Other approaches

aftershock sequence (cluster) starts and end time (duration) and used to discriminate the earthquake catalog in terms of aftershock clusters and backgrounds.

Batac and Kantz [51] described the spatio-temporal separation of earthquakes with the use of inter-event time∆t and inter-event distance∆rbetween the pair of successive events.

They defined a characteristic value∆r^∗and observed that after∆r>∆r^∗, the distributions of ∆r follows similar behavior as like randomly shuffled sequence. It means events are uncorrelated (and independent to each other) when∆r>∆r^∗. This method is applied to the regional earthquake catalog of Philippines.

2.9 Other approaches

2.9.1 Hidden Markov Model (HMM) based declustering approach

Wu [52] developed a hidden Markov model (HMM) for earthquake declustering. The model consists of a sequence of observationsO₁,O₂, ...,O_nand a sequence of hidden states h₁,h₂, ...,h_n. The distribution of observationsO_ionly depends on the hidden stateh_i, while the hidden state sequence h₁,h₂, ... builds a Markov chain. It uses forward-backward al- gorithm to compute the conditional distributionP(h_i|O₁, ...O_n)and the likelihood function P(O₁, ...O_n)of the model and Viterbi algorithm computes the most likely hidden state se- quence. The earthquake catalog used by the author belongs to the period 1926–1995, in the rectangular area 33°–39°N and 131°–140°E, with magnitudes greater than 4.0 and depth less than 100 km. The model highlights the difference between Gutenberg-Richter law, obtained from clustered and declustered data. Ebel et al. [53] further used the concept of a hidden Markov model to describe non-Poissonian earthquake clustering and forecasting method for California.

2.9.2 Seismic density index based grid network

Wang et al. [54] introduced a "seismic density index" which depends on the number of events, their magnitudes, and locations of the surrounding population of earthquakes. This multi-dimensional index (metric) quantifies the degree of clustering and seismic energy released in a region. In this method, a grid with interval∆in longitude and latitude is defined and for each grid node, specify an outer circle centered on the node of maximum radiusRand inner minimum radiusR_min. For the j^th node, only those earthquakes with epicenters inside the circle will contribute to the index. Those outside are assigned to a separate node in due course. If distancer_{i j} between j^thgrid node and the j^th earthquake epicenter is determined then seismic density index for a node jat timet, denoted byI_j,t is defined by

I_j,t=

n i=1

∑

M_i

∆mln(r_{i j}),R_min<=r_{i j}<=R. (2.34)

wherenrepresents the number of events,∆mis the difference between the threshold magni- tude and the maximum magnitude andM_iis the magnitude ofi^th event within the annulus.

2.9.3 Schuster spectrum-based approach

Adar and Avouac [55] proposed a framework to detect the unknown periodicities present in an earthquake catalog using the Schuster spectrum pvalues. The p-values of Schuster spectrum provides information about the change in the seismicity rate whether periodic or not using a sinusoidal function. It is the probability that the timing of events in a catalog varies according to a sine-wave function of periodT,t_k time of event numberk, then its associated phaseθ_k=2πt_k/T. If the catalog of times is converted into a 2D walk, by making successive unit-length steps, in directions according to these phases. The distanceDbetween the start and end points of this walk, the probability (Schuster p-value), that a distance greater than or equal toDcan be reached by a uniformly random 2D walk, is

p=exp(−D²/N) (2.35)

where N is the number of events in the catalog. This indicates a probability of the null hypothesis that event–times distribution arises from a uniform seismicity rate. Further, it helps to make a judgment about the accuracy of a declustering catalog. This method is demonstrated on Nepalese seismicity with annual variations of the seismicity rate of amplitude up to 40%. and relative amplitude response of the seismicity at these periods is less than 18%.

2.9.4 Multi point (m)-Morisita index based approach

Golay et al.[56] introduced the concept of Morisita index to quantify the degree of clustering by analyzing spatial patterns present in the datasets. For this, a dataset is covered by a regular grid of changing the size and then Morisita index measures how many times it is more likely to randomly select two sampled points from the same quadrant. The classical Morisita index I_δ for a quadrant sizeδ, is defined as

I_δ =Q

Q

∑

i=1

n_i(n_i−1)

N(N−1) (2.36)

2.9 Other approaches 27 Here, length of the diagonal is taken as quadrant sizeδ,Qis the number of quadrats necessary to cover the study area,n_iis the number of points in thei^th quadrant, andN is the total number of points. The shape of quadrants either square or rectangular. Initially,I_δ is determined by taking a relatively small quadrant size and then increased until it reaches to a chosen value. After that, a plot is drawn relating everyI_δ to its matchingδ. The plot is used to analyze the patterns as: (i) If the points are randomly distributed over the study area, each computedI_δ fluctuates around the value of 1. (ii) If the points are clustered, the number of empty quadrants at small scales increases the value of the index (I_δ >1) and, finally, (iii) if the points are dispersed, the index approaches 0 at small scales. The multi-point Morisita index (m-Morisita) version considersmpoints withm≥2.

