S U M M A R YThis paper studies the potential and gravity changes caused by dislocations in spherically symmetric earth models. We define dislocation Love numbers to describe the elastic deformation of the earth raised by point sources. We discuss the shear and tensile dislocations, which can be expressed by four independent components: a vertical strike-slip, a vertical dip-slip, a tensile opening on a horizontal plane, and a tensile opening on a vertical plane.The results for a homogeneous earth model agree very well, at least within lo, with those predicted from flat-earth theory. The far-field results indicate no larger than 10per cent difference within 10". It makes little difference whether we use the theory on a sphere or that for a flat earth in the near field, while it is reasonable t o use the spherical theory for global calculation. We proceed to calculations with a radially heterogeneous earth model (Model 1066A). The results are as a whole similar to those for a homogeneous sphere. In some cases, however, the difference between the two becomes significant. For example, the locations of the nodal lines of the gravity change differ significantly between the two models. This indicates that the vertical layering can cause considerable effects on the deformation fields.

Analytic expressions derived from a simplified model are invaluable because they often provide deep insight into geophysical phenomena. Such is the case with potential and gravity changes. Here we present expressions of potential and gravity changes caused by faulting on a finite rectangular plane buried in a homogeneous half‐space. The expressions can be used to evaluate coseismic changes in surface gravity and geoid height. Observed gravimetric data combined with our formulae thus provide constraints on the fault geometry and magnitudes of dislocation.

S U M M A R YBased on the authors' previous work, co-seismic deformations for a spherical symmetric earth model are summarized and reformulated. Unified expressions presented herein accommodate physical deformations: displacement, potential, gravity, geoid and strain changes. The corresponding Green's functions are derived by combining spheroidal and toroidal deformations. Sign errors in previous publications are corrected in these new formulas. These expressions are developed basically for a deformed earth surface because most traditional geodetic measurements are performed on the terrain surface. However, through development of space geodetic techniques, such as the satellite gravity missions, co-seismic gravity changes can be detected from space. In this case, the above dislocation theory (e.g., the co-seismic gravity change) cannot be applied directly to the observed data because the data do not include surface crustal deformation (the free air gravity change). Correspondingly, the contribution by the vertical displacement part must be removed from the traditional expressions. For this purpose, we present the corresponding expressions applicable to space observations. Some numerical technical problems are discussed. In addition, a smoothing technique is necessary to damp the high-frequency contribution so that the theory can be applied reasonably. Global co-seismic deformations caused by the 2004 Sumatra-Andaman earthquake (M9.3) are studied as an application of the new Green's function. That earthquake caused a global deformation detected by GPS, strain metres and even a satellite gravity mission. These global deformations are calculated based on the derived Green's functions and the seismic-wave derived earth model. A segment-summation scheme is used considering the slip distribution on a limited fault plane. The results are useful for interpreting observed deformations, especially those in the far field. The earthquake reveals global co-seismic deformations and effects of spherical curvature and the earth's layered structure. Comparisons between results for a spherical earth mode and a half-space model show a large discrepancy at an epicentral distance of about 1000 km, implying that effects of spherical curvature and layer structure are considerably large. In addition, the theoretical results are compared with the real observed strain steps, horizontal displacements and gravity changes caused by that earthquake. Good agreement validates the results of the current theoretical work. Finally, we discuss the application the above theory to the GRACE data through several case studies.

S U M M A R YWe introduce a new method by which to compute global post-seismic deformation (PSD) in a spherically symmetric, self-gravitating viscoelastic earth model. Previous methods are based on simplified earth models that neglect compressibility and/or the continuous variation of the radial structure of Earth. This is because the previous mode summation technique cannot avoid intrinsic numerical difficulties caused by the innumerable poles that appear in a realistic earth model that considers such effects. In contrast, the proposed method enables both of these effects to be taken into account simultaneously. We carry out numerical inverse Laplace integration, which allows evaluation of the contribution from all of the innumerable modes of the realistic earth model. Using this method, a complete set of Green's functions is obtained, including functions of the time variation of the displacement, gravity change, and the geoid height change at the surface for strike-slip, dip-slip, horizontal and vertical tensile point dislocations. As an earth model, we employ the preliminary reference earth model (PREM) and a convex viscosity profile. Further, we investigate the effects of fine layering of the viscoelastic structure and compressibility on a time-series of PSD using the Green's function for a dip-slip fault. The result indicates that the effect of increasing number of layers is saturated at several tens of layers even when compressibility is taken into account and that the effect of compressibility is detectable with modern observational techniques for a shallower large earthquake (Mw ∼ 8).As an application, the PSD due to the Sumatra-Andaman Islands earthquake (Mw = 9.3) is estimated. We show that the rate of post-seismic vertical displacement and gravity change is possibly detected in the far field where the epicentral distance exceeds 400 km.

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