S-Transform Based P-Wave and S-Wave Arrival Times Measurements Toward Earthquake Locating

2011 2nd International Conference on Control, Instrumentation and Automation (ICCIA)

Samaneh Azadi, Ali Akbar Safavi

Abstract— Measuring physical characteristics of the earth Knowledge about seismic waves and their different arrival using direct or indirect methods can be useful to achieve times can be used to extract information about geological knowledge about geological structures. This could help in structure of the earth and locate earthquakes. One important solving many geotechnical problems that scientists are issue in seismology is finding P-wave and S-wave arrival encountered with. Two important factors in seismology are P- times. There are different methods to do it like a fractal- wave and S-wave arrival times. This paper focuses on based algorithm [1], discrete wavelet transform [2] and L2 proposing a useful tool and method for these measurements. For this purpose, three time-frequency analysis tools are norms for a short- and long-term moving time window [3]. applied to two seismic data related to earthquakes happened in The method that is used in this paper is time-frequency Canada. By comparing the results, it is showed that S- representation, since it helps study a signal in both time and transform is the best tool for measuring P-wave and S-wave frequency domains. arrival times. Then the time-frequency representation of the In this paper after a review on different kinds of seismic first seismic data which is the result of applying hyperbolic S- waves, three different tools for time-frequency transform is showed and used to find P-wave and S-wave representation of two seismic waves are employed. These arrival times. Also, a peak based method to find instantaneous tools are Short Time Fourier Transform, Continuous frequency is introduced and applied to the seismic data. Wavelet Transform [4] and S-Transform [5]. Then, these

Keywords— P-wave, S-wave, Short Time Fourier Transform methods are compared with each other from the view of (STFT), Continuous Wavelet Transform (CWT), S-Transform, measurement objectives. At last because of the advantages Instantaneous Frequency. of S-transform over the other ones, this tool is used to find P-wave and S-wave arrival times. Besides, a peak based I. INTRODUCTION method [6] is used to obtain instantaneous frequency of the N geophysical exploration, some of the most important seismic wave which enables more researches on geological I physical characteristics of the earth can be measured by structures. The structure of this paper is as follows. special tools. These characteristics do not have always a direct relation with the aim of measurement. Therefore, there II. SEISMIC WAVES is always a need to indirectly measure the characteristic Seismic waves are elastic waves carrying energy caused variable. by earthquake activity in the earth or an explosion. They Although there is a lot of improvement on the field of travel through the earth and are recorded on seismographs. physical models and analysis of geotechnical problems, They are used to determine the internal structure of the earth there are still many problems that need a deeper and can be studied in seismology and geophysics. understanding and more advanced numerical approaches for There are several kinds of seismic waves which travel simulating underground layers motion caused by earthquake. through different regions. The two main types are body It contains problems like resonance in structures, simulation waves and surface waves. The first type contains P-waves of soil layers motion and achieving characteristics of and S-waves and the second type contains Rayleigh waves earthquakes. and Love waves. One of the tools used in these measurements and issues is the seismograph which records some earthquake waves. A. Body Waves These records show seismic waves and can be analyzed by Body waves travel through the earth’s inner layer [7]. P- computer methods toward measuring some critical variables. wave (primary wave) has the most velocity among all Seismic wave has two main features. First, it has non- seismic waves and arrives at a seismic station sooner than stationary amplitude because at the beginning of the the others. P-waves move the particles in the same direction earthquake, the energy is produced and at the end, it is of wave propagation. S-wave (secondary wave) is the second diminished. Second, it has non-stationary frequency, which wave that can be recognized in an earthquake and is slower means that at different times the wave frequency changes. than P-wave. It moves rock particles up and down or perpendicular to the direction of wave propagation. Different speed of P-wave and S-wave can be used to Samaneh Azadi, School of Electrical and Computer Engineering, Shiraz locate an earthquake as below: University, Shiraz, Iran (email: [email protected]). Ts: arrival time of S-wave Ali Akbar Safavi, School of Electrical and Computer Engineering, Shiraz University, Shiraz, Iran (email: [email protected]). Tp: arrival time of P-wave.

