_{1}

^{*}

In the present study, the temporal behavior of 2001 Bhuj aftershock sequence in Kachchh region of western peninsular India is studied by the modified Omori law. The Omori law parameters
*p*,
*c* and
*K* are determined with the standard errors by the maximum likelihood estimates using
*ZMAP* algorithm in
*MatLab* environment. The entire aftershock sequence is analyzed by diving it into three separate series with respect to time to weigh up the bigger earthquake of magnitude
*M* 5.7 occurring on March 7, 2006 at Gedi fault. This study helps to understand the cumulative effect of the aftershocks generated by this bigger earthquake of the mainshock sequence. The results of this analysis are discussed with other studies of the different earthquake sequence for the different parts of the world and suggest that all the three series of Bhuj aftershock sequence follow the Omori relation. Values of parameter p vary significantly from series 1 to series 3,
* i.e.*,
*p*-value varies significantly with time. Similarly, other two Omori law parameters
*K* and
*c* are also found to change significantly with time. These parameters are useful to describe temporal behavior of aftershocks and to forecast aftershock activity in time domain. Aftershock decay rate provides insight into stress release processes after the mainshock, thus helping to understand the heterogeneity of the fault zone properties and evaluate time-dependent seismic hazard analysis over the region.

Earthquakes are the deadliest natural hazards amongst all other natural hazards that cause lots of casualties and damage of property. Kachchh region of western India is one of the most earthquake prone zones of the world and witnessed three large earthquakes including 2001 Bhuj earthquake of M_{w} 7.7 at the beginning of the 21^{st} century. Many researchers have studied this devastation earthquake and its impact by different approaches [

In earthquake physics, the magnitude distribution of Gutenberg-Richter relationship reflects fundamental properties of how earthquakes grow and stop and the time-dependence of earthquake occurrence in strict way is generally ignored in standard seismic hazard assessment, though the time-dependent earthquake clustering is well known. In seismological studies, the Omori law, proposed by Omori in 1894 is one of the basic empirical laws [

In the present study, the aftershock sequence of 2001 Bhuj earthquake of M_{w} 7.7 for Kachchh region of western peninsular India is studied by the modified Omori law. The entire aftershock sequence from January 26, 2001 to the December 31, 2014 is categorized in three series and parameter p along with c and K is determined for all three series. As stated earlier, many researchers have addressed the 2001 Bhuj earthquake (M_{w} 7.7) with different scientific approaches but the aftershock activity over the Kachchh region is less attended. Here, an attempt is made to examine Omori’s decay law for long ongoing aftershock sequence of 2001 Bhuj earthquake (M_{w} 7.7). The results presented in this study are very useful to understand temporal behavior of aftershocks and to forecast aftershock activity in the future. It has significant implications on seismic hazard analysis since the study gives idea on stress release processes after the mainshock.

In the present study, the Omori’s law is tested against earthquake catalogs for the aftershock sequences of the January 26, 2001 Bhuj earthquake. Catalogue of India Meteorological Department (IMD) and Institute of Seismological Research (ISR) are used and verified with catalogue of International Seismic Centre (ISC), United States Geological Survey (USGS) and Geological Survey of India (GSI). In this study, the aftershocks from next hour of large earthquake on January 26, 2001; 08:46 hours local time to December 31, 2014; 24:00 hours local time with magnitude of M ≥ 1.5 are used. Most importantly, records of digital seismographs installed immediately after the mainshock by India Meteorological Department (IMD) near mainshock epicenter and adjoining area within Kachchh region are used extensively in this study. Afterwards, on the establishment of ISR seismological observatories, earthquake data recorded by the ISR network is included to weigh lower magnitude aftershocks and to achieve high quality completeness of catalogue. Total 11,334 aftershocks of M ≥ 1.5 are used in the study. A general observation regarding the Omori law decay parameter shows no correlation with cutoff magnitude. Initially, three different calculations performed on the basis of diurnal, weekly and monthly distribution of events as the period immediate after the mainshock is considered the most intense period of seismic activity. Later on month wise and year wise frequency of aftershocks is analyzed. It is important to note, however, that the main shock of larger size of M_{w} 7.7 is expected to produce aftershocks for years, even decades and sometimes an aftershock produced at any time may be large as each aftershock in turn that generates its own aftershocks which we can call secondary aftershocks and thus the long-term aftershock risk should be specially emphasized.

