Bulletin of the Seismological Society of America, Vol. 98, No. 3, pp. 1147–1161, June 2008, doi: 10.1785/0120070134

Surface Rupturing and Buried Dynamic-Rupture Models Calibrated with
Statistical Observations of Past Earthquakes
by Luis A. Dalguer,* Hiroe Miyake, Steven M. Day, and Kojiro Irikura

Abstract

In the context of the slip-weakening friction model and simplified asperity models for stress state, we calibrate dynamic rupture models for buried andsurface-rupturing earthquakes constrained with statistical observations of past earthquakes. These observations are the kinematic source models derived from source inversions of ground-motion and empirical source models of seismic moment andrupture area. The calibrated parameters are the stress-drop distribution on the fault
and average stress drop. We develop a set of dynamic rupture models that consistof asperities and surrounding background areas. The distribution of dynamic stressdrop outside the asperity is characterized by a fraction of the stress drop on the asperity. From this set of models, we identify dynamic fault models with defined stressdrop characteristics that satisfy the observations. The selected dynamic fault models
show that surface-rupturing earthquakes are characterized by a large area of negative
stress-drop surrounding the asperities, while buried earthquakes present positive orzero stress drop. In addition, the calibrated fault models that match the observations
show that the average stress drop is independent of earthquake size for buried earthquakes, but scale dependent for surface-rupturing earthquakes. This suggests that, inthe context of our parameterization, buried earthquakes follow self-similarity scaling,and surface-rupturing earthquakes break this self-similarity. We apply the calibrated
dynamic models to simulate near-source ground motion consistent with observations
that suggest that buried earthquakes generate stronger ground motion than surfacerupturing earthquakes at high frequency. We propose possible mechanisms that satisfy
this observation, as follows: buried rupture has a hypocenter location below the asperity; this can produce strong directivity of the slip velocity function toward the freesurface. That effect, in addition to a reduced fault area and low fracture energy duringrupture, may be significant in enhancing high-frequency ground motion. On the other
hand, surface-rupturing earthquakes have a shallow hypocenter, large fracture energy
on the asperities, and enhanced energy absorption due to large areas of negative stressdrop in the background area. These characteristics of large earthquakes inhibit severe
directivity effects on the slip velocity function directly toward the free surface, reducing the high-frequency ground motion.

Introduction
A physical understanding of the earthquake rupture
process can improve our capability for predicting groundmotion, and therefore our assessment of seismic hazard.
Numerical models of the dynamic rupture of earthquakes
provide a convenient framework for incorporating physicalconstraints on the source physics into ground-motion simulations. These models usually idealize the earthquake rupture
as a propagating shear crack on a frictional interface em*
Now at Institute of Geophysics, ETH Hoenggerberg, CH-8093 Zurich,
Switzerland.

bedded in a linearly elastic continuum. This idealization
has proven to be a useful foundation for analyzing and simulating natural earthquakes (e.g., Andrews, 1976; Das andAki, 1977; Day, 1982a,b; Olsen et al., 1997; Oglesby et al.,
1998; Dalguer et al., 2001; Peyrat et al., 2001; Day et al.,2008), and we adopt it here.
The main difficulty with the use of dynamic models tosimulate realistic earthquake ground motion is the lack of
information to realistically parameterize the friction modeland the state of stress in the crust. It is currently rather difficult to incorporate observational constraints into dynamic

1147

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models. In the context of slip-weakening friction models,
several attempts have been made to investigate what frictional parameters can be constrained with observations.
For example, Ide and Takeo (1997), Bouchon et al. (1998),
Guatteri and Spudich (2000), Pulido and Irikura (2000), Ide(2002), Mikumo et al. (2003), Zhang et al. (2003), and others
studied strong ground-motion records of large earthquake toinvestigate whether the critical slip distance and fracture energy can be inferred from these observations. Theirs results
show that the fracture energy is better constrained than thecritical slip distance as pointed by Guatteri and Spudich(2000). The qualitative analysis of theses studies suggeststhat these parameters are scale dependent to the earthquake
rupture process. Other attempts to constrain dynamic parameters use observations of earthquake scaling, as, for example, Abercrombie and Rice (2005). These authors argue thatobservations suggest that static stress drop is also scale dependent. They used the scale-dependence source parameters
to constrain possible models of dynamic rupture.
Guided by a characterized source model, which consistsof asperities and background areas (e.g., Somerville et al.,
1999; Miyake et al., 2003), Dalguer et al. (2004) estimate
the ratio between the dynamic stress drops on the asperity
and background areas of the rupture surface, for surface rupturing and buried earthquakes, using the empirical model forearthquake kinematic parameters proposed by Somervilleet al. (1999) as a constraint. The empirical model of Somerville et al. (1999) defines statistical slip characteristics in theasperity as a function of the average total slip. Dalguer et al.
(2004) use quasi-dynamic modeling to estimate dynamic
parameters consistent with this kinematic characterization.
They infer that surface-rupturing earthquakes are characterized by a large area of negative stress drop surrounding theasperities, while buried earthquakes are better characterized
by nonnegative stress drop.
In this study, we formulate a set of dynamic rupture
models that combine the stress-drop constraints of Dalguer
et al. (2004) with further constraints from empirical source
scaling of seismic moment and rupture area. We refer to theresulting empirically constrained dynamic models as calibrated dynamic models. We then calculate ground motions
from the calibrated models and compare results with observations that Somerville (2003) and Kagawa et al. (2004)
interpret to indicate generic ground-motion differences between buried (Mw 6.7–7.0) and surface-rupturing (Mw 7.2–7.6) earthquakes. Their interpretation is that the former generate systematically stronger high-frequency ground motion
than the latter. The simulations capture some of the groundmotion differences cited by Somerville (2003) and Kagawa
et al. (2004), and we propose possible mechanisms to account for those differences.
For the calibration phase (i.e., in the search to find model
parameters ensuring consistency with empirical scaling relations), we use quasi-dynamic simulations (simulations with
fixed rupture velocity) calculated with inelastic-zone fault
models (Dalguer and Day, 2006). For the ground-motion

L. A. Dalguer, H. Miyake, S. M. Day, and K. Irikura

simulation phase, we use fully dynamic, spontaneous rupture
modeling, applying the staggered-grid split node (SGSN)
method recently developed by Dalguer and Day (2007).
The SGSN method was implemented in the message-passing
interface (MPI) finite difference code of the TeraShake platform (Olsen et al., 2006), which is scalable to thousands ofprocessors (Dalguer et al., 2006), enabling high-performance
execution for large-scale dynamic rupture models.

