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Earth and Planetary Science Letters 195 (2002) 17^27
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Crustal gravitational energy change caused by earthquakes in
the western United States and Japan
Taro Okamoto a; *, Toshiro Tanimoto b
a
Department of Earth and Planetary Sciences, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro, Tokyo 152-8551, Japan
b
Institute for Crustal Studies, University of California, Santa Barbara, CA 93106-1100, USA
Received 31 July 2001; received in revised form 6 November 2001; accepted 6 November 2001
Abstract
We study the cumulative change of the coseismic, crustal gravitational energy caused by earthquakes occurring in two
regions with active tectonics: the western USA during 1992 to 1999, and Japan during January 1997 to August 2000.
We use regional earthquake source mechanism catalogues available for these regions in order to increase spatial
coverage and resolution. The catalogues, however, contain very shallow earthquakes for which the conventional normal
mode approach has difficulties in calculating the gravitational energy change under slow convergence in the mode
summations. To overcome this problem, we employ a `direct' solution approach which allows to generate very accurate
kernels for the integration of the coseismic gravitational energy change. In the western USA, the Coast Ranges and the
Transverse Ranges show systematically positive gravitational energy changes while the margin of the Basin and Range
shows negative changes. In Japan, the northeastern part shows predominantly positive changes while the southwestern
part shows negative to weak positive changes. These systematic patterns well correspond to the regional tectonic
regime: gravitational energy gain for compressional tectonics and energy loss for extensional tectonics. The total
(global), cumulative crustal gravitational energy change caused by the earthquakes occurring in the western USA is
+6.1U1017 J, and it is +2.7U1017 J by the earthquakes in Japan. These energy changes are positive in both regions
which indicates the predominance of the compressional stress state with respect to the earthquake occurrence. ß 2002
Elsevier Science B.V. All rights reserved.
Keywords: earthquakes; gravity sliding; energy; seismotectonics; crust; deformation
1. Introduction
Gravitational energy change in the earth can be
induced by earthquakes because they generate
permanent deformation in the earth. Dahlen [1]
pointed out that this coseismic gravitational en-
* Corresponding author. Tel.: +81-3-5734-3303;
Fax: +81-3-5734-3537.
E-mail address: [email protected] (T. Okamoto).
ergy change can be much larger ^ roughly three to
four orders of magnitude larger ^ than the radiated seismic wave energy which is usually referred
to as `seismic energy'. This gravitational energy
change by an earthquake is very nearly balanced
by the elastic energy change, and has nothing to
do with the total released energy. However, this
does not mean nothing happens: there occurs a
very large amount of energy transfer between
gravitational energy to elastic energy inside the
earth.
0012-821X / 02 / $ ^ see front matter ß 2002 Elsevier Science B.V. All rights reserved.
PII: S 0 0 1 2 - 8 2 1 X ( 0 1 ) 0 0 5 7 6 - 3
EPSL 6072 22-1-02 Cyaan Magenta Geel Zwart
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T. Okamoto, T. Tanimoto / Earth and Planetary Science Letters 195 (2002) 17^27
The sign of the coseismic gravitational energy
change can be either positive (energy gain) or negative (energy loss) depending on the mechanism of
the earthquake source. Chao et al. [2] showed,
however, that the cumulative behavior of the
earthquakes was strongly systematic: there was
a signi¢cant trend of energy loss in the global
gravitational energy change by earthquakes. The
energy release rate is about 2 TW (Terra Watt)
which amounts to almost 5% of the total heat
£ow [2].
Based on these theoretical and observational
backgrounds, Tanimoto and Okamoto [3] studied
the gravitational energy change of the crust and
found a systematic spatial pattern in the cumulative crustal gravitational energy change. Regions
of extensional tectonics, such as ridges, show a
systematic gravitational energy loss in the crust.
On the other hand, regions of compressional tectonics, such as subduction zones, typically show
energy gain. Our result roughly indicates that
compressional tectonics shows a systematic, overall uplift while the extensional tectonics shows an
overall subsidence of the crust. This has an intuitive appeal: under horizontal compression thrust
earthquakes tend to occur which raise the crust
and increase its gravitational energy, while under
horizontal extension normal faulting earthquakes
tend to occur which decrease the gravitational energy of the crust.
