4. Attenuation parameters for regional earthquakes in the northern Indian Ocean –
4.3 Regression analysis for Q value
The following well-established attenuation model (Atkinson and Mereu, 1992) is fitted with observed Fourier amplitudes by using the multiple linear regression technique. Regression analysis was undertaken in MS Excel. Note that, local site amplification term has been omitted since all recordings are on rock sites.
[ ( )]A fx i j [ ( )] . . ( )S f G A fi n (4.1)
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Where, [ ( )]A fx i j is the acceleration amplitude of Fourier spectra at jth station due to ith event for a given frequency f. [ ( )]S f i is the acceleration amplitude of Fourier spectra of ith event at the source for a given frequency f. G and ( )A fn are, respectively, geometric attenuation factor and anelastic whole path attenuation factor, having with following relations;
0
0
0
1.5
1.5 2.5
1.5
2.5 2.5
1.5
G R for R D
RR
G for D R D
D
R D
G for R D
D R
(4.2)
( ) exp
A fn fR Q (4.3)
Here, D, R, f, Q and are crustal thickness, epicentral distance, wave frequency, wave transmission quality factor, and shear wave velocity at the source region, respectively. R0, the reference distance, is equal to 1 km. Geometric damping or spreading parameter given in equation (4.2) for distant earthquakes accounts for attenuation of seismic waves at R-1 rate up to 1.5D distance due to spherical spreading of direct waves near the source area, zero attenuation between 1.5D and 2.5D due to compensatory effect of post-critical reflections and refractions from Moho and Conrad discontinuities and R-0.5 rate cylindrical attenuation after 2.5D due to multiple reflection and refraction of body waves dominating Lg phase in the shear window of the trace (Lam et al, 2000a; 2000b; 2000c). However, there is a general agreement that, unlike in continental crusts, Lg is not effective or totally absent to traverse in a thinner oceanic crust due to incompatibility associated with the wave field and seismic structure which governs physical features such as thickness and composition, i.e., presence of sediments (Knopoff et al, 1979; Kennett, 1985; 1986; Zhang and Lay, 1995). In light of this argument, it is refrained to relate the estimated Q value in this study as Lg Q, since all the dominant wave paths in the region are through a thinner oceanic crust spanning thousands of kilometres in size. Empirical studies based on recorded data analysis generally indicate fixed distance ranges for the geometric attenuation function, rather than distance ranges based on crustal thickness as denoted above. For instance, Atkinson (2004b) derives 70 and 140 km as best fitting hinge points for the geometric attenuation of southeastern Canadian region. Perhaps the values may show a reasonable consistency with the average crustal thickness of the region (40-50 km) especially for shallow crustal earthquakes. However, the applicability of equation (4.2) to account for geometric damping has been verified in numerous intraplate crustal regions (Lam et al, 2000b;
2009; Wilson and Lam, 2003). In some cases more rapid attenuation instead of R-1 near the source region (Atkinson, 2004b; Sonley and Atkinson, 2006) and negative attenuation instead of
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zero attenuation (Atkinson, 2004b) have been reported for stable intraplate regions such as southeastern Canada. In the present study, spreading parameters are retained the same as denoted in equation (4.2) to be consistent with other typical stable regions like Australia, Hong Kong and Singapore. Furthermore, uncertainties exist about the exact spreading rate of near- source attenuation for the subject region owing to lack of near-field records and thereby the rate accounted in equation (4.2) is more relied for the study. However, please note that a more reliable form of G for the northern Indian Ocean region is later derived using a separate spectral analysis which is described in Chapter 5. Crustal thicknesses, D values, near the source areas in the oceanic crust in the subject region are determined using the global crustal model called CRUST2.0 in which average crustal thicknesses all over the world are compiled to represent in 20 x 20 tiles. A value of 10 km is inferred for most of the parts in northern Indian Oceanic crust by referring to the data from CRUST2.0 and this value is also consistent with the hypocentral depth values given in catalogues in the database.
Equation (4.3) denotes attenuation of waves as a result of energy dissipation along the whole travel path during wave propagation through the rock medium. This energy dissipation entails heating of the heterogeneous medium and rearrangement/dislocation of particles during vibration of the medium, which are considered as permanent losses of energy. Moreover, whole path attenuation or seismic absorption depends on wave frequency in a manner in which high frequency waves are diminishing more rapidly than low frequency waves and hence the decaying has an exponential form, as noted in equation (4.3). The reason may be explained by comparing the situation with frequency dependent amplitude decaying nature of wave motion in an elastic medium due to viscous damping, in which decaying rate increases with the number of wave cycles in a unit length (frequency). Importantly, whole path attenuation can be dominant in distance earthquakes in which the amount of attenuation largely depends on the crustal quality of the travelled medium. If quality of the rock is high, the wave propagation is good and vice versa. Hence, the wave transmission quality of the rock, parameterized as Q (or, Q0 equals to Q at 1 Hz frequency), is a key parameter to be estimated correctly for long-distant events i.e., for regions like Sri Lanka.
[ ( )]S f i, which denotes FAS at the source for a given frequency, depends significantly on the energy release i.e., seismic moment/magnitude of the event and the typical stress drop level (
) of the subject region which can generally be considered to remain independent of the magnitude of the event, except for smaller magnitudes (Mw4.3) in which has shown an increasing trend with the magnitude (Atkinson, 2004b). Furthermore, stable continental regions have shown comparably higher stress drop values associated with hard rock conditions, than that of active regions located close to plate boundaries or subduction zones (Chen and Atkinson,
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2002; Castro et al, 1990). During the regression analysis the magnitude dependency (which comes through the source term) of observed FAS at the recorded station has been nullified by employing an assumed second order polynomial relationship between the source term and the magnitude which is shown in equation (4.4).
1 2 2 3
log[ ( )]S f iC M( 4) C M( 4) C (4.4)
where M is the magnitude in mb and C1 to C3 are constants to be determined from the regression.
Equation (4.4) decouples magnitude dependency of the regression equation and hence enables to perform the analysis for the whole dataset simultaneously over the entire magnitude range without the need to calculate for each magnitude separately (Joyner and Boore, 1981; Atkinson, 2004b). Besides, this decoupling technique is advantageous, since it minimizes errors in regression coefficients which can arise due to magnitude uncertainties associated especially with newer events, and it also curtails the total number of analysis steps of the regression process. Substituting terms of equations (4.2), (4.3) and (4.4) in (4.1) and taking logarithms give the following attenuation equation for regressions;
1 2 2 3 log
log[ ( )] ( 4) ( 4) log[1.5 ] 0.5log[ ] [ ]
x ij 2.5R f e
A f C M C M C D R
D Q
(4.5)