Telesca et al. [57] applied the concept ofm-Morisita index to describe the clustering (spatial domain) for swiss seismicity. The authors also justified the properties of the whole catalog, aftershock catalog, and aftershock-removed catalog obtained from the different benchmark declustering algorithms. They demonstrated the multi-fractal behavior of whole and aftershock depleted catalog and found similar to each other. They also concluded that the whole catalog is more spatially clusterized than the removed one and clustering increases with the increase of the threshold magnitude.

2.9.5 A 3D nearest-neighbor distance (NND) based approach

Baiesi and Paczuski [58] introduced a distance formula by considering occurrence timet_i, hypocenter (φ_i, λ_i, d_i), and magnitudem_iofi^thevent. The objective here, is to identify for each earthquake jits possible parent, which is an earlier closest earthquakeito jamong all earlier events. The distance metricη_{i j} between eventiand j>iis defined as

ηi j =







t_{i j}(r_{i j})^df10^{−b mi}, t_{i j} >0;

∞, t_{i j} >0.

(2.37)

wheret_{i j}=t_j−t_iis the inter-event time in years, which is positive if earthquakeioccurred before event jand negative. Otherwise,r_{i j} is the spatial distance between eventiand jin kilometers,d_f is the fractal dimension (which is related to hypocenter distribution) andbis the parameter of Gutenberg-Richter law. Here in the analysis,d_f =1.6 andb=1 is taken.

Zaliapin et al. [59, 60] extended the work by representing the distanceη in the form of its space and time component normalized by the magnitude of parent eventi:

T_{i j} =t_{i j}10^{−b mi/2} (2.38)

R_{i j}= (r_{i j})^df10^{−b mi/2} (2.39) From Eq.(2.38) and Eq. (2.39), it is readily seen thatη_{i j}=T_{i j}R_{i j}, or equivalently, log₁₀η_{i j}= log₁₀T_{i j}+log₁₀R_{i j}. The author also demonstrated the joint 2-D distribution of (T,R) for a stationary homogeneous Poisson process which is closely approximated by the Weibull distribution. It is unimodal and is concentrated along the line log₁₀T_{i j}+log₁₀R_{i j}=constant.

They did the cluster analysis based on the significant deviations of observed nearest neighbor distanceη from the results expected in the absence of clustering.

2.9.6 An approach based on Single link cluster analysis

Frohlich and Davis [61] reported a simple method to measure the degree of clustering and categorize the group of events in a set ofNearthquakes. In this approach events, are linked to their nearest neighbors to make event sub-groups. This process is repeated and each sub-group linked to its nearest neighbor recursively until (N−1) links are found to join all the earthquakes. Knowledge of these links is utilized to evaluate earthquake properties in a spatio-temporal domain and make a judgment about the various earthquake nests, isolated events, aftershocks sequences, seismic quiescence, etc. They defined a 2D space-time distance between earthquakeiand jas

d_{i j} = q

r_{i j}² +C²(t_j−t_i)² (2.40) whereC is constant and taken 1 Km/day. The application of SLC is demonstrated on the following: (i) a global catalog of 2178 earthquakes having a magnitude of 5.8 or greater reported by the International Seismological Centre (ISC) between 1964 and February 1986, and (ii) sets of earthquakes having a magnitude of 4.9 and greater as reported by the ISC, occurring in Central America and in Aleutians.

2.9.7 An approach based on multi-resolution wavelet analysis

Telesca et al. [62] applied a multi-resolution wavelet approach to analyze two seismic sequences occurred in Italy, Umbria-Marche and Irpinia, during 1986–2001. They considered a series of time intervals between successive seismic eventsτ_i,i=1, ...L, whereLis the length of series and defined a standard deviation of the wavelet coefficientσ_wav(m). They analyzed the temporal evolution of seismicity by varying theσ_wav(m)with respect to time. It is given by: