Wave-equation migration velocity analysis for VTI media using optimized implicit finite difference |
Therefore, I use an optimized implicit finite difference propagator first developed by Shan (2006). In this optimized implicit finite difference scheme, the anisotropic parameters and contribute to the wave-equation implicitly via two finite difference parameters and . Tables of and with respect to sets of discrete and values are precomputed before propagation by minimizing the difference between the true dispersion relationship and its rational series approximation at different wavenumbers. This table-driven, implicit finite difference method handles lateral variations and is accurate up to in a vertical transverse isotropic (VTI) medium.
When perturbing the wave-equation around its current state, the finite difference coefficients and are perturbed. These perturbations are then translated into the anisotropic parameters and using the chain rule. Tables of numerical derivatives of the finite difference coefficients and with respect to the anisotropic parameter are also precomputed from the previous coefficients tables.
Finally, I test this implicit finite difference implementation by 2D and 3D impulse responses for vertical velocity and . The results verify the theoretical understanding of the WEMVA operator for anisotropic models.
Wave-equation migration velocity analysis for VTI media using optimized implicit finite difference |