Multifractality in seismic sequences of NW Himalaya.pdf

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ORIGINAL PAPER Multifractality in seismic sequences of NW Himalaya Ashutosh Chamoli R. B. S. Yadav Received: 29 June 2013 / Accepted: 27 August 2013 Ó Springer Science+Business Media Dordrecht 2013 Abstract Multifractal behaviour of interevent time sequences is investigated for the earthquake events in the NW Himalaya, which is one of the most seismically active zones of India and experienced moderate to large damaging earthquakes in the past. In the present study, the multifractal detrended fluctuation analysis (MF-DFA) is used to understand the multifractal behaviour of the earthquake data. For this purpose, a complete and homogeneous earthquake catalogue of the period 1965–2013 with a magnitude of completeness M w 4.3 is used. The analysis revealed the presence of multifractal behaviour and sharp changes near the occurrence of three earthquakes of magnitude (M w ) greater than 6.6 including the October 2005, Muzaffarabad–Kashmir earthquake. The multifractal spectrum and related parameters are explored to understand the time dynamics and clus- tering of the events. Keywords Multifractal Multifractal detrended fluctuation analysis (MF-DFA) Himalaya Earthquakes 1 Introduction Earthquake phenomenon is a complex process, which incorporates occurrence of different processes or structures at different scales resulting from nonlinear interactions between its components. Such processes can be explained as a result of self-organized criticality (Turcotte 1997; Mandal et al. 2005; Sornette 2006), in which system evolves to marginally stable or critical state when perturbed. The scaling behaviour of seismicity can give details A. Chamoli (&) CSIR-National Geophysical Research Institute, Hyderabad 500007, India e-mail: [email protected] R. B. S. Yadav Department of Geophysics, Kurukshetra University, Kurukshetra 136119, India 123 Nat Hazards DOI 10.1007/s11069-013-0848-y

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ORI GIN AL PA PER

Multifractality in seismic sequences of NW Himalaya

Ashutosh Chamoli • R. B. S. Yadav

Received: 29 June 2013 / Accepted: 27 August 2013� Springer Science+Business Media Dordrecht 2013

Abstract Multifractal behaviour of interevent time sequences is investigated for the

earthquake events in the NW Himalaya, which is one of the most seismically active zones

of India and experienced moderate to large damaging earthquakes in the past. In the

present study, the multifractal detrended fluctuation analysis (MF-DFA) is used to

understand the multifractal behaviour of the earthquake data. For this purpose, a complete

and homogeneous earthquake catalogue of the period 1965–2013 with a magnitude of

completeness Mw 4.3 is used. The analysis revealed the presence of multifractal behaviour

and sharp changes near the occurrence of three earthquakes of magnitude (Mw) greater than

6.6 including the October 2005, Muzaffarabad–Kashmir earthquake. The multifractal

spectrum and related parameters are explored to understand the time dynamics and clus-

tering of the events.

Keywords Multifractal � Multifractal detrended fluctuation analysis (MF-DFA) �Himalaya � Earthquakes

1 Introduction

Earthquake phenomenon is a complex process, which incorporates occurrence of different

processes or structures at different scales resulting from nonlinear interactions between its

components. Such processes can be explained as a result of self-organized criticality

(Turcotte 1997; Mandal et al. 2005; Sornette 2006), in which system evolves to marginally

stable or critical state when perturbed. The scaling behaviour of seismicity can give details

A. Chamoli (&)CSIR-National Geophysical Research Institute, Hyderabad 500007, Indiae-mail: [email protected]

R. B. S. YadavDepartment of Geophysics, Kurukshetra University, Kurukshetra 136119, India

123

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of interaction between nonlinear constituent processes. Different power laws govern the

distribution of the statistical parameters such as the distribution of released energy

[Gutenberg-Richter law of Gutenberg and Richter (1944)], fractal distribution of the faults

(Kagan 1992), aftershocks decay rate (Utsu et al. 1995; Yadav et al. 2011, 2012a).

