Implementation of the Kirchhoff integral for elastic waves in staggered‐grid modeling schemes

Abstract

Implementation of boundary conditions in finite‐difference schemes is not straightforward for the elastic wave equation if a staggered grid formulation is used. Reverse time migration of VSP data requires a proper description of the recording surface so as not to excite false P‐ and S‐waves. Such contributions may cause artifacts in the imaging procedure. The boundary conditions for the elastic stress tensor can be implemented numerically in a staggered coarse grid modeling scheme by using band‐limited spatial delta‐functions and band‐limited first‐order derivatives of these spatial delta‐functions. A representation theorem for elastic waves is derived to test the implementation of the spatial part of the boundary condition. The implementation is tested in a 2-D numerical experiment for a closed, but curved, boundary S enclosing a volume V. The test condition is that within the volume V, the difference between the forward modeled field and the retropropagated field should be equal to zero. Both P‐ and S‐waves are properly recovered in a 2-D reverse time modeling example. The numerical artifacts related to the proposed spatial approximation of the boundary condition are found to be negligible.