2011 2nd International Conference on Control, Instrumentation and Automation (ICCIA) S-Transform Based P-Wave and S-Wave Arrival Times Measurements Toward Earthquake Locating 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. 978-1-4673-1690-3/12/$31.00©2011 IEEE 241 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) <k (x) <(x k) for k = (4) B. Surface Waves < is shifted to cover the whole real line. These waves move along the earth’s surface and arrive 1 x b 3) <a,b (x) <( ) for a,b R (5) after body waves. Love wave is the fastest 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 <m,k (x) (6) standard technique for time-frequency localization. mk f f where To calculate the continuous-time STFT, the function is cm,k (x) ³ f (x)<m,k (x) dx (7) multiplied by a window function which is non-zero for only a short period of time and then the Fourier transform of the In wavelet transform, the frequency resolution becomes resulting signal is taken [8]. arbitrarily good at low frequencies while the time resolution g iwt becomes arbitrarily good at high frequencies. STFT x (w,T) ³ x(t)g(t T)e dt (1) A very popular choice for g is a Gaussian function. Although wavelet is a powerful tool in time-frequency analysis, it has some weaknesses. To overcome some of 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 242 (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.