1 1 1) <(x) should decay to zero at rf . Ts-Tp= ( )*d s - wave speed p - wave speed f Where “d” is the distance from earthquake source. ³ <(x) dx 0 (3) The above statements are acceptable for epicentral distances f up to 1000 km. 2) surface wave and a a causes horizontal motion. Rayleigh wave moves the particles It is the defenition of general family of continous up and down and side-to-side in the same direction of 1 wave’s movement. This wave causes the most shake in an wavelets. is used for the sake of normalization. a earthquake and can be much larger than the other waves. In the above properties <(x) is called mother wavelet.

III. . TIME-FREQUENCY DISTRIBUTION Wavelets described in (5) are redundant wavelets [4] but beside this family, we can have discrete dyadic family, too. A. Short Time Fourier Transform Every ƒ(x) in L2 (R) can be expressed as a wavelet series: Short Time Fourier Transform (STFT) or windowed f f Fourier transform is a Fourier related transform and a f (x) ¦¦cm,k

To calculate the continuous-time STFT, the function is cm,k (x) ³ f (x)

these shortcomings, a new time-frequency representation We have two problems while using STFT as the time- called S-transform has been recommended which was first frequency distribution. First, according to the uncertainty published in 1996 and has been used in several practical principle [8]: applications [9] , [10] , [11] , [12]. 1 ΔtΔf t (2) 2 C. S-Transform Where denotes duration of the filter and denotes Defenition of continuous S-transform is as below [5]: the bandwidth of the filter. It denotes that the resolution in time and frequency cannot be arbitrarily small. Second, f f (7t)2 f 2 i2Sft STFT treats the whole signal with one constant window and s(7, f ) h(t) e 2 e dt (8) ³f 2S does not easily allow a good time-frequency distribution. More precisely, since the frequency of a signal is directly proportional to the length of its cycle, it follows that for Two one-dimensinal functions can be produced from (8) high-frequency spectral information, the time interval should which are called “voice” and “local spectrum”. be relatively small to give better accuracy, and for low “voice” is a function of time with a constant frequency frequency spectral information, the time interval should be ( f f 0 ) and “local spectrum” is a function of frequency relatively wide to give complete information. Therefore the with a constant time ( 7 70 ). most desirable is to have a flexible time-frequency window There are different methods to obtain S-transform. One of that automathically narrows and widdens properly. This is them is using the defenition of wavelet transform, exactly what wavelet transform does in providing time- multiplying it by two factors and replacing “d ” with inverse frequency representations of signals. of frequency [5]: f B. Continuous Wavelet Transform s(7, f ) e i2SftW(7, f ) (9) 2S Wavelet transform is a time-frequency representation which overcomes the shortcoming of STFT. Where f Some properties of wavelets ( <(x)' s ) are[4]: 1 t 7 W (7, d) h(t) <( ) dt (10) ³ d f d

