results are deviated from one study to another because of un-optimized settings of parameters for different earthquake catalogs. These methods were demonstrated mostly for real and benchmark catalog of California to determine the aftershock cluster and background events in a region.
3.2 Proposed Tetra-stage clustering model
The block diagram of proposed Tetra-stage cluster identification model for declustering an earthquake catalog is shown in Fig.3.1. The model consists of four stages and considers occurrence time, event’s location, magnitude and depth information as obtained from the catalog. The step-by-step procedure to obtain the clustered aftershocks and independent backgrounds are described in the following subsections:
Fig. 3.1.A Tetra-stage cluster identification model for seismic catalogs.
3.2.1 Earthquake catalog
Let an earthquake catalog EN×D is used in the analysis. It represents the occurrence of seismic activities in a particular region by considering the event’s time, position, focal depth and magnitude. Mathematically, it is represented as
EN×D=
e11 e12 · · · e1D e21 e22 · · · e2D ... ... . .. ... eN1 eN2 · · · eND
(3.1)
whereNis the total number of events andD=4 feature vector dimension of catalogEN×D.
3.2.2 Depth thresholding
Initially, shallow and deep focus (SF and DF) earthquake events are categorized by putting a threshold on depth to original catalogEN×Das follows-
If(Depth[ei∈E]>dthr)⇒
ei∈DF else,ei∈SF
(3.2)
wheredthris chosen 70 Km [74].N1number of events is considered as SF events andN2for DF events. eiis a row vector that equals to theith row ofE.
3.2.3 Shallow events
The event belongs to either deep focus or shallow focus category, they are analyzed inde- pendently due to the difference in behavior of seismicity. First, shallow focus events are considered for further analysis in the following sub-sections:
3.2.4 Mainshock identification
The events having a high magnitude in the shallow catalog are considered to be the main- shocks which are well separated in space as well as in time. These can be identified by putting a magnitude threshold in the catalog. Selection of number of mainshocks depend on the region of interest but it should be optimum for effective and accurate classification of output. The characteristics of these selected mainshocks are used as a cluster centroids to
3.2 Proposed Tetra-stage clustering model 39 build the model. These centroids are represented by
MC×D=
m11 m12 · · · m1D m21 m22 · · · m2D
... ... . .. ... mC1 mC2 · · · mCD
(3.3)
whereM∈EN1×Drepresents theCnumber of selected mainshocks that are considered as cluster centroids (obtained from the shallow seismic events).
3.2.5 Temporal cluster identification
The temporal cluster identification is carried out using temporal clustering and time zone based clustering as follows:
1. Temporal clustering:- The shallow events are classified intoC clusters (due to C number of centroids) with a single iteration distance algorithm. The steps involved are given in the algorithm 1.
Algorithm 1:Temporal clustering
Input :Time information of the mainshocksτj,j=1,2, ...,C∈MC×D Time information of shallow eventsti,i=1,2,3...N1∈EN1×D Output :C-time based clusters from the shallow events
1 fori=1 :N1do
2 for j=1 :Cdo
3 Calculate: For each event calculate the Euclidean distance (2-norm) from the cluster centers.
4 d(ti,τj) = sD=1
∑
d=1
(ti,d−τj,d)2
5 Assign: label is assigned to each event w.r.t nearest cluster center for which distance is minimum
6 d(ti,τj) =arg min
k d(ti,τk) k∈1,2,3...C;
7 end
8 end
9 Return to step 2:i=i+1;j= j+1;
2. Time zone based clustering:- The events that belong to each of theC clusters are then grouped into two categories according to the time zone (w.r.t. to mainshock).
The events located far away from the mainshock and nearby from the occurrence time
of the mainshock belong to a regular time zone (RTZ) and danger time zone (DTZ) respectively.
In order to achieve the above classification, each cluster which is obtained from the temporal clustering is divided into three sub-clusters using the same algorithm given in 1. The mainshock and two extreme points of the respective cluster represent the sub-cluster centers. Here, C1 and C2 are defined that represents the cluster of events belonging to background and danger time zone respectively.