The complexity of such systems cannot be dealt appropriately by first-order statistics

such as probability density functions, and thus, different approaches are developed to

understand the second-order information to investigate temporal fluctuations of the seismic

sequences giving additional information about the correlation properties. Earthquakes can

be considered as spatiotemporal point processes with power law behaviour in characteristic

properties (energy, occurrence rate, etc.). Some of the approaches to investigate second-

order information are the Allan factor (Allan 1966), the Fano factor (Lowen and Teich

1995) and the detrended fluctuation analysis (Peng et al. 1995), which have been used to

understand the clustering behaviour in time and space. In these approaches, the time

dynamics of a seismic process is defined by a single scaling exponent or fractal dimension

assuming the monofractal behaviour. The hypocenter distribution data suggest that the

changes in fractal dimension could be a good precursor parameter for earthquakes warn-

ings as it is a measure of the degree of clustering of seismic events. A change in fractal

dimension corresponds to the dynamic evolution of the states of the system. The complex

systems encompass different scaling in the subsets and thus require multifractal approaches

to understand the dynamics at different scales. In these approaches, different scaling

exponents characterize different segments of the seismic sequence to study the time var-

iation of the scaling. The concept of multifractals generalizes the use of higher-order

moments of statistical distribution. The generalized fractal dimension is constant for

homogeneous fractals or monofractals and different for heterogeneous fractals termed as

multifractals. The application of multifractal formalism is useful for the signals with

continuous variations in a quantity such as seismic interevent times of earthquakes.

A number of studies have been carried out in various seismically active regions of the

world for understanding the multifractal behaviour in earthquake dynamics (Hirata and Imoto

1991; Hirabayashi et al. 1992; Sunmonu and Dimri 1999, 2000; Dimri 2000; Sunmonu et al.

2001; Dimri 2005a, b; Telesca et al. 2005; Sornette 2006). Temporal clustering is observed in

earthquake events, which is related to aftershocks (Telesca and Lapenna 2006). The NW

Himalaya is an earthquake-prone region with complex geological structure (Shanker et al.

2007; Yadav et al. 2010, 2012b; Chamoli et al. 2011). The seismic activity in this region is

associated with either main boundary thrust (MBT) or main central thrust (MCT) which are

north-dipping mega thrusts running with the strike of Himalayan tectonic belt. This is one of

the most active regions of the world, where recently an earthquake of magnitude Mw 7.6 had

occurred on 8 October 2005, near Muzaffarabad, Pakistan, causing loss of more than 73,000

lives. The fractal behaviour in the seismicity of the NW Himalaya is also reported in previous

studies (Teotia et al. 1997; Roy and Mandal 2009; Teotia and Kumar 2011). In the present

study, the seismic sequences of the NW Himalaya is analysed by multifractal detrended

fluctuation analysis (MF-DFA), which handles the nonstationary behaviour of the earthquake

data in a robust and effective manner (Kantelhardt et al. 2002). The estimated scaling

coefficients are interpreted in terms of clustering of the earthquake events.

2 Earthquake data

A homogeneous and complete seismicity database is one of the most important require-

ments to assess the multifractal behaviour of earthquake sequences in this region. The

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earthquake data used in this study are compiled mainly from two sources: (1) a homo-

geneous and complete earthquake prepared by Yadav et al. (2012b) (time span 1900–2010)

and (2) earthquake data from NEIC of USGS (http://neic.usgs.gov/neis/epic/epic-global.

htm) to complete the data up to the present (i.e. June 2013). The first data is compiled from

different seismological sources such as International Seismological Summary (ISS),

International Seismological Centre (ISC), NEIC of USGS, HRVD CMT catalogue and

other local catalogues of Oldham (1883), Tandon and Srivastava (1974), Chandra (1978)

and Bapat et al. (1983). The magnitude of completeness (Mc) of the catalogue is also

calculated as Mw 4.3. We considered the catalogue events greater than this Mc from 1965

onwards to have a smooth and stable multifractal spectrum because events before 1965

appear to be sparse. The data used cover a region within a polygon geographically with

vertices of longitude and latitude as (71.5�E, 36�N), (71.5�E, 30�N), (85�E, 28�N) and

(85�E, 35�N) similar to the earlier study of Teotia and Kumar (2011) (Fig. 1). The analysis

is also repeated for interevent time series of a sub-region surrounding the epicentre of

Muzaffarabad earthquake (October 08, 2005) to investigate the behaviour for a single

earthquake separately and thus check the effect of merging of different seismic regimes in

the data. The other major earthquakes in the study region lack sufficient number of events

in space and time, and thus, it is difficult to test the multifractal behaviour separately. Thus,

two interevent time series of magnitude[4.3 are analysed covering I. whole study region

and II. sub-region surrounding the epicentre of Muzaffarabad earthquake.

The series I of length 2,767 is shown in Fig. 2a. The series II of length 1,356 is shown in

Fig. 2b. This sub-region for series II shown in Fig. 1 covers the data within a polygon of

vertices of longitude and latitude as (71.5�E, 36�N), (71.5�E, 32�N), (76�E, 32�N), (76�E,

36�N). Both the series clearly show irregular dynamics consisting of abrupt bursts. The

behaviour of these structures can be assessed by multifractal analysis.

