1. Introduction

Attenuation is one of the fundamental properties of seismic waves from which the material and physical conditions in the Earth’s interior can be inferred (Aki, 1980). The decay of seismic wave amplitude with distance defines the attenuation of the medium. Seismic attenuation is usually considered to be a combination of two mechanisms – intrinsic absorption and scattering loss. As seismic wave propagate through the Earth’s interior and finally arrives at seismic stations on the surface, its energy (amplitude) decays due to conversion of elastic energy to heat or other forms of energy (intrinsic attenuation) as well as energy redistribution in the heterogeneities present in the lithosphere (scattering). The measurement and interpretation of elastic wave attenuation is important for studying the medium through which seismic wave propagates.

Attenuation of seismic wave is described by a dimensionless parameter called the quality factor Q (Knopoff, 1964) that expresses the wave amplitude decay that occurs when a wave propagates through a medium the inverse of thequality factor (Q^-1) is known as the attenuation factor, which is proportional to decay of amplitude with passages of time or source- receiver distance. The attenuation quality factor (Q) is a combination of intrinsic quality factor, Qi and scattering quality factor Qs.

There are various methods to estimate the attenuation quality factor of seismic wave in a region. It is difficult to measure attenuation from a high frequency seismogram using a deterministic approach, because large number of parameters is required to adequately explain a high frequency seismogram. In order to overcome this difficulty Aki (1969) initiated the application of a statistical approach to study of high frequency seismic waves. All portions of a seismogram cannot be treated entirely statistically, because their characters are determined by a particular nature of the path between the source and station (Aki, 1969). Aki (1969), developed a model for the rate of coda decay and suggested that the seismic coda waves of local earthquakes are backscattered body waves from numerous randomly distributed heterogeneities in the earth’s crust and may be treated by a statistical method. The longer the wave travels the greater the variety of heterogeneities they encountered. The later portions of a seismogram therefore may be considered as a result of some kind of averaging over many samples of heterogeneities, thus suggesting a statistical treatment in which a small number of parameters characterize the average properties of the heterogeneous medium (Aki and Chouet, 1975). One of the most promising means of studying seismic scattering and attenuation in the crust and lithosphere is analysis of the coda of local earthquake (Aki, 1969). Coda is the end part of a seismogram of locally recorded earthquake following the body and surface waves containing a very complex and gradually decaying signal. The coda wave amplitude on a seismogram is explained by two backscattering models (Aki and Chouet, 1975).

The first is the single scattering model, according to which waves are backscattered S waves and generated when S wave encounters different heterogeneity present in the propagating media. This model assumes that the scattering wave field is weak and does not produce secondary scattering when it encounters another scatterer. The law of energy conservation is violated because of oversimplification of this model. This model assumes that earthquake source and the seismic station are located at the same point in an unbounded, homogenous, and isotropic medium containing a random but uniform spatial distribution of heterogeneities. This model was further modified by Sato (1977), considering a finite distance separating the source and receiver.

The second model is the multiple scattering models (Aki and Chouet, 1975; Gao et al., 1983a), which assumes secondary scattering and consider the seismic energy transfer as a diffusion process. Aki (1969) suggested that coda Qc-1 is more closely related to intrinsic rather than scattering attenuation. Within the frame work of the single scattering theory Qc-1 represents an effective attenuation, including the contribution of both intrinsic and scattering loss ( Akinci et al., 2000). Several authors (e.g. Aki and Chouet, 1975; Sato, 1977; Rautian and Khalturin, 1978; Hermann, 1098; Roecker et al., 1982; Frankel, 1991; Woodgold, 1992; Gupta, 1998) observed that the coda wave model of Aki (1969), extended by Aki and Chouet (1975) and Sato (1977) is the easiest way to estimate the attenuation, backscattering and source spectrum.

