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

Geomatics, Natural Hazards and Risk ISSN: 1947-5705 (Print) 1947-5713 (Online) Journal homepage: http://www.tandfonline.com/loi/tgnh20 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 Submit your article to this journal Article views: 113 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=tgnh20 Download by: [203.128.244.130] Date: 15 March 2016, At: 00:40 Geomatics, Natural Hazards and Risk, 2016 Vol 7, No 2, 493À521, http://dx.doi.org/10.1080/19475705.2014.941418 A stochastic model for earthquake slip distribution of large events S.T.G RAGHUKANTH* and S SANGEETHA Department of Civil Engineering, Indian Institute of Technology, Madras 600036, India Downloaded by [203.128.244.130] at 00:40 15 March 2016 (Received 13 January 2014; accepted July 2014) 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 Introduction Large-magnitude earthquakes (Mw > 7) occur frequently in active regions like Himalaya and northeast India Even in the Indian shield, Gujarat region also experiences such 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 (Mw D 7.7) caused severe fatalities and affected the economy of the Gujarat region Recently, Raghukanth (2011) developed the earthquake catalogue for India and ranked the 48 urban agglomerations in India based on seismicity The maximum possible magnitude in a control region of radius 300 km around the 24 urban agglomerations lies in between Mw D 7.1 and Mw D 8.7 This necessitates the estimation of the seismic input (design ground motion) in an accurate fashion for such large events to reduce the damages to structures Cases where the recorded strong motion data are not available, the source mechanism models where in the earthquake slip distribution and medium properties can be modelled analytically are preferred to simulate ground motion for such large events These models require the earthquake forces to be specified 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 ground motions can be computed for an Earth medium for a given slip distribution on the rupture plane These models provide reliable ground motion predictions if the fault and its slip distribution are known Specifying the slip distribution on the rupture plane for future events is the most challenging problem in mechanistic models To address this issue, there have been efforts to obtain spatial distribution of slip on the rupture plane by inverting ground motion records of the past earthquakes (Hartzell & Heaton 1983; Hartzell & Liu 1995; Ji et al 2002; Raghukanth & Iyengar 2008) *Corresponding author Email: raghukanth@iitm.ac.in Ó 2014 Taylor & Francis Downloaded by [203.128.244.130] at 00:40 15 March 2016 494 S.T.G Raghukanth and S Sangeetha 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 be modelled by stochastic approaches only These techniques require very few parameters to characterize the slip field Much effort has been made by the previous investigators in this direction (Somerville et al 1999; Mai & Beroza 2002; Lavallee et al 2006; Raghukanth & Iyengar 2009; Raghukanth 2010) Without going into the details regarding time-dependent stresses on the fault plane, few parameters have been identified from the slip distribution of past events The slip distribution is modelled as a random field with a specified power spectral density (PSD) A total of 15 slip distributions with the magnitude of the events ranging from 5.66 to 7.22 have been 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 large events This puts a serious limitation on the random field model developed by the previous investigators for simulating slip distribution for large events Due to advances in instrumentation, several large events have been recorded by the broadband instruments operating around the world These data have been processed and slip models for 45 large events are available in the literature Since large events are of concern to engineers, it would be interesting