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Spatial Statistical Technique in relating earthquake epicentres with structural features

Standard deviation ellipse
When the set of point locations comes from a particular geographic phenomenon that has a directional bias, standard deviation ellipse is used instead of standard distance circles. Similarly, the geological structural features control earthquake epicenters, thereby showing a directional bias in their distribution. Under these circumstances the standard distance circle will not be able to reveal the directional bias of the process (Lee and Wong 2000). The deviation along the major axis will be the direction of maximum spread of the points and the minor axis will be the direction of minimum spread of the points. 

Methodology
The methodology adopted in this study can be described in the following steps. All spatial statistical analysis were performed using Arc View (AV) scripts provided by Lee and Wong (2000). Rose diagram were plotted using Geo Tools (Ver. 1.0) Extension developed by T. Thatcher (available at http://www.dtmgis.com/GeoTools.htm):

• The map of Himachal Pradesh and its surrounding areas is digitised and the epicenters of the recorded earthquakes of more than 4 magnitudes in the region were plotted (total 253 events). From the geological map of the Himachal Pradesh the faults and thrusts (after Saraf, et. al. 2002) are digitised and taken for analysis. (Source: USGS Earthquake epicenter data with magnitude³ 4 occurring between 1839-1994).
  
• Various spatial statistical operations were performed like the calculation of mean center, weighted mean center, median center, standard distance, standard deviation ellipse etc. on the epicenter data. The horizontal location accuracy of earthquake events has been accepted assuming to be the best made available by USGS.
  
• Quadrat analysis and the nearest neighbour analysis were performed on the epicenter dataset to find out the spatial pattern in the point distribution.
  
• Finally, the spatial auto correlation coefficients like the Geary’s ratio (C) and Moran’s (I) were calculated to measure and test how clustered/ dispersed the positions of epicenters were in space with respect to their attribute values i.e. the magnitude of the earthquakes.
  

Results and Dissussion
The statistical operations for central tendency of the distribution for the data of earthquake epicenters in and around Himachal Pradesh were done with the help of Arc View 3.2 GIS software. The spatial mean and the median center for the distribution of the epicenters were calculated (Fig. 1). Both spatial mean and the median center when calculated with weightage given to the magnitude of the earthquake for that particular epicenter did not show much variance in their locations, thereby indicating that the epicenters with contrasting magnitude i.e. very low or very high compared to the magnitude of the majority of the earthquake epicenters do not lie scattered at far distant places. It, thereby, shows that the entire area is continuously being struck by earthquakes of varying magnitude that seem to be controlled by the same structural disturbances. Otherwise, the weighted mean center and the median center could have been at significant distance apart.

After performing quadrat analysis and the nearest neighbor analysis, it has been observed that the entire dataset of epicenters shows a distinct statistical clustering. Further spatial autocorrela-tion parameters like Geary’s C and Moran’s (I) were calculated to measure the similarity of the characteristics of the locations of the epicenters and the results are compared with the Standard Numeric scales for Geary’s and Moran’s Index (Table 1).

Table 1: Standard Numeric scales for Geary’s and Moran’s Index.

Spatial Pattern	Geary’s C	Moran’s I
Clustered pattern in which adjacent points show similar characteristics	0 < C < 1	I > E (I)
Random pattern in which points do not show particular pattern of similarity	C ≈ 1	I ≈ E (I)
Dispersed / uniform pattern in which adjacent points show different characteristics	1 < C < 2	I < E (I)

E (I) = (-1) / (n-1), with n denoting the number of points in distribution.
(Source: Lee and Wong, 2001)
The observed values for the above two parameters are: 
Geary’s C = 0.233047 
Moran’s (I) = 0.8311, E (I) = - 0.00381679

Both the C and I, when compared to the above-mentioned table suggest that the data show clustered pattern in which the epicenters with similar magnitude lie close to each other.