800
Bulletin of the Seismological Society of America, Vol. 95, No. 3, pp. 800–817, June 2005, doi: 10.1785/0120030171
Teleseismic Body Waves from Dynamically Rupturing Shallow Thrust
Faults: Are They Opaque for Surface-Reflected Phases?
by Deborah Elaine Smith, Brad T. Aagaard, and Thomas H. Heaton
Abstract
We investigate whether a shallow-dipping thrust fault is prone to wave-
slip interactions via surface-reflected waves affecting the dynamic slip. If so, can
these interactions create faults that are opaque to radiated energy? Furthermore, in
this case of a shallow-dipping thrust fault, can incorrectly assuming a transparent
fault while using dislocation theory lead to underestimates of seismic moment?
Slip time histories are generated in three-dimensional dynamic rupture simulations
while allowing for varying degrees of wave-slip interaction controlled by fault-
friction models. Based on the slip time histories,
P
and
SH
seismograms are calcu-
lated for stations at teleseismic distances. The overburdening pressure caused by
gravity eliminates mode I opening except at the tip of the fault near the surface;
hence, mode I opening has no effect on the teleseismic signal. Normalizing by a
Haskell-like traditional kinematic rupture, we find teleseismic peak-to-peak displace-
ment amplitudes are approximately 1.0 for both
P
and
SH
waves, except for the
unrealistic case of zero sliding friction. Zero sliding friction has peak-to-peak am-
plitudes of 1.6 for
P
and 2.0 for
SH
waves; the fault slip oscillates about its equilib-
rium value, resulting in a large nonzero (0.08 Hz) spectral peak not seen in other
ruptures. These results indicate wave-slip interactions associated with surface-
reflected phases in real earthquakes should have little to no effect on teleseismic
motions. Thus, Haskell-like kinematic dislocation theory (transparent fault condi-
tions) can be safely used to simulate teleseismic waveforms in the Earth.
Definition of Terms
To avoid confusion about terminology we first wish to
clearly define how we will be using the following terms.
When we refer to other articles that may have different def-
initions, we will enclose them in quotes.
1.
Dislocation theory
. Application of shear double couples
to approximate a jump in displacement across a fault sur-
face. There is no true cut in the medium; the double cou-
ples simply stretch the elastic medium to create high
strains across an infinitesimally thin zone.
2.
Haskell-like rupture
. It can be applied to either a dislo-
cation or a true cut boundary. Traditionally, it is a prop-
agating pulse of slip where slip at any point is a ramp
function in time. There is no reslipping.
3.
Dynamic rupture
. A finite-element model with a true cut
(fault) and specified tractions or slips. Reslipping may or
may not occur, depending on the evolution of the dy-
namic rupture simulation.
Introduction
It has been suggested that “dislocation” models may
seriously underestimate seismic energy for shallow-angle
thrust events (Brune, 1996; Shi
et al.
, 1998). It is hypothe-
sized that seismic energy may be partially trapped in the
hanging wall during shallow-angle thrust events; hence, tele-
seismic measurements would measure only some fraction of
the total signal. This idea springs, in part, from reported
discrepancies between near-field (regional) and far-field
(teleseismic) measurements of radiated energy where the
near-field estimates have sometimes far exceeded the far-
field estimates (Shoja-Taheri and Anderson, 1988; Sing and
Ordaz, 1994). Although this reported discrepancy was re-
solved for the strike-slip Hector Mine earthquake (Boat-
wright
et al.
, 2002), it remains to be seen if it can be resolved
for shallow, thrust-fault events. The suggestion that energy
may be trapped in the hanging wall of shallow, thrust-fault
events raises concerns regarding a transparent fault boundary
condition when using “dislocation” theory to model earth-
quake sources. A transparent fault boundary is a rupture
plane that allows seismic energy to pass through unimpeded;
as a result, there is no interaction between the earthquake
rupture and the seismic phases (which reflect off the Earth’s
surface and then propagate back down through a dipping
rupture) in terms of reslipping. Reslipping is where radiated
Teleseismic Body Waves from Dynamically Rupturing Shallow Thrust Faults: Are They Opaque for Surface-Reflected Phases?
801
1
2
3
Earth's Free Surface
Fault Surface
Figure
1.
Cartoon illustrating how an earthquake
rupture might make a fault appear opaque to propa-
gating waves: (1) seismic energy radiated from rup-
ture, (2) energy reflects off of free surface, and (3)
energy reflects off opaque fault boundary, trapping
energy in the overhanging wedge.
waves from the main slipping episode cause additional slip
on the same rupture interface. In contrast, for a fault bound-
ary with wave-slip interactions that produce reslipping, the
surface-reflected phases (e.g.,
pP
,
sP
,
sS
) interact with the
rupture, potentially trapping energy in the overhanging
wedge of a thrust fault. So we ask, do wave-slip interactions
on a shallow-dipping thrust fault result in a fault surface that
is opaque to radiated energy, leading to reduced teleseismic
ground motions? If earthquakes do indeed have opaque fault
boundaries that decrease the teleseismic ground motions and
if scientists do not take this effect into account (i.e., they
assume a transparent fault for traditional Haskell-like dis-
location modeling of teleseismic data), then scientists could
possibly underestimate the seismic energy released and the
seismic moment.
At teleseismic distances, most of the radiated energy is
low-frequency energy (
1 Hz), where the zero-frequency
component equals the moment. In practice, seismic moment
is more often used to estimate the size of earthquakes from
teleseismic waveforms; therefore, in this study of teleseismic
ground motions we will address the concern about Haskell-
like dislocation theory in the context of seismic moment.
In a series of foam-rubber block experiments intended
to simulate earthquake ruptures, Brune (1996) observed sig-
nificant mode I motion (opening) in addition to shear (mode
II and mode III) displacements. The mode I opening resulted
in considerable wave-slip interactions and an opaque bound-
ary that trapped radiated energy in the overhanging wedge.
Figure 1 schematically shows what such an opaque boundary
might look like for a dip-slip fault. Other work that has also
seen significant mode I opening are the more realistic 2D
lattice models of Shi
et al.
(1998). Brune (1996) suggested
that if modeling of earthquakes took into account this hy-
pothesized opaque boundary, reported discrepancies be-
tween near-field and far-field radiation estimates might be
resolved. In particular, near-field estimates of radiated en-
ergy for thrust faults have often greatly exceeded the far-
field (teleseismic) estimates (Shoja-Taheri and Anderson,
1988; Boatwright
et al.
, 2002). Also the near-field estimates
of apparent stress, that is, seismic energy divided by mo-
ment, can exceed the far-field estimates of apparent stress
(Sing and Ordaz, 1994); Brune hypothesizes this is caused
by an opaque boundary trapping the radiated energy close
to the fault.
Hence, we investigate whether ruptures on a shallow-
dipping thrust fault actually produce fault boundary opacity
(mode I opening and mode II/mode III shear displacement
opacity) and reduce teleseismic signals by examining tele-
seismic
P
and
SH
body waves. Using dynamic rupture sim-
ulations, we generate earthquake ruptures while allowing
different levels of wave-slip interactions. We found that re-
alistic overburdening pressure eliminates mode I opening in
the ruptures except at the very shallow surface (see Discus-
sion for more details). We then compute teleseismic body
waves by using these simulated earthquake ruptures. If
wave-slip interactions produce fault boundaries that are
opaque to teleseismic waves, then one might expect ruptures
with the greatest interactions to generate the smallest-
amplitude teleseismic motions.
