Two and one‐half dimensional Born inversion with an arbitrary reference

Multidimensional inversion algorithms are presented for both prestack and poststack data gathered on a single line. These algorithms both image the subsurface (i.e., give a migrated section) and, given relative true amplitude data, estimate reflection strength or impedance on each reflector. The algorithms are “two and one‐half dimensional” (2.5-D) in that they incorporate three‐dimensional (3-D) wave propagation in a medium which varies in only two dimensions. The use of 3-D sources does not entail any computational penalty, and it avoids the serious degradation of amplitude incurred by using the 2-D wave equation. Our methods are based on the linearized inversion theory associated with the “Born inversion.” Thus, we assume that the sound speed profile is well approximated by a given background velocity, plus a perturbation. It is this perturbation that we seek to reconstruct. We are able to treat the case of an arbitrary continuous background profile. However, the cost of implementation increases as one seeks to honor, successively, constant background, depth‐only dependent background, and, ultimately, fully lateral and depth‐dependent background. For depth‐only dependent background, the increase in CPU time is quite modest when compared to the constant‐background case. We exploit the high‐frequency character of seismic data ab initio. Therefore, we use ray theory and WKBJ Green’s functions in deriving our inversion representations. Furthermore, our algorithms reduce to finding quantities by ray tracing with respect to a background medium. In the constant‐background case, the ray tracing can be eliminated and an explicit algorithm obtained. In the case of a depth‐only dependent background, the ray tracing can be done quite efficiently. Finally, in the general 2.5-D case, the ray‐tracing procedure becomes the principal issue. However, the robustness of the inversion allows for a sparse computation of rays and interpolation for intermediary values. The inversion techniques presented here cover the cases of common‐source gather, common‐receiver gather, and common‐offset gather. Zero offset is a special case of the last of these. For offset data, the reflection coefficient is angle‐dependent, so parameter extraction is more difficult than in the zero‐offset case. Nonetheless, we are able to determine the unknown angle pointwise and derive parameter estimates at the same time as we produce the image. For each reflector, this estimate of the output is based on the Kirchhoff approximation of the upward‐scattered data. Thus, it is constrained to neither small discontinuities in sound speed at the reflector nor to small offset angle as would be the case for a strict “Born approximation” of the reflection process. The prestack algorithms presented here are inversions of single gathers. The question of how best to composite or “stack” these inversions is analogous to the question for any migration scheme and is not treated here.