fig.3.

Step 4:In any random signal, Bias or shift is present. So the Raw Signal

is detrended which removes the mean by using modified periodogram is

shown in Fig.4.

Step 5: Power Spectral Analysis for the raw signal at different

frequency components is Estimated and shown in Fig.5.

Step 6:FIR band pass frequency Spectrum isplotted. It limits the signal

bandwidth and gives normalized frequency in the range 0.02-0.3 as

shown in Fig.6.

Step 7:FIR Band Pass filtered signal is obtained by removing noise in a

particular frequency and 8th order FIR signal is used for the detrended

seismic signal as shown in Fig.7.

Step 8: After BPF, FFT spectrum in Normal Frequency vs. Magnitude

dB representation is shown in Fig.8.

Step 9: Spectral estimation of modified periodogram is calculated and

observed in the MATLAB shown in Fig.9.


205

The maximum peak frequency is observed at 0.0529 on X-axis and

16.5 on Y-axis.Only X-axis normalized frequency is considered. The

frequency is calculated by the equation.

𝑤 =2𝜋𝑓

𝑓𝑠 (10)

Here𝑓𝑠 = 500 and w=0.09594*π by replacing these values in the eq

(10) tonal frequency value is 𝑓 = 23.985 Hzobserved in Fig.9

Fig.1.Synthetic signal. Fig.2.Mod. periodogram spectral

analysis.

Fig.3. Raw seismic signal. Fig.4.Detrended raw Seismic signal.

Fig.5.Raw signal spectrum. Fig.6.FIR Band Pass filter o/p.


206

Fig.7.FIR Band Pass filtered signal. Fig.8.Sspectrum of seismic

signal after BPF.

Fig.9.Modified periodogram spectral analysis.

4. CONCLUSION

Many statistical techniques are used in different ways to predict

the probability of earthquakes. In this paper, the earthquake

occurrences predicted by using modified periodogram algorithm which

provides a trade-off between spectral resolution and spectral

masking.


207

5. REFERENCES

[1] P.Shebalin, V.Keilis-Borok, A.Gabrielov, I.Zaliapin, D.Turcotte, “Short-

term Earthquakeprediction by reverse analysis of lithosphere dynamics”,

pp.64-75, Dec.2005.

[2] Neeti Bhargava, V.K.Katiyar, M.L.Sharma and P.Pradhan, “Earthquake

Prediction Through Animal Behaviour”,Indian journal of Biomechanics,

pp.159-165, Mar.2009.

[3] Sajjad Mohsin and Faisal Azam, “Computational seismic algorithmic

comparision for Earthquake prediction”, International Journal of Geology,

Vol.5, pp.53-59, 2011.

[4] Arvind Kumar, Vivek Walia, Surinder singe, Bikramajit Singh Bajwa,

Sandeep Mahajan,Sunil Dhar and Tsanyao Frank Yang, “Earthquake

precursory studies at Amritsar Punjab,India using radon measurement

techniques”, International Journal of Physical sciences,Vol7 (42), pp.5669-

5677, Nov.2012.

[5] Kuo-Liang Wen, Tzay-Chyn Shin, Yih-Min Wu, Nai-Chi Hsiao, and Bing-

Ru Wu,“Earthquake Early Warning Technology progress in Taiwan”,

Institute of Geophysics,pp.202-210, Jun.2009.

[6] Robert J.Geller, “Earthquake Prediction: a critical review”, pp.425-450.

[7] Monson H.Hayes, “Statistical Digital Signal Processing and Modelling”,

John Wiley and Sons, Inc, 1996.

[8] Gary F.Margrave, “Numerical Methods of Exploration Seismology with

algorithms inMATLAB”, Jul.2003.

[9] Wail A. Mousa and Abdullatif A. Al-Shuhail, “Processing of seismic

reflection dataUsing MATLAB”,Morgan and Claypool publishers, 2011.

[10] Petre Stoica and Randolph Moses, “Spectral Analysis of Signals”,

Prentice Hall, Inc, 2005.


208

analysis of seismic waves using modified periodogram...

Documents