Methodology
We compute synthetic teleseismic
P
and
SH
body waves
in a two-step process shown in Figure 2. We generate slip
time histories using 3D finite-element dynamic rupture sim-
ulations and then calculate the teleseismic waveforms cor-
responding to these slip time histories with the method of
Yao and Ji (1997) and Yao and Harkrider (1983). We can
map finite-element solutions into point-dislocation time his-
tories to compute teleseismic distances, because in an elastic
space, either a stress boundary condition on a sliding surface
(like our finite-element dynamic ruptures) or a displacement
boundary condition on a box bounding the sliding surface
(like our teleseismic computations) uniquely determines the
solution (Achenbach, 1973, section 3.2). We will refer to the
stress boundary condition as a dynamic representation and
the displacement boundary condition as a dislocation rep-
resentation.
In our dynamic ruptures, the friction model, the elasto-
dynamic equations, and an initial stress state determine the
stress on the sliding surface,
s
(
X
1
,
X
2
,
X
3
0,
t
), and from
this we can calculate the displacement jump,
D
(
X
1
,
X
2
,
t
),
across the surface of the fault (Fig. 3). The dislocation rep-
resentation, which we use for our forward calculations of
teleseismic ground motions, begins with specifying the dis-
placement jump,
D
(
X
1
,
X
2
,
t
), across the surface of the fault.
If we equate
D
(
X
1
,
X
2
,
t
)to
D
(
X
1
,
X
2
,
t
), then our dislocation
representation (used to calculate our teleseismic ground mo-
tions) has an identical elastodynamic field as the finite-
element dynamic rupture simulation. In other words, the two
representations, dynamic and dislocation, produce the same
stresses and displacements everywhere, except inside an in-
finitesimally thin box bounding the fault. Because we map
the irregularly spaced dynamic rupture time histories into a
coarser, regularly spaced grid of dislocation time histories,
802
D. E. Smith, B. T. Aagaard, and T. H. Heaton
Fault Friction Models
Zero Sliding
Friction
Slip- and Rate-
Weakening Friction
Constant Sliding
Friction
Slip- and Rate-
Weakening Friction
(Pulselike)
Seismogram
Constant Sliding
Friction (Cracklike)
Seismogram
3-D Dynamic
Rupture Simulations
Slip Histories Output
from Simulations
(dynamics dictate shape).
Kinematic Slip Histories Used
in Teleseismic Calculations
Teleseismic Displacement
Seismograms
Zero Sliding Friction
(Cracklike)
Seismogram
Zero Sliding Friction
Slip History
Constant Sliding
Friction Slip History
Slip- and Rate-
Weakening Friction
Slip History
Haskell-Like
Kinematic
(Pulselike)
Seismogram
Haskell-Like Kinematic
Slip History
Traditional Slip History
(analytical function dictates shape)
Use Same Final Slip,
Peak Slip Rate, and
Rupture Velocities in
Haskell-Like Model
Figure
2.
Flowchart showing how the teleseismic seismograms for the four models
are made. For the three dynamic models, we start with frictional constitutive laws used
in the 3D dynamic rupture simulations: (1) zero sliding friction, (2) constant sliding
friction, and (3) slip- and rate-weakening friction. The 3D dynamic rupture simulations
produce slip histories. The resultant slip histories are kinematic descriptions of slip on
the fault during rupture. Yet at the same time, they account for dynamic effects by
expressing reslipping episodes or other modifications to their shape. In addition to the
three slip histories that are a result of our dynamic rupture simulations we also produce
analytically a traditional Haskell-like slip history. The Haskell-like slip history is de-
signed to have parameters similar to the slip- and rate-weakening friction model, but
without reslipping. Hence, the traditional kinematic Haskell-like model slip history
should be completely transparent to reflected seismic energy. Then we take all four
kinematic slip histories and use them as input for teleseismic calculations to produce
our resultant displacement seismograms as seen in the Results section.
the two solutions are not exactly the same close to the fault.
However, the two solutions converge far away from the fault
so the teleseismic results for the two representations are
equivalent. This allows us to compute the teleseismic body
waves for the different degrees of wave-slip interactions as-
sociated with different friction models.
The unique aspect of this approach is that it takes ad-
vantage of the strengths and avoids the weaknesses of each
modeling method. Strengths include the following: (1) Dy-
namic rupture simulations incorporate full fault dynamics,
friction, fracture energy, radiated waves, etc., enabling us to
explicitly account for wave-slip interactions, and (2) dislo-
cation theory has a rich analytical background that has paved
the way for fast and accurate computation of teleseismic
signals. Weaknesses that we avoid include the following:
(1) It is not feasible to construct a dynamic rupture grid that
has a sufficiently large number of elements to calculate both
the dynamic motions on a sliding fault and the teleseismic
body wave for periods down to 2 sec, and (2) dislocation
theory is purely kinematic and cannot explicitly incorporate
fault physics.
Dynamic Rupture Modeling
Calculation Method
The solution techniques for solving our dynamic earth-
quake ruptures in finite-element simulations have been dis-
cussed in a few articles (Aagaard, 1999; Aagaard
et al.
,
2001). The dynamic elasticity equation is solved within a
three-dimensional domain discretized with linear tetrahedral
elements. Friction models control the evolution of the stress
field under the constraint that friction produces reasonable
rupture dynamics. See Aagaard
et al.
(2001) for more detail
regarding the constraints on the friction model. The resulting
earthquake ruptures have realistic depth distributions of slip,
Teleseismic Body Waves from Dynamically Rupturing Shallow Thrust Faults: Are They Opaque for Surface-Reflected Phases?
803
X
1
X
2
X
3
13
+
τ
13
−
τ
Net traction across surface boundary = 0.
Specifically,
.
13
+
τ
13
−
τ
=
Figure
3.
Comparison of dislocation versus dy-
namic representations. In the dynamic representation
(1) we compute the stress boundary condition on an
actively slipping fault,
s
(
X
1
,
X
2
,
X
3
0,
t
), that is
compatible with friction models, elastodynamic equa-
tions, and the assumed initial stress. (2) From
s
(
X
1
,
X
2
,
X
3
0,
t
), we can calculate the displacement on
either side of the fault,
u
(
X
1
,
X
2
,
X
3
0,
t
) and
u
(
X
1
,
X
2
,
X
3
0,
t
). (3) This determines the dis-
placement jump,
D
(
X
1
,
X
2
,
t
), where
D
(
X
1
,
X
2
,
t
)
u
(
X
1
,
X
2
,
X
3
0,
t
)
u
(
X
1
,
X
2
,
X
3
0,
t
).
(3) From
D
(
X
1
,
X
2
,
t
), we can compute
s
(
X
1
,
X
2
,
X
3
,
t
) everywhere in the media. In the dislocation repre-
sentation, (1) we specify a displacement jump,
D
(
X
1
,
X
2
,
t
), across the width of an infinitesimally thin, lin-
early elastic box enclosing the fault. We do this by
applying a singular body force that stretches the linear
elastic material within the box, a finite amount,
D
(
X
1
,
X
2
,
t
), over the infinitesimal box width. (2) We can
calculate the stress caused by this displacement jump,
s
(
X
1
,
X
2
,
X
3
,
t
), everywhere. If
D
(
X
1
,
X
2
,
t
)
D
(
X
1
,
X
2
,
t
), then
s
(
X
1
,
X
2
,
X
3
,
t
)
s
(
X
1
,
X
2
,
X
3
,
t
) by the
uniqueness theorem (Achenbach, 1973, section 3.2),
including
. That is, the dy-
sss
s
13
13
13
13
namic and kinematic solutions are mathematically
identical everywhere except inside an infinitesimally
thin box surrounding the rupture. Inside this box, the
solutions are inherently different. For the dynamic
representation there is a cut in the material, whereas
in the dislocation representation, there is a singular
body force that stretches the elastic material. But
again, the solutions are identical outside the infinites-
imally thin box surrounding the linearly elastic ma-
terial; hence, when comparing the two representations
one must “think outside the box.”
rupture velocities, average final slip, and peak particle ve-
locities.