to examine these slip distributions In this paper, stochastic characterization of slip distribution is explicitly developed for large events Important properties of the random field are estimated from the PSD of slip distribution Empirical equations for estimating the slip field from magnitude are developed in this paper Slip database of large events Inversion for earthquake sources is fundamental to understand the mechanics of earthquakes The extracted slip models can be used to understand the damages in the epicentral region Much effort has been made by seismologists in developing methods to extract slip distribution on the rupture plane from ground motion records After the occurrence of a large event, the Incorporated Research Institutions for Seismology data management centre reports the broadband velocity data recorded by the Global Seismic Network (GSN) The preliminary earthquake slip distribution is determined from this data by several research groups In case of local strong motion data, global positioning system and ground deformation measurements become available, these records are combined with the GSN data to obtain the spatial distribution of slip on the rupture plane Several such slip maps for large events are available 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 observatory, California Institute of Technology (http://www.tectonics.caltech.edu6 ), are used to develop the model The methodology for obtaining the rupture models is based on Ji et al (2002), and is uniform for all the events The compiled database from these two website consists of 45 rupture models coming from 33 earthquakes in the magnitude range of Mw 7À9.15 from various seismic zones in the world These slip maps have been derived by the inversion of low-pass filtered ground motion data The location of the epicentre, average slip, total seismic moment, faulting mechanism and dimensions of the fault plane of the 45 slip models are reported in tables and The slip database consists of 36 thrust events, normal faulting mechanism and strike-slip earthquakes The epicentres of these large events along with 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 S no Kepulauan Antofagasta, Chile Solomon Islands Pisco, Peru PagaiIsland, Indonesia Benkulu, Indonesia Southern Java, Indonesia Kuril Islands Kuril Islands Northern California India Honshu, Japan Kashmir, Pakistan Kuril Islands Chile Kuril Islands New Britian Region Bhuj, India Peru North Sumatra Location 106 076 126 116 016 066 036 036 066 076 086 106 106 076 116 016 016 046 086 096 096 096 096 116 046 306 036 176 266 236 286 286 156 246 166 086 086 176 156 136 136 016 156 126 126 126 126 146 94 95 95 00 01 01 05 05 05 05 05 05 05 06 06 07 07 07 07 07 07 07 07 07 Date (m6 d6 yy) 43.77 ¡23.34 44.66 ¡05.50 23.42 ¡16.26 02.09 02.09 41.29 07.92 38.28 34.54 34.54 ¡09.28 46.59 46.24 46.24 ¡08.47 ¡13.39 ¡02.62 ¡04.44 ¡04.52 ¡02.62 ¡22.25 Latitude ( ) 147.32 ¡70.29 149.30 151.78 70.23 ¡73.64 97.11 97.11 ¡125.95 92.19 142.04 73.59 73.59 107.42 153.27 154.52 154.52 157.04 ¡76.60 100.84 101.37 101.38 100.84 ¡69.89 Longitude ( ) Table Slip models used in this study RV RV RV RV RV RV RV RV SS SS RV RV RV RV RV N N RV RV RV RV RV RV RV Mech (RV6 N6 SS) Downloaded by [203.128.244.130] at 00:40 15 March 2016 8.36 8.14 7.81 7.5 7.6 8.4 8.68 8.5 7.2 7.25 7.19 7.6 7.64 7.9 8.3 8.1 8.1 8.1 7.9 8.5 8.5 7.94 7.81 Mw 3.89EC021 1.82EC021 5.82EC020 2.00EC020 2.82EC020 4.47EC021 1.17EC022 6.31EC021 7.10EC019 8.40EC019 6.80EC019 2.82EC020 3.24EC020 7.94EC020 3.16EC021 1.59EC021 1.59EC021 1.59EC021 1.12EC021 7.94EC020 4.47EC021 4.47EC021 9.12EC020 5.82EC020 M0 (Nm) (continued) 1 1 1 2 1 2 2 2 1 Ref Geomatics, Natural Hazards and Risk 495 Turkey El Mayor-Cucapah, Mexico Kepulauan, Indonesia Honshu, Japan Honshu, Japan Vanuatu Islands Haiti Maule, Chile Sulawesi, Indonesia Padang, Indonesia Simeulue, Indonesia Tibet, China East Sichuan, China Location 116 026 036 056 056 116 096 096 106 016 026 026 046 106 