A stochastic model for earthquake slip distribution of large events

This paper presents a stochastic model to simulate spatial distribution of slip on the rupture plane for large earthquakes (Mw > 7). A total of 45 slip models coming from the past 33 large events are examined to develop the model. The model has been developed in two stages. In the first stage, effective rupture dimensions are derived from the data. Empirical relations to predict the rupture dimensions, mean and standard deviation of the slip, the size of asperities and their location from the hypocentre from the seismic moment are developed. In the second stage, the slip is modelled as a homogeneous random field. Important properties of the slip field such as correlation length have been estimated for the slip models. The developed model can be used to simulate ground motion for large events

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A stochastic model for earthquake slip distribution

of large events S.T.G Raghukanth & S Sangeetha

To cite this article: S.T.G Raghukanth & S Sangeetha (2016) A stochastic model for earthquake

slip distribution of large events, Geomatics, Natural Hazards and Risk, 7:2, 493-521, DOI:

10.1080/19475705.2014.941418

To link to this article: http://dx.doi.org/10.1080/19475705.2014.941418

© 2014 Taylor & Francis

Published online: 01 Aug 2014

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A stochastic model for earthquake slip distribution of large events

Department of Civil Engineering, Indian Institute of Technology, Madras 600036, India

(Received 13 January 2014; accepted 1 July 2014)

This paper presents a stochastic model to simulate spatial distribution of slip onthe rupture plane for large earthquakes (Mw> 7) A total of 45 slip modelscoming from the past 33 large events are examined to develop the model.The model has been developed in two stages In the first stage, effective rupturedimensions are derived from the data Empirical relations to predict the rupturedimensions, mean and standard deviation of the slip, the size of asperities andtheir location from the hypocentre from the seismic moment are developed In thesecond stage, the slip is modelled as a homogeneous random field Importantproperties of the slip field such as correlation length have been estimated for theslip models The developed model can be used to simulate ground motion forlarge events

1 Introduction

Large-magnitude earthquakes (Mw> 7) occur frequently in active regions like laya and northeast India Even in the Indian shield, Gujarat region also experiencessuch large events Due to their intensity and the geographical extent of the damage,large earthquakes pose the highest risk to the society The 2001 Kutch earthquake(MwD 7.7) caused severe fatalities and affected the economy of the Gujarat region.Recently, Raghukanth (2011) developed the earthquake catalogue for India andranked the 48 urban agglomerations in India based on seismicity The maximum pos-sible magnitude in a control region of radius 300 km around the 24 urban agglomera-tions lies in between MwD 7.1 and MwD 8.7 This necessitates the estimation of theseismic input (design ground motion) in an accurate fashion for such large events toreduce the damages to structures Cases where the recorded strong motion data arenot available, the source mechanism models where in the earthquake slip distributionand medium properties can be modelled analytically are preferred to simulate groundmotion for such large events These models require the earthquake forces to be speci-fied in terms of spatial distribution of slip on the rupture plane Hartzell et al (1999)and Raghukanth and Iyengar (2009) have demonstrated that surface level groundmotions can be computed for an Earth medium for a given slip distribution on therupture plane These models provide reliable ground motion predictions if the faultand its slip distribution are known Specifying the slip distribution on the ruptureplane for future events is the most challenging problem in mechanistic models Toaddress this issue, there have been efforts to obtain spatial distribution of slip on therupture plane by inverting ground motion records of the past earthquakes (Hartzell

Hima-& Heaton1983; Hartzell & Liu 1995; Ji et al.2002; Raghukanth & Iyengar 2008)

*Corresponding author Email:raghukanth@iitm.ac.in

Ó 2014 Taylor & Francis

Vol 7, No 2, 493
521, http://dx.doi.org/10.1080/19475705.2014.941418

Several such finite slip models are available in various journals and research reports.The obtained slip distribution of past events exhibit higher complexity which can bemodelled by stochastic approaches only These techniques require very few parame-ters to characterize the slip field Much effort has been made by the previous investi-gators in this direction (Somerville et al.1999; Mai & Beroza2002; Lavallee et al.

