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Auxiliary Material
Satellite gravimetry unravels seismic source processes of the 2011 Tohoku-Oki earthquake, by
Shin-Chan Han, Jeanne Sauber, and Riccardo Riva
1. Formulation of coseismic gravitational potential change from a seismic double-couple source
Analogous to the computation of geodetic displacements and long-period seismic waves
[Kanamori and Given, 1981; Pollitz, 1996], coseismic gravitational potential changes can be
expressed on the basis of a summation of normal modes on a spherically layered Earth. The
normal modes approach simultaneously solves for the conservation of linear momentum, where
body and surface forces compensate each other, and for perturbations in the gravitational
potential, due to mass redistribution and density changes (as described by Poisson's equation and
the continuity equation). The relation between stress and strain is assumed to be linear. We
analytically solve these fundamental equations by means of a spectral approach [Piersanti et al.,
1995]. The radial symmetry of the Earth model allows us to reduce the problem to a system of
first-order differential equations, dependent on the harmonic degree and valid inside each
homogeneous layer. By imposing continuity constraints on deformation, stress and gravity
potential at each layer boundary, the solution can be propagated from the free surface to the fluid
outer core. The presence of an ocean is modeled by means of an incompressible solid layer in
the limit of vanishing rigidity (shear modulus of 10-3 Pa) which accounts for both topographic
changes of the ocean surface and the solid-earth response to changes in the ocean load. We have
verified that the ocean surface follows the perturbed geoid, which verifies that the top layer
correctly reproduces the behavior of an ocean at rest.
For a point double-couple source located at the pole and striking in the prime meridian direction,
the following five spherical harmonic coefficients describe the permanent change in the
gravitational potential field at degree l:

1
Cl,0    K 0 ds y5 a  M 0 sin 2 sin  ,
2 n



Cl,1   K1 ds y5 a  M 0 cos  cos  ,
n



Sl,1   K1 ds y5 a  M 0 cos2 sin  ,
n


1
Cl,2   K 2 ds y5 a  M 0 sin 2 sin  ,
2 n

(S1a)
(S1b)
(S1c)
(S1d)