A shallow thrust-fault plane that breaks the surface with
a23
dip is placed in the middle of the simulation domain
as shown in Figure 4. Earthquakes on this fault plane have
a nominal rake angle of 90
for all the ruptures. In each case,
the hypocenter is located midway (14 km) along-strike and
13 km down-dip. Material properties vary as a function of
depth in a piecewise linear fashion as illustrated in Figure 5.
Going from deeper to shallower, the major change in ma-
terial properties occurs at approximately 6 km depth where
the velocities and density steeply decrease to simulate soft-
basin structure. The spatial resolution, 170 m at surface and
660 m at 7 km depth, yields a 2-sec lower bound on the
period of the radiated waves.
The dynamic ruptures can be divided into two classes
based on the type of friction model that controls the evolu-
tion of the stress field. The first class includes cracklike rup-
tures that are controlled by slip-weakening friction. Slip
weakening has a long history of being used to study fric-
tional sliding in earthquakes, including Ida (1972), Andrews
(1976a), and Burridge
et al.
(1979). Sliding friction drops
to a nominal value after slip progresses to some amount
D
0
as shown in Figure 6. Note that the slip-weakening param-
eter,
D
0
2
E
G
/(
r
max
r
min
), where
E
G
is the fracture
energy,
r
max
is the maximum frictional stress, and
r
min
is the
minimum frictional stress. The ruptures with slip-weakening
friction are cracklike because healing phases are initiated
when the rupture reaches the boundaries of the fault. In one
scenario (constant sliding friction), the minimum value of
sliding stress is sufficiently larger than zero to arrest slip
when healing phases arrive and the shear traction drops be-
low this minimum sliding stress. Note that in this case, when
the friction drops to this minimum value, the fault has a
constant friction boundary condition, not a constant stress
boundary condition. The minimum sliding friction provides
an upper limit on the traction magnitude. In another scenario
(zero sliding friction), the final minimum sliding stress is
zero, creating a traction-free sliding surface, which allows
the fault slip to overshoot and then oscillate about its equi-
librium position.
The second class of dynamic ruptures includes pulselike
60.00km
60.00km
North
24.00km
23
°
18.00km
28.00km
25.00km
H
X
Z
Y
Figure
4.
Geometry of the shallow-thrust
fault that breaks the surface of the finite-
element grid. It has a rake of 90
. The hypo-
center,
H
, is located 14 km along-strike and
13 km down-dip.
804
D. E. Smith, B. T. Aagaard, and T. H. Heaton
0
1
2
3
4
5
6
7
30
20
10
0
Depth (km)
V
P
V
S
ρ
Velocity (km/s), Density (g/cm
3
)
Figure
5.
Material properties as a function of
depth for the source region. The symbols represent
the following:
q
is density,
V
S
is
S
-wave velocity, and
V
P
is
P
-wave velocity. The values decrease close to
the surface to simulate basin structure.
0
1
D/D
o
Frictional Stress
σ
min
σ
max
fracture
energy
Figure
6.
A slip-weakening friction model. The
frictional stress reduces from maximum to a mini-
mum value as slip increases to
D
0
. The shaded region
is associated with the fracture energy.
0
1
2
3
4
0
1
2
3
4
D/D
o
V/V
o
σ
max
σ
post
σ
min
Frictional Stress
Figure
7.
A slip- and rate-weakening model. The
dark line represents a typical trajectory as a function
of time of the frictional stress for a point on the fault.
At first frictional stress decreases with increasing slip
up to a critical slip,
D
0
. Then frictional stress increases
as velocity falls below a critical velocity,
V
0
.
Table 1
The Variation as a Function of Depth of Initial Shear Stress,
Shear Stress at Failure, Shear Stress due to Frictional Sliding
during Rupture, and Fracture Energy for the Four Models
Depth
(km)
Sliding
Stress
(MPa)
Failure
Stress
(MPa)
Fracture
Energy
(MJ/m
2
)
Initial
Stress
(MPa)
Zero sliding friction
(cracklike)
0.0
0.00
0.75
0.18
0.11
3.0
0.00
3.04
0.80
2.05
6.0
0.00
7.65
1.50
4.05
Constant sliding friction
(cracklike)
0.0
0.25
1.00
0.18
0.14
3.0
1.00
4.04
0.80
2.51
6.0
1.00
8.65
1.50
4.95
Slip- and rate-weakening
friction (pulselike)
0.0
0.50
1.00
0.12
0.19
3.0
1.00
5.76
1.80
3.42
6.0
1.00
12.00
2.89
6.75
The values vary approximately linearly between 0 and 3 km depth,
linearly between 3 and 6 km depth, and are uniform below 6 km depth.
Initial shear stress and shear stress at failure increase with depth because
of the increasing overburdening pressure of the rock. Initial shear stress is
less than the shear stress at failure; therefore, a stress asperity is needed to
start the rupture. We vary the stress drop with depth to produce distributions
of slip without a clear depth dependence.
stress is what we have termed
r
max
. Last, there are differ-
ences in the stress and fracture energies in each friction
model in an attempt to produce similar source parameters
(final slip and average rupture speed) for the dynamic rup-
tures. Varying stress controls the final slip, and varying frac-
ture energy controls the average rupture speed.
In these formulations of the slip-weakening and slip-
and rate-weakening friction models, we assume that the
physical processes controlling the changes in stress on the
fault during sliding yields a friction stress that is independent
of the normal stress. However, when we allow fault opening,
the friction stress does go to zero when the fault has zero
ruptures that are controlled by slip- and rate-weakening fric-
tion. The slip- and rate-weakening applied here is based on
the models of Madariaga and Cochard (1996) and Madariaga
et al.
(1998). Shear stress initially drops as slip occurs and
then rises as the slip rate drops below a given threshold,
V
0
, as shown in Figure 7. In this case,
r
post
r
min
2/3
(
r
max
r
min
), where
r
post
is the final frictional stress, and
V
0
0.10 m/sec. The ruptures with slip- and rate-weakening
friction are pulselike because the healing phases occur spon-
taneously and trail behind the leading edges of the rupture,
producing much narrower actively slipping regions than the
cracklike ruptures (Heaton, 1990). A more detailed descrip-
tion of the friction models can be found in Aagaard
et al.
(2001).
In Table 1 frictional stress values and fracture energy
values are given at three depths, 0 km, 3 km, and 6 km. They
vary linearly between the depths of 0–3 km, linearly between
the depths of 3–6 km, and are uniform below 6 km depth.
The sliding stress is what we have termed
r
min
and the failure
Teleseismic Body Waves from Dynamically Rupturing Shallow Thrust Faults: Are They Opaque for Surface-Reflected Phases?
805
normal stress. Although traditional friction model formula-
tions in dynamic rupture simulations use a coefficient of
friction to allow dynamics variations in normal stresses to
influence the friction stress, they do not include normal
stresses that increase with depth corresponding to the over-
burden pressures. Because the overburden pressure over
most of the fault is about two orders of magnitude greater
than dynamic normal stress changes during the rupture, dy-
namic normal stress changes would not significantly alter
the frictional sliding stresses, except very close to the ground
surface. In other words, if we reformulated our friction mod-
els using a coefficient of friction, dynamic normal stress
changes that are small compared with the overburden pres-
sure would result in correspondingly small changes to the
friction stress. At seismogenic depths these stress changes
would be negligible, so we chose not to include them.