2006; Raghukanth & Iyengar 2009; Raghukanth 2010) Without going into thedetails regarding time-dependent stresses on the fault plane, few parameters havebeen identified from the slip distribution of past events The slip distribution is mod-elled as a random field with a specified power spectral density (PSD) A total of 15slip distributions with the magnitude of the events ranging from 5.66 to 7.22 havebeen analysed by Somerville et al (1999) The total number of large events included

in the database is two Mai and Beroza’s (2002) slip database includes 11 largeevents This puts a serious limitation on the random field model developed by theprevious investigators for simulating slip distribution for large events Due to advan-ces in instrumentation, several large events have been recorded by the broadbandinstruments operating around the world These data have been processed and slipmodels for 45 large events are available in the literature Since large events are of con-cern to engineers, it would be interesting to examine these slip distributions In thispaper, stochastic characterization of slip distribution is explicitly developed for largeevents Important properties of the random field are estimated from the PSD of slipdistribution Empirical equations for estimating the slip field from magnitude aredeveloped in this paper

2 Slip database of large events

Inversion for earthquake sources is fundamental to understand the mechanics ofearthquakes The extracted slip models can be used to understand the damages in theepicentral region Much effort has been made by seismologists in developing meth-ods to extract slip distribution on the rupture plane from ground motion records.After the occurrence of a large event, the Incorporated Research Institutions for Seis-mology data management centre reports the broadband velocity data recorded bythe Global Seismic Network (GSN) The preliminary earthquake slip distribution isdetermined from this data by several research groups In case of local strong motiondata, global positioning system and ground deformation measurements becomeavailable, these records are combined with the GSN data to obtain the spatial distri-bution of slip on the rupture plane Several such slip maps for large events are avail-able in the published literature In this study, the source models of large events,reported by Chen Ji (http://www.geol.ucsb.edu6 faculty6 ji6 ) and tectonics observa-tory, California Institute of Technology (http://www.tectonics.caltech.edu6 ), areused to develop the model The methodology for obtaining the rupture models isbased on Ji et al (2002), and is uniform for all the events The compiled databasefrom these two website consists of 45 rupture models coming from 33 earthquakes inthe magnitude range of Mw 7
9.15 from various seismic zones in the world Theseslip maps have been derived by the inversion of low-pass filtered ground motiondata The location of the epicentre, average slip, total seismic moment, faultingmechanism and dimensions of the fault plane of the 45 slip models are reported in

tables 1and2 The slip database consists of 36 thrust events, 2 normal faulting anism and 7 strike-slip earthquakes The epicentres of these large events along with

plate boundaries as reported by Bird (2003) are shown infigure 1 Slip fields of somelarge earthquakes are shown infigure 2(a)
(d) It can be observed that the slip distri-butions exhibit high complexity which cannot be modelled through simple mathe-matical functions Although the slip is continuous, the fault geometry for Kashmirand Mexico events is not planar The source model of Mexico event consists of slipdistribution on four planes, whereas Kashmir event consists of two rupture planes.

3 Scaling laws for source dimensions

The first step in characterizing the slip models is to understand the relationshipbetween magnitude or seismic moment and the rupture dimensions These relationsare fundamental to develop source models for simulating ground motions due tolarge events In figure 3, the length, width and area of the fault plane as reported

in the source inversion are shown as a function of seismic moment The mean value

of the slip is estimated from its spatial distribution on the rupture plane and its tion with seismic moment is shown infigure 3 In the same figure, a straight line ofthe form

where Y is the source dimension, is also fitted to the data The regression constantsfor L, W, D and fault area are reported intable 4along with the standard error Itcan be observed that the slope C1for all the four parameters lies in between 0.28 and0.55, respectively The theoretical relation between seismic moment (M0) and thesource dimensions is given by (Aki & Richards1980)

where L and W are the length and width of the fault and D is the average slip.m is therigidity of the medium surrounding the fault If stress drop remains constant,increase in the seismic moment occurs due to proportionately equal changes in L , W

Figure 1 Large earthquakes used in this study (lines
plate boundaries from Bird (2003))

Figure 2 (a) Slip distribution of 2008 Kashmir earthquake (Mw7.6) (b) Slip distribution of

2010 El Mayor-Cucapah, Mexico earthquake (Mw7.2) (c) Slip distribution of 2010 Indonesiaearthquake (M 7.82) (d) Slip distribution of 2010 Maule, Chile earthquake (M 8.9)

and average slip D The self-similar scaling can be expressed as M0/ L1 6 3, M0/ W

1 6 3, M0/ (LW )2 6 3and M0/ D1 6 3 Assuming the slope from the self-similar scaling,the intercept can be found from the data The obtained empirical equation assumingself-similarity is shown infigure 3along with the data The self-similar scaling equa-tions are reported intable 5along with the standard error The obtained slope fromthe data (C1) for all the quantities is of the same order indicating self-similar scaling

