We present an analytical approach to compute the curvature effect by the new analytical solutions of coseismic deformation derived for the homogeneous sphere model. We consider two spheres with different radii: one is the same as earth and the other with a larger radius can approximate a half-space model. Then, we calculate the coseismic displacements for the two spheres and define the relative percentage of the displacements as the curvature effect. The near-field curvature effect is defined relative to the maximum coseismic displacement. The results show that the maximum curvature effect is about 4 per cent for source depths of less than 100 km, and about 30 per cent for source depths of less than 600 km. For the far-field curvature effect, we define it relative to the observing point. The curvature effect is extremely large and sometimes exceeds 100 per cent. Moreover, this new approach can be used to estimate any planet's curvature effect quantitatively. For a smaller sphere, such as the Moon, the curvature effect is much larger than that of the Earth, with an inverse ratio to the earth's radius.

SUMMARY: In this study, we devised a new set of analytical foundation solutions to compute the internal co-seismic displacement and strain changes caused by four independent point sources (strike-slip, dip-slip, horizontal tensile and vertical tensile) inside a homogeneous spherical earth model. Our model provides constraints on the deformation properties at depth and reveals that the internal co-seismic deformation is larger than that on the surface. The deformation near the source is convergent with our formulae. For the internal deformation at radial section plane, the patterns of horizontal displacements ${u_\theta },{u_\phi }$ and strain changes ${e_{{ rr}}},{e_{\theta \theta }},{e_{\phi \phi }},{e_{\theta \phi }}$ caused by strike-slip and tensile sources appear symmetric at the equidistance above and below the source. Their amplitudes are not identical but with a small discrepancy actually. Unlike these, the patterns of radial displacements ${u_r}$ for strike-slip and tensile sources exhibit point symmetry with the equidistance from the source. Also, the corresponding amplitudes are slightly different. The displacements ${u_\theta },{u_\phi }$ and strain changes ${e_{{ rr}}},{e_{\theta \theta }},{e_{\phi \phi }},{e_{\theta \phi }}$ caused by dip-slip also show the same properties as ${u_{ r}}$ of the strike-slip source. The magnitudes of the displacements and strain changes depend on the source types. The curvature effect on the near-field surface deformations is small, and it increases with the studied depth. However, for the far-field deformation caused by the strike-slip source (ds = 20 km), the curvature effect can be as large as 77 per cent when the epicentral distance approximates to 1778 km.

Postseismic deformation observed by terrestrial Global Navigation Satellite System (GNSS) and seafloor GNSS-Acoustic techniques (GNSS-A) provides essential constraints on the depth-dependent viscoelastic structure of subduction zones. In this study, we collect and process decadal postseismic observations following the 2011 Tohoku-oki M_w9.0 earthquake, including 232 onshore GNSS stations and six offshore GNSS-A sites. After removing the interseismic velocity terms, we extract the postseismic deformation signals mainly driven by viscoelastic relaxation during the period from 3 to 9 years after the earthquake. The inversion is primarily constrained by horizontal displacements, which have higher accuracy than vertical observations. We adopt a radially layered viscoelastic Earth model with lateral heterogeneity between continental and oceanic domains based on the Burgers rheology and half-space dislocation theory. Using the least-squares principle, we invert for the optimal viscoelastic structure under the strong constraint of fixed mantle viscosity. The optimal continental and oceanic crustal elastic thicknesses are 24.4 km and 37 km, with minimum horizontal Root-Mean-Square errors (RMS) of 5.68 cm and 6.81 cm, respectively. The mantle viscosity shows significant depth-dependence and obvious land–ocean differences. These results verify the critical role of joint land and seafloor geodetic constraints and provide a refined viscoelastic structure model for subduction zones.