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The Gorkha Earthquake that occurred on 25
^{th} April 2015 was a long anticipated, low angle thrust-faulting shallow event in Central Nepal that devastated the mountainous southern rim of the High Himalayan range. The earthquake was felt throughout central and eastern Nepal, much of the Ganges River plain in northern India, and northwestern Bangladesh, as well as in the southern parts of the Plateau of Tibet and western Bhutan. Two large aftershocks, with magnitudes 6.6 and 6.7, occurred in the region within one day of the main event, and several dozen smaller aftershocks occurred in the region during the succeeding days. In this study, we have analyzed the 350 aftershocks of the 2015 Gorkha Earthquake of M
_{w} 7.8 to understand the spatial and temporal distribution of b-value and the fractal correlation dimension. The b-value is found to be 0.833 ± 0.035 from the Gutenberg-Richter relation by the least squares method and 0.95 ± 0.05 by the maximum likelihood method, indicating high stress bearing source zone. The spatial and temporal correlation dimension is estimated to be 1.07 ± 0.028 and 0.395 ± 0.0027 respectively. Spatial correlation dimension suggests a heterogeneous distribution of earthquake epicenters over a linear structure in space, while the temporal correlation dimension suggests clustering of aftershock activity in the time domain. The spatial variation of the b-value reveals that the b-value is high in the vicinity of the mainshock which is due to the sudden release of stress energy in the form of seismic waves. The spatial distribution of correlation dimension further confirms a linear source in the source zone as it varies from 0.8-1.0 in most of the region. We have also studied the temporal variation of b-value and correlation dimension that shows positive correlation for about first 15 days, then a negative correlation for next 45 days and after that, a positive correlation. The positive correlation suggests that the probability of large magnitude earthquakes decreases in response to increased fragmentation of the fault zone. The negative correlation means that there is a considerable probability of occurrences of large magnitude earthquakes, indicating stress release along the faults of a larger surface area
[1]. The correlation coefficient between b-value and the correlation dimension is estimated to be 0.26, which shows that there is no significant relation between them.

An earthquake is a sudden violent shaking of the surface of the earth, resulting from the sudden release of energy within Earth’s crust in the form of seismic waves. An aftershock is an earthquake that occurs after the mainshock in the same region but generally of a smaller magnitude. Aftershocks are the results of the adjustment of crust around the displaced fault plane which is caused by the main shock. The earthquakes occur along the fault zones or material heterogeneities present in the crust. The distribution of faults or fractures in the crust possesses a fractal structure [

The Gutenberg-Richter law ( log 10 N = a − b M , where M is the magnitude and N is the number of earthquakes of magnitude greater than or equal to M) relates the magnitude of the earthquake with the total number of earthquakes for a given region and time period of at least that magnitude [

There are different types of fractal dimension depending on the various methods of measurement, namely, Euclidian dimension (d), similarity dimension (D_{s}), Hausdorff dimension (D_{H}), box-counting dimension (D_{0}), information dimension (D_{1}) and correlation dimension (D_{2}). The box-counting dimension (D_{0}), also known as the capacity dimension, measures the space-filling properties of a fracture with respect to changes in grid scale [_{2}) measures the degree of clustering of a set of points [_{2} varies from 0 - 2 [_{2} = 1 indicates a line source or linear structure and when all events are homogeneously distributed over a two-dimensional embedding space, D_{2} = 2 [

On 25^{th} April 2015, a large shallow earthquake of magnitude 7.8 occurred in the Gorkha region of central Nepal. The aim of this research is to find the b-value and the fractal correlation dimension in space and time for the aftershocks of the 2015 Nepal Earthquake and to understand the correlation between both of them. This study helps to understand the variation in stress accumulation in different regions of Nepal. For this, we have analyzed the 350 aftershocks of magnitude greater than or equal to 4.0, compiled by the United States Geological Survey (USGS), occurred in the period 25^{th} April 2015 to 28^{th} February 2017. The spatial and temporal fractal correlation dimension is estimated using the [_{2} after the occurrence of the 2015 Gorkha earthquake of M_{w} 7.8 has been examined, and thus the correlation between the two of them is analyzed. We have also studied spatial variation in b and D_{2}.

Heim and Gansser [

1) Terai Zone (Gangetic Plain)

2) Sub-Himalayan Zone (Siwaliks)

3) Lesser Himalayan Zone

4) Higher Himalayan Zone (Greater Himalaya)

5) Tibetan-Tethys Zone

In between the Lesser Himalaya and the Greater Himalaya, there is a Main Central Thrust Zone, which is bounded by Main Central Thrust (MCT) I and Main Central Thrust (MCT) II [

Due to the presence of the Main Frontal Thrust (MFT), Main Boundary Thrust (MBT) and Main Central Thrust (MCT), Nepal had experienced several damaging earthquakes. The earthquakes of 1833, 1869 and 1988 occurred along the same segment of the Main Himalayan Thrust (MHT) as that of the 2015 earthquake (_{s}) 8.0. The 1988 Earthquake that occurred in Nepal was of magnitude (M_{w}) 6.9 and caused the death of 709 persons while thousands got injured.

