Fourier-Domain Green Function for an Elastic Semi-Infinite Solid under Gravity, with Applications to Earthquake and Volcano Deformation

Sylvain D. Barbot, & Yuri Fialko

Published 2010, SCEC Contribution #1335

We present an analytic solution in the Fourier domain for an elastic deformation in a semi-infinite solid due to an arbitrary surface traction. We generalize the so-called Boussinesq's and Cerruti's problems to include a restoring buoyancy boundary condition at the surface. Buoyancy due to a large density contrast at the Earth's surface is an approximation to the full effect of gravity that neglects the perturbation of the gravitational potential. Using the perturbation method, and assuming that the effect of gravity is small compared to the elastic deformation, we derive an approximation to the space-domain Boussinesq's problem that accounts for a buoyancy boundary condition at the surface. The Fourier- and space-domain solutions are shown to be in good agreement. Numerous problems of elasto-static or quasi-static time-dependent deformation relevant to faulting in the Earth's lithosphere (including inelastic deformation) can be modeled using equivalent body forces and surface tractions. Solving the governing equations with the elastic Green function in the space domain can be impractical as the body force can be distributed over a large volume. We present a computationally efficient method to evaluate the elastic deformation in a three-dimensional half space due to the presence of an arbitrary distribution of internal forces and tractions at the half-space surface. We first evaluate the elastic deformation in a periodic Cartesian volume in the Fourier domain, then use the analytic solutions to the generalized Boussinesq's and Cerruti's problems to satisfy the prescribed mixed boundary condition at the surface of the half space. We show some applications for magmatic intrusions and faulting. This approach can be used to solve elasto-static problems involving spatially heterogeneous elastic properties (by employing a homogenization method) and time-dependent problems such as nonlinear viscoelastic relaxation, poroelastic rebound, and non-steady fault creep under the assumption of spatially homogeneous elastic properties.

Citation
Barbot, S. D., & Fialko, Y. (2010). Fourier-Domain Green Function for an Elastic Semi-Infinite Solid under Gravity, with Applications to Earthquake and Volcano Deformation. Geophysical Journal International, 182, 568-582.