Gravity Variation Estimation of the 2011 Tohoku Earthquake

Journal of the Korean Society of Surveying, Geodesy, Photogrammetry and Cartography.
2015.
Dec,
33(6):
497-506

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

- Received : November 11, 2015
- Accepted : December 12, 2015
- Published : December 31, 2015

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Gravity variations due to the 2011 Tohoku (M9.0) earthquake, which occurred at the plate boundaries near the northeastern coast of Japan, were estimated through the GRACE spherical harmonic (Stokes) coefficients derived from the CSR. About -5 μGal gravity variation by the GRACE data was found in the back-arc basin area with respect to a reference gravity model. The mean gravity variations in the back-arc basin area and the Japan Trench area were -4.4 and -3.2 μGal in order. The small negative gravity variations around the Japan Trench area can be interpreted by both crustal dilatation and the seafloor topography change in comparison with the large negative gravity variations in the back-arc basin area by co-seismic crustal dilatation of the landward plate. From the results of the gravity variations, vertical displacements generated from relatively short wavelength caused by the earthquake were estimated by use of multi-beam bathymetric measurements obtained from JAMSTEC. The maximum seafloor topography changes of about ±50 m were found at west side of the Japan Trench axis by the earthquake. The seafloor topography change by the megathrust earthquake can be considered as the results of the landslide of the seafloor throughout the landward side.
^{−8}
m/sec
^{2}
) precision level.
Since GRACE launched in 2002, it has been utilized as a method to measure time-variable gravity variations by megathrust earthquake. The case of using GRACE was firstly investigated in co-seismic gravity variations by mass redistribution from the 2004 Sumatra-Andaman earthquake (M9.1) (
Han ., 2006
). They observed ±15 μGal gravity variations induced by the earthquake, which was referred; that the large gravity decrease as crustal dilation in the Andaman Sea of the east of the Sunda Trench, while the large gravity increase as vertical displacement and crustal dilatation in the southwestern area of the Sunda Trench. The gravity variations by the 2010 Chile earthquake, which occurred in the central Chile, was observed by GRACE (
Han ., 2010
;
Heki and Matsuo, 2010
). Several studies have described gravity variations by processing the time series of monthly GRACE gravity solutions by optimizing the signal over the earthquake region by the 2011 Tohoku earthquake (
Han ., 2011
;
Matsuo and Heki, 2011
;
Cambiotti and Sabadini, 2013
;
Wang ., 2012
;
Han ., 2013
). Following to
Han . (2014)
, the post-seismic gravity increase by 6 μGal was observed, which indicates 40~50% of the co-seismic gravity variation, within a couple of years after the 2011 Tohoku-Oki earthquake by GRACE. That is, the GRACE satellite measurements enable to monitor gravity variations by generating signals related to the megathrust earthquake, where the ground truth observations, such as gravity anomalies are limited or absent after the earthquake.
The objective of the study is to estimate gravity variation due to the 2011 Tohoku earthquake at convergent plate boundaries that characterized by the major subduction zones. In this study, the latest gravity model, Global Geopotential Model 03 Combination (GGM03C) (
Tapley ., 2007
) which is released to the public by combination GRACE satellite mission with gravity information before March 11, 2011 was adopted as a reference gravity model to compute gravity variation by the megathrust earthquake. In order to estimate gravity variation related to the megathrust earthquake, the one 2-year differenced pair and two 1-year differenced pairs from the cumulative gravity variations during 3 consecutive years before and after the earthquake were computed to eliminate hydrological and seasonal signals.
For accomplishing changes of gravity and seafloor topography, related to the megathrust earthquake in this study, an individual change was independently performed without analysis of correlation between these two changes. Gravity variations, caused by the megathrust earthquake, accompany the changes of seafloor topography at short wavelength at the Japan Trench axis, which is a plate boundary of subduction zone. A comparison of available shipborne depth measurements before and after the megathrust earthquake was performed to verify significant seafloor topography change at the Japan Trench area in this study.
C_{lm}
,
S_{lm}
in the GRACE Level-2 gravity field products are defined in the following expansion of the Earth’s exterior geopotential (
Bettadpur, 2007
):
where
V
is the gravitational potential;
r
is the radius of point P;
ϕ
is the colatitude of point P;
λ
is the longitude of point P;
GM
is the gravitational constant of the Earth;