Stress-Drop Calibration
Stress-Drop Distribution on the Fault
Inversion of near-source ground motion (e.g., Wald andHeaton, 1994; Sekiguchi and Iwata, 2002) has revealed thatfault-surface slip distributions of past earthquakes are highly
heterogeneous and complex at all observable scales, withlocalized patches of large slip conventionally being referred
to as asperities. This kinematic information can provide
information relevant to earthquake dynamics, in the formof estimates of fault-plane stress change (e.g., Bouchon,
1997; Ide and Takeo, 1997; Day et al., 1998; Dalguer et al.,2002), which are also highly heterogeneous. Details of theabsolute stress fields are currently not measurable, but it
is likely that much of the heterogeneity in stress change isattributable to heterogeneities in the initial preearthquake
stress state. A dynamic rupture simulation that begins froman initial stress state and friction coefficient distribution thattogether are consistent with the inferred fault-plane stresschanges will, of course, reproduce the final slip distributionof the event. Thus, using stress change information from pastearthquakes in this fashion may potentially put useful constraints on dynamic models, and hence on their groundmotion predictions.
Since our purpose is not to reproduce a single past event,rather to predict ground-motion characteristics of futureearthquakes, it is appropriate to use statistical characteristics
of past earthquakes to constraint stress-drop distributions.
We use the characteristic slip models proposed by Somervilleet al. (1999). These authors analyzed kinematic images fromsource inversions of past earthquakes and proposed statisticalproperties, such as (1) the average of combined asperity area
is 0.22 times the total rupture area and (2) the ratio between
the average asperity slip and average total slip is Dasp =D  2:0. In previous work (Dalguer et al., 2004), we analyzed a series of forward dynamic rupture models, with fixedrupture velocity (quasi-dynamic models), and from themproposed ratios of stress-drop distribution on the fault(Δσb =Δσa ), where Δσb and Δσa are, respectively, the stressdrop on the background area and asperity, that satisfy theforegoing empirical rules of Somerville et al. These ratios
provide a simplified characterization of the stress-drop distributions of the asperity-source models. Here, we briefly
summarize those results.
We represent buried earthquakes (those with no surface
rupture) by a set of circular fault models (and checked that

Surface Rupturing and Buried Dynamic-Rupture Models Calibrated with Statistical Observations

1149

circular and rectangular faults, for buried earthquake, resultin the same estimation of the metrics of interest), and surfacerupturing earthquakes by a set of rectangular fault models(the latter allowing us to take into consideration the generally
larger aspect ratio of surface-rupturing earthquakes). In eachcase, we consider both single- and multiple-asperity models.
The rectangular faults consist of earthquakes with L ≥ W max ,where L and W are, respectively, the length and width of thefault, and W max is assumed to be 20 km, representing thebrittle crust of the earth. The fault models are calculatedfor L up to 20W max . Forward dynamic rupture simulations
are carried out for different stress-drop ratios (Δσb =Δσa )in the range from 0:2 to 0.2. These simulations used a fixedrupture velocity of 0.8 times the S-wave velocity and thesimple slip-weakening friction model in the form given by
Andrews (1976) with a critical slip-weakening distanceDc  0:4 m. The material properties are represented by
the P-wave velocity V p  5:543 km=sec S-wave velocity
V s  3:2 km=sec and density ρ  2:8 gr=cm3.
Figure 1 shows the average slip ratio (Dasp =D) plottedwith the stress-drop ratio (Δσb =Δσa ) for the set of dynamic
solutions of the circular faults (buried earthquakes) and rectangular fault models (surface-rupturing earthquakes). Considering up to four asperities, the dynamic fault models thatmatch the kinematic source characteristics of Somervilleet al. (1999) have stress-drop ratios in the range from
0:1 to 0.1 for buried earthquakes and from 0:15 to
0.05 for surface-rupturing earthquakes, as indicated by thearrows in Figure 1. Based on the data presented by Somerville et al. (1999), we assume that the number of asperities
increases with earthquake size, with the average number ofasperities being 2.6 (as proposed by Somerville et al.), fromwhich a reasonable inference is that buried faults have (onaverage) less than 2.6 asperities and surface-rupturing faults
have (on average) more than 2.6 asperities. As a consequence, the stress drop surrounding the asperities is zeroor positive for buried earthquakes and negative (with amplitude more than 10% of the stress drop of the asperity) for
surface-rupturing earthquake. The difference in stress-drop
distribution between buried and surface-rupturing earthquake is marked, and this systematic difference may be relevant to the physics of the rupture mechanism. For example,
the large area of apparent negative stress drop surrounding
the asperities during large earthquakes may actually reflect
energy losses from a large area of damage zone off fault
where energy is dissipated, or a change in frictional parameters at shallow depth to favor velocity strengthening.
However, here we take a phenomenological approach, using
the results to calibrate the asperity/background stress-drop
ratio within the context of our simple parameterization ofground-motion simulation models.

(a)

(b)

Stress-Drop Scaling with Earthquake Size
Empirical source scaling of seismic moments and rupture area provides a guideline to constrain the variability ofthe macroscopic rupture parameters, such as seismic
moment, rupture area, and average stress drop with earthquake size. Several scaling models, derived from past earthquakes, were presented in the literature (e.g., Kanamori andAnderson, 1975; Scholz, 1982; Wells and Coppersmith,
1994; Somerville et al., 1999; Mai and Beroza, 2000; Irikura
and Miyake, 2001; Hanks and Bakun, 2002; Scholz, 2002;
Irikura et al., 2004). Here we use the empirical model proposed by Irikura et al. (2004) based on Irikura and Miyake(2001). These authors proposed three-stage scaling relationship between seismic moment and rupture area, as shown in

3.2

Dasp/D

2.8
2.6
2.4

3.2

1 Asp
2 Asp
3 Asp
4 Asp

2.2

1 Asp
2 Asp
3 Asp
4 Asp

3
2.8
Dasp/D

3

2.6
2.4
2.2

2

2

1.8

1.8

1.6
1.4
−0.2−0.15 −0.1−0.05 0 0.05 0.1 0.15 0.2∆σb /∆σa1.6
1.4−0.2−0.15 −0.1−0.05 0

Figure 1.

