Wave-equation migration velocity analysis for VTI media using optimized implicit finite difference

Introduction

Building an anisotropic model requires accurate wave field descriptions at all angles, both in simple and complex geological settings. Therefore, instead of using travel times and ray paths, we use the wave fields as the carrier of the model information (Li and Biondi, 2011a; Li et al., 2012; Li, 2012). Many choices for wavefield propagators may be considered. On one hand, the one-way wave propagators (Li, 2012) excel in speed but lose their accuracy rapidly with the increasing propagation angle. On the other hand, the two-way wave propagators (Li et al., 2012) are more accurate in modeling waves at large angles but their computational costs are less affordable.

Therefore, I use an optimized implicit finite difference propagator first developed by Shan (2006). In this optimized implicit finite difference scheme, the anisotropic parameters $\eta$ and $\delta$ contribute to the wave-equation implicitly via two finite difference parameters $\alpha$ and $\beta$ . Tables of $\alpha$ and $\beta$ with respect to sets of discrete $\eta$ and $\delta$ 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 $60^\circ$ in a vertical transverse isotropic (VTI) medium.

When perturbing the wave-equation around its current state, the finite difference coefficients $\alpha$ and $\beta$ are perturbed. These perturbations are then translated into the anisotropic parameters $\eta$ and $\delta$ using the chain rule. Tables of numerical derivatives of the finite difference coefficients $\alpha$ and $\beta$ with respect to the anisotropic parameter $\eta$ 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 $v_v$ and $\eta$ . 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