a
_{e}
is mean equatorial radius, which is the value of reference ellipsoid: 6378136.3 m;
is fully normalized associated Legendre polynomials;
C_{lm}
and
S_{lm}
are geopotential spherical harmonic coefficients; and
l
and
m
are degree and order.
The
C
_{20}
spherical harmonic coefficients, those of derived from GRACE, are still noisier than those of derived from Satellite Laser Ranging (SLR) data (
Bettadpur, 2008
). The GRACE
C
_{20}
spherical harmonic coefficients are replaced by the SLR-derived value (
Cheng and Tapley, 2004
). In the GRACE data processing, the origin of reference frame is selected to be at the instantaneous center of mass of the Earth (
Bettadpur, 2007
). Therefore, the summation in Eq. (1) starts at degree 2, since changes in the location of the center of mass of the Earth cannot be obtained from the GRACE data.
A gravity field functional
F
can be computed from the spherical harmonic coefficients:
where
l
_{max}
is the maximum degree of the spherical harmonic expansion of the gravity solution.
X_{l}
=
a_{e}
for the geoid;
for free-air gravity anomaly; and
for free-air gravity disturbance.
X
_{0}
, which is the degree zero term, is zero for GRACE data processing. The degree one term is zero in the free-air gravity anomaly of Eq. (2) due to the (
l
− 1) term.
For geophysical interpretation in Eq. (2), gravity variations with respect to a reference gravity field are generated by removing from the GRACE monthly spherical harmonic coefficients:
where
and
are geopotential spherical harmonic coefficients of a reference gravity field.
C_{lm}
(
t
) and
S_{lm}
(
t
) can be used in Eq. (2).
W
) given by the recursion relations. An isotropic averaging filter on the sphere is defined by the following function:
where
ψ
is the spherical distance between the points (
ϕ
,
λ
) and (
ϕ'
,
λ'
), and cos
ψ
= cos
ϕ
cos
ϕ'
+ sin
ϕ
sin
ϕ'
cos(
λ
−
λ'
).
The weighting coefficients
W_{l}
in the spectrum of the function
W
(
ψ
) of Eq. (4) is defined by one dimensional Legendre transform:
where
P_{l}
(cos
ψ
) is the Legendre polynomials.
The isotropic Gaussian filter is composed of the averaging function:
where the averaging radius
r
=
Rψ
is the distance called a half-width of the filter at
. The filter cofficients
W_{l}
are computed with recursion relations (
Wahr ., 1998
):
C
_{20}
(the degree 2 order 0) spherical harmonic coefficients for the Earth’s dynamic oblateness were replaced by the values, determined from SLR measurements due to large uncertainties in the GRACE coefficient
C
_{20}
(
Cheng and Tapley, 2004
). The degree 1 spherical harmonic coefficients (
C
_{10}
,
C
_{11}
,
S
_{11}
), related to the movement of geocenter, were not taken into account due to less contributions to local gravity variations. The de-striping filtering and the 600 km width of Gaussian smoothing filtering were implemented to reduce the noise that caused by longitudinal stripes in the short wavelength (high degree and order) of spherical harmonic coefficients (
Wahr ., 1998
;
Swenson and Wahr, 2006
).
The cumulative gravity variations with respect to a reference gravity model, GGM03C (Tapley et al ., 2007 ), using the GRACE data before and after the 2011 Tohoku earthquake are April ~ July 2009 (a), April ~ July 2010 (b), and April ~ July 2011 (c). Gravity variations by subtracting the 2009 from the 2010 (d), the 2009 from the 2011 (e), the 2010 from the 2011 (f).
The one 2-year differenced pair (2011 and 2009) and two 1-year differenced pairs (2010 and 2009, 2011 and 2010) were computed from differences between the cumulative gravity variations -
Figs. 1(a)
,
1(b)
, and
1(c)
. Those
Figs. 1(d)
,
1(e)
, and
1(f)
present gravity variations of those by differencing, between 2010 and 2009, between 2011 and 2009, and between 2011 and 2010, respectively. The differences of gravity variations between 2010 and 2009 in
Fig. 1(d)
are a little, while the gravity variations in 2011 with respect to 2009 and 2010 show remarkable negative anomalies in the back-arc basin in
Figs. 1(e)
and
1(f)
. The gravity variations in
Fig. 1(e)
of 2011 with respect to 2009 are much smaller than those in
Fig. 1(f)
of 2011 with respect to 2010 in the back-arc basin area; the result could be explained by those relatively smaller cumulative gravity variations of 2009 year in
Fig. 1(a)
than those of 2010 year in
Fig. 1(b)
in the East Sea around Japan. Therefore, it can be considered due to mass change in the seawater, during the same period (April ~ July) in different years. In
Figs. 1
and
2
, the red star indicates the epicenter of the 2011 Tohoku earthquake, and the contour interval is 0.1 μGal.
The gravity variations before and after the 2011 Tohoku earthquake by averaging two differenced gravity variations (Figs. 1(e) and 1(f) ) in a different time. The red lines represent the tectonic plate boundaries.
The gravity variations by the 2011 Tohoku earthquake were presented by averaging these two variations of