These previous works used the Harvard centroid moment tensor (HCMT) catalogue of earthquakes [4] in computing the global variations in
the gravitational energy change. The lower limit
of the magnitudes in this catalogue is a slightly
over 5 in moment magnitude (MW ). On the other
hand, regional earthquake catalogues usually contain many, smaller earthquakes recorded by regional dense seismological networks. For example, the catalogues for Japan discussed later
include more than 3000 earthquakes while the
HCMT catalogue contains 234 earthquakes during the same period (January 1997 to August
2000). Thus, based on such regional catalogues,
the coverage and resolution of the spatial variation in the coseismic gravitational energy change
can be increased, although the larger earthquakes
Fig. 1. Maps of the studied regions. The names of several
geologic provinces and plates are shown. (a) The western
USA. (b) Japan. NA, the North America plate; OK, the
Okhotsk microplate.
mainly contribute to the total amount of the energy change.
In this paper, we study two regions with active
tectonics: the western USA and Japan (Fig. 1).
EPSL 6072 22-1-02 Cyaan Magenta Geel Zwart
T. Okamoto, T. Tanimoto / Earth and Planetary Science Letters 195 (2002) 17^27
19
2. Regional earthquake catalogues
Fig. 2. Epicenter distribution for the western USA. The number of earthquakes is 786.
We use several regional earthquake catalogues
available for these areas, which are brie£y discussed in Section 2. One of the characteristics of
these catalogues is that they contain very shallow
earthquakes. In Section 3 we describe a method
for accurate calculation of the energy change for
these very shallow earthquakes, followed by a discussion about the gravitational energy change in
these two regions in Section 4.
We use two catalogues for the western USA
(Fig. 2). For the southern part we use the catalogue of seismic source mechanism determined
from the broadband waveforms recorded at the
TERRAscope stations [5]. For the northern part
we use the `Analyst reviewed moment tensor catalogue' by UC Berkeley Seismological Laboratory
[6^9] determined by a surface wave spectral inversion method and a three-component complete
waveform inversion method.
For Japan, we use the `NIED Seismic Moment
Tensor Catalogue' (NIED : National Research Institute for Earth Science and Disaster Prevention)
by NIED FREESIA Project [10^12]. The moment
tensors are determined from broadband waveforms recorded by FREESIA/KIBAN seismic network (Fig. 3).
The catalogues for the western USA include a
total of 789 events during 1992 to 1999. The
depths of the events range from 0 km to 45 km
with most events being shallower than 30 km. The
moment tensor catalogues for Japan include 3082
events during 1997 (January) to 2000 (August).
The depths range from 5 km to 400 km. Note
Fig. 3. Epicenter distribution for Japan. (a) Source depth 933 km. The number of earthquakes is 2358. (b) Source depth s 33
km. The number of earthquakes is 724.
EPSL 6072 22-1-02 Cyaan Magenta Geel Zwart
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T. Okamoto, T. Tanimoto / Earth and Planetary Science Letters 195 (2002) 17^27
Fig. 4. Magnitude distribution of the earthquakes used in this study. (a) For the western USA (786 earthquakes in total). (b) For
Japan (3082 earthquakes in total).
that the depths in the NIED catalogues are determined for several prescribed discrete intervals and
5 km is the shallowest depth assigned in the determination.
Figs. 4 and 5 show the magnitude and depth
distributions, respectively. In both regions all
events (but one in the western US) have magnitudes equal to or larger than 3, and in contrast to
the HCMT global catalogue, events with magnitudes smaller than 5 are dominant. Also note the
dominance of the shallow events: many events
occur at depths shallower than about 10 km
with peaks at 8 km in the western USA and at
5 km in Japan. This shallowness of the events is
one of the characteristics of the regional catalogues because of the higher accuracy in the depth
determinations due to the dense seismic stations
close to the sources and the high frequencies used
for analysis. There is a di¤culty, however, in calculating accurate gravitational energy change for
these shallow earthquakes by the conventional
normal mode approach [2,3,13,14], and one of
the contributions of this paper is the alternative
method which will be discussed in Section 3.
3. Direct solution for depth kernel
The coseismic gravitational energy change of
the crust is
Z
b ugdV
…1†
vE g ˆ 3
VC
Fig. 5. Depth distribution of the earthquakes used in this study. (a) For the western USA (785 earthquakes). (b) For Japan
(2358 earthquakes). Only events with depths shallower than 33 km are shown.