036 036 036 036 036 036 106 146 206 206 126 126 166 306 306 076 126 276 276 046 256 096 116 116 116 116 116 236 07 08 08 08 08 08 09 09 09 10 10 10 10 10 11 11 11 11 11 11 11 Date (m6 d6 yy) Longitude ( ) ¡69.89 95.96 81.47 103.32 103.32 122.1 99.87 99.87 166.18 ¡72.57 ¡72.90 ¡72.72 ¡115.28 100.12 142.84 142.34 142.34 142.86 142.86 142.80 43.51 Latitude ( ) ¡22.25 02.77 35.49 31.00 31.00 01.27 ¡00.72 ¡00.72 ¡13.05 18.44 ¡36.12 ¡35.84 32.30 ¡3.480 38.44 38.32 38.32 38.10 38.10 38.10 38.72 RV RV N SS RV RV SS RV RV SS RV RV SS RV RV RV RV RV RV RV RV Mech (RV6 N6 SS) Note: 1: www.geol.ucsb.edu; 2: www.tectonics.caltech.edu; Faulting mechanism: RV À reverse, SS À strike slip, N Ànormal 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 S no Table (Continued ) Downloaded by [203.128.244.130] at 00:40 15 March 2016 7.78 7.4 7.14 7.9 7.97 7.3 7.6 7.6 7.6 8.9 8.8 7.2 7.82 7.4 9.1 9.1 9.1 9.1 7.13 Mw 3.98EC020 1.41EC020 5.80EC019 7.94EC020 1.01EC021 1.00EC020 2.82EC020 2.82EC020 2.82EC020 3.50EC019 2.51EC022 1.78EC022 7.10EC019 6.03EC020 1.41EC020 5.01EC022 5.01EC022 5.01EC022 5.01EC022 3.55EC022 5.60EC019 M0 (Nm) 2 2 2 2 2 1 1 1 Ref 496 S.T.G Raghukanth and S Sangeetha 10 11 12 13 14 15 16 17 18 19 20 21 22 23 S no 255 240 168 168 65 300 380 416 102 98 112 76 126 240 315 200 224 300 192 240 400 560 312.5 Length, L (Km) 121 156 112 100 41.6 190.4 260 320 35 42 72 35 54 162.5 132 35 40 80 210 190 368 159.5 130 Width, W (Km) 234.87 96.32 67.57 27.68 357.89 173.49 255.67 119.38 67.19 67.35 14.71 294.34 175.09 152.82 173.59 702.21 356.32 147.38 58.14 36.93 55.99 90.23 39.37 Mean slip, (cm) 15 15 14 12 15 20 16 12 15 8 15 12 12 16 20 12.5 Subfault size, dx (Km) 11 13 14 10 5.2 13.6 20 16 3.5 12.5 12 5 10 10 10 16 14.5 10 Subfault Size, dz (Km) 54 226 240 82 308.5 326 325 221 118 24 3206 343 331 289 220 42 42 305 318 323 324 323 319 Strike ( ) Dip ( ) 76 18 18 32 51 15 10 88 80 70 29 29 10 15 57.89 58 25 066 206 30 15 15 12 19 Table Source dimensions and orientation of the fault plane Downloaded by [203.128.244.130] at 00:40 15 March 2016 123.4 90.9 100.3 63.5 75.7 54.2 117.2 90.1 362.3 198.7 90.2 102.9 124.9 83.5 99.4 246.2 ¡97.9 85.4 59.4 96.8 94.4 110.1 98.1 Rake ( ) (continued) 0.309 0.374 0.188 0.168 0.027 0.571 0.988 1.331 0.036 0.041 0.081 0.021 0.068 0.390 0.416 0.070 0.090 0.240 0.403 0.456 1.472 0.893 0.406 Area (1.0EC05Ã sq.km) Geomatics, Natural Hazards and Risk 497 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 S no 375 162 152 140 260 315 120 54 48 91 45 260 570 171 270 171 475 475 475 475 625 45 Length, L (Km) 200 126 112 45 28 40 56 45 45 60 22.5 187 180 21 100 104 200 200 200 200 280 45 Width, W (Km) 21.89 87.95 15.27 33.21 376.12 278.51 45.27 158.25 177.76 87.34 144.93 405.48 229.28 93.60 72.51 47.62 1219.88 1219.88 1219.88 1254.31 782.86 90.39 Mean slip, (cm) 15 10 10 15 6 13 30 15 25 25 25 25 25 Subfault size, dx (Km) 9 5 5 2.5 17 15 10 20 20 20 20 20 Subfault Size, dz (Km) Strike ( ) 355.08 302 206 229 229 93 193 72 346 836 2576 90 17.5 18 3556 3126 131 322 190 199 198 198 198 201 248 Table (Continued ) Dip ( ) 16.56 20 48 33 33 22 58 51 40 706 556 45 18 18 456 756 60 7.5 11 10 10 10 10 45 Downloaded by [203.128.244.130] at 00:40 15 March 2016 95.7 85.6 90.5 287.3 136.8 120.1 88.1 44.2 122.2 59.7 34.9 113.3 108.6 ¡6.3 99.7 80.4 67.2 67.2 67.2 67.1 89.5 65.1 Rake ( ) 0.750 0.204 0.170 0.063 0.073 0.126 0.067 0.024 0.022 0.055 0.010 0.486 1.026 0.036 0.270 0.083 0.950 0.950 0.950 0.950 1.750 0.020 Area (1.0EC05Ã sq.km) 498 S.T.G Raghukanth and S Sangeetha Geomatics, Natural Hazards and Risk 499 Downloaded by [203.128.244.130] at 00:40 15 March 2016 Figure Large earthquakes used in this study (linesÀplate boundaries from Bird (2003)) plate boundaries as reported by Bird (2003) are shown in figure Slip fields of some large earthquakes are shown in figure 2(a)À(d) It can be observed that the slip distributions exhibit high complexity which cannot be modelled through simple mathematical functions