Figs. 1(e)
and
1(f)
, related to the earthquake. The anomalies of the gravity variations reach about –5.0 μGal in the back-arc basin area, locating in the eastern margin of East Sea (Sea of Japan).
Table 1
presents the statistics of gravity variations in two test areas, as shown in
Fig. 2
. By comparing with gravity variations in the two test areas, the mean gravity variations in the back-arc basin area (denoted as #1) and at the Japan Trench area (denoted as #2) were –4.4 and –3.2 μGal in order. At the Japan Trench area, the gravity variations estimated by the GRACE data, before and after the earthquake are relatively smaller than those in the back-arc basin area.
Statistics of gravity variation comparisons in two test areas
As a result, the large negative gravity variations in the back-arc basin area can be interpreted as co-seismic crustal dilatation of the landward plate, while the small negative gravity variations at the Japan Trench area can be interpreted as both crustal dilatation and the seafloor topography change. Also, these gravity variations estimated from the GRACE data by the megathrust earthquake present a similar result (
Matsuo and Heki, 2011
).
KR11_05_leg
2; denoted as the gray dots in
Fig. 3(d)
) were computed at the Japan Trench, based on the quick survey from Research Vessel Kairei (KR) of JAMSTEC between March 22 and 23, 2011 after the earthquake.
(a) a 5×5 arc-second JAMSTEC shipborne gridded model after the earthquake, (b) a 5×5 arc-second JAMSTEC shipborne gridded model before the earthquake, (c) the seafloor topography changes on JAMSTEC shipborne locations after the earthquake denoted as the black dots in (d), Location map with JAMSTEC shipborne tracks (d) after earthquake and (e) before earthquake. The contour interval is 100 m in (a) and (b), respectively.
The shipborne grids before and after the earthquake around the Japan Trench area are delineated by a blue box in
Figs. 3(d)
and
3(e)
. Those were estimated by interpolating with 276,872 JAMSTEC shipborne depth measurements of
Fig. 3(d)
after the earthquake; and 374,391 JAMSTEC shipborne depth measurements of
Fig. 3(e)
before the earthquake, based on the General Mapping Tools (GMT, http://gmt.soest.hawaii.edu) (Wessel and Smith, 1998) routine “surface.” The routine is continuous curvature surface gridding algorithm with a tension factor of 0.25 to produce a 5×5 arc-second spacing in latitude and longitude. The blue boxes in
Figs. 3(d)
and
3(e)
are the boundary of the Japan Trench area used for examining the seafloor topography changes by the earthquake. The black stars in
Figs. 3(d)
and
3(e)
present the epicenter of the earthquake. The attributes listed for this and subsequent maps include the amplitude range (AR = minimum and maximum values), amplitude mean (AM), amplitude standard deviation (ASD), and amplitude unit (AU).
The point-by-point differences between two the shipborne grids before and after the earthquake could be calculated by subtracting the shipborne grid data of before the earthquake in
Fig. 3(b)
from the shipborne grid data of after the earthquake in
Fig. 3(a)
. The seafloor topography changes at the Japan Trench were estimated by interpolating the point-by-point differences between JAMSTEC shipborne grids before and after earthquake on JAMSTEC shipborne tracks. In fact, the seafloor topography before and after the earthquake shows significant change of the maximum ±50 m at the Japan Trench, denoted as a black dotted circle in
Fig. 3(c)
.
In addition, the shipborne grids before and after the earthquake and the seafloor topography change between the models were compared along two profiles: Profile #1 and #2 (black lines in
Figs. 3(a)
,
3(b)
, and
3(c)
). The results of bathymetry comparisons before and after the earthquake for Profiles #1 and #2 presented in
Fig. 4
show evident change of the seafloor topography at the Japan Trench between 143.90° and 144.10° E (denoted as the green box in
Figs. 4(a)
and
4(b)
).
Bathymetry comparisons before and after the 2011 Tohoku earthquake for Profiles #1 (a) and #2 (b), respectively, as shown in Figs. 3(a) , 3(b) , and 3(c) , at the Japan Trench between 143.90° and 144.10° (denoted as the green boxes) in the insets. Comparisons of bathymetry change along Profiles #1 (c) and #2 (d).
The bathymetry (red dot) after the earthquake is shallower than that (black dot) before the earthquake at the Japan Trench axis, as shown in
Figs. 4(a)
and
4(b)
because of submarine landslide throughout landward side which of following to the megathrust earthquake. In
Figs. 4(c)
and
4(d)
, the maximum seafloor topography changes about ±50 m is indicated as a black dotted circle in
Fig. 3(c)
following to the west side of Japan Trench axis around 144° E in
Figs. 4(a)
and
4(b)
. The results of seafloor topography changes at the Japan Trench, as shown in
Figs. 4(c)
and
4(d)
, are similar to the results of
Kim and Lee (2015)
at profiles #4 and #5 in Fig. 8(c).