0.05 0.1 0.15 0.2∆σ b /∆σaAverage slip ratio (Dasp =D) plotted with the stress-drop ratio (Δσb =Δσa ) for the dynamic solution of the buried earthquake (a)and surface-rupturing earthquakes (b) calculated by Dalguer et al. (2004). The arrow specifies the band of the stress-drop ratios that lies on
the characterized slip model proposed by Somerville et al. (1999).

1150

L. A. Dalguer, H. Miyake, S. M. Day, and K. Irikura

Figure 2. This scaling model is empirically constrained witha statistical analysis of kinematic source models estimatedfrom waveform inversion of strong-motion records forcrustal earthquakes (Somerville et al., 1999; Miyakoshi,
2002; Asano et al., 2005) and earthquake catalogs (Wells
and Coppersmith, 1994). The analyzed data range frommoment magnitude Mw 5.7 to Mw 7.6. The first scaling lineof this empirical model, for moderate-size earthquakes, is expressed by the empirical equation proposed by Somervilleet al. (1999):
S  2:23 × 1015 M02=3

for M0 < 7:5 × 1025 dyne cm:(1)
The second line, proposed by Irikura et al. (2004) for larger
earthquakes, is represented byS  4:59 × 1011 M01=2

for M0 ≥ 7:5 × 1025 dyne cm;(2)and the third line, also proposed by Irikura et al. (2004) forthe largest earthquakes, is represented byS  5:30 × 1025 M0

for M0 ≥ 7:5 × 1027 dyne cm;(3)
where S is the fault area in km2 and M0 is the seismic
moment in dyne centimeters.
Our target is to constrain the variability of the average
stress drop with the empirical scaling model of Irikura et al.
(2004) shown in Figure 2. For that, we perform 3D simula-

5

Rupture Area (Km 2 )

10

4

10

6

Mw

7

8

Wells and Coppersmith (1994)
Somerville et al. (1999)
Somerville (2003)
Hanks and Bakun (2002)
Miyakoshi (2002) and Asano et al. (2005)
Somerville et al. (1999)
Hanks and Bakun (2002)

tions of dynamic faulting for a series of single- and multipleasperity source models of rectangular faults for surface
rupturing and buried earthquakes. We expect that the stress
distribution will affect the moment/area relation principally
through the average stress drop, and we will initially examine
the effect without introducing the stress-drop distribution differences (asperity/background ratios) inferred previously.
However, in a later section, we will verify that adding thatcomplexity does not alter the conclusion. The material andfriction law characteristics are the same as the models described in the previous section. Our simulations consist ofvertical strike slip faults. For buried earthquakes, the fault
is located at 3-km depth and has one asperity. Fault models
with fault area (L × W) 12:5 × 12:5, 17 × 17, and 20 × 20 in
km2 are used. For surface-rupturing earthquakes, the faulting
breaks the free surface, and the faults have a width (W) of
20 km, corresponding to the maximum W that represents thebrittle crust of the earth. The fault length L (in kilometers)
and number of asperities (nasp) for each model areL; nasp  20; 1, (40, 2), (100, 3), (200, 4), (300, 4),
and (400, 4).
Models with stress drop on the asperity (Δσa ) equal to
8 Mpa (1.76 Mpa), 10.5 Mpa (2.31 Mpa), 15 Mpa (3.3 Mpa),
and 20 Mpa (4.4 Mpa) are used for surface-rupturing andburied earthquakes. The values in parentheses are the corresponding average stress drop for each model. The stress dropon the background area is zero for all the models. The ratio
between the combined asperity area and total rupture area isequal to 0.22. Notice that buried faults are now modeled bysquare faults, and not by circular fault as in the previous section, so that all faults (buried and surface rupturing) can becharacterized by the ratio L=W.
Figure 3 shows the final slip distribution (for the case ofmodels with stress drop on the asperity equal to 10.5 Mpa) ofsome of the faults calculated with the dynamic rupture simulation. Using the set of dynamic rupture models, we verified
that for the same fault rupture area, the relation
M01 Δσ1≈
M02 Δσ2

Irikura et al. (2004)(4)

is satisfied, where M0 and Δσ are, respectively, the seismic
moment and average stress drop, and the subscripts 1 and 2
denote two different earthquakes (earthquake 1 and 2).
Equation (4) permits estimating average stress drops
consistent with the empirical scaling model. This estimationsuggests a variation of average stress drop as shown in Figure 4 and summarized as follows:

3

10

2

10

1

10
24
10

25

10

26

10

27

10

28

10

29

10

Moment (Dyne−cm)
Figure 2.

Empirical scaling models of seismic moment and rupture area, proposed by Irikura et al. (2004) (thick black solid line
and thick black dashed line), Hanks and Bakun (2002) (thin graydashed line), and Somerville et al. (1999) (thick gray solid line andthick gray dashed line).

Δσ ∼ 1:5 Mpa

for L ≤ 1:5W max ;Δσ ∼ 1:5 Mpa to 5:3 Mpa
for 1:5W max < L ≤ 10W max ;Δσ ∼ 5:3 Mpa

for L > 10W max :

(5a)
(5b)
(5c)

Surface Rupturing and Buried Dynamic-Rupture Models Calibrated with Statistical Observations

1151

Figure 3. (a) Final slip distribution of some rectangular asperity models calculated with dynamic rupture simulation for the case withstress drop in the asperity Δσa  10:5 Mpa. MAI1 to MAI3 are buried earthquakes, and the rest correspond to surface-rupturing earthquakes. The white dashed lines border the asperity area, and the star is the hypocenter. (b) Final slip distribution of the largest fault models of
surface-rupturing earthquakes, calculated with dynamic rupture simulation for the case with stress drop on the asperity Δσa  10:5 Mpa.
The white dashed lines border the asperity area, and the star is the hypocenter.