Our traditional kinematic rupture is also pulselike and
has slip similar to traditional Haskell models (Haskell,
1964), hence, the Haskell-like label. Using a slip time history
that follows the integral of Brune’s (1970) far-field time
function, the Haskell-like traditional kinematic rupture is
constructed to approximately match only the pulselike slip-
and rate-weakening friction model (Fig. 2). When (1) the
final slip and slip onset times (rupture velocities) are
matched between the Haskell-like traditional kinematic rup-
ture and the slip- and rate-weakening friction rupture and
(2) the rise times of the Haskell-like traditional kinematic
rupture are set to match the peak slip rates in the dynamic
slip- and rate-weakening friction rupture, then the only dif-
ference between the two ruptures is the shape of their slip
time histories. It is the shape of their respective time histories
that will produce dissimilar wave-slip interactions, where the
Haskell-like traditional kinematic rupture is transparent
(shape is fixed a priori, and there are no wave-slip interac-
tions) and the slip- and rate-weakening rupture allows wave-
slip interactions.
Zero sliding friction allows the most wave-slip inter-
actions because it allows for complete shear-traction decou-
pling once the critical,
D
0
, is achieved. Constant sliding fric-
tion permits the second most wave-slip interactions. The
dynamic rupture allowing the least wave-slip interactions is
the pulselike slip- and rate-weakening friction. Last, the
Haskell-like traditional kinematic rupture is completely
transparent (no interactions) by construction.
These friction models (cracklike constant sliding fric-
tion, cracklike zero sliding friction, and pulselike slip- and
rate-weakening friction) control the development of the
stress field and the rupture dynamics while an earthquake is
underway. A uniform strain field provides the driving force
for the earthquake rupture. This produces slip with no sys-
tematic depth dependence, which is compatible with kine-
matic source inversions (Heaton, 1990; Somerville
et al.
,
1999). This is in contrast with the assumptions of constant
static stress drop, or constant dynamic stress drop, both of
which would produce large systematic variations in slip with
depth in our layered half-space. Using the coordinate system
in Figure 4, we apply the following tectonic strain field,
e
t
,
to simulate tectonic strain:
t
e
C
yy
tt
t
e
e
v
e
(1)
xx
zz
yy
ttt
e
e
e
0,
xy
yz
xz
where
V
is Poisson’s ratio and
C
is a constant. The values
of
C
are:
4
1.69
10
for zero sliding friction (cracklike)
4
C
2.06
10
for constant sliding friction (cracklike)
4
2.81
10
for slip- and rate-weakening friction (pulselike).
Note, different values of
C
are associated with different lev-
els of shear stress to create similar final slips among the three
models. We also include the overburden pressure,
z
1
ggg
e
e
e
q
(
s
)
gds
xx
yy
zz
3
k
2
l
0
(2)
ggg
e
e
e
0.
xy
yz
xz
The strain fields given in (1) and (2) produce both normal
and shear tractions on the fault interface. The shear tractions
are in the direction of desired slip and are the driving stress
for the earthquake rupture. Initially, they are somewhat be-
low the failure level (Table 1), but after the rupture starts,
the concentration of shear stress ahead of the leading edge
of the rupture increases the total shear stress above the fail-
ure level. A circular stress asperity (an area of high-shear
tractions, 2% larger than the failure stress) of radius 2 km,
located at the desired hypocenter, initiates the rupture. The
earthquake starts at the asperity, spreads out in an elliptical
fashion over the rest of the fault, and then arrests at the
boundaries of the fault. The fault boundaries, which deter-
mine the spatial extent of the earthquake, are created by
tapering the initial shear traction to a small value. Figure 8
shows the spatial distribution of the initial fault traction, in-
cluding the stress asperity and tapered edges. Table 1 gives
the control points for the nominal depth dependence of the
initial shear stress.
Slip History Results
We aim for similar source parameters in the dynamic
ruptures, but small differences exist in the final slip, rupture
velocities, and peak slip velocities. Consequently, we scale
the slip histories so that all four scenarios have a seismic
potency of
P
4.68
10
8
m
3
, where
P
D
̄
A
,
D
̄
is average
slip and
A
is area (Heaton and Heaton, 1989; Ben-Zion and
Zhu, 2002). By design, the traditional Haskell-like kinematic
model shares the same slip rate and slip onset times (rupture
velocities) as those in the slip and rate-weakening friction
rupture. Table 2 summarizes the slip-history information of
the four ruptures.
806
D. E. Smith, B. T. Aagaard, and T. H. Heaton
Initial Shear Stress
0
7
14
21
28
18
12
6
0
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Initial Normal Pressure
Distance Along Strike (km)
Distance Down Dip (km)
Shear Stress (MPa)
0
20
40
60
80
100
120
140
160
Normal Pressure (MPa)
0
7
14
21
28
Distance Along Strike (km)
Distance Down Dip (km)
18
12
6
0
Distance Down Dip (km)
Figure
8.
Schematic illustration of the initial shear tractions and normal pressures
on the thrust fault. The increase in normal pressure as a function of depth is caused by
the overburden pressure. The shear traction has an asperity of higher stress to initiate
the earthquake and edges tapered to a lower stress to keep the rupture confined within
a predetermined boundary.
Table 2
The Slip History (Average Final Slip and Maximum Final Slip), Slip-Rate History (Average Peak
Slip Rate and Maximum Peak Slip Rate), and Potency Statistics for the Four Models
Final Slip
Peak Slip Rate
Potency (m
3
)
Avg. (m)
Max. (m)
Avg. (m/sec)
Max. (m/sec)
Zero sliding friction
(cracklike)
0.94
1.62
0.47
2.22
4.68
10
8
Constant sliding friction
(cracklike)
0.94
1.47
0.37
2.02
4.68
10
8
Slip- and rate-weakening
friction (pulselike)
1.00
2.13
0.65
2.79
4.68
10
8
Slip- and rate-weakening
friction with opening
(pulselike)
1.00
2.06
0.65
2.87
4.68
10
8
Haskell-like kinematic
(pulselike)
1.00
2.14
0.65
2.74
4.68
10
8
Note that the source parameters for slip- and rate-weakening friction and the Haskell-like model closely
match. The very slight differences between these two pulselike models arises from the interpolation between the
grids; the slip and peak slip rate distributions are almost exactly the same. The parameters have been normalized
such that the potencies are all,
P
4.68
10
8
m
3
.
In preparation for our teleseismic calculations, we in-
terpolate the slip histories from dynamic rupture models
(nonuniform grid) to a grid of 18
28 points (uniform grid),
and low-pass filter the time histories using a fourth order
Butterworth filter with a corner frequency of 0.5 Hz. Be-
cause most of the radiated energy is at periods longer than
2.0 sec, this removes only a little energy near the high-
frequency limit of the simulations and slightly reduces the
peak slip rates. For example, the cracklike average peak slip
rates are reduced by 1–6%, and the pulselike average peak
slip rates are reduced by 10–25%.
To visualize the down-dip and along-strike variations,
we plot the slip time histories for a series of points in two
cross sections, AA
, which runs down the dip of the fault,
and BB
, which runs along the strike of the fault. Figure 9
shows the locations of these two cross sections, and Figure
10 shows the slip time histories along the cross-sections. Slip
parallel to the strike is minimal for all points on the fault,
except in the case of zero sliding friction. For zero sliding
friction, the fault oscillates symmetrically about AA
, and
there is significant strike-parallel slip for points on either side
of AA
.
Rupture in the mode II direction (the direction of the
AA
profile) reaches a rupture speed of 63–83% of
V
S
, where
V
S
varies as a function of depth depending on the material
properties. Despite larger fracture energies over most of the
fault, the slip- and rate-weakening friction’s mode II rupture
is slightly faster than the other dynamic ruptures (constant
sliding friction and zero sliding friction) due to the slip- and
rate-weakening friction’s faster slip rates for comparable
slip. Rupture in mode III (the direction of the BB
profile)
accelerates rapidly and then approaches a stable speed within
Teleseismic Body Waves from Dynamically Rupturing Shallow Thrust Faults: Are They Opaque for Surface-Reflected Phases?
807
Hypocenter
14 km
13 km
B
B'
A
A'
Figure
9.
