Internal co-seismic displacement and strain changes inside a homogeneous spherical Earth

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.