1152

L. A. Dalguer, H. Miyake, S. M. Day, and K. Irikura
Mw

6

5

10

7

8

Irikura et al. (2004)
4

Rupture Area (Km 2)

10

L ~ 10Wmax
3

10

L ~ 1.5Wmax

2

10∆σ ~ 1.5MPa

1.5MPa < ∆σ< 5.3MPa ∆σ~ 5.3MPa

1

10 24
10

10

25

26

10

27

10

28

10

29

10

Moment (Dyne−cm)

Figure 4. Average stress-drop variation, calculated from the dynamic rupture models, that lies on the empirical scaling models ofseismic moment and rupture area proposed by Irikura et al. (2004).
Equation (5a) is consistent with the self-similar scaling
model proposed by many authors (e.g., Hanks, 1977), thatis, the stress drop is independent of the earthquake size.
But this self-similarity is sustained for buried earthquakes
only. Equation (5b), which represents surface-rupturing
earthquakes, breaks this self-similarity, and the stress drop
becomes earthquake-size dependent. Scale-dependent earthquake were proposed by many authors (e.g., Kanamori et al.,
1993; Abercrombie, 1995). Equation (5c) is a projection of
our numerical model, because we were not able to model
earthquake larger than L  20W max . The few points givenby the model with L > 10W max suggest that the stress drops
turns out to be independent of the earthquake size, but thisneed to be verified with further research.
We must acknowledge that there are uncertainties in thiscalibration procedure that are large but difficult to quantify
due to the fact that we subsume the complex mechanics oflarge earthquakes into only a few simple dynamic parameters. For example, and just to suggest one possibility,
some or all of the slope change around L ∼ 1:5W max may
be actually be due to slip or distributed shear below thebrittle-ductile transition rather than systematic stress-drop
variations.

Hypocenter Location and Fracture Energy
Effect of Hypocenter Location
Hypocenter location may play an important role in
influencing ground-motion generation and rupture propagation. Mai et al. (2005) analyzed the relationship of hypocenter location to slip distribution for more than 50 earthquakes. They found that most of these earthquakes have
hypocenters located in regions of large slip or very close
to those regions, suggesting an association of hypocenter

location with asperities, that is, with regions of high stressdrop. We explore the effects of the hypocenter location on
the peak slip velocity (using slip velocity as a simple wayof quantifying effects on ground-motion amplitudes). For
that purpose, two buried faults located at 3-km depth fromthe free surface are used. One fault has the dimensions L×
W  25 × 17 km, and one asperity with 9.5-km length and9.5-km width. The other fault has L × W  12:5 × 17 km,
and a narrow asperity with 5-km length and 9.5-km width.
The two models have stress drop 10.0 Mpa on the asperity,and zero stress drop outside the asperity. The critical slip distance for the first model is 0.4 m, and for the second model
0.2 m (with a corresponding factor of 2 reduction in fractureenergy for the second model). The material properties andfriction conditions are the same as for the models describedin the previous sections.
Figure 5 shows the peak slip velocity distribution forthese two models. The upper figures are the results of thefirst fault model with different hypocenter location as shownby the black star. The model with deeper hypocenter generates the largest peak slip velocity due to the directivity effectof rupture propagation toward the free surface. For the second model, we use only a deep hypocenter, as shown in thebottom figure. In this latter model, the peak slip velocity isfurther enhanced as a result of reduced energy absorption due
to the smaller critical slip distance. This simple dynamic rupture model indicates that a deep hypocenter location can produce directivity leading to concentration of the slip velocity
function toward the free surface. That effect, in addition to a
reduced fault area and low fracture energy during rupture,
may be significant in enhancing high-frequency groundmotion. We investigate the fracture energy effect in the next
section.
Effect of Fracture Energy
As a exploratory step before applying the calibrated
models to ground-motion simulation, we consider separately
the effect of fracture energy, now using the spontaneous rupture model. Theoretical and empirical studies of fractureenergy (e.g., Ide and Takeo, 1997; Bouchon et al., 1998;
Guatteri and Spudich, 2000; Ide, 2002; Tinti et al., 2005;
Mai et al., 2006) suggest that fracture energy is scale dependent, varying with the spatial scale of the earthquake rupture.
Mai et al. (2006) analyzed scaling properties of fractureenergy derived from dynamic rupture models of past earthquakes, concluding that the fracture energy scales markedly
differently for surface ruptures than it does for buried rupture. Their study suggests that a large earthquake consumes
more fracture energy as the rupture expands and reaches thefree surface, compared with a confined rupture.
We explore the effects of the fracture energy, again using
the peak slip velocity as a measure of ground-motion excitation. For that, one surface fault is used. The fault has the dimensions L × W  100 × 20 km, and four asperities as
distributed in Figure 5. The fault model has stress dropSurface Rupturing and Buried Dynamic-Rupture Models Calibrated with Statistical Observations

1153

Figure 5. Peak slip velocity distribution calculated from dynamic rupture simulation for two buried-fault models with stress drop in theasperity area Δσa  10:0 Mpa, and zero stress drop outside the asperity. The critical slip distance for the upper fault models is 0.4 m and
0.2 m for the bottom fault model. The star represents the hypocenter location. The white dashed lines border the asperity area.

10.0 Mpa on the asperity, and zero stress drop outside theasperity. We simulate two ruptures, with high and low fracture energy, respectively. The fracture energy is varied byvarying the slip-weakening distance (Dc ), using Dc  0:4
and 0.8 m, respectively, for the low- and high-fracture energy
model. The material properties and other frictional parameters are the same as in the models described previously.
As shown in Figure 6, the large-fracture energy model(bottom figure) has overall lower slip velocities, as expected,
and diminished directivity effects on the slip velocity function due to large energy absorption in the fracture development. This effect on the slip-velocity function may besignificant in reducing the high-frequency ground motion.