Diagram showing the locations of the
two cross sections AA
and BB
.BB
only traverses
the right half of the fault because slip is symmetric
with respect to the fault’s center line.
a few kilometers of the hypocenter. In the absence of fracture
energy, mode III rupture should be faster than mode II rup-
ture relative to the shear-wave speed, but in our models frac-
ture energy is included to stabilize the numerical solution.
Fracture energy, which slows rupture propagation, affects
mode III more strongly than it does mode II (Andrews,
1976b; Day, 1982; Madariaga
et al.
, 1998); as a result, rup-
ture in the mode II direction in our models propagates ap-
proximately 20% faster than the mode III rupture.
In the AA
slip time histories, the final slip exhibits a
small increase as the rupture approaches the free surface.
This is a result of a static effect that is accentuated by dy-
namics. Static offsets from a constant stress-drop fault in an
elastic half-space produces the largest displacements where
the fault breaks through to the free surface (Mansinha and
Smylie, 1971; Parsons
et al.
, 1988; Okada, 1992; Madariaga,
2003). The fault-rupture dynamics accentuate this trend. At
any one instant in time, only a fraction of the fault is slip-
ping. If the entirety of the slipping zone is below the free
surface, then the fault rupture at that moment behaves like
a buried fault whose slip tapers to zero at the edges. How-
ever, as the rupture progresses upward from the hypocenter
to the free surface, the slipping zone will eventually include
the free surface. When it does, the rupture at that instant in
time acts like a fault that breaks through to the free surface
where slip is a maximum at the free surface. There is also a
difference in the depth distribution of slip between the crack-
like and pulselike ruptures. As seen in the AA
slip time
histories of Figure 10, the pulselike ruptures have a greater
percentage of slip partitioned near the surface and less at
depth than the cracklike ruptures. This occurs because the
pulselike ruptures have faster slip rates than the cracklike
ruptures, which result in more energetic breakout phases at
the free surface.
Zero sliding friction overshoots and then oscillates
about the final slip along both profiles AA
and BB
because
all locations on the fault rupture are oscillating. These os-
cillations tend to be complex near the surface and more si-
nusoidal at depth with a period of 12 sec. Additionally, it
has multiple modes of oscillation including along-dip and
along-strike modes. Because energy cannot be dissipated
through frictional sliding, it is dissipated through radiation
damping, wherein energy is transferred from the source to
the far field as the material oscillates and radiates far-field
elastodynamic waves. In this case, the radiation damping of
the fundamental mode of zero sliding friction is 6%. Note
this is substantially smaller than the radiation damping of
58% for a uniform spherical cavity oscillating radially in a
Poissonian whole space (Achenbach, 1973). This difference
in radiation damping results from the differences in geom-
etry between a dip-slip fault oscillating about its equilibrium
position and the radial oscillation of a uniform cavity.
Whereas the behavior of zero sliding friction is highly un-
realistic, its teleseismic ground motions are important as our
end-member rupture that permits the most wave-slip inter-
actions; hence, if opacity is a problem, zero sliding friction
should accentuate its effects.
Figure 10 also shows the long slip durations (up to 14
sec) that occur for the cracklike constant sliding friction rup-
ture. The long-duration slip takes place because healing
fronts in the cracklike ruptures initiate only when the rupture
reaches one of the fault’s boundaries. Then the healing fronts
sweep in toward the center of the fault, terminating slip
along the way. Therefore, one sees an extended period of
slip toward the center and more pulselike slip behavior to-
ward the edges of the fault. Since the ruptures have only a
pP
reflected phase delay of 3 sec, an
sP
delay of 4.5 sec,
and an
sS
delay of 6 sec near the hypocenter, we know that
the cracklike ruptures, with their long slip intervals, allow
for ample interactions between the rupture and radiated seis-
mic waves.
On the other hand, the pulselike ruptures generally have
slip durations of less than 5 sec. In the case of slip- and rate-
weakening, the healing front spontaneously follows the rup-
ture front in accordance with its friction model. As a result,
the width of the rupture at a given point in time is much
smaller than what we find in the cracklike ruptures, so the
slip duration is shorter. With this shorter slip duration, there
is less opportunity for interactions between the rupture and
radiated energy; that is, slip- and rate-weakening would have
less chance of being opaque than either cracklike rupture.
The Haskell-like traditional kinematic rupture is transparent
with no interaction by design.
Teleseismic Waveform Modeling
Calculation Method
Using the slip histories as point-dislocation sources dis-
tributed on a fault-plane grid in a continuous elastic medium,
808
D. E. Smith, B. T. Aagaard, and T. H. Heaton
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
10
20
30
40
7
6
5
4
3
2
1
0
1
Depth (km)
Time (s)
A-A' Cross-Section
0
2
4
6
8
10
12
14
Horizontal Distance from Center (km)
B-B' Cross-Section
010203040
Time (s)
= 1.0 m
= 1.0 m
Zero Sliding Friction (Cracklike)
Constant Sliding Friction (Cracklike)
Slip- and Rate-Weakening Friction (Pulselike)
Haskell-Like Kinematic (Pulselike)
Figure
10.
Slip histories are plotted for a series of grid points along AA
and BB
.
Only the slip component perpendicular to strike is shown because any slip parallel to
strike is minimal for points along the cross sections. There is approximately uniform
slip with depth. Slip does increase some as the rupture approaches the free surface.
Zero sliding friction’s slip history oscillates about equilibrium with a period of 12 sec
and radiation damping of 6%. The cracklike models, zero sliding friction and constant
sliding friction, have long slip durations (up to 14 sec), and the pulselike models, slip-
and rate-weakening friction and Haskell-like traditional kinematic rupture, have shorter
slip durations (less than 5 sec). Therefore, the cracklike models allow more interactions
between the rupture and radiated energy than the pulselike models.
we create synthetic seismograms of teleseismic
P
and
SH
body waves for a range of epicentral distances, 30
D
90
, and azimuths. The dynamic rupture-displacement jump
time histories have been mapped into a regular grid of point-
dislocation time histories, 18
28 points. The top row of
points occurs 0.5 km down-dip from the free surface so there
are no point dislocations exactly at the free surface. All other
boundary conditions (the Earth’s free surface, displacements
going to zero amplitude at infinite distances, material lay-
ering) are matched between the elastic continua and the dy-
namic ruptures. Resultant teleseismic
P
and
SH
body waves
are calculated by using the method of Yao and Ji (1997) and
Teleseismic Body Waves from Dynamically Rupturing Shallow Thrust Faults: Are They Opaque for Surface-Reflected Phases?
809
Yao and Harkrider (1983) that computes the response of a
layered Earth to each point dislocation with
1 sec and
*
T
P
4 sec, for the attenuation. Note
T
*
travel time/
*
T
S
quality factor (
Q
) and the amplitudes of the teleseismic body
waves are attenuated by the following relationship:
A
A
0
e
p
fT*
, where
f
is frequency (Lay and Wallace, 1995, sec-
tion 3.7). The earth structure through which the waves are
propagated is iasp91 (Kennett and Engdahl, 1991), a radially
stratified Earth model.
A test of whether wave-slip interactions produce suffi-
cient fault opacity to affect the teleseismic signal is to com-
pare the teleseismic peak-to-peak displacement amplitudes
of the different ruptures; namely, if opacity is created by
wave-slip interactions (reslipping), then one might expect
that the dynamic rupture that allows the most wave-slip in-
teractions should have the smallest peak-to-peak amplitudes,
and the dynamic rupture that allows the least wave-slip in-
teractions (transparent) should have the largest peak-to-peak
amplitudes. If there is no trend, then the fault is not signifi-
cantly opaque to teleseismic radiation.
