A stochastic model for earthquake slip distribution of large events

Geomatics, Natural Hazards and Risk

ISSN: 1947-5705 (Print) 1947-5713 (Online) Journal homepage: />
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
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Date: 15 March 2016, At: 00:40


Geomatics, Natural Hazards and Risk, 2016
Vol. 7, No. 2, 493À521, />
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 1 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.

1. 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:
Ó 2014 Taylor & Francis


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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.

2. 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 (6 faculty6 ji6 ) and tectonics observatory, California Institute of Technology (6 ), 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 1 and 2. The slip database consists of 36 thrust events, 2 normal faulting mechanism and 7 strike-slip earthquakes. The epicentres of these large events along with


1
2
3
4

5
6
7
8
9
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 1. 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)

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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
8
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
1
2
1
1
2
2
1
1
2
1
2
2
1
2
1
2
2
2
1
1
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 1. (Continued )

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7.78
7.4
7.14
7.9
7.97
7.3
7.6
7.6
7.6
7
8.9
8.8
7.2
7.82
7.4
9.1
9.1
9.1
9.1
9
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
1

2
1
2
2
2
2
2
1
2
2
1
1
1
1
1
1
2
1

Ref.

496
S.T.G. Raghukanth and S. Sangeetha


1
2
3
4
5

6
7
8
9
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,
<D>
(cm)
15
15
14
12
5
15
20
16
6
7
8
4
9
12
15
8

8
15
12
12
16
20
12.5

Subfault size,
dx
(Km)
11
13
14
10
5.2
13.6
20
16
5
7
8
3.5
9
12.5
12
5
5
10
10

10
16
14.5
10

Subfault Size,
dz
(Km)
54
4
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
8
10
88
80
70
29
29
10
15
57.89
58
25
066 206 30
15
15
12
19

Table 2. Source dimensions and orientation of the fault plane.

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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,
<D>
(cm)
15
9
8
10
10
15
8
6
6
7
3
13
30

3
15
8
25
25
25
25
25
5

Subfault size,
dx
(Km)
8
9
8
9
4
5
4
5
5
5
2.5
17
15
3
10
8
20

20
20
20
20
5

Subfault Size,
dz
(Km)
Strike ( )
355.08
5
302
206
229
229
93
193
72
346
836 2576 6 90
17.5
18
3556 3126 131
322
190
199
198
198
198

201
248

Table 2. (Continued )

Dip ( )
16.56
20
7
48
33
33
22
58
51
40
706 556 45
18
18
456 756 60
7.5
11
10
10
10
10
9
45

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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

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Figure 1. Large earthquakes used in this study (linesÀplate boundaries from Bird (2003)).

plate boundaries as reported by Bird (2003) are shown in figure 1. 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.
3. 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 3. 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 4 along with the standard error. It
can be observed that the slope C 1 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

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500

Figure 2. (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).


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Geomatics, Natural Hazards and Risk

501

Figure 3. 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 3
, M0 / (LW )26 3 and M0 / D 16 3. 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 3 along with the data. The self-similar scaling equations are reported in table 5 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 2 that slip along the edges of the
rupture plane is zero or very small compared to the mean slip. In such cases, the


502

S.T.G. Raghukanth and S. Sangeetha

length and the width of the reported slip models will overestimate the true rupture
dimensions. Estimating the exact source dimension from the slip distribution is difficult. To circumvent this problem, there have been techniques developed based on
empirical 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 times
less than the average slip, the entire row or the column is removed from the rupture
distribution. Mai and Beroza (2000) defined the effective source dimensions based on
autocorrelation function. To estimate the effective length and width, marginal slip
distributions are derived by summing the slip in both the along-strike and down-dip
directions. The autocorrelation function is estimated and the width of this function is
computed as (Bracewell 1986)

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1
Wa ¼
DÃDj0

Z

1
¡1

DÃDdx

(3)

where Wa is the autocorrelation width and à is the convolution operator. The effective length and width of the rupture plane are estimated from the marginal slip distribution in along- strike and down-dip directions. In a similar fashion, effective length
and width have been estimated for all the slip models. These are reported in table 3
for all the 45 slip models. In figure 4, a comparison between the effective area and
the area of fault dimensions used in the earthquake source inversion is shown. The
ratio between effective dimensions to original dimensions has been estimated. The
ratio between effective length to original length lies in between 0.51 and 0.95 with a
median 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. The
effective area of all the events lies in between 24% and 88% of the original source
dimensions. Empirical equations to predict effective source dimensions from seismic
moment are derived from the data. The coefficients are reported in table 4. The fitted
equations are shown along with the data in figure 5. The self-similar scaling relations
by constraining the slope are also shown in figure 5. The effective source dimensions
increase with increase in the seismic moment. Since the effective area is less than the
original source dimensions, the slip on the fault plane has to be increased to conserve
the seismic moment. The average effective slip variation with moment is shown in
figure 5(d). The standard deviation around the mean value also increases with
increase in the seismic moment.
4. Asperities on the rupture plane
After deriving the equations for estimating the source dimensions, the next step is to
understand the regions of concentration of large slip relative to the mean slip on the

rupture 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 the
literature have been empirical and based on personal judgement. Somerville et al.
(1999) define asperities as rectangular regions whose average slip is 1.5 times more
than 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 maximum slip is used to define asperities. The subfaults on which the ratio lies in between
0.33  D6 Dmax  0.66 are taken as large asperity. The regions where the ratio


Geomatics, Natural Hazards and Risk

503

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Table 3. Effective source parameters.