1. Introduction

A megathrust earthquake has been frequently occurred at convergent plate boundaries of major subduction zones, where the oceanic crust subducts under the continental crust at deep-sea trench. The 2011 Tohoku earthquake (M9.0) occurred on the subduction zone along the Japan Trench, where the Pacific plate subducts beneath the North American plate near the northeastern coast of Japan. The earthquake of the fourth gigantic earthquakes since 1900 hit offshore the northeastern area of Honshu Island on March 11 at 05:46:23 UTC according to the United States Geological Survey (USGS, http://earthquake.usgs.gov/earthquakes/eqinthenews/)
The megathrust earthquake produces intense and large scale of the mass redistribution, which resulted in variations in the Earth gravity field. Gravity variations induced by the megathrust earthquake are essential to better understand the shape and evolution of the Earth as fundamental dynamic processes, related to change of gravity field inside Earth. The gravity variation by megathrust earthquake is caused by (i) the crustal dilatation due to density change and (ii) deformation of the seafloor topography based on an assumption that there are no horizontal density variations in oceanic crust under the ocean bottom (
Pollitz, 2006
).
The megathrust earthquake ruptured the fault as large as 500 km × 200 km between the northeastern Honshu and the Japan Trench (
Ammon ., 2011
;
Ozawa ., 2011
). The study about surface deformation caused by the megathrust earthquake was focused on the land around the epicenter of the earthquake. The co-seismic deformation by GPS data mainly directed toward off the Pacific coast of northeast Honshu. The co-seismic displacements - up to 5.3 m in the eastward in horizontal displacements and up to 1.2 m in vertical displacements - were detected by GPS along coast of Tohoku region after the earthquake (
Ozawa ., 2011
). The motions in the horizontal and vertical displacements of about 4.2 m eastward movement and about 0.5 m downward respectively were appeared along the coast of northeast Honshu (
Lee, 2012
).
Before the launch of the Gravity Recovery and Climate Experiment (GRACE), the mapping of variations in the Earth’s gravity field on global scale was limited. The Earth’s gravity field variations on global scale have been determined, since the launch of GRACE satellite gravity mission in March, 2002. The GRACE gravity solutions have improved the knowledge for maps of the large scale gravity field variations; those were generated from mass change within the Earth, and also on or above its surface, such as ground water storage change, polar ice sheets, sea level change, and atmospheric and oceanic mass change. However, the satellite gravity measurements from the GRACE originally measures the global mass variations in the solid earth-oceanatmosphere system (
Tapley ., 2004b
) but also allows for measuring co-seismic gravity change by earthquake with μGal (=10
2. Gravity Recovery from GRACE

