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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 1 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 5 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. 3 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). 4 Figure S2. Spatial patterns of the three excitation functions at three depths. The absolute scale is dependent on the scalar moment. 5 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). 6 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. 7 Figure S5. Synthetic coseismic gravity change across the epicenter in the dip direction of the example double-couple source located at 0N and 0E 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. 8 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. 9 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]. 10 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’. 11 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. 12 References Aki, K. and P.G. Richards (2002), Quantitative Seismology. Second Edition, University Science. Books, Sausalito, CA. Alterman, Z., H. Jarosch, C. Pekeris (1959), Oscillations of the Earth, Proc. Roy. Soc. London, A, 252: 80-95. Bassin, C., G. Laske, and G. Masters (2000), The current limits of resolution for surface wave tomography in North America, Eos Trans. AGU, 81, F897. Duputel, Z. L. Rivera, H. 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