Absorbing Boundary Conditions for 3D Anisotropic Media

Abstract

Seismic methods of subsurface exploration are based on mechanical wave propagation and the numerical modeling of these phenomena is a worthy tool that can be applied as a complement. Since small regions of Earth’s crust are studied, it is necessary to consider absorbing boundary conditions for solving the wave equations efficiently. Therefore, this work presents a derivation of low-order absorbing boundary conditions at the artificial boundaries of the computational domain with the purpose of minimizing spurious reflections. Laboring on a surface S, which separates disturbed and undisturbed regions of the domain, the equations for the absorbing boundary conditons are derived from kinematic conditions, considering continuity of the displacements across S and dynamic conditions, using momentum equations of the wave fronts arriving normally to S and expressions for the strain energy density along S. The arguments to obtain non-reflecting artificial boundaries are carried out for the more general case, through the generalized Hooke’s law. In this way, an isotropic medium is included in this derivation. The performance of these absorbing boundary conditions is illustrated for different models of effective anisotropy -vertically and tilted transversely isotropic media- and, obviously, for isotropic media. The numerical simulations use these absorbing boundary conditions to propagate waves in anisotropic media using an iterative domain decomposition finite element procedure that is implemented in machines with parallel architecture.