Adapting elastic wavefield extrapolation to laterally varying HTI media

Richard Andrew Bale, Gary F. Margrave

Elastic wavefield extrapolators have been derived for HTI media which vary only in the depth direction, based upon eigensolutions to the Kelvin-Christoffel equation. Here, we extend elastic wavefield extrapolation to laterally heterogeneous media, using PSPI and NSPS type pseudodifferential operators, as has previously been done for scalar wavefield extrapolation. As in that case, we observe that forward extrapolation with PSPI and reverse extrapolation with NSPS (or vice versa) better recovers the input, than does use of either algorithm in both directions. Thus, NSPS and PSPI appear to be adjoint operators when applied in opposing directions of extrapolation.

Elastic wavefield extrapolation in HTI media can be formulated in two alternative ways. The first is based upon extrapolation of a continuous displacement-stress vector by eigen-decomposition into wave-modes, propagation through each homogeneous layer, and recomposition at a new depth. The second is based upon extrapolation of the wave-modes within homogeneous layers, and application of "interface-propagators" to cross layer interfaces. For laterally invariant media, the two approaches give identical results, with the interface-propagator method having a performance advantage. In the laterally heterogeneous case, the two approaches can give different results. To be specific, the latter method suffers polarization errors when formulated for the same efficiency gain as is possible in the homogeneous case. Nevertheless, for media with gradual continuous changes of anisotropic symmetry axis, this more efficient approach gives acceptable results.