Estimation of Coda Wave Attenuation Quality Factor from Digital Seismogram Using Statistical Approach

The main objective of the paper is to estimate seismic wave attenuation from the decay rate of coda wave amplitudes of digital seismogram of local earthquakes (epicentral distance < 100 km) following statistical approach. Three earthquake events have been selected for estimation of coda wave attenuation quality factor (Qc). The coda wave of 9 seismograms from 3 local earthquakes recorded digitally by three seismic stations in the region have been analyzed for this purpose at ten central frequencies (1, 1.5, 2, 3, 4, 6, 8, 12, 16 and 18Hz) and three lapse time windows of 20, 30 and 40 seconds duration. Single backscattering model proposed by Aki (1969) and extended by Aki and Chouet (1975) and Sato (1977) is employed to determine Qc values. The mean values of the estimated Qc vary from 31+12 (at 1Hz) to 1974+211 (at 18Hz) for 20 seconds coda window length. For 30 seconds coda window length Qc vary from 46+16 (at 1Hz) to 1977+ 256(at 18Hz). Similarly for 40 seconds coda window length Qc vary from 97+ 22 to 2552 +312. It is observed that Qc value for the study area is frequency dependent and increase with increase in frequency. Moreover the observed Qc values increase with increasing lapse time at all frequency bands. The estimated Qc values show a frequency dependent relationship of the form Qc=Qof , where Q0 is Qc at 1Hz and n represents degree of frequency dependence. The frequency dependent empirical attenuation relationship for 20, 30 and 40 seconds window lengths are obtained as Qc = 21.49+1.17 f , Qc = 48.6+1.11 f 1.29+0.06 and Qc = 88.86+1.12 f 1.19+0.06 respectively.


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 pr opagates thro ugh a med ium the inverse of the quality 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, Q i and scattering quality factor Q s .
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 prop-2 Jwngsar Brahma: Estimation of Coda Wave Attenuation Quality Factor from Digital Seismogram Using Statistical Approach erties 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 Q c -1 is more closely related to intrinsic rather than scattering attenuation. Within the frame work of the single scattering theory Q c -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, Q c 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. Q c 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 Q c values are calculated for different lapse time windows of coda envelope to investigate the lapse time dependence of Q c as well as the possibility of depth dependence of seismic attenuation.

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. 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 Pwave first arrival. An example for MND station is shown in Figure 2a and Figure 2b.

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: i. Scattering is a weak process and doesn't produce any secondary (multiple) when it encounters anther scatter. This so call Born approximation, which violates the energy conservation law but has been accepted in various physical problems and used successfully in high frequency seismic waves analysis. ii. As the coda waves arrive long time after arrival of all direct waves (p, s and surface waves) the source and receiver are assumed to be placed at one point (for coincident).
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 A(ω,t) = C(ω). t -1 exp(-ωt/2Q c ) (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 A(r, ω, t) = C (ω)[K(r, x)] exp(-ωt/2Q c (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 K(r, x) = 1/r.1/x. ln(x+1/x-1) (3) By taking the natural logarithms of equation (2) and rearranging terms, we obtain the following equation: ln[ A(r, ω,t)/ [K( r,x)]] = ln[ C(ω)]-(ωt/2Q c ) (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 In order to study the frequency and lapse time dependence of Q c , we used following scheme to analyze the data: i. The seismograms are filtered for narrow frequency bands centered at f c = 1, 1.5, 2, 3, 4, 6, 8, 12, 16 and 18 Hz respectively using 8 -pole Butterworth band pass filter. The ten frequency bands (bandwidth 0.67f c , where f c is the central frequency) with filter parameters are given in Table 2.
ii. The beginning of coda amplitude starts at 2t s , where t s is the travel time of S -wave measured from the origin time of the earthquake (Rautian and Khalturin, 1978).
iii iv. Once the set of A(r, ω, t)/[K(r,x)] and the coda intervals are obtained beginning at 2ts, Q c could be easily estimated from the slope of the linear fit of equation (4), ln [ A(r, ω,t)/ [K(r,x)] vs. t, the lapse time.
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.

Results and Discussion
The Q c values are estimated filtering the coda waves of 9 waveforms in frequency band centered at 1, 1.  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 = Q 0 f n , where Q 0 is the quality factor at 1Hz and n is the frequency dependent coefficient. For 20 seconds coda window Q 0 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 .  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 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 .   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.

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.