In the present study, Qc is estimated following single scattering model of Aki and Chouet (1975) extended by Sato (1977). The events used in this study are listed in Table 1. Qc is measured as a function of frequency using the digital waveform data of local earthquakes. A study is made of elastic wave attenuation to develop an attenuation relationship. The coda Qc values are calculated for different lapse time windows of coda envelope to investigate the lapse time dependence of Qc as well as the possibility of depth dependence of seismic attenuation.

2. Data

In this study, we selected only 3 earthquakes given in the Table 1 recorded by local seismic network during the three local seismic stations namely Tezpur (TZR), Dokmok (DMK) and Seijusa (SJA).The broadband seismic stations are equipped with CMG-3T / CMG-3ESP seismometer (3-component) and REFTEK 72A series data acquisition system. The sampling rate of the signal is 100 samples per second. The locations of the seismic stations as well as the epicenters of the earthquake events are shown in Figure 1. The selection of the data set is made on the basis of the following criteria:year 2001. These events are recorded by

i. Epicentral distance of the events less than 100 km

ii. Higher signal to noise ratio.

iii. Clear record of coda wave

Out of two horizontal components of seismograms only one component is utilized for this purpose for which S-wave arrival is earlier compared to the other. Total 9 horizontal component seismograms are selected for this purpose. The hypocentral parameters of the selected events are listed in Table 1.

Figure 1. Figure showing location of 3 earthquakes (Red balls) used in the present study and the recording seismograph stations (black triangles) namely Tezpur (TZR), Dokmok (DMK) and Seijusa (SJA)

The signal to noise ratio of each event is estimated. For this purpose the noise and signal power spectra are determined using Fast Fourier Transformation (FFT) algorithm using about 2 seconds long time series before and after the P- wave first arrival. An example for MND station is shown in Figure 2a and Figure 2b. Table 1. Hypocentral parameters of the events used in this study

Figure 2a. shows the original seismogram and two time windows (2 sec before and after P arrival) used in the evaluation

Figure 2b. Shows the corresponding frequency spectra designated as Noise (N)[black], Signal (S)[blue] and S/N ratio [red]

Figure 3. Figure showing a digital seismogram recorded by DMK seismic station. Different coda window length used in this study are shown along with the discrete window length of 2.56 sec used for smoothening the coda envelope by calculating RMS values of coda wave amplitudes with a sliding window of half of the discrete window length i.e. 1.28 seconds

3. Method and Data Analysis

The coda Q method was introduced by Aki(1969) and has been extended by different researchers e.g. Aki and Chouet (1975), Sato (1977) and Rautian and Khalturin (1978). We have utilized the single back scattering model developed by Aki (1969) and extended by Aki and Chouet (1975) and Sato (1977) for estimation of coda wave attenuation quality factor, Q_c.

This model is based on the following assumptions:

According to Aki (1969), Aki & Chouet (1975) and Sato (1977), the time dependence of root mean square coda wave amplitude, A(ω,t), on a bandpass-filtered seismogram can be written as

(1)

where Q_c is the attenuation quality factor as a function of frequency, t^-1 is a correction factor for the geometrical spreading , and C(ω) takes into account these terms of source and site amplification. This model is believed to be more appropriate for small local earthquake than multiple – scattering model (Ibanez et al., 1990).

Sato (1977) developed the model where root mean square (rms) coda wave amplitude at lapse time t may be written as

(2)

where, x = t/t_s ( t_s is the travel time of S wave) and r is station-source distance; K(r, x) is a function of x and r, defined as

(3)

By taking the natural logarithms of equation (2) and rearranging terms, we obtain the following equation:

(4)

For narrow bandpass-filtered seismograms, C(ω) is constant. Therefore, by using a linear regression of terms on the left side of equation (4) vs t, Q_c can be determined from the slope of the fit, which is equal to -ωt/2Q_c

Table 2. Parameters of band-pass filter showing central frequencies with respective low and high cut off frequencies Low cutoff (Hz) Central frequency (Hz) High cutoff (Hz) 0.67 1.0 1.33 1.00 1.5 2.00 1.33 2.0 2.67 2.00 3.0 4.00 2.67 4.0 5.33 4.00 6.0 8.00 5.33 8.0 10.67 8.00 12.0 16.00 10.67 16.0 21.33 12.00 18.0 24.00