Fig. 1 Map showing seismicity (1965–2013) and tectonics of the study region in NW Himalaya. The threemain earthquake events of Mw [ 6.6 are plotted with stars. The polygon in the upper left corner of the mapshows sub-region near to the 08 October, 2005—Muzaffarabad earthquake

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3 Methodology

The multifractal detrended fluctuation analysis (MF-DFA) is used to study the multifractal

behaviour of interevent time series. The method is recommended to characterize the

intermittent and long-range variability of the data for different ranges of temporal or spatial

scales. Different workers have applied the method in diverse scientific fields (Kantelhardt

et al. 2002; Telesca and Lapenna 2006; Ihlen 2012). A summary of the used approach is as

follows:

1. The interevent time x(i), where i = 1, 2, 3, …, N is assumed and converted to a

random walk process by subtracting the mean value xm and calculating the cumulative

sum using:

yðiÞ ¼Xi

k¼1

x kð Þ � xm½ � ð1Þ

then, the series y(i) is converted to Ns nonoverlapping segments of equal length scale s.

2. The linear trends of each Ns segments are calculated by least-square fitting and the

variance is calculated using:

Fig. 2 Interevent time series of the seismicity from 1965 onwards and magnitude greater than Mw 4.3 fora whole region (series I) and b sub-region surrounding the epicentre of Muzaffarabad earthquake (series II)

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F2 s; vð Þ ¼ 1

s

Xs

i¼1

y v� 1ð Þsþ if g � yv ið Þ½ �2; ð2Þ

for each segment m = 1,…,Ns. The analysis is also repeated from back to forth to utilize the

remaining segment due to segmenting as suggested by Telesca and Lapenna (2006).

3. The average qth-order fluctuation function is then calculated for q = 0 using:

Fig. 3 Calculated fluctuation functions for a whole region (series I) and b sub-region surrounding theepicentre of Muzaffarabad earthquake (series II). The multifractal behaviour is indicated by different slopesfor q = -1.5, 0 and 1.5

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FqðsÞ ¼1

2Ns

X2Ns

v¼1

F2 s; vð Þq2

h i( )1q

: ð3Þ

If the series x(i) follows long-range power law, the Fq(s) increases with s as:

Fq sð Þ � shq : ð4Þ

The value at h0 is calculated using

F0 sð Þ ¼ exp1

4Ns

X2Ns

v¼1

ln F2 s; vð Þ� �

( )� sh0 : ð5Þ

Fig. 4 Variation in generalized Hurst exponent hq with q (-1.5 to 1.5) for a whole region (series I) andb sub-region surrounding the epicentre of Muzaffarabad earthquake (series II)

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The hq is the generalized Hurst exponent and characterizes the long-/short-range

dependence structure in the series (Kantelhardt et al. 2002; Chamoli et al. 2007; Chamoli

2010). In Eqs. 4 and 5, the variation in hq with q is useful to understand different scaling of

small and large fluctuations. For multifractal time series, the scaling behaviour of the large

fluctuations is characterized by the values of hq for positive q values. The scaling of the

small fluctuations is characterized by the values of hq for negative q values.

The multifractal behaviour can be identified looking into the scaling of calculated

fluctuation function with scales. The fluctuation function is calculated for scales from 10 to

N/4 for series I and 10 to N/2 for series II. The fluctuations are calculated for q 2�1:5; 1:5½ � and three cases of q = -1.5, 0 and 1.5 are plotted in Fig. 3a, b for series I and

series II, respectively. The figure clearly shows different slopes hq pointing to different

scaling of small and large interevent fluctuations. It is also observed that slopes are rela-

tively more stable for positive q values than negative q values. The multifractal behaviour

Fig. 5 Multifractal spectrum of the interevent time series for a whole region (series I) and b sub-regionsurrounding the epicentre of Muzaffarabad earthquake (series II). A quadratic fit is also shown by black line

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is also well established from Fig. 4a, b, in which generalized Hurst exponent hq is plotted

with q for series I and II, respectively. Fig. 4 illustrates the decreasing trend of hq with q,

which is a typical characteristic of multifractal behaviour. Thus, Figs. 3 and 4 reveal the

presence of multifractal behaviour in the two interevent time series.