S. no.

Leff
(Km)

Weff
(Km)

Aeff
1.0EC04 Ã sq.km

Deff
1.0EC03Ã(cm)

s (D)
(cm)

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26

27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45

184.5
192.4
132.1
99.9
43.6
175.0
302.8
263.3
79.6
52.3

56.9
60.4
96.6
221.5
281.2
182.6
173.4
241.8
100.1
125.7
220.3
353.0
193.4
230.9
153.9
84.4
115.2
191.7
243.6
90.3
39.7
37.3
77.3
37.5
178.5
483.6
118.6
213.3
118.6
282.4

282.4
282.4
287.6
484.7
29.2

95.9
121.4
89.3
74.6
36.7
166.6
190.2
175.8
22.2
23.2
36.1
28.6
34.8
154.9
123.6
33.3
27.1
76.1
101.1
96.2
157.7
146.5
105.4
137.9

84.5
77.1
27.6
24.3
30.6
32.4
32.0
34.3
43.3
17.4
178.6
170.2
13.4
68.1
83.6
162.5
162.5
162.5
144.4
194.3
36.2

1.7706
2.3368
1.1794
0.7452
0.1605
2.9151
5.761
4.6315

0.177
0.1215
0.2056
0.1729
0.3364
3.4333
3.4772
0.6095
0.4712
1.8412
1.0124
1.2096
3.4759
5.1724
2.0386
3.184
1.3019
0.6507
0.3182
0.4668
0.7461
0.2933
0.1274
0.1281
0.3355
0.0655
3.1885
8.236
0.1596
1.4518

0.481
4.5908
4.5908
4.5908
4.1549
9.4194
0.1059

0.4491
0.2934
0.1206
0.0679
0.7798
0.1297
0.1531
0.6073
0.0405
0.0661
0.0622
0.6137
0.1157
0.0649
0.2997
0.8065
0.1395
0.1099
0.3732
0.0001
0.0003
0.2191

0.0142
0.1063
0.1912
0.0001
0.0188
0.2003
0.4845
0.0675
0.4502
0.4564
0.2337
0.2378
0.6108
0.2057
0.0576
0.0711
0.1587
3.2795
3.2795
3.2795
1.8181
1.8136
0.2992

243.10
96.24
58.34
30.42
347.90
186.86

273.07
200.46
77.72
106.79
27.07
254.38
181.95
78.77
103.61
455.48
339.42
97.42
136.66
79.70
134.75
92.36
54.57
39.41
75.16
23.65
36.40
361.89
250.51
58.52
153.47
154.06
90.63
114.97
321.43
154.92

121.92
70.11
48.62
1347.45
1347.45
1347.45
1526.18
764.28
93.10


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504

S.T.G. Raghukanth and S. Sangeetha

Figure 4. Comparison between effective and original rupture area.
Table 4. Scaling relations of slip models log10(Y) D C0 C C1 log10(M0).
Y
L
W
A
D
sD
Leff
Weff
Aeff
Deff
ALA

AVLA
ACA
RDmax
RLA
RVLA

C0
¡3.58
¡3.64
¡7.39
¡5.87
¡5.49
¡3.52
¡4.34
¡8.02
¡6.50
¡8.33
¡6.75
¡7.45
¡4.7
¡1.39
¡3.97

C1

s(e)

0.28
0.27
0.55

0.38
0.36
0.27
0.29
0.57
0.40
0.56
0.47
0.53
0.31
0.11
0.24

0.19
0.23
0.34
0.36
0.31
0.19
0.21
0.31
0.45
0.31
0.27
0.29
0.35
0.37
0.41

505

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Geomatics, Natural Hazards and Risk

Figure 5. Effective source dimensions and effective 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; (e) standard deviation of slip
versus moment.


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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 (ACA) is also shown in figure 6(c). It can
be observed that asperities increase with increasing seismic moment. Empirical equations between log(A) and log(M0) are fitted to the data and the constants are found
by regression analysis. These are reported in table 4 along with the standard error in
the regression. The large asperities occupy about 10%À55% of the effective rupture
area, whereas the area of very large asperities is about 2%À40% of Aeff for all the 45
events. The combined area of asperities lies in between 12% and 78% of the effective
area. 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 and
the intercept is obtained from the regression on the data. It can be observed that the
area of very large asperities deviates from self-similarity, whereas large asperities are
closer to the self-similar relations.


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507

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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 0 and 0.5, whereas Hz lies in between 0 and 1.
Hx D 0 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 0 denotes the hypocentral location at the
top edge of the fault and Hz D 1 denotes the bottom edge of the rupture plane. These
non-dimensional quantities are shown in figure 7 as a function of magnitude. It can be
observed that Hx and Hz do not show any pattern with Mw. The histograms are also
shown in figure 7. 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 4 and 5. Figure 9 shows the closest distance to asperities
Table 5. 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

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508

Figure 7. Normalized hypocentre position in (a) along-strike and (b) down-dip directions.


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509

Figure 8. 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 do 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.

5. 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


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510

S.T.G. Raghukanth and S. Sangeetha

Figure 9. 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 2 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

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Z


Sðkx ; kz Þ ¼



1
¡1

Z

1
¡1

Dðx; zÞ e

¡ ikx x ¡ ikz z


2