Application of the Calibrated Dynamic Models
to Simulate Near-Source Ground Motionof Surface-Rupturing and Buried Earthquakes
Given the type of earthquake (buried or surface rupture)and the size of the earthquake (quantified by rupture area,
seismic moment, and/or moment magnitude), we form a dynamic stress-drop distribution calibrated to the Δσb =Δσaratio shown in Figure 1 and the average stress-drop scaling
shown in Figure 4. Then we apply the calibrated dynamic
models to study the differences of faulting and near-source
ground motion of surface-rupturing and buried earthquakes,
with the goal of exploring possible explanations of recent
observations of Somerville (2003) and Kagawa et al.

(2004). These authors suggest that buried earthquakes (buried rupture with Mw 6:7–7:0) generate stronger groundmotion than surface-rupturing earthquakes (large surface
rupture with Mw 7:2–7:6) around a period of 1 sec. In thecontext of the calibrated dynamic model, we explore themechanism of surface rupturing and buried earthquakes
and propose a parameter set for each type that is consistent
with the observed differences of near-source ground motion.
The effects of hypocenter location and fracture energy noted
in previous sections help guide this exploration.
We use two buried and two surface-rupturing fault
models with the parameters specified in Table 1 and shownin Figure 7. The P-wave velocity (V p ), S-wave velocity (V s ),and density are 6:0 km=sec, 3:464 km=sec, and 2:67 gr=
cm3 , respectively. The rupture nucleates artificially in a circular patch with radius R located inside the first asperity ofeach model, and then it spontaneously propagates along thedefined fault. The values of Se and Dc shown in Table 1 wereselected, from results of many dynamic rupture simulations,for their consistency with the difference of ground motion
between surface-rupturing and buried earthquakes, as observed by Somerville (2003). The hypocenter locations, as
shown in Figure 6, were also intentionally selected a posteriori for this purpose. A grid size of 100 m is used for thesimulations. With this numerical resolution, we adequately
model wavelengths larger than six time the grid size, permitting to represent waves up to 5.0 Hz.

1154

L. A. Dalguer, H. Miyake, S. M. Day, and K. Irikura

Table 1Model Parameters for the Dynamic Rupture Simulations
Buried (3-km Depth)
Model Name

L × W km2 Asperity size (km2 )N° asperities
Dc (m)Δσa (Mpa)
Δσb (Mpa)
Se (first asperity) (Mpa)
Se (rest) (Mpa)
Gc (first asperity) (J=m2 )Gc (other asperities) (J=m2 )Grid size (km)Nucleation radio R (km)Time step (sec)

Surface Rupturing

B1

B2

S2

S3

12:5 × 17:0
6:0 × 7:8
1
0.2
6.82
0.0
6.82
2.4
1:36 × 106
—
0.1
1.3
0.006525:0 × 17:0
9:6 × 9:6
1
0.3
6.82
0.0
6.82
2.4
2:05 × 106
—
0.1
1.5
0.006575:0 × 20:0
10:5 × 10:5
3
1.6
17.8
1:78
17.8
3.56
28:5 × 106
17:1 × 106
0.1
2.5
0.0065

100:0 × 20:0
10:5 × 10:5
4
1.6
21.8
2:18
21.8
4.36
34:9 × 106
20:9 × 106
0.1
2.5
0.0065

Dc is critical slip distance, Se is strength excess (static strength minus initial shear stress), ΔσaandΔσb are, respectively, dynamic stress drop on the asperity and outside the asperity, and Gc is
the surface fracture energy.

Recall that the model calibration was done using quasidynamic simulations, and by doing separate calibrations of
the ratio Δσb =Δσa and of the average stress drop. We now
check whether the full spontaneous rupture simulations
based on the calibrated parameters remain in agreement withthe empirical parameter estimates. The moment magnitude(Mw ), stress-drop ratio (Δσb =Δσa ), the average slip ratio(Dasp =D), and the ratio between the combined asperity area
and total rupture area (Aasp =A) of each model are specified inTable 2. The slip ratio, as well as the asperity-area ratios, arein good agreement with the empirical targets shown in Figure 1. The simulated earthquakes also still follow the empiri-

cal moment-area scaling relation as shown in Figure 8. Thus,
the buried and surface-rupturing earthquakes simulated fromthe proposed calibrated dynamic models maintain these statistical properties of real past earthquakes.
Figure 9 shows the rupture-time contours from the 3D
spontaneous dynamic rupture simulation of the four models
described in Figure 7 and Table 1. The results of the buried
earthquakes (B models) show smooth or well-defined rupture
velocity (subshear) toward the free surface. However, for thesurface-rupturing earthquakes, rupture propagation is more
complex as a result of interactions of the rupture with multiple asperities, low-stress regions, and the free surface. When

Figure 6. Peak slip velocity distribution calculated from dynamic rupture simulation for two surface-rupturing earthquakes, respectively,
with low fracture energy (Dc  0:4 m, upper figure) and large fracture energy (Dc  0:8 m, bottom figure). The fault models have stressdrop in the asperity area Δσa  10:0 Mpa, and zero stress drop outside the asperity. The star represents the hypocenter location. The whitedashed lines border the asperity area.