EPSL 6072 22-1-02 Cyaan Magenta Geel Zwart
T. Okamoto, T. Tanimoto / Earth and Planetary Science Letters 195 (2002) 17^27
where b and g are the initial density and initial
gravitational acceleration before the deformation,
respectively, u is the coseismic static displacement
induced by an earthquake, and VC denotes the
volume of the crust. Eq. 1 represents one of the
largest terms (i.e. ¢rst order in displacement) in
the energy change due to an earthquake, and is
nearly balanced by the elastic energy change (i.e.
the work done against the initial stress) [1,15].
Our discussions are mainly based on Eq. 1 because, as discussed later, the crust seems to be a
good unit for discussing the tectonics.
For the SNREI (spherically symmetric, non-rotating, perfectly elastic, isotropic) earth model
(e.g. [16]), the crustal gravitational energy change
(Eq. 1) may be expressed as
Z Z
XZ R
lm
2
vE g ˆ
b …r†ur …r†g…r†r dr
sin a da
l;m
Z
0
2Z
RC
0
dP Y m
l …a ; P †
…2†
m
where ulm
r …r†Y l …a ; P † is the static radial displacement, RC denotes the bottom of the crust and
m
Ym
l …a ; P † ˆ Pl …cos a †exp…im P † is the spherical
harmonics with Pm
l …cos a † being the associated
Legendre function. The integration with respect
to a and P leaves only the radial deformation
(i.e. angular degree l = 0), and Mrr is the only
relevant moment tensor component under the assumption Mrr +Maa +MPP = 0 that is usually made
in the moment tensor analysis in order to exclude
isotropic component from the solution (e.g. [4,9]).
Thus the formula further reduces to
Z R
vE g ˆ M rr
K…r; rs †dr
…3†
RC
where we have de¢ned the depth kernel
K…r; rs †r4Zr2 b …r†ur …r; rs †g…r†:
…4†
ur …r; rs † is the static radial displacement with
l = m = 0 for a source at r = rs .
The normal mode summation approach has
been used so far for the calculation of the above
energy change [2,3,13,14] because of the advan-
21
tages in global analysis. There is, however, a dif¢culty in calculating accurate energy change for
very shallow earthquakes because of the slow convergence or the truncation error in the normal
mode summation. For example, we used normal
modes up to the radial order number 140 (corresponding to an eigenperiod of only 7 s) and we
still had some spurious oscillations in the radial
function ur …r; rs † [14]. Clearly, more accurate approach is required for very shallow earthquakes
because the displacement shows an abrupt (discontinuous) change at the source depth.
We advocate here a direct approach for the
evaluation of the static displacement ur …r; rs † instead of the normal mode approach. We brie£y
describe the method below because it has not been
widely used so far. The original full theory developed for the earth's oscillation is described by
Takeuchi and Saito [17].
In the case of the radial deformation (l = m = 0),
the relevant variables are the radial displacement
ur and the traction crr which can be expressed in
terms of two radial functions y1 and y2 (note
Y00 = 1):
ur ˆ y1 …r†; c rr ˆ y2 …r†:
…5†
The ¢rst-order radial equations for y1 and y2
are
U 323 W 1
dy1
1
ˆ 32
y1 ‡
y2
dr
U ‡ 43 W r
U ‡ 43 W
…6†
"
#
dy2
bg
3U W
4W 1
y1 3
ˆ 34
3
y2
dr
r U ‡ 43 W r2
U ‡ 43 W r
…7†
where U(r) and W(r) are the bulk and shear moduli, respectively.
We ¢rst integrate the system Eqs. 6 and 7 from
an initial radius close to the center of the earth
upward to the earth's surface. The solution for
homogeneous sphere is used as the initial condition for this ¢rst integration. Second, we integrate
the system from the source depth r = rs with initial
conditions speci¢ed by the source mechanism
(i.e. moment tensor). The two initial values are
[17]
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T. Okamoto, T. Tanimoto / Earth and Planetary Science Letters 195 (2002) 17^27
Fig. 6. Depth kernels calculated for various source depths.
y1 …rs † ˆ
1
M rr ;
4Z…U ‡ 43 W †r2s
…8†
y2 …rs † ˆ
3U
M rr :
4Z…U ‡ 43 W †r3s
…9†
The ¢nal solution can then be obtained by a
linear combination of the two solutions with a
coe¤cient that satis¢es the boundary condition
(crr = 0) at the earth's surface. Thus, the system
needs to be solved only two times, and no eigenfunction summation is involved so that no truncation error occurs.
