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Study of source parameters of small to moderate and large earthquakes is important to understand the differences and similarities between dynamic ruptures of different earthquakes and clarifying the scaling relations. In the present study, we have characterized source parameters and presented new and revised empirical relationships between various source parameters for Kachchh region of Gujarat, India to facilitate to draw first-order conclusions regarding the trends in the region. We have studied total 202 aftershocks of shallow-focus (hypo central depth less than 40 km) and moderate magnitude recorded over the Kachchh region during January 2001 to December 2012 by different seismological observatories of India Meteorological Department. We have adopted the spectral technique for source parameter estimation, where S-wave displacement spectra are considered and applied Fast Fourier Transform (FFT) to compute displacement spectra. We have followed the Brune’s source model for our estimation and the estimated values of source parameters show close approximation to the global values. While derived empirical relations between different source parameters, they demonstrate direct or inverse proportion to linear or power scale. Interrelation between seismic moment, rupture parameters, corner frequency and radiated seismic energy can be summarized as
,
,
and
E_{R}
μ
M
_{0}
and E_{R}
μ
M_{w}
for our analysis. Stress drop distribution over the Kachchh region is very scattered and due to its peculiar behavior, it is difficult to derive its empirical relation with other source parameters. Sufficient accuracy on measuring source parameters like corner frequency stress drop, rupture dimensions, radiated seismic energy etc. helps to understand earthquake processes in the region. This is the first ever attempt to establish empirical relation between different source parameters for Kachchh region for longer aftershock sequence and they are useful to assess future earthquake potential over the region.

Hazard analysis involved with seismic activity is based on the estimation of the future earthquake potential in a given region. The future earthquake potential of a fault is evaluated from estimates of fault rupture parameters which are directly related to earthquake magnitude. Source parameters of small to moderate and large earthquakes are important for understanding the differences and similarities between dynamic ruptures of small and large earthquakes and clarifying the scaling relations. However, it is often difficult to accurately determine source parameters of small earthquakes because relatively high-frequency seismic waves excited by small earthquakes are easily scattered and attenuated along the path. Many researchers have published their studies on source parameters for different regions of the world [_{0}, moment magnitude M_{w}, stress drop Δσ, corner frequency f_{c }and rupture area A. We have interrelated some of these parameters on the basis of assumptions of similarity and scaling and spectral source theories.

For the present study, we have used total 202 aftershocks of magnitude M_{w} ≥ 4.0 (except some events of 3.5 < M_{w} < 4.0) for the aftershock sequence of January 26, 2001 Bhuj earthquake. The data include shallow-focus (hypo central depth less than 40 km), continental intraplate earthquakes from January 26, 2001 to December 31, 2012 recorded by different seismological observatories of India Meteorological Department (IMD). The epicentral locations of all earthquakes under the study are shown in

There are two approaches to describe source parameters. The simplified approach describes the seismic source by limited number of parameters such as the origin time and location, initial rupture, magnitude, intensity or acceleration of measured ground shaking and sometimes fault plane solution. These are easily obtainable parameters and provide quick information to the public and concern authorities. Nevertheless, they are fundamental for different research, they are not sufficient to describe the true nature and geometry or the energy release by the source. The second approach deals with the detailed analysis of a given event i.e., analyzing near and far-field waveforms and spectra of seismic waves. It provides important information on energy distribution at the source of the seismic event in the frequency domain. We have adopted the second approach i.e. the spectral technique for source parameter estimation and followed the Brune’s [

trace mode from the three-component seismogram. An example of the three-component seismogram recorded at station Bhuj for moderate earthquake of magnitude M_{w} ~ 5.5 is shown in