Surface Rupturing and Buried Dynamic-Rupture Models Calibrated with Statistical Observations
Table 2
Characteristic Parameters of Dynamic Rupture Simulations
Buried (3-km Depth)

Model nameMw
Δσb =ΔσaDasp =D
Aasp =A

B1
6.3
0.0
2.1
0.26

B2
6.6
0.0
2.3
0.22

Surface Rupturing

S2
7.4
0:1
1.9
0.22

S3
7.5
0:1
1.9
0.22

the rupture front is between asperities, the rupture velocity
sometimes reduces to almost zero, but then, when it reaches
an asperity, the rupture recovers and accelerates, reaching tosupershear rupture velocity after crossing the asperity (aqualitative behavior that has been documented in previous
simulations, e.g., Day, 1982b). When the rupture approaches
the free surface, the rupture becomes incoherent and the rupture velocity is not well defined, perhaps because (in themodels) the entire near-surface region has low stress drop,
and rupture through this zone is largely being driven fromthe asperities below. This characteristic of the rupture velocity in the surface-rupturing earthquake simulations, if it applies to real earthquakes, may reduce the classic rupture
directivity effect on the ground motion, especially very close
to the fault (Olsen et al., 2008).
Figure 10 shows the final slip and the peak slip velocity
from the dynamic rupture simulations. The final-slip amplitudes are quite different between the surface-rupturing andburied earthquakes (as expected, simply from the stress-drop
scaling we used to match the empirical moment-area relations, and the smaller fault surfaces of the buried ruptures).
However, the peak slip velocities from the buried earthquakes are comparable in amplitude to those of the surface-rupturing earthquakes. The differences are mainly in
B1

5

Rupture Area (Km 2 )

17km

17km

4

10

25km

12.5kmthe directionality of the peak slip velocities. For the buried
earthquakes, the peak slip velocity grows toward the free surface, following a relatively simple rupture directivity effect.
For the surface-rupturing earthquakes, the largest peak slip
velocities concentrate mainly on the asperity areas. The directivity effect on the slip velocity of the surface-rupturing
earthquakes was inhibited by the high fracture energy on
the asperities, the negative background stress drop (which
produces enhanced energy absorption), and the near-surface
rupture incoherence described earlier.
Finally, we examine the ground motion predicted by the
dynamic models. Figure 11 shows the location of the receivers where velocity ground motion is simulated. Figure 12
shows low-pass and high-pass filtered peak ground velocity
at the receivers located along the lines 1, 2, and 3, as specified in Figure 11. The low- and high-pass filters each with
cutoff frequency at 1 Hz. The low-frequency ground velocity
is much higher for the surface-rupturing simulations, as
would be expected, because those simulations have higher
moments and slip. However, the buried simulations haveground velocity comparable in amplitude to that of the surface-rupturing simulations, and in some cases even larger.
The high-frequency fault-normal component for the buried
earthquakes is especially strong. The highest amplitudes
are on the trace of the fault, and gradually diminish as the sitemoves away from the trace of the fault. Away from the fault,
the parallel component of the buried earthquakes also develops high-frequency amplitudes comparable to those of thesurface-rupturing earthquake. Another interesting characteristic, seen in this figure, is the variation of the amplitude of
the peak velocity along the strike direction of the surfacerupturing earthquakes. This variation of amplitude does
not show the classic rupture directivity effect in the high-

10

B2

S2

1155

6

Mw

7

8

Wells and Coppersmith (1994)
Somerville et al. (1999)
Somerville (2003)
Hanks and Bakun (2002)
Miyakoshi (2002) and Asano et al. (2005)

Irikura et al. (2004)

Characterized
earthquake models
3

10

20km
75km

2

10

S3
20km

1

10 24
10

25

10

26

10

27

10

28

29

10

Moment (Dyne−cm)

100km

Figure 7. Fault models for dynamic rupture simulation of buried earthquakes (B models) and surface-rupturing earthquakes (Smodels).

10

Figure 8.

Identification of the calibrated dynamic models(black stars) on the empirical source-scaling model of Irikura et al.
(2004).

1156

L. A. Dalguer, H. Miyake, S. M. Day, and K. Irikura3

Mw=6.3

Aasp/A = 0.26

B2

Mw=6.6

Aasp/A = 0.22

1
2
3

10

B1
4

3

2

1

1 2 3

2

Z(km)

0

20

5

3 4

6

20
21

14.5 16
.517
18

27

26.528

21.5
24

.5

.5

20

16.5

15

12.5
13.5

5
11.

9

3

5

11. 5 1
3.
12.5 5

6.5

2

4.
5
5.5

19

5

60

25.5

3.5

26
24.5 2

19

23
20

40
X(km)

18.5

7 6 .5
4

19.5

19

2201

18

.5
17 6
1

13

11

20

18.5

14

17

0

3

2.5

12

1.5

67

2

18

Aasp/A = 0.22

Mw=7.5
8. 5

2.5

5
5
4

10
7.5

1

11

11

9
6

2

5

20

3

0.

3.5
.5
54.5
6

10

2.
25

13

.5
17
.5
15 14

2

S3

0

9

9.5
8.5
7

5

0
1 .52
3
2 .5
3. 4

Aasp/A = 0.22

Mw=7.4

7.5

5

1.

10

6

10

3.5 4

3

5

8

3

2

4

Z(km)

4

2

S2

20
Z(km)

5 6

2

2

4
5

20
0

4
3
1

1
1

6
4
2

10

4
3

5 4 3

Z(km)

0

28

80

Figure 9.

Rupture-time contours from the 3D spontaneous dynamic rupture simulation of the buried (B models) and surface-rupturing (Smodels) earthquakes. The black dashed line represents the asperities and the star the hypocenter location.

frequency band. As mentioned earlier, the high fracture energy on the asperity, negative stress drop outside the asperity,and incoherence of rupture combine to reduce the rupture
directivity effect on the ground motion.
Figure 13 shows the comparison of time history and response spectra of the normal component of the ground velocity from the surface-rupturing and buried earthquakes, atreceiver, as specified in Figure 11. Notice that the receivers
are located at different locations for the different events. The
locations were selected to illustrate the maximum groundmotion for each event. As shown by this figure, the buried
events have spectral peaks that are shifted toward a shorter
period, relative to the surface-rupturing events. The velocity
response spectra show clearly that the high-frequency groundmotion of the buried events is higher than that of the surfacerupturing events. On the other hand, the low frequency
ground motion of the surface-rupturing events is higher thanthat of the buried events, consistent with the higher momentand slip of the former. These results are qualitatively consistent with the observations reported by Somerville (2003).