Although the slip is continuous, the fault geometry for Kashmir and Mexico events is not planar The source model of Mexico event consists of slip distribution on four planes, whereas Kashmir event consists of two rupture planes Scaling laws for source dimensions The first step in characterizing the slip models is to understand the relationship between magnitude or seismic moment and the rupture dimensions These relations are fundamental to develop source models for simulating ground motions due to large 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 variation with seismic moment is shown in figure In the same figure, a straight line of the form log10 ðY Þ ¼ C0 þ C1 logðM0 Þ (1) where Y is the source dimension, is also fitted to the data The regression constants for L, W, D and fault area are reported in table along with the standard error It can be observed that the slope C for all the four parameters lies in between 0.28 and 0.55, respectively The theoretical relation between seismic moment (M0) and the source dimensions is given by (Aki & Richards 1980) M0 ¼ mLWD (2) where L and W are the length and width of the fault and D is the average slip m is the rigidity 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 S.T.G Raghukanth and S Sangeetha Downloaded by [203.128.244.130] at 00:40 15 March 2016 500 Figure (a) Slip distribution of 2008 Kashmir earthquake (Mw 7.6) (b) Slip distribution of 2010 El Mayor-Cucapah, Mexico earthquake (Mw 7.2) (c) Slip distribution of 2010 Indonesia earthquake (Mw 7.82) (d) Slip distribution of 2010 Maule, Chile earthquake (Mw 8.9) Downloaded by [203.128.244.130] at 00:40 15 March 2016 Geomatics, Natural Hazards and Risk 501 Figure Source dimensions and mean slip as a function of seismic moment: (a) area of the rupture plane versus moment; (b) rupture length versus moment; (c) rupture width versus moment; (d) mean slip versus moment and average slip D The self-similar scaling can be expressed as M0 / L 16 3, M0 / W 16 , M0 / (LW )26 and M0 / D 16 Assuming the slope from the self-similar scaling, the intercept can be found from the data The obtained empirical equation assuming self-similarity is shown in figure along with the data The self-similar scaling equations are reported in table along with the standard error The obtained slope from the 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 seismic moment for large events 3.1 Effective source dimensions In earthquake source inversion, fault dimensions are generally chosen large to map the entire rupture It can be observed from figure that slip along the edges of the rupture plane is zero or very small compared to the mean slip In such cases, the Geomatics, Natural Hazards and Risk 507 Downloaded by [203.128.244.130] at 00:40 15 March 2016 4.1 Location of hypocentre and asperities Another important aspect which affects the near-field ground motion is the location of hypocentre and asperities on the fault plane This information is important to understand the crack propagation during the rupture which can be further used to develop dynamic rupture models The distance of the hypocentre from the edges of the fault in along-strike (Hx ) and down-dip (Hz ) directions normalized by length and width of the fault is computed for all the 45 slip models Due to symmetry, the normalized distance in along-strike direction (Hx ) lies in between and 0.5, whereas Hz lies in between and Hx D indicates that hypocentre is located on the edge of the fault, whereas Hx D 0.5 indicates the centre of the fault Similarly, Hz D denotes the hypocentral location at the top edge of the fault and Hz D denotes the bottom edge of the rupture plane These non-dimensional quantities are shown in figure as a function of magnitude It can be observed that Hx and Hz not show any pattern with Mw The histograms are also shown in figure The Hx is distributed with a mean of 0.30 and a standard deviation 0.13, whereas for Hz, these two moments are 0.52 and 0.23, respectively