## 1. Introduction

[2] The motion of Earth-orbiting satellites is governed primarily by spatial and temporal variations of Earth's gravity field. The Gravity Recovery and Climate Experiment (GRACE) satellite mission has been providing valuable data that reflect both mass distribution and redistribution within the Earth system by detecting the changes in distance between two proof masses, identical satellites orbiting Earth at 500 km mean altitude. Since their launch in March 2002, extensive analyses of time-variable gravity have resolved hydrological mass fluxes across large river basins [*Tapley et al.*, 2004b], global mean ocean mass variations [*Chambers et al.*, 2004], ocean tides [*Ray et al.*, 2003], ice sheet mean mass fluxes [*Luthcke et al.*, 2006; *Velicogna and Wahr*, 2006], and solid-Earth mass movements and density changes [*Han et al.*, 2006], to name but a few applications of this remarkable and growing data set. The global spherical harmonic (SH) analysis of the GRACE satellite tracking data has been the principal approach to generate monthly mean geopotential fields [*Tapley et al.*, 2004a]. Instrumental and other system errors yield a theoretical limit on the accuracy of the solutions. Additional modeling errors, such as aliasing errors [*Han et al.*, 2004] require special processing in order to approach this limit. Various spatial smoothing techniques have been developed to mitigate errors in the ill-determined SH coefficients at higher degree and order [*Wahr et al.*, 1998; *Davis et al.*, 2004; *Han et al.*, 2005; *Swenson and Wahr*, 2006; *Kusche*, 2007]. All of those postprocessing techniques are to be applied to the monthly mean gravity field maps or SH coefficients, the so-called level 2 (L2) products.

[3] Spherical harmonics are nonlocalized, global spherical basis functions [*Freeden and Michel*, 1999] and the effective bandwidth of SH expansions of typical smoothing windows grows fast in response to the progressive restriction of such windows to spatial regions of interest as a result of the Heisenberg uncertainty principle [*Percival and Walden*, 1993]. Independently from the GRACE community, a method to constrain regional contributions to global SH spectra has been developed in the context of planetary tectonics [*Simons et al.*, 1997] and used to detect the incomplete rebound of the Canadian Laurentide ice sheet [*Simons and Hager*, 1997]. The windows constructed by this method were axisymmetric and obeyed a useful but ad hoc criterion to achieve a balance between spatial and spectral concentration. *Wieczorek and Simons* [2005] quantified the concentration criterion and derived by optimization the shape of ideally concentrated but still isotropic window functions.

[4] The principle is simple. Seeking a band-limited function that is optimally concentrated within a spherical cap extending over the colatitudes 0 ≤ *θ* ≤ *θ*_{0} amounts to maximizing the ratio of the energy of the function within the region compared to the entire sphere. We denote this ratio

where θ is colatitude, ϕ is longitude, and *h*(θ) is an azimuthally invariant window given by the band-limited zonal SH expansion

where *Y*_{l0} is a properly normalized real spherical harmonic of degree 0 ≤ *l* ≤ *L*_{h} and order *m* = 0 on the unit sphere Ω = (θ, ϕ) (see *Wieczorek and Simons* [2005] for further details). The desired coefficients *h*_{l} are found by diagonalizing a square and symmetric “localization kernel,” as follows:

where the elements *D*_{ll′} are integrals of products of Legendre functions. These can be computed accurately by numerical integration, or, in the axisymmetric polar cap case, analytically without great effort [*Simons et al.*, 2006; *Simons and Dahlen*, 2006]. In that case, they define a matrix that is tridiagonal, which lends itself easily to diagonalization.

[5] *Simons et al.* [2006] extended the above ideas to nonaxisymmetric windows optimally concentrated within an arbitrarily shaped boundary. Their methods are expected to be well-suited for the analysis of time-variable gravity fields from GRACE since each of the time-variable signals appears only associated with its own particular geographical regime and usually displays characteristic temporal behavior and intensity. Time-dependent geophysical signals tend to originate in geographically confined regions, while satellite measurement errors are relatively uniformly distributed over the globe. At the same time, the noise affecting individual SH geopotential coefficients grows significantly with increasing degree, and thus care must be taken to limit the bandwidth of any localizing window so as to minimize spectral leakage effects. By localizing the global SH fields to the area (spatially as well as within the appropriate spectral range) where the signal is expected to appear with most of its energy, the signal-to-noise ratio (SNR) can be significantly enhanced.

[6] An earthquake-triggered gravity change exemplifies perfectly the type of phenomenon that is better analyzed by spatiospectral localization since its power attenuates rapidly away from the epicenter and thus results primarily in regional anomalies. The great Sumatra-Andaman earthquake (M_{w} > 9.0) on 26 December 2004 ruptured the seafloor by several to tens of meters along the Java/Sunda trench (over 1300 km in length) within 7–8 min [*Ammon et al.*, 2005]. It permanently changed Earth's gravity field [*Sabadini et al.*, 2005] and disturbed the distance between the two GRACE satellites, normally separated by approximately 220 km. These minute changes in intersatellite distance were measured with the onboard K-band microwave ranging (KBR) instrument [*Tapley and Reigber*, 2005]. *Han et al.* [2006] studied the coseismic deformation near the subduction zone from the satellite-tracking data directly (thus not from the L2 products of the global SH modeling) and documented, for the first time, evidence for crustal dilatation as a result of the Sumatra-Andaman earthquake.

[7] In this study, we show the power of the localization method of *Wieczorek and Simons* [2005] in unlocking observational evidence of the Sumatra-Andaman earthquake directly from the L2 monthly time series of GRACE global SH geopotential coefficients. The intuitive ease by which the method affords the extraction of geophysical signal by postprocessing of the L2 solutions should be welcomed by the science community at large. We show that the coseismic gravity changes processed from the monthly global fields are resolved with almost the same spatial resolution as the regional inversion method [*Han et al.*, 2006] that, however, requires greater efforts. We quantify how large the effects of the earthquake are in the time series of individual SH harmonic coefficients after windowing. These measurements are subsequently analyzed on the basis of a seismic model based on elastic dislocation theory [*Okada*, 1992; *Okubo*, 1992] by considering various effects such as the vertical displacements of the seafloor and Moho topography, expansion of the crust and compression of the mantle.