On 25th April 2015, a severe earthquake struck near the city of Kathmandu in central Nepal. It killed about 9000 people, and thousands were injured. At 6:11 am (UTC time), the initial shock of magnitude (Mw) 7.8 was registered. Its epicenter was located about 34 Km east-southeast of Lamjung and 77 km northwest of Kathmandu. The depth of the focus was 15 km and the cause of the earthquake was the thrust faulting in the Indus-Yarlung suture zone. This earthquake released the seismic energy accumulated because of the compressional pressure between the Eurasian plate and the Indo-Australian plate. As the Indian plate subducts the Eurasian plate at an average rate of 4 - 5 cm annually, the height of the Himalayan mountain is increased by more than 1 cm every year.

A total of 361 aftershocks of the 2015 Nepal earthquake were recorded on the United States Geological Survey (USGS) Global Seismic Network (GSN). The geographical area considered for the aftershocks was from 26.096˚ to 30.581˚ in latitude and from 79.849˚ to 88.330˚ in longitude and the period considered was from 25th April 2015 to 28th February 2017. There are however, we analyzed only 350 aftershocks which were having magnitudes greater than or equal to 4.0.

First, we have made a magnitude histogram to show the decay of the frequency of occurrences of magnitude ≥ 4 aftershocks (

The b-value of these aftershocks is estimated using the Guttenberg and Richter (1944) relation:

log 10 N = a − b M (1)

where N is the number of earthquakes of magnitude greater than or equal to M, “a” and “b” are constants. Seismicity level is indicated by the constant “a”, and the amount of stress accumulated in a region is indicated by the constant “b” known as b-value. These constants are determined by applying linear regression over the log_{10}N Vs M, using the method of least squares (

A higher b-value means that there is less number of larger earthquakes as compared to smaller earthquakes, whereas a lower b-value means that there are more numbers of larger earthquakes as compared to smaller ones. Thus, a low b-value indicates higher stress in a region [

[

b = ( log 10 e ) / ( M a − M c ) (2)

where M_{a} is the average magnitude of all events, and M_{c} is the minimum magnitude of all the events. But since, the number of aftershocks is quite small, the b-value calculated by the maximum likelihood method is not a good estimate.

The b-value is estimated to be 0.833 ± 0.035 by the method of least squares whereas 0.95 ± 0.05 by the maximum likelihood method.

To study the spatial variation in the b-value, we have chosen a small geographical area (84.3˚ to 87˚ longitude and 27.3˚ to 28.5˚ latitude). This area is divided into 36 grids of 0.3˚ × 0.3˚ and b-value is found for each of the grids having a minimum occurrence of 10 events. Mapping of b-value is done by interpolating the b-values (

Spatial distribution of earthquake epicenters can be represented by a self-similar fractal structure, and the scaling parameter is known as the fractal dimension D [

C p ( r ) = ∑ i = 1 k H ( r − ‖ x p − x i ‖ ) (3)

where k is the total number of events in the catalogue, r is the radial distance to

be considered, x_{i} is the location of i^{th} earthquake event, and thus x_{p} is the location of the point at which correlation sum is evaluated. The double bars (||) represent the scalar distance which is to be considered between two events. H is the Heaviside function that is 0 when the argument is negative and is 1 otherwise. Therefore, correlation sum simply tells the number of points within a distance “r” from the point x_{p}. The correlation integral is related to the standard correlation function as [

C ( r ) ~ r D (4)

Thus, the following fractal model can be used to characterize the spatial distribution of earthquakes (using Equation (4)):

log 10 C ( r ) = C + D log 10 r (5)

where C and D are constants and D is the fractal dimension, more strictly, correlation dimension. For the estimation of the correlation fractal dimension (D_{2}), the generalized correlation integral C(r) is used [

D 2 = lim r → 0 log 10 C ( r ) log 10 r (6)

where C(r) is the correlation integral and r is the radius of sphere of investigation. The correlation integral is given by:

C ( r ) = lim N → ∞ 1 N 2 ∑ i = 1 N ∑ j = 1 N H ( r − ‖ x i − x j ‖ ) (7)

The estimator of correlation integral is correlation sum and is calculated as shown:

C ( r ) = 1 N 2 ∑ i = 1 N ∑ j = 1 N H ( r − ‖ x i − x j ‖ ) (8)

where N is the total number of earthquakes considered, x are the coordinates of the epicenters and H is the Heaviside step function.

H ( x ) = { 0 for x < 0 1 otherwise (9)

Thus, the function H evaluates the number of points which have an inter-event distance less than or equal to r and C(r) evaluates the probability that a distance will separate two earthquake events less than r.