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(t7)2 f 2 B F i2Sf (t7) B F 2 J HY J HY <((t 7) f ) e 2 e (11) X (7 t,{J HY ,J HY , O HY}) ( )(7 t ] ) 2 B F J HY J HY The two factors have important roles in the difference J B J F ( HY HY ) (7 t ] )2 2 (16) between wavelet and S-transform. B F O HY 2J HY J HY f 1) is normalization factor which normalizes the 2S U is used as a set of parameters that control the shape of localizing window to have unit area. As a result, the window. Here, it contains B (backward taper parameter), amplitude response is not dependant on frequency and is the J HY same as the amplitude of Fourier transform, wheras the F 2 J HY (forward taper parameter) and O HY (positive curvature amplitude of wavelet transform diminishes with increase in parameter). frequency. Translation by ] in the defenition of X, ensures that the 2 2 2 t f t 2 f i2Sft 2) e 2 e is phase factor. In (9) to (11), e 2 is peak of occurs at T=t and is defined as i2Sft translated while e is stationary. It causes the real and B F 2 2 (J J ) O HY imaginary components of the spectrum, given by S- ] HY HY (17) 4J B J F transform, be localized independantly. The reference for the HY HY phase information will be time t=0. It is the same as Fourier transform but in contrast to the wavelet transform in which So the hyperbolic window is assymetrical at f=0 and it the reference is the center in time of the analyzing wavelet becomes more symmetrical as frequency increases [13]. [5]. The resolution of the beginning time of events on the S- transform can be improved by using a narrower window. IV. THE EARTHQUAKE DATA Although it results in a wide window in frequency domain A seismic data has been used which is related to an and a worse frequency resolution, it does not make problem earthquake happened in Saguenay on 25th of November, in many applications. 1988 at 23 46’ 0” and the results are for the seismic signal If we use an asymmetric window in a shape that the slope in vertical direction. Table Ι contains more information on in the forward direction is greater than backward direction, the seismic data and Fig. 1 indicates acceleration of the this problem can be solved [13]. seismic wave. Besides, another seismic data has been used which is related to an earthquake happened in Miramichi on 1) The Hyperbolic Window 31th of March, 1982. The arrival times of this earthquake are The window used as a solution to the mentioned problem shown in Table ΙΙΙ. is better to be more symmetrical at high frequencies since it TABLE I causes better frequency resolution and be more asymmetrical MORE INFORMATION OF SEISMIC DATA at low frequencies. Sample/second Data type Unit To have such a window, hyperbolic window has been 50 Acceleration cm/ defined ( ZHY ). First Gaussian window and the resulted S- transform were introduced as below [13],[9]: f 40 S(7, f ,V ) ³ h(t)g(t 7)ei2Sft dt (12) 30

f ) 2 Where 20

1 i2Sft g(t) e :Gaussian window (13) 10 V 2S Now, instead of gaussian window for the sake of more 0 accurate representation, we can use hyperbolic window: -10

f Acceleration(cm/s i2Sft -20 S(7, f , U) ³ h(t)ZHY (7 t, f , U) e dt (14) -30 f 0 4 8 12 16 20 24 28 32 36 40 Where time(s) 2 B F 2 2 Fig. 1. The seismic wave 2 f f X (7t,{J HY,J HY,O HY}) (15) ZHY F B e 2 2S (J HY J HY )

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V. P-WAVE AND S-WAVE ARRIVAL TIMES MEASUEMENTS In this section, in order to measure P-wave and S-wave arrival times, three kinds of time-frequency distributions are compared with each other. Hyperbolic S-transform of the first seismic data is compared with CWT using 3D plots and with STFT using 2D plots. At last, advantages of S- transform to the other ones are found out. In Section VII, hyperbolic S-transform is applied to the first data in order to find and show P-wave and S-wave arrival times. Fig. 5. Applying ST to the seismic data By comparing Fig. 2 and Fig. 3, it can be found that the whole frequency band cannot be detected when STFT is As it is clear from Fig. 4 and 5, in contrast to S-transform, employed to analyze the seismic data. the amplitude of CWT diminishes as frequency increases. This note is also true according to the formulas of S- 1) transform and CWT. Certainly, the amplitude response of ST is invariant to the frequency due to the presence of 0.5 Gaussian function, while the normalization of CWT 0.45 diminishes the higher frequency components [5]. 0.4 P-wave and S-wave arrival times related to the 0.35 earthquakes happened in Saguenay and Miramichi are found 0.3 by STFT, CWT, ST and compared with each other in Tables 0.25 ΙΙ and ΙΙΙ, respectively. 0.2 0.15 TABLE II

0.1 ARRIVAL TIMES OF THE EARTHQUAKE IN SAGUENAY FOUND

normalized frequency BY DIFFERENT METHODS 0.05

0 2 4 6 8 10 12 14 16 18 20 Real Time ST CWT STFT time(s) P-wave arrival 0.56 0.56 0.54 0.48 Fig. 2. Applying STFT to the seismic data (using Gaussian window) time (s) S-wave arrival 14.3 14.26 14.58 14.24 time (s) 0.45 0.4 TABLE III 0.35 ARRIVAL TIMES OF THE EARTHQUAKE IN MIRAMICHI FOUND