The hypocentre is approximately located at the centre of the fault in down-dip direction To understand the relationship between the hypocentre and the location of the asperity, the closest distance to the asperity from the hypocentre is determined from the data In figure 8, the variation of the closest distance to large and very large asperities from hypocentre is shown as a function of seismic moment The histograms of these distances are also shown in the same figure The closest distance increases with increase in the seismic moment Large asperities are located close to the hypocentre, whereas very large asperities are located at approximately 24 km from the hypocentre The regions of maximum slip are located approximately at a distance of 50 km from the hypocentre Empirical equations between distance and moment are derived from the data with and without constraining the slope, and constants are reported in tables and Figure shows the closest distance to asperities Table Scaling relations of slip models assuming self-similarity log10(Y) D C0 C C1 log10(M0); C1 is fixed Y L W A D sD Leff Weff Aeff Deff ALA AVLA ACA RDmax RLA RVLA C0 ¡4.70 ¡5.03 ¡9.74 ¡4.87 ¡4.84 ¡4.85 ¡5.16 ¡10.02 ¡5.16 ¡10.54 ¡10.88 ¡10.36 ¡5.47 ¡6.02 ¡5.84 C1 s(e) 0.33 0.33 0.67 0.33 0.33 0.33 0.33 0.67 0.33 0.67 0.67 0.67 0.33 0.33 0.33 0.19 0.24 0.36 0.37 0.31 0.20 0.21 0.32 0.45 0.33 0.33 0.31 0.35 0.42 0.42 S.T.G Raghukanth and S Sangeetha Downloaded by [203.128.244.130] at 00:40 15 March 2016 508 Figure Normalized hypocentre position in (a) along-strike and (b) down-dip directions Downloaded by [203.128.244.130] at 00:40 15 March 2016 Geomatics, Natural Hazards and Risk 509 Figure Scaling of the closest distance to asperities with seismic moment: (a) large asperity; (b) very large asperity; (c) location of maximum slip normalized by maximum distance to the farthest subfault on the plane, Rmax as a function of moment magnitude It can be observed that they not show any pattern with Mw The histograms are also shown in the same figure It is interesting to note that asperities are not located randomly on the rupture plane but they lie close to the hypocentre Power spectral density (PSD) The above empirical equations can be used to fix up the rupture dimensions, average slip and location of hypocentre for a given seismic moment However, to simulate ground motion by source mechanism model requires complete slip distribution on the rupture plane One requires representation of the slip field in terms of mathematical functions The previously derived equations provide information on the average properties of the slip field The possibility of representing the slip field in terms of Downloaded by [203.128.244.130] at 00:40 15 March 2016 510 S.T.G Raghukanth and S Sangeetha Figure Closest distances to asperities and Dmax normalized by maximum distance to the farthest subfault on the plane simple mathematical expressions is ruled out due to the randomness in the derived rupture models It can be observed from figure that the obtained slip component of large events are erratic which can be attributed to randomness observed in ground motion records More number of parameters are required to characterize completely the spatial distribution of slip The only way to model the slip distribution is through stochastic approaches where a few parameters are sufficient to explain the complex data The two striking features of the slip models are randomness and non-stationarity in their spatial distribution Assuming the slip as a homogeneous random field, the spatial mean and standard deviation are computed for all the 45 slip models For estimating the further statistics, the slip field is standardized as Dðx; zÞ ¼ Dðx; zÞ ¡ h D i sD (4) In second-order analysis, the variation of random field models at two different locations is characterized either in space by an autocorrelation function or in the Geomatics, Natural Hazards and Risk 511 wave-number domain by a