The GRACE launched on March 2002 is a joint satellite mission of National Aeronautics and Space Administration (NASA: http://www.nasa.gov/) and Deutsches Zentrum für Luft-und Raumfahrt (DLR: http://www.dlr.de/). The purpose of GRACE is to map and to understand signals related to mass variations in the Earth system with unprecedented accuracy and resolution in the form of time-varying gravity field (
Wahr ., 1998
). The GRACE makes continuous measurements of the change in distance between two satellites, which are co-orbiting as a polar orbit at 500 km altitude and 89.5° inclination, spaced from each other apart approximately 220 km, and connected by accurate inter-satellite system with the K-band microwave ranging (KBR) satellite-to-satellite tracking (SST) system (
Seeber, 2003
;
Tapley ., 2004a
;
Bettadpur, 2007
). Each satellite is consisted of Global Positioning System (GPS) receivers, attitude sensors, and high precision accelerometers. Those GPS receivers determine the position of each satellite, while those 3-D accelerometers in each satellite are used to measure the non-gravitational acceleration and to remove its effects from the satellite-to-satellite distance measurements.
The Level-2 gravity field products derived from the GRACE are currently provided by different research groups: Geo Forschungs Zentrum (GFZ: http://gfz-potsdam.de), Center for Space Research (CSR: http://www.csr.utexas.edu/grace/asdp.html), and Jet Propulsion Laboratory (JPL: http://www.jpl.nasa.gov) to produce monthly gravity solutions. The estimates of spherical harmonic coefficients developed by these groups are up to degree between 50 and 120 with every 30 days spatial resolution.
The GRACE gravity field products provided by the Level-2 processing use an a-priori static gravity field in the processing, and there are important differences in Release-01 (RL-01) solutions from different processing centers. Those dynamic parts of the a-priori static gravity field include the ocean tides, the solid Earth tides, and the atmospheric and ocean variability. The different official solutions are highly comparable (
Schmidt ., 2008
) in the Release-04 monthly time-variable gravity field solutions because of standardized background models.
The spherical harmonic coefficients of the geopotential
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- 2.1 Filtering of GRACE level-2 products

Each gravity field of the GRACE data from the CSR is comprised of a set of spherical harmonic coefficients with the degree and order of 60. The GRACE monthly mean gravity solutions exhibit increased errors in high frequencies that obscure the derivation of geophysical signals (
Rangelova, 2007
). Either Spatial averaging, or smoothing of the GRACE data is necessary to reduce the contribution of noisy short wavelength components of the gravity field solutions. However, as the smoothing radius decreases, the errors of stripes with north-south orientation imply correlations in the gravity field coefficients. The De-striping filter developed by
Swenson and Wahr (2006)
is applied to identify the geophysical signature of the correlations between spherical harmonic coefficients that is manifested in the striped pattern in GRACE maps. The Gaussian smoothing in
Swenson and Wahr (2006)
is applied after de-striping filter. The degree of smoothing is controlled by fitting a low order polynomial to the limited interval of coefficients of odd, or even degrees, in order to reduce aliasing errors, which related to the spatial correlations.
The isotropic Gaussian smoothing filter, which is introduced by
Jekeli (1981)
and developed for time-variable gravity field of the GRACE data by
Wahr . (1998)
, is described as the weights (filter coefficients
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3. GRACE Data Processing for Estimating Gravity Variation

In order to compute gravity variation induced by the 2011 Tohoku earthquake, we used Release-04 (RL-04) monthly mean estimates of the spherical harmonic coefficients from the GRACE with the degree and order of 60, obtained from the CSR, University of Texas at Austin (UT). As a reference gravity model in this study, the GGM03C (
Tapley ., 2007
), which is a combination of the Global Geopotential Model 03 Satellite (GGM03S) with terrestrial (ocean and land) gravity information with the degree and order of 360, was used to compute gravity variation caused by the earthquake. The GGM03S - complete to degree and order of 180 - is derived from the GRACE 47 months data from January 2003 to December 2006 (January 2004 excluded) and is unconstrained by any other information (
Tapley ., 2007
).
The method of
Han . (2006)
was implemented in the study for detecting gravity variations that can be estimated by removing the annual hydrological variations between 1- and 2-year differences from stacking over several months before and after the earthquake
The GRACE monthly estimations of the spherical harmonic coefficients of the maximum degree and order of 60, obtained from the CSR, were used to recover gravity for three months (April and July, June in each year but excluded missing June data of 2011 year) in three consecutive years (2009, 2010, and 2011), whenas before and after the earthquake occurred on March 11, 2011. The
4. Results and Discussions

- 4.1 Gravity variation by the earthquake

The
Figure 1
shows the cumulative gravity variations around Japan;
(a)
April to July 2009,
(b)
April to July 2010, and
(c)
April to July 2011 for three continuous years; and the differences between the cumulative gravity variations by subtracting
(d)
the 2009 from the 2010,
(e)
the 2009 from the 2011, and
(f)
the 2010 from the 2011 by the 2011 Tohoku earthquake. The gravity variations of 2011 year (
Fig. 1(c)
) after the earthquake show significant anomalies in a back-arc basin of the eastern margin of East Sea (Sea of Japan). The signals with hydrological and seasonal effects but not associated with earthquake were removed from the differences between the cumulative gravity variations of three years.
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Statistics of gravity variation comparisons in two test areas