0.3 BY DIFFERENT METHODS

0.25 Real Time ST CWT STFT

0.2 P-wave arrival 0.015 0.010 0.020 0.020 0.15 time (s)

0.1 S-wave arrival 0.485 0.470 0.525 0.460

normalized frequency time (s) 0.05

0 0 2 4 6 8 10 12 14 16 18 20 Consequently, in order to have better results in the time(s) measurement, S-transform is used as a tool for time- Fig. 3. Applying ST to the seismic data 2) frequency representation of the seismic data. A new concept, called instantaneous frequency, is described in the following section.

VI. ESTIMATING THE INSTANTANEOUS FREQUENCY Instantaneous frequency defined as derivative of the phase of a signal, describes the variations of signal frequency contents with time. “IF” is a concept used in many applications such as communication, heart beat signals, radar, sonar, seismology and etc.

Fig. 4. Applying CWT to the seismic data (using daubechies2) There are different methods used to obtain instantaneous frequency such as differentiation of the phase, counting zero crossing, phase-locked loop techniques, etc [14].

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Another useful tool for IF estimation in signal analysis is A. Application of IF in seismic processing time-frequency destribution. Since S-transform is one of the When an earthquake happens, some waves travel through best tools for time-frequency analysis, it can be used for IF the earth and as mentioned before, different waves arrive at estimation. different times. It can be used to get information about “IF” of a monocomponent signal z(t) can be estimated by geological structure of the region. And also comparison using the highest part of the “ridge” [6]. between the IF of the received signal with that of a source can be used to get information about the amount of Z(t) ae j)(t) (18) dispersion and absorbing nature of layers in the way. When analyzing signal based on normal amplitude cannot help us, Zˆ(t) arg [maxf TFDz (t, f)] (19) IF can be useful [14].

WhereTFDz (t) is the signal z(t) time-frequency distribution. VII. RESULTS And if a signal is multicomponent, it can be modeled as sum of two or more monocomponent signals [6] (each with A. Applying Hyperbolic S-Transform to The Seismic its own IF) : Signal M M j)m(t) x(t) ¦¦Z(t) am (t)e (20) m 11m 0.45 Here, IF of a signal has been found as an example based 0.4 on the peak of time-frequency representation resulted from S-transform. This example is present in [15] but IF in this 0.35 P-wave reference is the result of other methods. Good accuracy of 0.3 arrival our methodology is seen: time 0.25 for 0 t d10 0.2 S-wave x(t) 3000sin(3St) 500sin(7St) arrival 0.15 time for 10 t d 20 0.1 x(t) 3000sin(5St) 500sin(7St) frequency normalized 0.05 for 20 t d 30 0 x(t) 3000sin(4St) 500sin(7St) 0 2 4 6 8 10 12 14 16 18 20 time(s) 0.5 B F 0.45 Fig. 8. Hyperbolic S-transform calculated at J HY =1.5, J HY =0.5, 0.4 2 0.35 and O HY =1.0 0.3 0.25

0.2 In Section II, it was mentioned that P-waves and S-waves

0.15 are body waves and have the most velocity in all kinds of

0.1

normalized frequency normalized seismic waves. 0.05 In the above figure, after using hyperbolic S-transform 0 0 5 10 15 20 25 30 time(s) B F 2 with J HY =1.5, J HY =0.5, and O HY =1.0, the arrival times of Fig. 6. Original IF P-wave and S-wave related to the first seismic data have

0.5 been determined. P-wave arrives at 0.56 second and S-wave

0.45 arrives at 14.26 second. 0.4 0.35

0.3

0.25

0.2

0.15

0.1 normalized frequency normalized 0.05

0 0 5 10 15 20 25 30 time(s) Fig. 7. Estimated IF using hyperbolic s-transform

Mean Square Error (MSE) which is calculated in this methodology is equal to 4.9644e-4.