Sl,2    K 2 ds y5 a  M 0 sin  cos  .
n

(S1e)
where K 0 , K1 , and, K 2 , of which explicit forms are given in Kanamori and Cipar [1974], are
functions dependent on the Earth’s interior structure (density and compressional and shear
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moduli) and evaluated at the seismic source depth, ds . For permanent change, the summation
over all overtones n is required, as also given in Eqs. (3) of Pollitz [1996], but for spheroidal and
toroidal displacements. y5 a  is the gravitational potential eigenfunction evaluated at the
Earth’s surface, computed as a part of the solution of the elasto-gravitational eigenproblem of the
free oscillation [Saito, 1967; Alterman et al., 1974; Aki and Richards, 2002]. The seismic
double-couple source is characterized by scalar seismic moment M 0 , dip angle  , and rake
angle  following the notation and spherical geometry given in Kanamori and Cipar [1974] and
Kanamori and Given [1981].
Spherical rotations of the coefficients given in Eqs. (S1a) - (S1e) need to be implemented to
account for an arbitrary fault strike, similarly done by Stein and Geller [1977] and Geller and
Stein [1977]. Furthermore, subsequent spherical rotations can be performed to compute the
coefficients excited at arbitrary spatial locations (latitude and longitude) of the centroid.
The harmonics Cl,0 , Sl,1 , and Cl,2 are excited by a dip slip (   90 ) component, while Cl,1 and
Sl,2 are excited by a strike slip (   0 ) component.
The three functions necessary to determine the five spherical harmonic coefficients representing
the coseismic gravity field on the surface,  K 0 ds y5 a  ,  K1 ds y5 a  , and
 K d y a ,
2
s
5
n
n
are dependent only on wavelength (i.e., spherical harmonic degree, l) and
n
source depth ds . Thanks to the spectral approach, they can be computed directly from the
fundamental equations and independently for each harmonic degree [Piersanti, et al., 1995;
Pollitz, 1996] rather than summing each of normal modes explicitly. This allows numerically
stable calculations up to high degrees.
Examples of those excitation functions, isotropic (zeroth order)
 K d y a  ,
0
s
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dipole (1st
n
order)
 K d y a , and quadrupole (2nd order)  K d y a , are provided in Figure S1.
1
s
5
2
n
s
5
n
They were evaluated at three depths in the upper crust, lower crust, and upper mantle of the
Earth model PREM. A significant depth dependence is found in the isotropic excitation
function.
The corresponding spatial pattern of each excitation function at each depth is shown in Figure
S2, truncated at degree 40. The absolute scale is dependent on the scalar moment. Only relative
amplitudes among various excitation functions can be compared. The gravity change from any
double-couple source at one of the three depths is simply a linear combination of those maps.
The coseismic gravity excitation, particularly the isotropic excitation (m=0), is shown to be
sensitive to the source depth at shallow layers (due to a depth-dependent mechanical property of
the Earth’s material). The gravity changes (up to degree and order 40, or equivalently a spatial
resolution of 500 km) from the same seismic source but excited at different depths are compared
in Figure S3. A centroid with a scalar moment M0 5.0  1022 N-m and strike 180, dip 15, and
2
rake 90 was used. The effect of isotropic excitation, mostly responsible for a large central
negative anomaly, reduces with increasing depths and the higher order non-isotropic modes
(dipole and quadrupole) become more pronounced in the total coseismic gravity change.
For a double-couple source with a given location, magnitude and focal mechanism, it is possible
to obtain the gravitational excitation coefficients from Eqs. (S1a)-(S1e) and from spherical
rotations of those coefficients for latitude, longitude and fault strike of the centroid in a
geographical coordinate system.
Those coefficients subsequently are used for orbit
determination to compute synthetic perturbation in the inter-satellite distance changes that are
being measured by the K-Band Ranging (KBR) instrument onboard the GRACE satellites.
2. Moment tensor components versus fault (double-couple) parameters
The gravitational potential coefficients are linear with respect to 5 independent moment tensor
components, while they are non-linear with respect to 4 double-couple parameters as shown in
Eqs. (S1a)-(S1e). When the GRACE data are inverted for the moment tensor components, the
fault parameters can, subsequently, be estimated from the moment tensor solution by considering
the full error covariance. We found that, instead of the two step procedure, the iterative linear
inversion with respect to the fault parameters is stable requiring only 3 iterations for
convergence, if strike and depth are fixed a priori so that one solves only 3 parameters, M0, dip
and rake, each time by varying strike and depth. In this case, both results must be identical.
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Figure S1.
Examples of three gravitational excitation functions at various depths,
 K 0 ds y5 a  (isotropic, m=0),  K1 ds y5 a  (dipole, m=1), and  K 2 ds y5 a 
n
n
n
(quadrupole, m=2).
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Figure S2. Spatial patterns of the three excitation functions at three depths. The absolute scale
is dependent on the scalar moment.
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Figure S3. Examples of synthetic gravity change from the double-couple seismic source at
different depths, but using the same magnitude and focal mechanism (M0 5.0  1022 N-m or Mw
9.1, strike 180, dip 15, and rake 90). The source is located within the upper crust (10 km),
lower crust (20 km), and upper mantle (30 km).
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Figure S4. Synthetic gravity change computed from various centroid moment tensor (CMT)
solutions, global CMT [Nettles et al., 2011], USGS CMT [USGS, 2011], USGS W-Phase CMT
[Hayes, 2011], and IPGS W-Phase CMT [Duputel et al., 2011]. The spatial location (red star)
and depth of the centroid and focal mechanism (moment magnitude / strike / dip / rake) for each
solution are given.
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Figure S5. Synthetic coseismic gravity change across the epicenter in the dip direction of the
example double-couple source located at 0N and 0E with a magnitude of M0 5.0  1022 N-m,
strike of 180 and rake of 90. (Left) Various dip angles at a fixed depth of 20 km. (Right)
Various depths in the lower crust with a fixed dip 10. The number in the parenthesis of each
test case reflects the ratio of the positive peak to negative peak, which is an indication of the
spatial pattern of the gravity change. At a certain depth (Left), the larger the dip angle, the more
pronounced the negative anomaly. With a fixed dip angle (Right), the shallower the depth, the
more pronounced the negative anomaly. Both depth and dip angles are most influential in the
overall spatial pattern of coseismic gravity change as well as magnitude, while the scalar
moment controls only magnitude. It is anticipated that a deeper seismic source with higher dip
and a shallower source with lower dip would produce a similar spatial pattern of the coseismic
gravity change.
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Figure S6. As for Figure 1, but for the GRACE data east of the epicenter. The green and
magenta lines in (a) indicate the ground tracks before and after the earthquake, respectively.
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Figure S7. Variance Reduction (VR) of the GRACE observations with seismic model
synthetics. VR is defined by 1 – var{data  model} / var{data} where var{} is an operator
calculating variance. (a) VR computed with four different seismic CMT solutions at various
depths. Although each CMT solution is provided with a specific depth, we tested the solutions at
other depths as well. The Earth model PREM delineates the lithospheric layers characterized
with density and elastic moduli. (b) We computed VR with GCMT by varying the elastic moduli
within the lower crust while keeping other Earth’s parameters the same as PREM. The most
plausible bulk and shear moduli pair is found to be 70 – 90 GPa and 35 – 50 GPa, respectively.
It agrees with the moduli in the lower crust of PREM and is intermediate between the middle and
lower crust of CRUST2 [Bassin et al., 2000].
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Figure S8. The GRACE observations of inter-satellite range change due to the earthquake and
synthetics after fitting the earthquake source parameters. The GRACE CMT solution at depth 17
km was used for computing synthetic ‘waveform’. They are displayed along latitude (top) and
over time (bottom). The concatenated data and synthetics on the bottom panel are continuous
only within each ‘waveform’.
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Figure S9. The predicted gravity changes from coseismic and postseismic (afterslip) slip models
from Ozawa et al. [2011]. The approximate coseismic and postseismic slip model parameters are
given as (38 / 143 / 24 km / 3.4  1022 N-m / 196 / 14 / 90) and (39 / 142 / 40 km / 4.5 
1021 N-m / 196 / 19 / 90) for (latitude / longitude / depth / M0 / strike / dip / rake),
respectively. The profiles in the third panel show the gravity change along the gray line in the
map.
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