Theoretical displacement spectrum d(f) according to Brune [

where, M_{0} is the seismic moment (N.m), _{c} is corner frequency (Hz), ρ is density at the source and taken 2700 kg/m^{3}, V_{p,s} is seismic-wave velocity of the corresponding phase and they are 6.2 km/sec for P-wave and 3.6 km/sec for S-wave. In equation (1), the term G (r,h) represents geometrical spreading, for S- wave, it has been considered to depend on distance and depth. For distance r and depth h, geometrical spreading G(r,h) = 1/r (for r < 100 km) and G(r,h) = 1/(100 × r)^{1/2} [

where, k accounts for kappa, the near-surface high frequency attenuation with the constant kappa having a value of 0.025 for Kachchh region [^{th} event recorded at j^{th} station at a distance of r and for frequency f can be written as [

where, SO_{i} and SI_{j} are source and site spectral function respectively and P(r_{ij}, f) is the propagation path term which can be expressed as,

The term G(r_{ij},h) is geometrical spreading and similar to G(r, h) of Equation (1) as discussed earlier and V_{s} is assumed to be 3.6 km/sec based on a velocity model of the Kachchh region [_{c} study of the Kachchh region by Mandal et al. [

In the coda Q_{c} study, Mandal et al. [_{c} estimates obtained using Aki and Chouet’s [

In a homogeneous half-space, M_{0} can be determined from the spectra of seismic waves observed at the Earth’s surface by using the relationship given by Brune [

where, R is epicentral distance (km) and Ω_{0} is long period amplitude level (m-sec).

From the spectral parameters, other parameters like moment magnitude, source radius, stress drop and radiated seismic energy are derived using Equation (3) to Equation (8) as follows,

Moment magnitude (Hanks and Kanamori, [

Rupture radius and area,

where, r is rupture radius (km), A is rupture area (sq∙km) and f_{c} is corner frequency (Hz).

Stress drop (Kellis-Borok, [

Radiated seismic energy (Richter, [

The results of our analysis are summarized in

Parameter | Lower bound value | Upper bound value |
---|---|---|

Moment Magnitude―M_{w} | 3.5 | 5.7 |

Seismic Moment―M_{0} (N∙m) | 2.0 × 10^{14} | 6.3 × 10^{17} |

Corner Frequency―f_{c} (Hz) | 0.624 | 8.187 |

Stress Drop―Δσ (bars) | 68.4 | 299.8 |

Rupture Radius―r (km) | 0.168 | 2.100 |

Rupture Area―A (sq km) | 0.088 | 13.847 |

Radiated Seismic Energy―E_{R} (J) | 2.0 × 10^{10} | 1.8 × 10^{13} |

Mean of seismic moment,

Standard deviation of seismic moment,

We have followed the same procedure as mentioned above to calculate mean and standard deviation in corner frequency and rupture radius. Stress drop values are highly deviate from the mean hence we have derived percentage error for stress drop which can be defined as,

Mean values of corner frequency, seismic moment, rupture radius and stress drop are 2.886 Hz, 2.3 × 10^{16} N.m, 0.597 km and 200.8 bar respectively whereas standard deviation in estimation of corner frequency, seismic moment and rupture radius are 1.296 Hz, 0.67 N∙m and 0.361 km respectively. As discussed, we have calculated an average % error in the estimation of stress drop as stress drop values are discrete and they found to be 3%.

In the process, we have further derived empirical relations between the estimated source parameters, which are described individually in following subsequence. The empirical relations between source parameters we have derived in the present study can be grouped in three different categories, i.e., empirical relations for seismic moment, empirical relations for rupture parameters and empirical relations for radiated seismic energy.

Seismic moment relations with corner frequency and rupture area are shown in _{0} can be computed from the source spectra of body and surface waves or it is derived from a moment tensor solution [_{0} from the spectra of seismic waves observed at the Earth’s surface by relationship presented in the Equation (6). It can be observed from _{0 }is decreasing with increasing in corner frequency. From our analysis we have found that_{w} 6.7 Niigata, Japan earthquake and found a relation of _{w} 5.65, 2011 Virginia earthquake. Kanamori and Rivera [_{0 }µ A^{3/2} scaling with stress drop ranging from 1 bar to 1000 bar. In addition, result obtained for Kachchh region in this study is closer to the relation given by Purcaru and Berckhemer [