In order to study the frequency and lapse time dependence of Q_c, we used following scheme to analyze the data:

In the next step, the lapse time dependence of Q_c is observed by increasing the coda duration step by step, measured from origin time. Each seismogram falling in the lapse time range of 20-90 seconds is analyzed starting at 2t_s. Figure 4a shows an example of original and band pass filtered seismogram (event no. 2 in Table 1) of a local earthquake recorded by the DMK (Dokmok) station and Figure 4b shows the plot of ln [ A(r, ω,t)/ [K(r,x)] vs. t, the lapse time for 30 sec coda window.

Figure 4 (a). Shows an example of original and band pass filtered seismogram (event no. 2 in Table 1) of a local earthquake recorded by the DMK (Dokmok) station and (b) Shows the plot of ln [ A(r, ω,t)/ [K(r,x)] vs. t, the lapse time for 30 sec coda window

4. Results and Discussion

The Q_c values are estimated filtering the coda waves of 9 waveforms in frequency band centered at 1, 1.5, 2, 3, 4, 6, 8, 12, 16 and 18Hz for lapse time window length of 20, 30 and 40 seconds. Total 270 Q_c measurements are obtained for 20, 30 and 40 seconds coda window length estimated from the linear regression of the ln[A(f, t)/K(r, x)] vs t plot. The Q_c values for all lapse time and frequency ranges are listed in Table 3a, Table 3b and Table 3c

The Q_c measurements estimated from 20 sec coda window length vary from 7 to 92 at frequency 1Hz and 562 to 5492 at 18Hz. The distributions of Q_c values with frequency are shown in Figure 5a. The mean value of Q_c observations (filled circles) using 20 seconds coda window length for all the stations are also shown in the Figure 5a which vary from 31 + 14 at 1Hz to 1974 + 209 at 18Hz. It is observed from the general trend (Figure 5a) that Q_c values follows a power law of the from Q_c = Q₀fⁿ, where Q₀ is the quality factor at 1Hz and n is the frequency dependent coefficient. For 20 seconds coda window Q₀ and n are 21.49+1.11 and 1.48+0.08 respectively and we obtain the frequency dependent attenuation relation as Q_c = 21.49 + 1.11 f ^1.48+0.08.

For 30 sec coda window length Q_c measurements vary from 14 to 115 at frequency 1Hz and 1065 to 3400 at 18Hz. The distributions of Q_c values with frequency are shown in Figure 5b. The mean value of Q_c observations (shown by dark filled circles in Figure 5b) vary from 46 + 22 at 1Hz to 1977 + 225 at 18Hz. The empirical attenuation relationship for 30 seconds coda window length is obtained as Q_c = 69.92 + 1.11 f ^123+0.058.

Table 3a. The Qc values for the three at different stations and lapse time coda window length of event no.1. Frequercy (Hz) Qc values at different stations and lapse time coda window length TZR DMK SJA 20Sec 30Sec 40Sec 20Sec 30Sec 40Sec 20Sec 30Sec 40Sec 1 7 14 24 15 37 80 32 115 80 1.5 14 33 67 27 79 178 59 332 178 2 25 70 148 45 153 370 119 599 370 3 58 184 354 120 354 834 345 600 834 4 99 305 718 206 474 1197 497 655 1197 6 149 379 1157 263 789 1349 545 637 1349 8 189 451 1065 382 830 1643 603 760 1643 12 325 675 1407 614 1205 1813 1065 1314 1813 16 482 938 2372 827 1552 2206 2036 1957 2206 18 562 1065 3143 921 1704 2377 2254 2103 2377