Synthetic Teleseismic Seismogram Results
To examine how sources from different depths affect
the solution, we plot
P
and
SH
teleseismic displacements
(Fig. 11) for a surface station at an azimuth of 45
and
D
60
. We first integrate the point time histories along-strike
and plot the resultant displacement seismograms. These are
shown with respect to depth in the top half of Figure 11.
This intermediate result illustrates the depth dependence of
the solution. We then integrate the time histories along-dip
(second spatial dimension), to produce the teleseismic
P
and
SH
displacement seismograms in the bottom half of Figure
11. These seismograms represent the solution for the entire
fault.
Inspecting Figure 11, we see that zero sliding friction
produces the seismograms with the largest peak-to-peak dis-
placement amplitudes. As expected, at teleseismic distances,
the pulselike ruptures, slip- and rate-weakening friction,
and Haskell-like traditional kinematic ruptures have similar
waveforms. However, the two cracklike ruptures (zero slid-
ing friction and constant sliding friction) are quite dissimilar,
especially for sources near the surface, but approach the
same shape (overlooking any time delays or subsequent os-
cillations about zero) for sources at depth. Note that near the
surface, the pulselike ruptures have larger peak-to-peak dis-
placement amplitudes than the cracklike constant sliding
friction rupture, and at depth, the pulselike ruptures have
smaller peak-to-peak displacement amplitudes than either
cracklike rupture. This phenomenon is simply a result of the
cracklike and pulselike ruptures partitioning their slip dif-
ferently as a function of depth (Fig. 10).
In Figure 12, we explore the radiation pattern of the
P
and
SH
body waves by plotting the teleseismic displacement
time histories for surface receivers at different azimuths. We
compute the response for azimuths every 15
for 13 epicen-
tral distances within the range, 30
D
90
; however,
Figure 12 shows the results for only
D
60
and azimuths
30
apart. A 30
separation gives a sufficient representation
of the azimuthal variation, and different distances simply
change the overall amplitude, not the shape of the waveform.
In Table 3, we use all these computed azimuths for compil-
ing the statistics.
One of the most important observations from Figure 12
is that constant sliding friction, slip- and rate-weakening fric-
tion, and Haskell-like traditional kinematic ruptures all have
nearly identical waveforms at any particular azimuth. The
cracklike constant sliding friction does have a time delay
relative to the two pulselike ruptures of approximately 2.0–
3.1 sec for
P
waves and 2.5–3.1 sec for
SH
waves; this is a
consequence of the cracklike constant sliding friction rupture
requiring a much longer period to complete its total slip than
the pulselike rupture (Fig. 10). The similar waveforms for
these ruptures indicate the seismograms are primarily con-
trolled by rupture geometry, earthquake size, and rupture
speed.
The quantified results in Table 3 provide additional in-
sight. We compare the peak-to-peak displacement ampli-
tudes of the three dynamic ruptures. Their amplitudes are
normalized by the peak-to-peak displacement amplitudes of
the Haskell-like traditional kinematic rupture to produce di-
mensionless numbers. The normalized peak-to-peak dis-
placement amplitudes are averaged over azimuth for each
epicentral distance and then averaged over distance. If the
assumption that a transparent fault in Haskell-like disloca-
tion theory leads to underestimates of seismic moment is
correct, then the dynamic ruptures with the greatest wave-
slip interactions would produce the smallest teleseismic
peak-to-peak displacement amplitudes. Furthermore, all the
dynamic ruptures would have normalized numbers less than
one.
Table 3 shows that the dynamic ruptures with wave-slip
interactions do not have numbers noticeably less than one
when normalized by the transparent, Haskell-like traditional
kinematic rupture. Within approximately one standard de-
viation, all the numbers are either one or greater than one.
Specifically, constant sliding friction has average values of
0.96 for
P
waves and 1.02 for
SH
waves, and slip- and rate-
weakening friction has average values of 1.08 for
P
waves
and 1.05 for
SH
waves. When the friction model is taken to
the extreme case to produce the greatest fault wave-slip in-
teractions, that is, zero sliding friction, the normalized peak-
to-peak displacement amplitudes are considerably greater
than one (1.65 for
P
waves and 2.03 for
SH
waves), just the
opposite of what we would expect if wave-slip interactions
produced opacity. In the next section, we demonstrate that
these increased values are a result of the traction-free sliding
surface, not opacity.
Hence, (1) there is no noticeable opacity, and (2) the
wave-slip interactions fail to trap a sufficient fraction of en-
810
D. E. Smith, B. T. Aagaard, and T. H. Heaton
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
5
10
15
20
25
30
35
40
45
5
010203040
Time (s)
Zero Sliding Friction (Cracklike)
Constant Sliding Friction (Cracklike)
Slip- and Rate-Weakening Friction (Pulselike)
Haskell-Like Kinematic (Pulselike)
010203040
7
6
4
3
2
1
0
Depth (km)
Time (s)
= 0.5
μ
m
= 5
μ
m
50
Summed Seismogram (Displacement)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
10
20
40
50
= 2
μ
m
= 20
μ
m
P Body-Waves
SH Body-Waves
Summed Seismogram (Displacement)
Figure
11.
Synthetic seismograms for
P
body waves and
SH
body waves at an azimuth
of 45
and
D
60
. The top portion shows the teleseismic response for point sources
integrated along-strike at each depth. The bottom portion shows response for the entire fault
rupturing, that is, integrated along-strike and along-dip. Zero sliding friction produces the
largest peak-to-peak displacement amplitudes. The two pulselike ruptures, slip- and rate-
weakening friction and Haskell-like traditional kinematic rupture, have similar waveforms.
The two cracklike ruptures, zero sliding friction and constant sliding friction, converge at
depth. All the ruptures, except zero sliding friction, have similar waveforms for the entire
fault rupturing in the bottom box. Constant sliding friction, however, has a time delay
because it requires more time to complete its slip.
ergy in the hanging wall to reduce the amplitude of the tele-
seismic ground motions. This indicates that Haskell-like dis-
location theory, with the assumption of a transparent fault,
will not lead to underestimates of seismic moment.
Discussion
Peak-to-Peak Displacement Amplitudes
The peak-to-peak displacement amplitudes we use to
measure the size of teleseismic ground motion have the fol-
Teleseismic Body Waves from Dynamically Rupturing Shallow Thrust Faults: Are They Opaque for Surface-Reflected Phases?
811
010 203040
Time (s)
Zero Sliding Friction (Cracklike)
Constant Sliding Friction (Cracklike)
Slip- and Rate-Weakening Friction (Pulselike)
Haskell-Like Kinematic (Pulselike)
0 10 203040
350
Azimuth (degrees)
Time (s)
= 0.5
μ
m
50
50
P Body-Waves
SH Body-Waves
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
300
250
200
150
100
50
0
= 10
μ
m
= 20
μ
m
(a)
(b)
Figure
12.
Synthetic seismograms for (a)
P
body waves and (b)
SH
body waves at
D
60
for azimuths 30
apart to display the radiation patterns. Constant sliding
friction, slip- and rate-weakening friction, and Haskell-like traditional kinematic rup-
ture all have nearly identical waveforms at any particular azimuth. The only significant
difference is that constant sliding friction has a time delay because it requires more
time to complete its slip. This similarity indicates that rupture geometry, earthquake
size, and rupture speed, not degree of wave-slip interactions, are primarily determining
the waveform shapes and sizes. When wave-slip interactions are taken to the extreme
(zero sliding friction), larger, not smaller, peak-to-peak displacement amplitudes are
generated. This is due to an effect other than opacity (see Discussion and Fig. 15).
lowing correspondence to seismic phases and crustal rever-
berations. By comparing seismograms in Figure 12 for the
P
body waves with those in Figure 13 (Green’s functions
for impulsive point sources), one can see that the small initial
positive peak at about 14–15 sec for azimuths 90–270
comes from the
P
phase, which is largest at azimuth 180
and has a significant signal for the 90
on either side. A
substantial negative peak occurs later in time, anywhere
from 15 to 21 sec, for all azimuths. This is due to a combined
effect of
pP
and
sP
. For the plotted azimuths 0–90
and 270–
330
,
pP
is the primary phase contributing to the negative
signal, and for azimuths 120–240
,
sP
is the primary phase
contributing to the negative signal. Because
sP
arrives later
than
pP
and the
sP
becomes increasingly dominant as a sta-
812
D. E. Smith, B. T. Aagaard, and T. H. Heaton
Table 3
Averaging the Dynamic Models’ Peak-to-Peak Displacement Amplitudes in
P
and
SH
over
Azimuth and Normalizing by the Transparent, Haskell-like Traditional Kinematic Model
Zero Sliding Friction
Constant Sliding Friction
Slip- and Rate-Weakening
Friction
Epicentral
Distance,
D
Peak-to-Peak
Averaged
over Azimuth
Stand.