PSD Assuming the slip field as ergodic, two-dimensional PSD (S (kx,kz)) is obtained from the slip distribution as Downloaded by [203.128.244.130] at 00:40 15 March 2016 Z Sðkx ; kz Þ ¼ ¡1 Z ¡1 Dðx; zÞ e ¡ ikx x ¡ ikz z dxdz (5) where kx and kz are the spatial wave numbers The obtained standardized slip field is transformed into the two-dimensional wave-number domain using zero-padded grids of size 1024 £ 1024 km The Nyquist wave number in both the directions depends on the size of the subfaults determined from the earthquake source inversion It can be observed from table that the size of the subfaults is not uniform along the length and width of the fault for all the 45 models Hence the highest wave number for which the slip model is valid will be different for all the events The two-dimensional PSD function computed from ‘equation (5)’ for Kashmir earthquake slip model shown in figure 2(a) is shown in figure 10 A random field is known as isotropic, if Figure 10 Two-dimensional power spectral density function for the slip model of Kashmir event shown in figure 2(a) 512 S.T.G Raghukanth and S Sangeetha the correlation is independent of direction (Vanmarcke 1983) It can observed from figure 10 that the correlation structure is different in along-strike and dip-directions indicating that the moment field is anisotropic In figure 11, the PSD functions at cross sections kx D and kz D are also shown There are several theoretical twodimensional correlation functions available in the literature Three correlation functions widely used in literature are Gaussian, exponential and von Karman (Mai & Beroza 2002) The expressions for the autocorrelation and PSD for these three random field models are as follows: Gaussian: Rðzx ; zz Þ ¼ e ¡ 2 zx z2z þ a2x a2z  Sðkx ; kz Þ ¼ ax az ¡ 1ða2x k2x þa2z k2z Þ e (6) Downloaded by [203.128.244.130] at 00:40 15 March 2016 Exponential: Rðzx ; zz Þ ¼ e ¡ qffiffiffiffiffiffiffiffi ffi 2 zx zz þ a2x a2z Sðkx ; kz Þ ¼ ax az ð1 þ a2x k2x þ a2z k2z Þ2 (7) von Karman: Rðzx ; zz Þ ¼ GH ðrÞ GH ð0Þ Sðkx ; kz Þ ¼ where GH ðrÞ ¼ rH KH ðrÞ ax az ð1 þ a2x k2x þ a2z k2z ÞHþ1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z2x z2z r¼ þ a2x a2z (8) (9) ax and az are the correlation lengths along x and z-directions, respectively H is the Hurst exponent and KH is the modified Bessel function of the first kind It can be observed that when H D 0.5 von Karman is identical to the exponential PSD The parameters ax and az of the three random fields are estimated from the slip field by minimizing the mean square error between the computed PSD and the expressions shown in equations (6)À(8) The fitted Gaussian, exponential and von Karman PSD at cross sections kx D and kz D are plotted in figure 11 along with the data for Kashmir earthquake Besides, the estimated parameters are also shown in the same figure The misfit associated with von Karman function is slightly lower than that obtained from exponential and Gaussian function, and hence the slip fluctuation can be modelled as a von Karman random field Similarly, the correlation lengths are estimated for all the slip models These are reported in table for Gaussian, exponential and von Karman PSD The correlation length ax is much larger than az which can be attributed to large length compared to the width of the fault Scaling laws for correlation lengths After estimating the spectral parameters from PSD, it remains to identify the patterns with the previous determined effective source dimensions and moment 513 Downloaded by [203.128.244.130] at 00:40 15 March 2016 Geomatics, Natural Hazards and Risk Figure 11 Comparison with the Gaussian, exponential and von Karman PSD at the cross sections kx D and kz D for the slip model of Kashmir event shown in figure 2(a) [...]... 