We use isotropic PREM [16] with the crust
thickness of 24.4 km as the SNREI model for
the calculation. We replace the surface water layer
in PREM with upper crustal material in order to
locate very shallow sources close to the earth's
surface. The displacement ur …r; rs † is calculated
for a unit Mrr so that the resultant kernel
K…r; rs † can be applied to any type of earthquakes (note the only source term in the energy
change (Eq. 3) is Mrr ). The kernels (Eq. 4) for
various source depths are shown in Fig. 6. The
discontinuous changes in the kernels just at the
source depths are clearly reproduced and no spurious oscillations are introduced. These are the
desired characteristics for the kernels, especially
for shallow earthquakes.
The sign of the gravitational energy change is
apparently controlled by the sign of Mrr as it is
multiplied to the integrated kernel (Eq. 3). The
integrated kernel, however, changes its sign at
shallow depth: at about 4 km for the model
used in this study (Fig. 7). Thus the shallow earth-
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T. Okamoto, T. Tanimoto / Earth and Planetary Science Letters 195 (2002) 17^27
quakes have less or opposite e¡ects on the crustal
gravitational energy change as compared with
deeper earthquakes.
4. Crustal gravitational energy change and
discussions
Figs. 8 and 9 plot the cumulative crustal gravitational energy change induced by earthquakes in
the western USA during 1992 to 1999, and in
Japan during January 1997 to August 2000. In
making these ¢gures, we ¢rst divided the area
into 1³U1³ cells. Second, in each cell, we summed
up the gravitational energy changes induced by all
earthquakes in the cell. Finally, we drew the colored maps by assigning a color to each cell according to the logarithm of the summed energy
change. The region with positive change (energy
gain) is indicated by red, and negative change
(energy loss) is indicated by blue. The cells with
no earthquakes are indicated by gray.
The gravitational energy change calculated as
above involves an integration within all of the
crust (Eqs. 1^3) so that it is the global, total
change induced by earthquakes rather than the
local energy change of the local crustal block or
cell. But, as found by [3], it re£ects the tectonic
stress state at around the earthquake. The overall
correspondence between the gravitational energy
change pattern and the compressional/extensional
tectonic modes can be seen in Figs. 8 and 9 in
¢ner scale than in the results by [3], because of
the regional detailed catalogues used here.
In general, the gravitational energy change is a
good indicator of the vertical displacement. The
earthquake generates coseismic distribution of uplift and subsidence around the source, but such
£uctuations are leveled o¡ in our calculations because of the involved global integration (Eq. 3).
Thus, our result roughly indicates that compressional tectonics shows a systematic, overall uplift
while the extensional tectonics shows a systematic,
overall subsidence of the crust. We consider that
the crust is a good unit for discussing such tectonics with respect to the gravitational energy
change caused by earthquakes because most of
the earthquakes occur inside the crust.
23
In the case of extremely shallow earthquakes, it
is possible for an area to have negative energy
change while Mrr of the earthquakes are positive
(horizontal compression), because extremely shallow earthquakes have less or opposite e¡ect on
the energy change as compared with deep earthquakes (Fig. 7). In that case, we may interpret
that the compressional stress state in the crust is
con¢ned to a very shallow portion.
4.1. The western USA
In the western USA (Fig. 8), a band of coseismic energy gain along the coast (or along the San
Andreas fault) can be clearly identi¢ed. This energy gain band is consistent with the compressional tectonics in the Coast Ranges [18]: the geologic
feature in the Coast Ranges indicates a transverse
shortening (compression) during roughly from 3.5
Myr to present, and it is in harmony with the
small boundary-normal component in the relative
motion at the Paci¢c^North America plate
boundary along the San Andreas fault [18].
We have almost no large earthquakes and thus
almost no gravitational energy change induced by
earthquakes inside the Basin and Range. However, at the margin of the province, particularly
at the southwestern margin along the Sierra Nevada, we have distinct region of gravitational en-
Fig.