Discussion and Conclusions
The main impediment to the use of spontaneous rupture
models in earthquake ground-motion prediction is probably
the limited state of our knowledge of the friction law andstress (and other) conditions on faults prior to rupture in large
events. In the long term, these limitations may be overcome
through detailed experimentation and modeling at the microscale, combined with model testing through full-scale simulation of ground-motion observations from many individual
earthquakes. In the short term, an alternative is to invoke
some of the spirit of kinematic ground-motion modeling,
but within the framework of dynamically consistent models
of rupture. We can assume simple parameterizations to describe the friction law and stress state, perform simulations
that sample the parameter space, and then calibrate thoseSurface Rupturing and Buried Dynamic-Rupture Models Calibrated with Statistical Observations

B1(Mw=6.3)
0Dasp/D=2.1

0

10

10
20
0

20
B2(Mw=6.6)
0Dasp/D=2.3

10

10
20
0

Dasp/D=1.9

20
S2(Mw=7.4)
0
10

Dasp/D=1.9

20
S3(Mw=7.5)
0

10

Z(km)

20
0

10

10
20

0

1157

0
1

2

20

40
X(km)

3

4

5

60
6

7

20

80
8

(m)
9

0

0

1

20
2

3

4

40
X(km)
5

60

6

7

8

80
(m/s)
9 10

Figure 10. Final slip (left) and peak slip velocity (right) from the 3D spontaneous dynamic-rupture simulation of the buried (B models)and surface-rupturing (S models) earthquakes. The dashed line represents the asperities and the star the hypocenter location.
parameters by identifying the parameter sets that give resultsthat conform, in a statistical sense, to key empirical observations from past earthquakes. Here we have followed thislatter general strategy, in the context of a two-parameter
slip-weakening friction model and a highly simplified asperity model for stress state. The key empirical constraints are(1) the asperity area and slip-ratio values that characterize
moderate to large earthquakes (as interpreted by Somervilleet al., 1999, from kinematic rupture inversions), and (2) the
Irikura et al. (2004) empirical scaling relation (for seismic
moment versus area).

There is of course substantial uncertainty in the resulting
inferences of the dynamic parameters, as well as a very highdegree of nonuniqueness, exacerbated by the requirement toselect very simple parameterizations. Future efforts in thisdirection will have to consider much more complete parameterizations of the rupture physics; we have already mentioned
stable sliding and/or distributed inelastic shearing beneaththe seismic zone as a possible alternative (or additional) factor to explain the moment-area scaling, for example. Additional observational constraints should be incorporated aswell. For example, we treated fracture energy as a free para-Line 34.0km

Line 2
Line 1 (On the trace of faulting)

1.2km

B1 B2
-10

0

10

S2
20

30

40

50

60

Fault
plane

S3
70

80

90

100km

Figure 11. Location of receiver for ground-motion simulation on the free surface. Along the lines 1, 2, and 3, peak velocity groundmotions are saved. The triangles specify location of receivers for the velocity waveform, respectively, for each fault model (B and S models).
The star is the projection of the hypocenter for all the fault models, where the origin of the axis along the strike of the fault is also located.

1158

L. A. Dalguer, H. Miyake, S. M. Day, and K. Irikura

S3

LINE 1
1

S2
1.5

Parallel

B1

B2

LINE 1

Parallel

0.5

Max. Velocity (m/s)

Max. Velocity (m/s)

101Normal0.5
01

Vertical

0.5
0
1.5

Normal

1
0.5
0
1.5

Vertical

1
0.5
0

0.5
−10

0

10

20

30

40

50

60

70

80

90

0

100

−10

0

10

20

30

LINE 2
1

1Max. Velocity (m/s)

Max. Velocity (m/s)

Normal0.5
0Vertical

0.5
0

1

1

−10

0

10

20

30

40

50

60

70

80

90

0

100

90

100

80

90

100

Vertical

−10

0

10

20

30

40

50

60

LINE 3
0.6 Parallel

Parallel

0.2

Normal0.4
0.2
0Vertical00.6

Normal

0.4
0.2
0
0.6

Vertical

0.4

0.4

0.2

0.2
0

80

NormalLINE 30

0.6

70Parallel

Distance (km)

0.4

0.8

100

0

0.2

0.6

90

0.5

0.4

0.8

80

0.5

Max. Velocity (m/s)

Max. Velocity (m/s)

0.6

70

0

Distance (km)
0.8

60

0.5

0

1

50

LINE 2

Parallel

0.5

1

40

Distance (km)Distance (km)

−10

0

10

20

30

40

50

60

70

Distance (km)Figure 12.

80

90

100

0

−10

0

10

20

30

40

50

60

70

Distance (km)Parallel, normal, and vertical component of the peak velocity ground motion along the line 1 (upper), line 2 (middle), and line
3 (bottom) as specified in Figure 10, from the 3D spontaneous dynamic rupture simulation of the buried (B models) and surface-rupturing (Smodels) earthquakes. The left column and right column correspond, respectively, to low-pass filtered and high-pass filtered velocity wave
forms, with cutoff frequency at 1 Hz.

Surface Rupturing and Buried Dynamic-Rupture Models Calibrated with Statistical Observations
1.5

B1 (N comp.)

1

1

0.5

0.5

0

0

−0.5
0
Velocity (m/s)

1.5

2

4

6

8 10 12−0.5
1.5

B2 (N comp.)

1

1

0.5

0.5

0

0

−0.5
0

S2 (N comp.)

S3 (Surface)
S2 (Surface)
B2 (Buried)
B1 (Buried)
14 16 18 20 22 24
S3 (N comp.)

4

6 8
Time(s)10 12

ζ = 5%
1.5

24 26 28 30 32 34
Time(s)

S3

1
0.5
0

−0.5
2

2
Spectral Velocity (m/s)

Velocity (m/s)

1.5

1159

0

B2

S2

B1
2

4
6
Period (s)810Figure 13.