15 March 2016 506 S.T.G Raghukanth and S Sangeetha Figure 6 Scaling of the size of asperities with seismic 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... figure 11 along with the data for Kashmir earthquake Besides, the estimated parameters are also shown in the same figure The misfit associated with von Karman function is slightly lower than that obtained from exponential and Gaussian function, and hence the slip fluctuation can be modelled as a von Karman random field Similarly, the correlation lengths are estimated for all the slip models These are reported... the slip models are randomness and non-stationarity in their spatial distribution Assuming the slip as a homogeneous random field, the spatial mean and standard deviation are computed for all the 45 slip models For estimating the further statistics, the slip field is standardized as Dðx; zÞ ¼ Dðx; zÞ ¡ h D i sD (4) In second-order analysis, the variation of random field models at two different locations... that the obtained slip component of large events are erratic which can be attributed to randomness observed in ground motion records More number of parameters are required to characterize completely the spatial distribution of slip The only way to model the slip distribution is through stochastic approaches where a few parameters are sufficient to explain the complex data The two striking features of. .. [203.128.244.130] at 00:40 15 March 2016 Geomatics, Natural Hazards and Risk Figure 11 Comparison with the Gaussian, exponential and von Karman PSD at the cross sections kx D 0 and kz D 0 for the slip model of Kashmir event shown in figure 2 (a) 514 S.T.G Raghukanth and S Sangeetha Table 6 Estimates of correlation lengths and Hurst exponents, H, for slip models listed in table 1 Downloaded by [203.128.244.130] at... self-similarity of earthquakes Empirical equations by constraining the slope in the regression assuming self-similarity have been derived from the data It can be observed from figures 3 and 4 that the source dimensions follow self-similarity The area of asperities and their location on the rupture plane also have been examined for all the slip models The asperities have been classified into large and very large. .. very large asperity based on the ratio of local slip to the average slip on the rupture plane It is observed that large asperities constitute 10%À55% of the effective rupture area, and very large asperities constitute about 2%À40% of the Aeff, respectively The area of asperities also increases with increase in the seismic moment Empirical equations have been developed to estimate the area of asperities... the area of very large asperities deviates from self-similarity, whereas large asperities are closer to the self-similar relations Geomatics, Natural Hazards and Risk 507 Downloaded by [203.128.244.130] at 00:40 15 March 2016 4.1 Location of hypocentre and asperities Another important aspect which affects the near-field ground motion is the location of hypocentre and asperities on the fault plane This... whereas very large asperities are located at approximately 24 km from the hypocentre The regions of maximum slip are located approximately at a distance of 50 km from the hypocentre Empirical equations between distance and moment are derived from the data with and without constraining the slope, and constants are reported in tables 4 and 5 Figure 9 shows the closest distance to asperities Table 5 Scaling... at 00:40 15 March 2016 Geomatics, Natural Hazards and Risk 519 Figure 14 (Continued) estimated from the autocorrelation width To conserve the seismic moment, slip on the fault plane is increased by the ratio between area and effective area and effective mean slip is obtained from the data Scaling laws to estimate source dimensions from seismic moment have been derived from the data The data also have ... 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... this paper Slip database of large events Inversion for earthquake sources is fundamental to understand the mechanics of earthquakes The extracted slip models can be used to understand the damages... 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