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B. Instantaneous Frequency of the first Seismic Wave [4] I. Daubechies, “ten lectures on wavelets”, PA: SIAM; 1992. [5] R. G. Stockwell, “Why Use the S-Transform”; Fields Institute 0.5 Communications, Northwest Research Associates, Colorado

0.45 Research Associates Division, 3380 Mitchell Lane, Boulder

0.4 Colorado USA 80301; 2006.

0.35 [6] J. Lerga, V. Sucic and B. Boashash.” An Efficient Algorithm for

0.3 Instantaneous Frequency Estimation of Non-stationary Multicomponent Signals in Low SNR”; Hindawi Publishing 0.25 Corporation, EURASIP Journal on Advances in Signal 0.2 Processing, volume 2011, Article ID 725189, 16 pages; 2011. 0.15 [7] C. W. Tseng,” Tutorial—Time-Frequency Analysis for 0.1

normalized frequency normalized Seismology”; Graduate Institute of Communication Engineering, 0.05 National Taiwan University, unpublished. 0 13 14 15 16 17 18 19 20 [8] N. E. Zabihi, H.R. Siahkoohi, ”single frequency seismic attribute time(s) based on Short Time Fourier Transform, Continuous Wavelet Fig. 9. Estimated IF near S-wave arrival time using hyperbolic S-transform Transform, and S Transform “, International Conference 0.5 and Exposition on Petroleum Geophysics; 2006. 0.45 [9] Y. H. Wang,” The Tutorial: S Transform”; Graduate Institute of

0.4 Communication Engineering, National Taiwan University.

0.35 [10] E. Sejdic, L. Stankovic, M. Dakovic, J. Jiang,” Instantaneous

0.3 Frequency Estimation using the S-Transform”, IEEE signal processing letters; 2008. 0.25 [11] S. Parolai, “Denoising of Seismograms Using the S Transform”, 0.2 Bulletin of Seismological Society of America, Vol. 99, No. 1, 0.15 pp. 226-234; 2009. 0.1

normalized frequency [12] T. I. Todorov, G. F.Margrave,” Variable-factor S-transform for 0.05 time-frequency decomposition, deconvolution, and noise 0 0 0.4 0.8 1.2 1.6 2.0 2.4 attenuation”, GeoCanada, Working with the Earth; 2010. time(s) [13] C. R. Pinnegar and L. Mansinha,” The S-Transform with Fig. 10. Estimated IF near P-wave arrival time using hyperbolic S- windows of arbitrary and varying shape”, Geophysics, VOL. 68, transform NO. 1; P- 381-385; January-February 2003. [14] A. Shafieezadeh, M.Motavalli. ”Assessing Different Methods of As mentioned in previous sections, instantaneous Deriving Instantaneous Frequency Including STFT, Wavelet, frequency of a seismic wave can be useful in determining EMD-HT and GPOF”, The International congress of civil engineering. the frequency dispersion, absorbing nature of rocks, [15] B. Boashash, “Estimating and Interpreting the Instantaneous detection of some hydrocarbons, some geological structures Frequency of a Signal-Part2: Algorithms and Applications”, and etc. proceedings of the IEEE, Vol. 80, No. 4; 1992.

VIII. CONCLUSION The aim in this paper was analyzing a seismic wave and measuring P-wave and S-wave arrival times. Therefore, three time-frequency distributions (STFT, CWT, ST) were used and compared with each other. The result of this comparison was finding some advantages of S-transform over the other ones. For the sake of having a better accuracy in the onset time while using S-transform, hyperbolic window was used. Besides, a method for estimating instantaneous frequency was introduced and applied to a simple example. Then by using the resulted time-frequency distribution of the seismic wave, P-wave and S-wave arrival times and the instantaneous frequency of the wave which is useful for future researches on geological structures were found.

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