The Omori law (1894) states that the main earthquake is immediately followed by a sequence of aftershocks whose frequency of occurrence decays at the rate proportional to t^{−p}, where p ~ 1. The simplest version of Omori’s Law can be written as,

However, in most of scientific studies, more commonly, a modified version of this law is used. In fact, it resembles more closely the original, empirically-derived model developed by Omori in the 1890s [

The modified version of Omori can be given as follow:

In this equation, N is the number of events following the mainshock, dN/dt is the occurrence rate of aftershocks with magnitudes greater than a lower cutoff M_{c}, t is time since a mainshock while, K, c and p are empirical constants. Shcherbakov et al. (2004) proposed a generalized Omori’s law for aftershocks combining Gutenberg-Richter b-value [_{given} as a function of time t, b-value, mainshock magnitude M_{mainshock} and the difference of mainshock-maximum aftershock magnitude Δm as follow,

Of the Omori law parameters, we can immediately observe that K may be related to aftershock productivity, c to the period of time after the mainshock during which the aftershock rate is roughly constant and p to the power law decay in the rate of aftershocks. This is helpful when it is necessary to calculate the number of aftershocks in a given time interval from the Omori law. The significance of equation 2 and equation 3 is the constant that defines the rate of decay i.e., the p-value by graphing a simple model of Omori’s Law. This allows us to see how the shape of an exponential decay curve looks like and how the shape changes with the value of p―the rate of decay. Actually, c is no longer a constant but scales with a lower magnitude cutoff and a mainshock magnitude. In this study, the entire aftershock sequence is divided into three series. First series, initial aftershocks from next hour to 365 days i.e. from January 26, 2001 to January 25, 2002; second series is from January 26, 2001 to March 7, 2006 up to the occurrence of earthquake of M 5.7; the maximum magnitude aftershock of the entire sequence which in turn again generated aftershocks of its own and finally for the entire aftershock sequence i.e. January 26, 2001 to December 31, 2014. The parameters in the modified Omori formula are estimated accurately by the maximum likelihood method.

As discussed earlier, the entire aftershock sequence is divided in three series; series 1 for first 365-days, includes aftershocks from the next hour of the occurrence of the mainshock, i.e. January 26, 2001 to January 25, 2002;

series 2 includes aftershocks from January 26, 2001 to the occurrence of maximum magnitude aftershock of the entire sequence M ~ 5.7 on March 7, 2006 and series 3 includes the aftershocks for the entire aftershock sequence i.e. from January 26, 2001 to December 31, 2014. The decay of aftershocks with Omori law is studied for all these three series and displayed in Figures 2-4 along with values of three parameters p, c and K.

It can be observed that _{w} 7.6 Tonga earthquake and the 631 km deep M_{w} 8.3 Bolivia earthquake occurred in 1994, are earthquakes those produced aftershocks [

have shown very low or no aftershock productivity [_{w} 7.7 is the example of large SCR earthquake and it is of shallow focused and the aftershock activity is still ongoing over the Kachchh region. A general observation regarding the value of p is that it shows no correlation with either the magnitude of the mainshock or the cutoff magnitude M_{c} [

Another two components of Omori law K and c provide important insight into aftershocks behavior. The K-value depends on the total number of events in the sequence and reflects the earliest part of the sequence and accounts for the observed fact that the earliest aftershocks do not follow a steady decay rate rather their rate increases in the first minutes to hours, then begins to decrease [

The analysis of aftershock sequence of the 2001 Bhuj earthquake (M_{w} 7.7) suggests that the seismic activity associated with some moderate events in 2006 had an influence on the future seismicity in the area, in particular on the temporal distribution of p-values observed for the aftershocks of 2001 Bhuj earthquake. The initial decay rate after the 2001 Bhuj mainshock was higher and later the decay rate is decreased significantly. Even though till the date the decay curve for 2001 Bhuj aftershock sequence is not found parallel to the time axis which suggests that more aftershocks are expected over the region and it may continue for a long. Using the temporal distribution of the aftershock sequence and the modified Omori’s law, the rate of aftershock occurrence can be forecasted and we can forecast the aftershock activity in the next 100 days, 365 days, 730 days and so on and compare it to the actually observed rates. The 2001 Bhuj aftershock sequence is well modeled by the modified Omori law and the p-value is found to be decreased from the initial to the later time of the sequence. The modified Omori’s law is the best option to describe aftershock activity and decay rates amongst exponential and other functions. The close examination of the decay curves presented in this study indicates that though the general decay trend is exponential, the plot of actual aftershocks occurring after 2001 Bhuj mainshock is distributed in a sinusoidal pattern which suggests the stress release mechanism of tectonics may be in a cyclical route. The results presented in this study are consistent with the existing knowledge about decay properties of aftershocks for different earthquakes recorded in different regions of the world.

The author is grateful to India Meteorological Department for providing valuable data and permission to publish the work and to Institute of Seismological Research for catalogue data. I am thankful to Dr. Jayanta Sarkar, Director, IMD, Ahmedabad for his kind co-operation.

Parul C.Trivedi, (2015) Application of Omori’s Decay Law to the 2001 Bhuj Aftershock Sequence for Kachchh Region of Western India. Open Journal of Earthquake Research,04,94-101. doi: 10.4236/ojer.2015.43009