In case of an infinite fractal distribution, the log 10 C ( r ) Vs log 10 r plot is a straight line whose gradient is the fractal correlation dimension. It has been found that for large values of “r”, the gradient is quite low. On the other hand, for small values of r, the gradient is artificially high. These two conditions are known as “saturation” and “depopulation” respectively. So the fractal dimension is estimated by fitting a straight line to a subjectively chosen straight part of the curve. If r_{n} and r_{s} are the distances of depopulation and saturation respectively, then they can be determined by the following formulae:

r n = R [ 1 N ] 1 / d , r s = R 2 ( d + 1 ) (10)

where R is the maximum inter-event distance, N is the total number of events and d is the embedding dimension. We can also safely start the scaling range from even r_{n}/3 [

| X i − X j | = cos − 1 ( cos θ i cos θ j + sin θ i sin θ j cos ( ϕ i − ϕ j ) ) (11)

where θ i and θ j are the latitude of the i^{th} and j^{th} event and ϕ i and ϕ j are the longitude of the i^{th} and j^{th} event respectively whereas for the estimation of the temporal correlation dimension ( D 2 t ), inter-event time between two aftershocks is computed as:

τ = t i − t j (12)

where t_{i} and t_{j} are the origin time of the i^{th} and j^{th} event respectively.

In this case, R is estimated to be 5.085˚ while r_{n} and r_{s} are found to be 0.272˚ and 0.84˚. Similarly the maximum inter-event time (T) is 730.37 days and t_{n} and t_{s} values considered are 2.087 days and 182.6 days respectively. For calculating R and T, the embedding dimensions are taken as 2 and 1 respectively.

By estimating the slopes of the ( log 10 C ( r ) Vs log 10 r ) and ( log 10 C ( τ ) Vs log 10 τ ) in their scaling range (

In order to study the spatial variation of the fractal correlation dimension, we have chosen the same geographical area (84.3˚ to 87˚ longitude and 27.3˚ to 28.5˚ latitude) in which the spatial variation of the b-value was analyzed. Here also, the area is divided into 36 grids of 0.3˚ × 0.3˚ and D 2 s is found for each of

the grids having a minimum occurrence of 10 events. Mapping of spatial correlation dimension is done, in the similar manner as it was done for b-values, by interpolating the D 2 s -values (

We have divided the whole earthquake catalogue, sorted by date and time, into 25 subsets between April 2015 and February 2017 and each subset consists of 14 aftershocks.

The b-value and spatial fractal correlation dimension is estimated for each subset. The time change of b-value and fractal dimension is plotted (

The b-value is proportional to the fractal dimension and is related to each other by the following equation [

D = 2 b (13)

Though Aki’s relation suggests a positive correlation between the fractal dimension and the b-value, some studies have shown a negative correlation between the two [

To see the correlation between the b-value and the fractal dimension ( D 2 s ), the fractal dimension is plotted against the b-value for 25 subsets (

The correlation coefficient is 0.26 which indicates that the correlation is not very significant. The mean value of the

For the first 10 days the correlation is positive (

In this research paper, we have analyzed the 2015 Nepal Earthquake Aftershock activity. A total of 350 aftershocks with magnitude greater than or equal to 4.0 have been considered for the analysis. The geographical region for the analysis is from 26.096˚ to 30.581˚ in latitude and from 79.849˚ to 88.330˚ in longitude. The aftershock data encompasses a period from 25^{th} April, 2015 to 28^{th} February, 2017. The b-value is computed by the maximum likelihood method as well as the least squares method. But since the number of aftershocks is quite less, the maximum likelihood method is not very reliable.

The b-value is estimated to be 0.833 ± 0.035 by least squares method and 0.95 ± 0.05 by maximum likelihood method. The b-value is identical within uncertainties to the b-value 0.83 ± 0.05 estimated from 2102 events of magnitude greater than or equal to 2.4, between 1995 and 2015, in the same area [

The spatial and temporal correlation dimensions have been estimated using the correlation integral method. For the scaling range, 0.272˚ ≤ r ≤ 0.84˚, the slope of the regression line is

The b-value and

The variation in the b-value depends on the levels of stress accumulated in the source zone or the degree of heterogeneity of the source medium [

The author S.M. would like to thank the Science Academies’ Summer Research Fellowship Programme, which is jointly conducted by IASc (Bengaluru), INSA (New Delhi) and NASI (Allahabad), for giving an opportunity to work as a Summer Research Fellow. This work is a part of the Summer Research performed by S.M. at the CSIR Fourth Paradigm Institute, Bengaluru.

The authors declare no conflicts of interest regarding the publication of this paper.

Minocha, S. and Parvez, I.A. (2020) Self-Organized Fractal Seismicity and b-Value of Aftershocks of the 2015 Gorkha Earthquake, Nepal. International Journal of Geosciences, 11, 562-579. https://doi.org/10.4236/ijg.2020.118030