Rupture area to corner frequency relation is shown in

^{10} J to 2.0 × 10^{13} J for Kachchh region for the magnitude range from 3.5 < M_{w} < 5.7. It has been observed that each earthquake has unique characteristics even if they are of same category and of same region. The radiated seismic energy is an important parameter that represents the dynamic characteristic of the earthquake mechanics and helps to understand differences between earthquakes. From _{0 }and M_{w}. Choy and Boatwright [

Stress drop results are listed in the

Earthquakes with higher stress drops will have more intense ground motions. Large earthquake populations reveal strong variations in stress drop but little in the way of systematic behavior or dependence on seismic moment [

Faultwise stress drop distribution can be described as follow:

Events located near Gora Dungar fault show lower stress drop values compared to other faults of Kachchh region.

Gedi fault having moderate stress drop events.

Western part of widely known North Wagad fault exhibits moderate stress drop values at the same time eastern part exhibits that of higher values. Northwest of South Wagad fault can be characterized by higher stress drop values.

Only few events with high stress drop values are observed at Kachchh Mainland fault (KMF) and that is again at extreme eastern end of KMF. While North of KMF and South of KMF are confined with high stress drop events.

This difference in stress drop from one fault to another can be explained with fault geometry, frictional strength of faults and crustal brittleness. There are several possible reasons for variation in stress drop over the Kachchh region. The inferred high heat flow for Kachchh region can be attributed to the presence of mafic instructive at lower crustal depths in the region [

In the present study, the source parameters of crustal earthquake for moderate magnitude occurred in Kachchh seismic zone were estimated using Brune’s theory by estimating corner frequency and the low frequency asymptote from the spectral technique. The estimated source parameters are summarized in

Seismic moment and hence the moment magnitude are inversely proportional to corner frequency for Kachchh region and in accordance with Gutenberg-Richter relation [

Seismic moment to rupture area relation from our analysis is identical to the average relation suggested by Abe [

Estimated stress drop values are found to be very scattered. Stress-drop behavior also depends on the tectonic characteristics of the region. Moreover, the properties of the earth’s crust change from one to another location. Thus, stress drop observations cannot extrapolate from high stress drop regions to moderate or low stress drop regions and vice versa even for earthquakes of similar magnitudes. Lower stress drop values estimated for the events are recorded near the fault while higher stress drop values are found for events recorded far away from the fault irrespective of magnitude.

An important outcome of our study is that the full range of magnitude we have considered is having power scale with rupture size. Radiated seismic energy ranges from 10^{10} J to 10^{13} J and exhibits linear relation with M_{0 }and M_{w}.

Sufficient accuracy on measuring source parameters like stress drop, rupture dimensions and radiated seismic energy helps to understand earthquake source processes. It has considerable implications for studies of earthquake rupture physics and seismic hazards for large earthquake. By applying proper conditions and considering uncertainties in locations, we can increase our ability to estimate ground motion from large earthquakes. We conclude that even though, the quantitative prediction of earthquake initiation is an extremely complicated task, by integrating knowledge of the propagation of seismic waves and strength of seismic shaking, the state of stress in the crust and detailed slip inversions combining geodetic methods, earthquake mitigation can be achieved and it is the best possible practical approach to save the loss of lives and properties.

The authors are grateful to India Meteorological Department for providing valuable data and permission to publish this work. We would like to extend our sincere thanks to Scientist-In-Charge, CSIR Fourth Paradigm Institute (formerly C-MMACS) for providing the facilities. We are thankful to Mr. Sandeep Agarwal for his help to solve Seisan related issues.

Parul C.Trivedi,Imtiyaz A.Parvez, (2015) Characterization of Source Parameters and Some Empirical Relations between Them for Kachchh Region, Gujarat, India: Implication of January 26, 2001 Bhuj Earthquake and Its Aftershock Sequence. International Journal of Geosciences,06,1127-1139. doi: 10.4236/ijg.2015.610088