Dev.
(%)
Peak-to-Peak
Averaged
over Azimuth
Stand.
Dev.
(%)
Peak-to-Peak
Averaged
over Azimuth
Stand.
Dev.
(%)
a.
P
body waves
31
1.71
10.5
0.97
4.9
1.07
2.6
45
1.69
9.6
0.97
4.6
1.08
3.0
60
1.65
9.4
0.96
4.6
1.08
3.3
75
1.61
9.9
0.95
4.9
1.09
3.8
89
1.55
12.1
0.93
4.6
1.10
4.2
Averaged over
D
1.65
0.96
1.08
b.
SH
body waves
31
2.04
8.2
1.02
1.2
1.05
0.7
45
2.04
7.5
1.02
0.9
1.05
0.6
60
2.03
6.4
1.02
0.9
1.05
0.4
75
2.03
5.1
1.02
0.8
1.05
0.2
89
2.02
4.0
1.02
0.7
1.05
0.1
Averaged over
D
2.03
1.02
1.05
Numbers
1 indicate the transparent fault assumption underestimates the moment. Numbers
1 indicate the
transparent fault assumption overestimates the moment. Numbers approximately 1 indicate the transparent fault
assumption has no effect on the moment estimate. (a) Statistics for
P
body waves. (b) Statistics for
SH
body
waves.
tion is moved closer to an azimuth of 180
, the negative peak
shifts later and later in time the closer a station is to an
azimuth of 180
. For azimuths 0–60
and 310–330
, one can
see a small positive peak at 18–22 sec (depending on the
model), that follows the first negative peak. It is due to a
small positive
sP
at those azimuths and crustal reverbera-
tions within the material layers.
Examining the
SH
body waves in Figure 12 and the
Green’s functions in Figure 14, the initial positive peak for
azimuths 30–150
and initial negative peak for azimuths
210–330
which occur at 15–18 sec, are simply due to the
direct
S
seismic phase. Then for slightly later times, 22–24
sec, a negative peak follows the positive one for azimuths
30–150
, and a positive peak follows the negative one for
azimuths 210–330
. This is due to a combined effect of
sS
and crustal reverberations. For azimuths 90–150
and 210–
270
,
sS
is the larger wave, whereas crustal reverberations
are the larger waves for the other azimuths.
Wave-Slip Interactions (Reslipping)
We have shown that the long-slip-duration cracklike
models, zero sliding friction and constant sliding friction,
would have the most opportunity for wave-slip interactions
that produce reslipping (where radiated waves cause the fault
to reslip, producing a partially opaque fault boundary). The
pulselike model, slip- and rate-weakening model, is less
likely to see significant wave-slip interactions (opacity) be-
cause of its shorter slip durations. We have also shown that
whatever wave-slip interaction exists, it has a small effect
on the teleseismic peak-to-peak displacement amplitudes,
except in zero sliding friction.
It is difficult to separate explicitly the effects of wave-
slip interaction between the rupture and reflected phases
from other dynamic effects. The important point is that we
generate ruptures that allow differing levels of wave-slip in-
teractions. Thus, this study focuses on whether Haskell-like
dislocation theory, which assumes a transparent fault bound-
ary, is an adequate description of the rupture process for
teleseismic signals generated by ruptures on a shallow-
dipping thrust fault where significant wave-slip interactions
may occur. Accordingly, our results indicate that Haskell-
like dislocation modeling is indeed adequate.
The zero sliding friction dynamic rupture produces tele-
seismic peak-to-peak displacement amplitudes considerably
larger than the other three ruptures, approximately 60%
larger in
P
and 100% larger in
SH
. This difference in peak-
to-peak displacement amplitudes is caused by dynamic ef-
fects associated with the oscillations. Figure 15 shows the
potency rate of the different ruptures as a function of time.
The Fourier amplitude spectrum of the potency rate is also
shown. Notice that the zero sliding friction rupture has the
same spectral amplitude as the other three ruptures for zero
frequency (this value is determined by the final potency) but
has a much larger spectral amplitude than any other rupture
for 0.06–0.18 Hz with a peak at 0.08 Hz. This peak corre-
sponds to the slip oscillation period of 12 sec. The peak-to-
peak displacement amplitudes measured for zero sliding fric-
tion in Table 3 are associated with this nonzero frequency,
which is why we obtain normalized numbers much larger
Teleseismic Body Waves from Dynamically Rupturing Shallow Thrust Faults: Are They Opaque for Surface-Reflected Phases?
813
350
Azimuth (degrees)
300
250
200
150
100
50
0
Time (s)
010
50
10
510
5
020
15
Time (s)
Time (s)
Point Source
in Half-Space
Full Fault
in Half-Space
Full Fault in
Layered Velocity Model
Figure
13.
The dip-slip component of the Green’s functions for the
P
waves. Be-
cause the rake is almost exactly 90
in our simulations, dip-slip is the appropriate
component to study to understand the effect of reflected phases in our seismograms.
On the left is the Green’s function for an impulsive point source at approximately 6.5
km depth in a half-space. Note that for the point source,
Q
is increased to reduce the
attenuation; this clearly delineates the three main phases,
P
,
pP
, and
sP
.
Q
is increased
by reducing
t
*
P
from 1.0 sec to 0.1 sec. In the center panel, the sum of the Green’s
functions for the entire fault rupturing in a half-space is plotted. In this case
t
*
P
1.0
sec, but it has little effect because the summation of the point sources also filters out
the high-frequency information. Notice in this center panel that the effects of
P
,
pP
,
and
sP
merge together. The panel on the right shows the summed Green’s functions
for the entire fault rupturing through the velocity model we use in the simulations. It
shows how material layering and associated crustal reverberations alter the signal.
than those for any other rupture. The oscillations with a spec-
tral peak of 0.08 Hz are a result of the unrealistic friction
model (sliding friction of zero), and they demonstrate the
substantial difference in behavior between zero sliding fric-
tion and the other ruptures.
Mode I Opening
Brune’s (1996) foam-rubber block experiments and our
dynamic rupture simulations have dramatically different
mode I behaviors. Specifically, the foam-rubber experiments
experience significant mode I opening; whereas our dynamic
ruptures slip only in the mode II and mode III directions.
Restricting slip to mode II and mode III is a valid assump-
tion, because when we allow mode I opening, we find it has
a negligible effect. Permitting mode I opening produces
slips, displacements, and velocities that are indistinguishable
from the case of preventing mode I opening, except close to
the free surface where there are some minute differences. In
addition, it produces indistinguishable teleseismic displace-
ments. Because we are interested only in teleseismic signals,
far away from the surface trace, including mode I opening
is not important.
