Self-Organized Fractal Seismicity and b-Value of Aftershocks of the 2015 Gorkha Earthquake, Nepal

The Gorkha Earthquake that occurred on 25 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 Mw 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 How to cite 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 Received: June 11, 2020 Accepted: August 22, 2020 Published: August 25, 2020 Copyright © 2020 by author(s) and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access S. Minocha, I. A. Parvez DOI: 10.4236/ijg.2020.118030 563 International Journal of Geosciences 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.


Introduction
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 [2]. These fractal structures in time, space and magnitude dimensions are observed not only in the distribution of faults but also in the seismicity of earthquakes [3]. In 1993, Kagan suggested that earthquakes do not occur on a single surface, but rather on a fractal structure of many closely related faults [4]. Also, the number of aftershocks decay follows a power law [5], suggesting a scaling property between the main shock and its aftershocks using  [6]. The relationship between the magnitude and the total released seismic energy ( ( ) 2 3 b N E E − = , where E is the total released seismic energy) also confirms the fractal nature of earthquakes. The Gutenberg-Richter law ( 10 log N a bM = − , 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 [7]. The constant b is known as the b-value, which represents stress regime and material heterogeneity of the earth's crust. The b-value is usually close to 1 over long time periods and large regions but statistically significant variations have been found for earthquake data for a short period and small geographical regions. [8] has derived a simple relation between the fractal dimension of the earthquake seismicity (D) and the b-value (D = 2b) which shows a positive correlation between the two. However, in contrary to this, a negative correlation had been noticed between the b-value and the fractal dimension by [9] for the earthquakes of the Tohoku 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 [9]. On the other hand, the correlation dimension (D 2 ) measures the degree of clustering of a set of points [12]. The correlation dimension has been found for both earthquake epicenters and hypocenters in numerous studies [13]. The correlation dimension is usually found by using the correlation integral method, as it is the simplest and most naïve way to estimate D [14]. In general, D 2 varies from 0 -2 [15]. When it is 0, all events cluster into a single point; D 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 [16]. The spatial and temporal fractal correlation dimension is estimated using the [14] algorithm while the b-value is estimated using the least squares method. The temporal variation of b and D 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 . Apart from the high elevations of the Himalayan mountain range (>5000 m), it also thickened the Indian crust to its present thickness of 70 Km [21].   Figure 5). This segment is MCT and is adjacent to segments northwest that ruptured in 1505 [25] and to the southeast that ruptured in the 1934 Nepal-Bihar earthquake [26]. The 1833 Earthquake that took place in Kathmandu was of magnitude (M 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.

Estimation of b-Value
The b-value of these aftershocks is estimated using the Guttenberg and Richter (1944) relation: 10 log N a bM = − (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 ( Figure 6). International Journal of Geosciences  It also shows that the aftershock activity decayed significantly with time.
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 [27]. [28] proposed the following relation to find b-value by the maximum likelihood method: 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.

Depth and Spatial Variation of the b-Value
To study the spatial variation in the b-value, we have chosen a small geographi-

Estimation of Spatial and Temporal Fractal Correlation Dimension
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 [2]. Though many fractal models have been proposed for the distribution of epicenters, the correlation dimension is the easiest and the efficient method to estimate the fractal dimension. The correlation dimension is calculated using the correlation integral algorithm proposed by [14]. The fractal correlation is written as [30]: where k is the total number of events in the catalogue, r is the radial distance to  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 [12]: Thus, the following fractal model can be used to characterize the spatial distribution of earthquakes (using Equation (4)): ( ) where C(r) is the correlation integral and r is the radius of sphere of investigation. The correlation integral is given by: The estimator of correlation integral is correlation sum and is calculated as shown: where N is the total number of earthquakes considered, x are the coordinates of the epicenters and H is the Heaviside step function. International Journal of Geosciences ter-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 ( ) 10 log C r Vs 10 log 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: 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 [31]. For estimating the spatial correlation dimension ( 2 s D ), the angular distance between two events is calculated by using a spherical triangle [9].
( ) ( ) 1 cos cos cos sin sin cos where i θ and j θ are the latitude of the i th and j th event and i φ and j φ are the longitude of the i th and j th event respectively whereas for the estimation of the

Spatial Variation of the Fractal Correlation Dimension
In order to study the spatial variation of the fractal correlation dimension, we

Temporal Variation in the b-Value and the Correlation Dimension
We have divided the whole earthquake catalogue, sorted by date and time, into The correlation coefficient is 0.26 which indicates that the correlation is not very significant. The mean value of the 2 s D b is estimated to be 1.9 which is in support with the Aki's relation. But it is statistically not a correct method to find the relation between any two quantities whose uncertainties vary and are not constant. Also the correlation coefficient is found to be decreasing as the number of subsets considered increases. It means that the correlation is significant for a shorter duration of aftershock activities just after the mainshock.
For the first 10 days the correlation is positive (Figure 11

Discussion and Conclusions
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 [35]. A low b-value suggests that large amount of stress accumulates in this region. An alternative view concerning the b variation before a major earthquake results from natural time analysis, which is a recent procedure to analyze complex time series such as seismicity and other complex systems like heart rate fluctuations displaying 1/f noise and fractal dynamics [36] [37]. Such an analysis of seismicity reveals [38] that the decrease of the b-value reflects an increase of the order pa-