It can be observed that the source dimensions linearly increases with increasing mic moment for large events

seis-3.1 Effective source dimensions

In earthquake source inversion, fault dimensions are generally chosen large to mapthe entire rupture It can be observed fromfigure 2that slip along the edges of therupture plane is zero or very small compared to the mean slip In such cases, the

Figure 3 Source dimensions and mean slip as a function of seismic moment: (a) area of therupture plane versus moment; (b) rupture length versus moment; (c) rupture width versusmoment; (d) mean slip versus moment

length and the width of the reported slip models will overestimate the true rupturedimensions Estimating the exact source dimension from the slip distribution is diffi-cult To circumvent this problem, there have been techniques developed based onempirical approaches Somerville et al (1999) defined the rupture dimensions based

on the slip distribution If the slip distribution along the edges of the fault is 0.3 timesless than the average slip, the entire row or the column is removed from the rupturedistribution Mai and Beroza (2000) defined the effective source dimensions based onautocorrelation function To estimate the effective length and width, marginal slipdistributions are derived by summing the slip in both the along-strike and down-dipdirections The autocorrelation function is estimated and the width of this function iscomputed as (Bracewell1986)

effec-for all the 45 slip models Infigure 4, a comparison between the effective area andthe area of fault dimensions used in the earthquake source inversion is shown Theratio between effective dimensions to original dimensions has been estimated Theratio between effective length to original length lies in between 0.51 and 0.95 with amedian value of 0.74 In down-dip direction, the median change in width is 0.76 and

it lies in between 0.43 and 0.96 for all the 45 rupture models used in this analysis Theeffective area of all the events lies in between 24% and 88% of the original sourcedimensions Empirical equations to predict effective source dimensions from seismicmoment are derived from the data The coefficients are reported intable 4 The fittedequations are shown along with the data infigure 5 The self-similar scaling relations

by constraining the slope are also shown infigure 5 The effective source dimensionsincrease with increase in the seismic moment Since the effective area is less than theoriginal source dimensions, the slip on the fault plane has to be increased to conservethe seismic moment The average effective slip variation with moment is shown in

figure 5(d) The standard deviation around the mean value also increases withincrease in the seismic moment

4 Asperities on the rupture plane

After deriving the equations for estimating the source dimensions, the next step is tounderstand the regions of concentration of large slip relative to the mean slip on therupture plane These regions are known as asperities There is no guideline available

to determine the threshold value of slip to define an asperity The approaches in theliterature have been empirical and based on personal judgement Somerville et al.(1999) define asperities as rectangular regions whose average slip is 1.5 times morethan the mean slip on the entire rupture plane In this study, the approach of Mai

et al (2005) based on the ratio of slip distribution on the rupture plane to the mum slip is used to define asperities The subfaults on which the ratio lies in between0.33  D6 D  0.66 are taken as large asperity The regions where the ratio

Table 3 Effective source parameters.

S no (Km) (Km) 1.0EC04sq.km 1.0EC03(cm) (cm)

Figure 4 Comparison between effective and original rupture area.

Table 4 Scaling relations of slip models log10(Y)D C0C C1log10(M0)

Figure 5 Effective source dimensions and effective mean slip as a function of seismicmoment: (a) area of the rupture plane versus moment; (b) rupture length versus moment;(c) rupture width versus moment; (d) mean slip versus moment; (e) standard deviation of slipversus moment.

(D6 Dmax) is greater than 0.66 is defined as a very large asperity A very large asperity

is always enclosed by large asperity In figure 6, the area of very large asperity(AVLA) and large asperity (ALA) are shown as function of seismic moment for all the

45 events The combined area of asperities (ACA) is also shown infigure 6(c) It can

be observed that asperities increase with increasing seismic moment Empirical tions between log(A) and log(M0) are fitted to the data and the constants are found

equa-by regression analysis These are reported intable 4along with the standard error inthe regression The large asperities occupy about 10%
55% of the effective rupturearea, whereas the area of very large asperities is about 2%
40% of Aefffor all the 45events The combined area of asperities lies in between 12% and 78% of the effectivearea It will be of interest to know how self-similar scaling relations model the data.The self-similar scaling relations are derived by constraining the slope to be 26 3 andthe intercept is obtained from the regression on the data It can be observed that thearea of very large asperities deviates from self-similarity, whereas large asperities arecloser to the self-similar relations

Figure 6 Scaling of the size of asperities with seismic moment