814
D. E. Smith, B. T. Aagaard, and T. H. Heaton
350
Azimuth (degrees)
300
250
200
150
100
50
0
Point Source
in Half-Space
Full Fault
in Half-Space
Full Fault in
Layered Velocity Model
Time (s)
010
510
5
010
5
020
15
Time (s)
Time (s)
15
20
25
Figure
14.
The dip-slip component of the Green’s functions for the
SH
waves. On
the left is the Green’s function for an impulsive point source at approximately 6.5 km
depth in a half-space. Note that for the point source,
Q
is increased to reduce the
attenuation; this clearly delineates the two main phases,
S
and
sS
.
Q
is increased by
reducing
t
*
S
from 4.0 sec to 0.4 sec. In the center panel, the sum of the Green’s functions
for the entire fault rupturing in a half-space is plotted. In this case
t
*
S
4.0 sec, but
it has little effect because the summation of the point sources also filters out the high-
frequency information. Notice in this center panel that the effects of
S
and
sS
merge.
The panel on the right shows the summed Green’s functions for the entire fault rup-
turing through the velocity model we use in the simulations. It shows how material
layering and associated crustal reverberations alter the signal.
The primary reason we do not see significant mode I
opening in our dynamic ruptures is that the overburden pres-
sure caused by gravity keeps the fault closed (except as the
rupture breaks the Earth’s surface). In the foam-rubber ex-
periments, the fault normal stress and shear stress are ap-
proximately equal, whereas in our simulations the normal
pressure is approximately 10 to 20 times greater than the
effective shear stress (true shear stress minus sliding fric-
tional traction) at seismogenic depths. This indicates that the
foam-rubber results are primarily representative of the shal-
low surface. A 2D lattice model study has also observed
mode I opening over a significant percentage of the fault
(Shi
et al.
, 1998). Although many of the parameters are more
representative of the Earth than foam-rubber block models,
based on the amount of slip relative to the model size, these
simulations appear to be applicable to depths of less than
1 km. We come to this order-of-magnitude estimate as fol-
lows: (1) Assume that the total displacement is about 1 m,
(2) notice that the total displacement in the simulation is
about one grid cell length (therefore, one grid cell is ap-
proximately equivalent to 1 m), and (3) then use this to scale
the simulation of 256 by 256-grid cells. This leads to a max-
imum fault length of 256 m, indicating this simulation has
tested only the shallow-fault cause. Thus, it seems that small
overburden pressures permit fault opening in the foam-
rubber and 2D lattice models, whereas larger overburden
Teleseismic Body Waves from Dynamically Rupturing Shallow Thrust Faults: Are They Opaque for Surface-Reflected Phases?
815
0.1
0.2
Frequency (Hz)
0
2
4
6
Spectral Amplitude (
10
9
m
3
)
0.00
0
510152025
Time (s)
0.0
1.0
-1.0
Potency Rate (
10
8
m
3
/s)
Zero Sliding Friction (Cracklike)
Constant Sliding Friction (Cracklike)
Slip- and Rate-Weakening Friction (Pulselike)
Haskell-Like Kinematic (Pulselike)
Potency Rate
Fourier Transform of Potency Rate
5
3
1
0.05
0.15
0.25
-0.5
0.5
1.5
Figure
15.
Potency rate for the entire fault rupturing (left) and the Fourier transform
of this potency rate, that is, the spectral amplitude (right). Note that the zero sliding
friction has a spectral amplitude peak at 0.08 Hz, which coincides with the oscillation
period of 12 sec that is seen in the slip histories.
pressures appropriate for seismogenic depths, suppress this
type of fault behavior.
There is one source of mode I opening, Weertman
waves, that we are unable to test because of the length scale
of our numerical simulations. Weertman waves are wrinkle-
like slip pulses produced in ruptures where there is a contrast
in material properties across the slip plane (Weertman, 1980;
Andrews and Ben-Zion, 1997; Cochard and Rice, 2000).
However, including Weertman waves should create little or
no opportunity for opacity in our dynamic ruptures because
Weertman waves are a short-length-scale phenomenon. That
is, only a small portion of the fault, directly behind the rup-
ture front, would be opaque at any instant. Although it re-
mains to be shown what length scale is needed to make faults
opaque to teleseismic radiation, it would be longer than what
is permitted by physics as we understand it.
On the other hand, a rupture with disparate material
properties on either side of the fault could experience other
effects that would change the teleseismic signals and slip
estimates. Several articles have discussed how the local dy-
namics change (Andrews and Ben-Zion, 1997; Harris and
Day, 1997; Rubin and Gillard, 2000; Ranjith and Rice, 2001;
Ben-Zion and Huang, 2002), but we have yet to study how
it affects teleseismic ground motions. This is an important
issue because subduction zones, which account for most of
the total earthquake moment on the Earth, typically have a
strong material property contrast across their faults.
Conclusion
Brune (1996) suggested that earthquakes experience
wave-slip interactions that produce opaque fault ruptures.
Therefore, we asked, can Haskell-like dislocation models of
earthquakes in which a transparent fault boundary is as-
sumed, lead to underestimates of the seismic moments of
shallow-dipping thrust fault events? Using 3D dynamic rup-
ture simulations for faults with varying levels of wave-slip
interactions, we numerically test this question and find no
evidence for opacity. We compare the teleseismic ground
motions of dynamic ruptures (with varying levels of wave-
slip interactions in an attempt to produce fault boundary
opacity) and a Haskell-like traditional kinematic reference
rupture (there are no wave-slip interactions; therefore, it has
a transparent fault boundary). If wave-slip interactions pro-
duce significant opacity to teleseismic waves, the models
with greater wave-slip interactions would produce smaller
peak-to-peak displacement amplitudes (which would lead to
smaller moment estimates). Instead, we find that peak-to-
peak displacement amplitudes of teleseismic
P
and
SH
body
waves are not affected by the level of wave-slip interactions.
In the unrealistic case of zero sliding friction, we see a dif-
ference in the peak-to-peak displacement amplitudes, but
this is due to an effect other than wave-slip interactions trap-
ping energy in the hanging wall of the fault. Specifically,
the oscillatory behavior of the zero sliding friction creates
far-field seismic radiation that increases the peak-to-peak
816
D. E. Smith, B. T. Aagaard, and T. H. Heaton
displacement teleseismic amplitudes relative to the other
models. In the frequency domain (Fig. 15), we see that it has
the same potency or zero frequency energy as the other mod-
els, but zero sliding friction has a nonzero frequency spectral
peak associated with the far-field radiation. Consequently,
opacity is not significant for teleseismic motion calculations;
it is fine to assume a transparent fault boundary when using
dislocation modeling for teleseismic data.
Our results should not be surprising given that Wald
and Heaton (1994), Wald
et al.
(1996), and Ji
et al.
(2002,
2003) have shown it is possible to approximately match
near-source seismograms, geodetic data, and teleseismic
seismograms simultaneously with the same finite-source
models. Because the models fit each of the different data sets
simultaneously, this suggests that the current use of dislo-
cation models can adequately represent the source. Further-
more, because the dislocation model can be used to deter-
mine radiated energy uniquely (Aagaard
et al.
, 2001), there
should be no inconsistency in the radiated energy from the
different data sets. Indeed, the reported discrepancy between
near-source and teleseismic estimates of radiated energy is
now being resolved through refinements of energy estimate
techniques for strike-slip faults (Boatwright
et al.
, 2002) and
may be resolved for dip-slip faults as well.
Acknowledgments
This material is based on work supported by National Science Foun-
dation Grant 0208494. We thank Chen Ji for modifying his code so we
could forward-model teleseismic ground motions for any given slip history.
We also thank James Brune, Pascal Favreau, Ruth Harris, Art McGarr, and
David Oglesby for their insightful and helpful reviews and Barbara Smith
for her editing. Access to the Hewlett-Packard V-Class computer, located
at the California Institute of Technology, was provided by the Center for
Advance Computing Research.
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