First-order systems for elastic and acoustic variable-tilt TI media

Abstract

ABSTRACT We derive and compare first-order wave propagation systems for variable-tilt elastic and acoustic tilted transversely isotropic (TTI) media. Acoustic TTI systems are commonly used in reverse-time migration. Starting initially with homogeneous vertical transversely isotropic (VTI) media, and then extending to heterogeneous variable-tilt TI media, we derive a pseudoacoustic 5×5 first-order system of differential equations by setting the shear-wave speeds to zero and simplifying the full-elastic system accordingly. This 5×5 system conserves a complete energy, but only when the anelliptic anisotropy parameter η>0. For η=0 (including isotropic media), the system allows linearly time-growing and spatially nonpropagating nonphysical solutions frequently taken for numerical noise. We modified this 5×5 acoustic first-order system by changing the stress variables to obtain a system that stays stable for η≥0. This system for homogeneous VTI media is generalized to heterogeneous variable-tilt TI media by rotating the stress and strain variables in the full elastic system before setting the shear-wave speeds to zero; the system obtained can be greatly simplified by combining the rotational terms, resulting in only one rotation and extra lower-order terms compared to the 5×5 first-order acoustic system for VTI media. This new system can be simplified further by neglecting the lower-order terms. Both systems (with and without lower-order terms) conserve the same complete energy. Finally, the corresponding 9×9 full elastic system for variable-tilt acoustic TI media can be used for the purposes of benchmarking.