7. The value of the integrated depth kernel
R
( R
RC K…r; rs †dr) plotted against the source depth. `Neutral
Depth' indicates the depth for which the earthquake has no
e¡ect on crustal gravitational energy change. Note that the
integrated value is dimensionless.
EPSL 6072 22-1-02 Cyaan Magenta Geel Zwart
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T. Okamoto, T. Tanimoto / Earth and Planetary Science Letters 195 (2002) 17^27
ergy loss during the studied period. Such energy
loss pattern matches the extensional tectonics in
the Basin and Range. The negative energy change
along the Golda ridge also re£ects the extensional
environment there.
There are relatively large energy gains at the
Transverse Ranges and at the northern end of
the San Andreas fault (Mendocino fracture
zone). The curved fault at these places might act
as the obstacles to the plate motion and cause
large compressional stresses that are relevant to
the relatively large gravitational energy gain at
these places.
The gravitational energy itself is believed to
have a role on the dynamics of the continental
deformation: the spatial di¡erences in the gravitational energy produce buoyancy forces that drive
the deformation of the continent (for the western
USA, see e.g. [19,20]). The lithospheric gravitational energy distribution in the western USA
[19] shows negative energy in the Coast Ranges
and Transverse Ranges, and positive energy in the
Sierra Nevada and the northern Basin and Range.
The crustal gravitational energy change caused by
earthquakes (Fig. 8) has roughly reversed pattern
when compared with the lithospheric gravitational
energy: positive changes in regions with negative
energy and negative changes in regions with positive energy. This again indicates the correlation
between the tectonics (here it is the gravitationally
driven lithospheric deformation) and the crustal
gravitational energy change.
On the other hand, the coseismic `lithospheric'
gravitational energy change shows almost reversed pattern when compared to the crustal gravitational energy change (i.e. energy gain in regions
with positive energy and energy loss in regions
with negative energy when compared with the
gravitational energy [19]). This is because the
depth kernel (Fig. 6) has negative amplitude below the source depth, and there are no deep earthquakes in the region (note the lithospheric energy
change is calculated by integrating Eq. 1) or Eq. 3
for whole lithosphere instead of whole crust).
Thus, in the western USA, earthquakes seem
not to contribute to the lithospheric gravitational
energy change because regions with positive energy would lose energy while regions with nega-
Fig. 8. Crustal gravitational energy change in the western
USA during 1992 to 1999. `Energy gain' is indicated by red,
and `energy loss' is indicated by blue. The cells with no
earthquakes are indicated by gray. The value of the largest
(absolute) energy change of all the changes assigned to the
cells is shown at the top of the map. The value assigned to
each cell indicates the gravitational energy change for whole
crust induced by the earthquakes in each cell. See text for
details.
tive energy would gain energy. We need more detailed (i.e. localized) analysis of the coseismic
gravitational energy change in order to study further such correlations.
4.2. Japan
In Japan, we ¢nd a distinct contrast in the energy change pattern between northeastern and
southwestern Japan (Fig. 9): regions with coseismic energy gain are dominant in northeastern Japan while weak energy gain or energy loss regions
dominate southwestern Japan. The dominance of
energy gain in and around northeastern Japan can
be related to the compressional stress state due to
the convergence between the North America plate
(or the Okhotsk plate [21]) and the Paci¢c plate.
EPSL 6072 22-1-02 Cyaan Magenta Geel Zwart
T. Okamoto, T. Tanimoto / Earth and Planetary Science Letters 195 (2002) 17^27
The relatively weak energy gain pattern in
southwestern Japan indicates a weak compressional tectonics or even somewhat extensional tectonics despite the convergent relative motion between the Philippine Sea and the Eurasian plates.
Actually, an opening of Okinawa Trough is believed to be taking place (e.g. [22]), and extensional crustal deformation in the central Kyushu has
been inferred based on triangulation data [23]: the
energy loss region in Kyushu and Okinawa
Trough is consistent with these extensional tectonic behaviors.