The other way to quantify the multifractal behaviour is in the form of multifractal

spectrum. The qth-order mass exponent sq can be computed from hq using:

sq ¼ qhq � 1: ð6Þ

The qth-order singularity exponent a and the qth-order singularity dimension f(a) can be

calculated as follows (Kantelhardt et al. 2002):

a ¼ dsq

dqð7Þ

f ðaÞ ¼ qa� sq: ð8Þ

Fig. 6 Temporal variation in a a0 (a at maximum f(a)), b coefficient A of parabolic fit, c coefficient B ofparabolic fit, d coefficient C of parabolic fit, e width of spectrum, calculated for series of whole region(series I shown in Fig. 2a). Earthquake events greater than Mw 6.6 (19 November 1996—Kashmir region ofMw 6.8; 08 October 2005—Muzaffarabad of Mw 7.6 and 25 August 2008—Xizang, China of Mw 6.7) areplotted by dots. The dashed line shows the calculated mean for 10 random shuffles of same series, which isdetailed in the text

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The plot of f(a) verses a gives information about existing different scaling exponents

and long-range correlation properties of the time series (Fig. 5a, b). The width and shape of

multifractal spectrum can be used to characterize and quantify such properties. A quadratic

function of the form f(a) = A(a - a0)2 ? B(a - a0) ? C is fitted to the spectrum to

quantify the width and shape of multifractal spectrum (Telesca and Lapenna 2006; Shimizu

et al. 2002). The regular behaviour of the process is quantified by a0 [the value of acorresponding to maximum f(a)]. The larger values of a0 imply the process will be rela-

tively more regular. The coefficient A gives a measure of concavity of the parabola, and the

shape of curve changes from bowing down for negative to up for positive values. The

coefficient B gives information about the symmetry in the shape, which is equal to zero,

positive and negative for symmetric, right-skewed and left-skewed shape, respectively.

The right-skewed and left-skewed shapes correspond to weighting of high-fractal expo-

nents and low-fractal exponents, respectively. The coefficient C governs the vertical shift

in the parabolic fit. The width of spectrum (W = max(a) - min(a)) is an important

parameter for interpretation because it gives the range of a and thus the degree of mul-

tifractality. These parameters (a0, A, B, C and W) are estimated to understand and quantify

the multifractal behaviour. The temporal variation in different parameters of multifractal

spectrum gives insight into the dynamic behaviour of the seismic sequence (Gamero et al.

1997; Martin et al. 2000; Telesca and Lapenna 2006). This is attained using a sliding

window of 900 and 400 events with a shift of 20 events for series I and II, respectively. The

values of window parameters are finalized to get the stable spectrum for different windows.

Fig. 7 Temporal variations in amin and amax calculated for series of whole region (series I shown inFig. 2a). Earthquake events greater than Mw 6.6 (19 November 1996—Kashmir region of Mw 6.8; 08October 2005—Muzaffarabad of Mw 7.6; 25 August 2008—Xizang, China of Mw 6.7) are shown by dots

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Different parameters (a0, coefficients A, B and C, width of spectrum, amin and amax) are

calculated for -1.5 B q B 1.5 for local spectrum of the windows and plotted against the

last timing of the individual window (Figs. 6, 7 for series I and Figs. 8, 9 for series II).

4 Results and discussion

The MF-DFA clearly shows the presence of a rich multifractal spectrum of both the time

series. The coefficients of parabolic fitting A, B and C are -0.32, 0.06 and 1.00 for series I

(Fig. 5a). For series I, the multifractal spectrum shows relatively long right tail (Fig. 5a),

which is also observed by the levelling of hq for positive q values (Fig. 4a). This shows that

Fig. 8 Temporal variation in a a0 (a at maximum f(a)), b coefficient A of parabolic fit, c coefficient B ofparabolic fit, d coefficient C of parabolic fit, e width of spectrum, calculated for the sub-region surroundingthe epicentre of Muzaffarabad earthquake (series II shown in Fig. 2b). Earthquake event (08 October2005—Muzaffarabad of Mw 7.6) is plotted by dot. The dashed line shows the calculated mean for 10 randomshuffles of same series, which is detailed in the text