Table 3b. The Qc values for the three at different stations and lapse time coda window length of event no.2. Frequercy (Hz) Qc values for different stations and lapse time coda window TZR DMK SJA 20Sec 30Sec 40Sec 20Sec 30Sec 40Sec 20Sec 30Sec 40Sec 1 7 16 30 13 30 56 91 90 165 1.5 18 46 88 23 62 111 131 117 168 2 35 97 192 40 99 183 273 200 203 3 81 233 499 97 217 334 396 326 314 4 118 310 655 147 309 437 283 345 414 6 174 392 702 242 539 704 423 725 1136 8 249 528 822 388 873 1061 783 1360 1369 12 446 941 1253 804 1376 1661 1831 2170 2071 16 734 1327 1586 1148 1740 2149 2592 2313 3291 18 883 1513 1773 1277 1905 2338 4480 3400 3701

Table 3c. The Qc values for the three at different stations and lapse time coda window length of event no.3 Frequercy (Hz) Qc values for different stations at different lapse time coda window TZR DMK SJA 20Sec 30Sec 40Sec 20Sec 30Sec 40Sec 20Sec 30Sec 40Sec 1 7 14 25 13 35 60 92 64 357 1.5 15 35 64 27 74 125 139 130 536 2 30 69 116 46 126 206 224 234 551 3 63 112 237 106 239 37 439 446 693 4 94 117 328 174 332 433 635 454 990 6 141 278 414 362 577 612 1217 659 1193 8 205 400 640 699 1056 895 1722 982 1316 12 368 664 1003 1429 1485 1305 2357 1539 1964 16 548 976 1425 1876 1876 1783 3104 2296 2941 18 632 1157 1697 2050 2135 2006 3697 2814 3558

Similarly, the distribution of Q_c estimates using 40 Sec window length is shown in Figure 5c. The mean value of Q_c estimates at 40 seconds vary from 97 + 34 at 1Hz to 2552 + 685 at 18Hz. The empirical attenuation relationship from these Q_c values are obtained as Q_c = 88.86 + 1.12 f ^1.19+0.06.

Figure 5a. Shows the distributions of Qc values with frequency for coda window lengths 20 seconds

Figure 5b. Shows the distributions of Qc values with frequency for coda window lengths 30 seconds

Figure 5c. Shows the distributions of Qc values with frequency for coda window lengths 40 seconds

From the above results it is observed that Q_c values obtained for the seismograms are highly frequency dependent. The Q_c values increases with increase in frequency. The high frequency dependent characteristics of the Q_c values may be due to different heterogeneity present in the propagating media. Aki(1980) observed that only highly fractured media can generate frequency dependent Q_c values. Moreover it is observed that Q_c value increases with increase in coda window length. The variation Q_c values with lapse time is plotted in Figure 6. The degree of frequency dependence ‘n’ is almost constant whereas Q_o values are increasing with increase in lapse time coda window length. The higher lapse time of coda window samples larger area of the earth’s crust covering the deeper part. The Q_c value increases with depth implies that attenuation is decreasing with depth. This may be due to the fact that homogeneity increases with depth.

Figure 6. Figure showing the variation of Qc values with lapse time of coda window length

5. Conclusions

This study suggests that Q_c values are frequency dependence in the media through which it propagates. The value of frequency dependence n (> 1) suggests that the media is not homogeneous. Since the media is non-homogeneities, intrinsic and scattering mechanism takes place causing the attenuation of the coda wave amplitudes. Attenuation decreases with depth may indicating that deeper part of the earth’s crust is comparatively homogeneous than the upper most crust.

ACKNOWLEDGEMENTS

Hearty thanks to Dr. Debajit Hazarika, Scientist – B, Wadia Institute of Himalayan Geology, Dehradun for the intensive support in data analysis and interpretation. I extent my thanks to all the Teaching staff of the Department of Mathematical Science, Tezpur University, Tezpur for their valuable support during my works. I would like to thank the Director, School of Petroleum Technology, Gandhinagar for his support in this work. Finally special thanks to all my family members for their valuable support in my works.