In Japan we have a somewhat complex pattern
of coseismic energy change : e.g. there are several
`isolated' cell(s) of energy loss such as the ones o¡
northern Honshu and in central Honshu. Such
short wavelength pattern is more apparent in Japan than in the western USA. We ¢nd that such
complexity may be partially attributed to the effect of the deep earthquakes. For example, the
energy loss in central Honshu is the result of the
contribution from deep earthquakes: the energy
change due to crustal earthquakes alone is positive which is in harmony with those of surrounding cells. Although the relatively short duration
(3 yr) of the catalogue can also cause some complexity due to temporal variation [14], our results
indicate that the deep earthquakes may cause effects on the pattern of the crustal gravitational
energy change in some circumstances.
4.3. Total gravitational energy change
The cumulative, total (global) crustal gravitational energy change caused by the earthquakes
occurring in the western USA is +6.1U1017 J
(for Fig. 8), and it is +2.7U1017 J by the earthquakes in Japan (for Fig. 9). Although the selection of the studied area is somewhat arbitrary, we
consider the positive change in Japan to be typical
for the compressional tectonics such as the subduction zones. Despite the major extensional deformation in the Basin and Range, the total positive change in the western USA also indicates the
dominance of the compressional stress state there
with respect to the occurrence of the earthquakes.
Also, we can integrate Eq. 1 for whole earth.
The cumulative whole earth energy change caused
25
Fig. 9. Crustal gravitational energy change in Japan during
January 1997 to August 2000. All earthquakes including the
deep ones are used.
by the earthquakes in the western USA is
33.2U1018 J, and it is 31.0U1018 J by the earthquakes in Japan. As discussed above (Section 4.1),
the depth kernel (Fig. 6) has negative value (subsidence) below the source depth. Thus the material below the source is compressed and loses gravitational energy for positive Mrr (or horizontal
compression). Because of the large amount of materials below the source, this e¡ect overcomes the
energy gain of the materials (mostly crust) above
the source, which results in a energy loss of whole
earth.
4.4. Structural models
There is a slight discrepancy between the earth
model we used in this study (slightly modi¢ed
EPSL 6072 22-1-02 Cyaan Magenta Geel Zwart
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T. Okamoto, T. Tanimoto / Earth and Planetary Science Letters 195 (2002) 17^27
PREM) and the local models used for the regional
catalogues : e.g. the NIED seismic moment tensor
catalogue for Japan uses a £at earth model with a
crust whose thickness is 33 km, in contrast to the
24.4 km crust in PREM. In order to check the
e¡ect of this discrepancy, we constructed a spherical model for Japan (i.e. we simply replaced the
top 425 km layer of the PREM with the structure
used for the NIED catalogue) and calculated the
cumulative energy change. We set RC = R324.4
(km) rather than R333.0 (km) because of the involved global integration. The resultant pattern is
almost identical to Fig. 9 with slight changes in
the amplitude. This indicates the patterns we obtained are relatively robust with respect to the
slight changes in the structural models, provided
that the globally averaged crustal thickness RC is
used in the integration (Eq. 3).
5. Conclusion
We study the cumulative change of the crustal
gravitational energy induced by earthquakes in
the western USA and Japan. The conclusions
are: (1) We adopt a `direct' approach in calculating the kernel for the evaluation of the gravitational energy change rather than the conventional
normal mode summation scheme, avoiding the
in¢nite summation (or the truncation error) so
very accurate kernels are obtained. (2) We ¢nd
systematic regional patterns that correspond to
the regional tectonic regime: gravitational energy
gain for compressional tectonics and energy loss
for extensional tectonics. For example, in the
western USA, the Coast Ranges and the Transverse Ranges show systematically positive gravitational energy change while the margin of the Basin and Range shows negative change. In Japan,
the northeastern part shows predominantly positive change while the southwestern part shows
negative to weak positive change. (3) The total
(global) crustal gravitational energy changes induced by the regional earthquakes are positive
for both regions which indicates the predominance of the compressional stress state with respect to the earthquake occurrence. (4) Some of
the short wavelength complex features in Japan
can be interpreted in terms of the contribution
of the deep earthquakes.
Acknowledgements
The authors are grateful to L. Zhu and D.V.
Helmberger of California Institute of Technology,
D. Dreger and others of UC Berkeley Seismological Laboratory, and E. Fukuyama and others of
NIED for their online earthquake catalogues. We
are also grateful to B.F. Chao and an anonymous
reviewer for their helpful comments on the manuscript. We used the GMT package [24].[RV]
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EPSL 6072 22-1-02 Cyaan Magenta Geel Zwart