3.2.6 Spatial based cluster identification
Spatial based cluster identification is carried out in two sub-stages. Initially, coordinate-based clustering is applied to divide the entire region into subzones according to the coordinates (Latitude, Longitude). After that, on each sub-zone, a coordinate threshold is applied which a portion of the maximum distance between a mainshock cluster center and furthest point in the respective cluster. The sub-stages are summarized as follows:
1. Coordinate based clustering:- Events in the dataset belonging to regular zone C1 and cluster time zone C2 undergo location based cluster identification separately. The distance-based algorithm discussed in the previous section is employed for clustering of the two datasets. The location of mainshocks is considered as cluster centers. The dimension of each event is two, which corresponds to longitude and latitude coordi- nates. Finally both datasets C1 and C2 are classified intoCclusters after applying the similarity measure between the coordinates.
2. Coordinate thresholding:- The events belong to each cluster are then classified into two categories based on a threshold. Events located far away from the location of mainshocks belongs to the background coordinate zone. Otherwise, events near to them belong to a clustered coordinate zone. The threshold parameter is determined as a portion of the maximum distance between the cluster center and the furthest point inside the cluster. For all shallow eventsei∈SF the thresholding process is as follows:
If(d(ei,cj)<[η×d(ei,Fj)])⇒
( ei∈RCZ
else,ei∈DCZ (3.4) whereFjis the furthest point in each cluster which has maximum distance from the respective cluster head. Theη is a threshold value generally lies between 0.1-0.5.
3.2 Proposed Tetra-stage clustering model 41 Table 3.1 Categorization of the shallow events
Category Cluster belongingness Event’s zone
1 C1 and C3 RTZ and RCZ
2 C1 and C4 RTZ and DCZ
3 C2 and C3 DTZ and RCZ
4 C2 and C4 DTZ and DCZ
The selection of optimum value of η depends on linearity and stationarity of the classification results.
3.2.7 Categorization
After temporal and coordinate based cluster identification, the shallow events in the catalog are classified into four categories as given in Table 3.1.
3.2.8 Magnitude based cluster identification
1. Magnitude based clustering:- Magnitude based clustering is applied to events which belong to categories 2 and 3. An event is classified as an aftershock if it belongs to the danger magnitude zone otherwise, it belongs to background event. Cluster analysis on this dataset (consisting of category 2 and 3) is performed by widely usedk-means algorithm [23],[64]. The pseudo code of this is outlined in Algorithm 2.
2. Magnitude thresholding:- The events belonging to category-1 (within C1 and C3, i.e., events which are located far away with respect to time and coordinate from the mainshocks) are categorized into aftershocks or background events based upon magnitude based thresholding. For all eventsei∈Category-1 the thresholding process is as follows:
If[mi>mav]⇒
( ei∈Aftershock
else,ei∈Background (3.5) wheremav= mean[m∈Category-4]. This value ofmavrepresents the average mag- nitude of aftershocks. Those events that are far from the mainshock with respect to coordinate and time are considered as aftershocks, if they have a higher magnitude thanmav.
Algorithm 2:Magnitude based clustering
Input :Event’s magnitudeme∈C14 and C23,P=2,D=1,MIter
Output :Classification: Events are categorized in terms of Regular magnitude zone (RMZ) or Danger magnitude zone (DMZ)
1 Initialization: Randomly initialize theP-cluster centroidrmin,rmax
2 fori=1 :MIterdo
3 for j=1 :length(me)do
4 Calculate: For each event calculate the Euclidean distance from the cluster centers.
5 d(mj,r) =
s D
∑
d=1
(mj,d−rP,d)2
6 Assign: label is assigned to each event w.r.t nearest cluster center for which distance is minimum
7 d(mj,r) =arg min
P d(mj,rP) k∈1,2;
8 end
9 Update: determine the cluster centroid by taking the mean of eachPcluster.
10 Return: step 2,i=i+1;
11 end
3.2.9 Deep events
Magnitude thresholding for deep events
Deep earthquakes which are known to generate relatively few aftershocks. Therefore deep events are generally one-third part of the total number of events. They are classified based on the magnitude thresholding as follows:
If([mi∈DF]>mav)⇒ (
ei∈ Deep Aftershock
else,ei∈Deep Background (3.6) wheremav=mean[m∈Category-4]. This value ofmavrepresents the average magnitude of aftershocks. Those events that are far from the mainshock with respect to coordinate and time are considered as aftershocks if they have a higher magnitude thanmav.