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- 4.2 Seafloor topography change at the Japan Trench

The results of the gravity variations in the two areas #1 and #2 estimated by the GRACE data are shown in
Fig. 2
. The seafloor topography changes can be estimated at shipborne bathymetric locations, measured by Japan Agency for Marine-Earth-Science And Technology (JAMSTEC, http://www.jamstec.go.jp), after the earthquake with assumptions of existing both effects on small crustal dilatation and large seafloor topography changes at the Japan Trench area (#2 in
Fig. 2
).
A bathymetric survey by JAMSTEC around the earthquake epicenter in the Pacific Ocean was conducted by operating a Sea Beam 2112.004 multi-beam echo sounder with those; the frequency of 12 kHz, the beam angle width of 2° by 2°, the range from 50 to 11,000 m, the depth resolutions of center beam with depth (m) × 0.2%, and the side beam with depth (m) × 0.5%. The seafloor topography changes before and after the earthquake on the JAMSTEC shipborne tracks (73,459 points of Survey ID:
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5. Conclusion

In this study, we have estimated gravity variations caused by the 2011 Tohoku earthquake, using the GRACE time-variable gravity field solutions. The map of the gravity variations by the megathrust earthquake shows the maximum variations of about –5.0 μGal in the back-arc basin area. The large negative gravity variations in the back-arc basin area can be interpreted as the co-seismic crustal dilatation of landward plate, while the small negative gravity variations at the Japan Trench area can be interpreted as both crustal dilatation and the seafloor topography change. The gravity solutions obtained from GRACE satellite mission are useful to detect the remarkable gravity variations, induced by mass redistribution of megathrust earthquake, which occurs at the major subduction zone.
The seafloor topography changes in trench axis on the subduction zone by shipborne depth measurements were estimated to reveal vertical displacements of the seabed by the megathrust earthquake. The seafloor topography from differences between bathymetry grid data before and after the earthquake presented the maximum change of ±50 m at west side of the Japan Trench axis because of the landslide of the seafloor throughout the landward side. As a result, the changes in seafloor topography, which was generated from relatively short wavelength by the megathrust earthquake at convergent plate boundaries of the major subduction zone, could be effectively estimated, only and only if the quick shipborne depth measurements after earthquake become available. In addition, the future research about the seafloor topography changes in the back-arc basin area of the East Sea (Sea of Japan) is needed to compare with that at the Japan Trench area by the megathrust earthquake.
Bettadpur S.
(2007)
Level-2 Gravity Field Product User Handbook
Center for Space Research, The University of Texas at Austin
GRACE 327-734

Bettadpur S.
(2008)
GRACE: Progress towards product improvement, and prospects for synergy with GOCE, 2008, Oral presentation
GGEO 2008 Symposium
Chania, Crete, Greece
June 23-27, 2008

Jekeli C.
(1981)
Alternative Methods to Smooth the Earth’s Gravity Field
Dept. of Geodetic Science and Surveying, Ohio State University
Columbus

Rangelova E.V.
(2007)
A Dynamic Geoid Model for Canada, Ph.D. dissertation
Department of Geomatics Engineering, University of Calgary
Calgary, Canada
242 -

Seeber G.
(2003)
Satellite Geodesy
Walter de Gruyter
Berlin, New York

Tapley B.D.
,
Ries J.
,
Bettadpur S.
,
Chambers D.
,
Cheng M.
,
Condi F.
,
Poole S.
(2007)
The GGM03 mean earth gravity model from GRACE
Eos Trans. AGU, Fall Meet. Suppl.
Abstract G42A-03
88
(52)

Citing 'Gravity Variation Estimation of the 2011 Tohoku Earthquake
'

@article{ GCRHBD_2015_v33n6_497}
,title={Gravity Variation Estimation of the 2011 Tohoku Earthquake}
,volume={6}
, url={http://dx.doi.org/10.7848/ksgpc.2015.33.6.497}, DOI={10.7848/ksgpc.2015.33.6.497}
, number= {6}
, journal={Journal of the Korean Society of Surveying, Geodesy, Photogrammetry and Cartography}
, publisher={Korean Society of Surveying, Geodesy, Photogrammetry and Cartography}
, author={Kim, Kwang Bae
and
Lee, Chang Kyung}
, year={2015}
, month={Dec}