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multifractality is not affected by the local fluctuations with large magnitudes. The coef-

ficients of parabolic fitting A, B and C are -0.34, 0.07 and 1.02 for series II (Fig. 5b). The

values of coefficients of parabolic fit for series I and II show that both series are behaving

similarly. It is inferred from these computed coefficients that seismic sequence is following

multifractal behaviour with more weighted high-fractal exponents. The relatively less

stable multifractal spectrum for negative q values (Fig. 3a, b) shows complex heteroge-

neity in small fluctuations. The temporal variations plotted in Figs. 6, 7, 8 and 9 reveal

strong variability in different parabolic parameters. For series I, the distinctive behaviour

of these parameters is observed as sharp changes near the occurrence time of three major

earthquake events of magnitude (Mw) [ 6.6 (19 November 1996—Kashmir region of Mw

6.8; 08 October 2005—Muzaffarabad of Mw 7.6; 25 August 2008—Western Xizang, China

of Mw 6.7). The a0 shows small jump just after the first event, a sudden jump after the

second event and sharp decrease after the third event (Fig. 6a). This suggests change in

local fractal dimension of most of the clusters just after the earthquake and energy release

with distributed seismicity. The coefficient A gives information about the bending of the

parabola and shows a pattern of jump after the first event, small jump followed by sharp

prominent decrease after the second event and very small jump followed by sharp

prominent decreasing pattern after the third event (Fig. 6b). The coefficient B shows no

prominent trend after the first event, sharp increase after the second event and the third

event, respectively (Fig. 6c), indicating change in skewness of parabola and thus changes

from dominant low-fractal exponents to dominant high-fractal exponents for the second

and the third events. The coefficient C shows a sudden decrease after the first, small

decrease followed by prominent sharp increase after the second and small jump followed

by sharp decrease after the third event, respectively (Fig. 6d). The width W is the most

important parameter because it gives a measure of range of scaling in the spectrum. The

W shows sharp jump after all the three events (Fig. 6e), which indicates the change from

homogeneous to heterogeneous dynamics. The amin (maximum value of a) shows decrease,

Fig. 9 Temporal variations in amin and amax calculated for the sub-region surrounding the epicentre ofMuzaffarabad earthquake (series II shown in Fig. 2b). Earthquake event (08 October 2005—Muzaffarabadof Mw 7.6) is plotted by dot

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increase and decrease pattern after the first, the second and the third event, respectively

(Fig. 7). The amax (maximum value of a) shows pattern of sharp prominent increase for the

first and the second events and small jump followed by prominent decrease for the third

event (Fig. 7). The variations in multifractal parabolic parameters for series II (Figs. 8, 9)

are showing similar variations in prominent patterns as observed for the second event for

series I after the Muzaffarabad earthquake. Thus, the results of series I are not likely to be

affected by different seismic regimes. The significance of the results is tested using 10

randomly shuffled version of the analysed series and then repeating the calculation of

temporal variations in parameters for each series. The average values of the parameters of

these 10 cases are plotted in the Figs. 6, 7, 8 and 9 by dashed lines. The difference in the

average values and multifractal parameters shows that the observed sudden changes are

realistic and significant. The MF-DFA method extracted the information in detail than

earlier studies using correlation integral method in the NW Himalaya in the sense that it

gave prominent signatures for more than one event in the region.

5 Conclusions

The MF-DFA analysis of the earthquake sequences in the NW Himalaya reveals multi-

fractal behaviour of occurrence of seismic events. The decreasing trend of hq with

q (Fig. 4) indicates that segments with small fluctuations (represented by negatives q)

values have a random-walk-like structure (more persistent structure), and the segments

with large fluctuations (represented by positive q values) have noise-like structure. The

singularity exponent (a) ranges from 0.73 to 2.4 for both the series I and II, which indicates

rich multifractal behaviour in the interevent time series. This further highlights different

scaling for short and long interevent intervals. The time dynamics of the seismic sequence

is characterized by different parameters of parabolic fit: a0, A, B, C, width of spectrum, amin

and amax. The sharp changes are observed in these parameters after the three earthquakes of

magnitude (Mw) [ 6.6 (Mw 6.8 on 19 November 1996; Mw 7.6 on 08 October 2005; Mw 6.7

on 25 August 2008). The changes mimic the clustering of seismic events in the region. The

change in local fractal dimension of most of the clusters after the earthquake and energy

release with distributed seismicity is reflected in the form of sudden jump of a0 after three

earthquakes. During all three events, there is a change from homogeneous to heteroge-

neous dynamic behaviour as inferred from variations in width (W). The results of MF-DFA

in this study show sharp variations than as observed in earlier seismicity studies in the

region and thus supported multifractal analysis of seismicity sequences.

Acknowledgments The authors are thankful to their respective institutes for the support. Authors thank toProf. V.P. Dimri for useful discussions. AC acknowledges the support from the project HEART-PSC0203for the study. Authors are also thankful to the two anonymous reviewers for their suggestions to augment themanuscript to the present form.

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