Time-history (left) and response spectral (right) of the normal component of the velocity ground motion located at receivers as
specified in Figure 11, from the 3D spontaneous dynamic rupture simulation of the buried (B models) and surface-rupturing (S models)
earthquakes.

meter in our explorations of buried versus surface-rupturing
dynamics, and it would be very desirable to obtain independent empirical constraints.
Within the context of our parameterization, we find asignificant difference in the stress-drop distribution between
our buried-rupture models, which would represent small- tomoderate-magnitude events, and surface-rupturing models,
which would represent mainly large earthquakes rupturing
the full seismogenic thickness. It is quite possible that a more
realistic friction model with, for example, strong velocity dependence might be able to reproduce the asperity area andslip-ratio constraints without requiring the negative stressdrop areas, and this should be explored in future work incorporating more general friction laws. Alternatively, the large
areas of apparent negative stress drop surrounding the asperities may be real, perhaps indicative of enhanced energy absorption mechanisms acting preferentially in large events, orpreferentially at shallow depth (e.g., friction transitional between velocity weakening and velocity strengthening or offfault rupture-induced damage).
Again, within the context of our parameterization, thelargest values of stress drop on the asperity of surfacerupturing earthquakes also provide us insight on the occurrence of large earthquakes. Because of the large amount ofnegative stress drop surrounding the asperities, large valuesof stress drop in the asperities appear to be necessary to allow
rupture to progress until it becomes a large earthquake, as
seen in the spontaneous dynamic-rupture simulations(Fig. 9). In addition, the dynamic models that match the empirical source-scaling model suggest that the average stressdrop for the buried earthquakes is independent of earthquakes size, indicating that this type of earthquake follows

self-similar scaling. On the contrary, large earthquakes characteristically suggest that the stress drop scales with theearthquake size, indicating that this type of earthquake
breaks the self-similarity.
The fracture energy turns out to be the key parameter inour efforts to suggest an explanation for the proposed differences in frequency content between small buried earthquakes
and large surface-rupturing earthquakes (Somerville, 2003).
In numerical experiments, we found that an elevated value offracture energy reduces high-frequency excitation (above
about 1 Hz), relative to simulations done with low fractureenergy. The resulting shift in spectral content between thehigh and low fracture energy simulations is qualitatively consistent with the Somerville et al. empirical comparison of
surface-rupturing earthquakes versus buried ruptures. The
contrast in frequency content between the two types of eventswas enhanced in our simulations by the relatively strong upward directivity in the low fracture energy, single-asperity
buried events, and the weakened high-frequency directivity
in the surface-rupturing events. The latter was due in partto our assumption of shallower hypocentral depth for thesurface-rupturing events, and perhaps in part to the incoherence of the rupture front that developed through its interaction with multiple, isolated asperities in a negative stressdrop background.
The importance of the fracture energy as a free parameter in our modeling points out the importance of obtaining
independent empirical constraints on this parameter (or,more generally, of employing more realistic physical assumptions in which energy dissipation is not treated simply
as a scale-invariant material property). The partial explanation we have proposed for observed spectral differences in1160

L. A. Dalguer, H. Miyake, S. M. Day, and K. Irikura

ground motion between smaller, buried events and large,
surface-rupturing events depends upon there being a systematic difference in fracture energy between the two eventclasses. This may be at least plausible, as several recent observational studies have suggested a relationship between
fracture energy and earthquake size (e.g., Ide and Takeo,
1997; Tinti et al., 2005; Mai et al., 2006). The fractureenergy may be a macroscale representation of bulk microfracturing, which would be expected to scale with event size(e.g., Yamashita, 2000; Poliakov et al., 2002; Dalguer et al.,2003a,b; Andrews, 2005; Rice et al., 2005). There is also
some independent evidence suggestive of higher fractureenergy in shallow rupture (e.g., Ide and Takeo, 1997). Rateand state-dependent friction models also predict enhanced
dissipation for very shallow rupture, where velocity-strengthening friction is expected to pertain (e.g., Marone andScholz, 1988). Enhanced energy absorption zones due to
off-fault damage are possible characteristics of larger faults.
A damage zone can be accumulated during the lifetime of a
fault, either as the result of dynamic stress change induced byrupture during an earthquake (e.g., Dalguer et al., 2003a,b;Andrews, 2005) or from quasi-static deformation during thelife of a shear fault (e.g., Vermilye and Scholz, 1998).
In this work we used a highly simplified parameterization of the rupture process, calibrating the parameters separately for different types and sizes of events so as to conformto empirical scaling relations. This approach provided some
useful insights into the origin of ground-motion differences
between surface-rupturing and buried earthquakes, but is unsatisfying from a physical perspective. It is a step in a longerterm program to model more realistic initial stress conditions
and a more complete thermo-mechanical description of the
rupture process, and to simulate large suites of events without predetermined constraints on individual event sizes. The
ground-motion scaling laws that emerge naturally from such
simulation suites may then be compared directly with statistics from past earthquakes to identify acceptable ranges ofthe model parameters. It is our hope that this approach willcontribute to an improved capability for the prediction ofground motion from future earthquakes.

Acknowledgments
We would like to thank the two reviewers of the article, Joe Andrews
and Benchun Duan, and the associate editor David D. Oglesby for constructive comments and very helpful reviews that led to improvements in the
manuscript. This work was partially supported by (1) the Study on Development of Simulation System and its Application for Catastrophic Earthquake and Tsunami Disaster Response in Mega-Cities Facing the Pacificin Special Project for Earthquake Disaster Mitigation in Urban Areas in
RR2002, by the Ministry of Education, Science, Sports, and Culture ofJapan; and (2) by the National Science Foundation (NSF), under Grants
Number ATM-0325033 and Number EAR-0623704, by the Southern California Earthquake Center (SCEC). The SCEC is funded by NSF Cooperative
Agreement EAR-0106924 and U.S. Geological Survey Cooperative Agreement 07HQAG0008. The SCEC contribution number for this article is 1104.

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Department of Geological Sciences
San Diego State University
5500 Campanile Dr.
San Diego, California 92182
ldalguer@moho.sdsu.edu
day@moho.sdsu.edu(L.A.D., S.M.D.)
Earthquake Research Institute
University of Tokyo
1-1-1 Yayoi, Bunkyo-ku
Tokyo 113-0032, Japanhiroe@eri.u‑tokyo.ac.jp(H.M.)
Disaster Prevention Research Center
Aichi Institute of Technology
1247 Yachigusa, Yakusa
Toyota, Aichi 470-0392, Japan
irikura@geor.or.jp(K.I.)
Manuscript received 24 May 2007