Wave-equation Migration from Topography

Seismic data acquired on topography usually require significant preprocessing before any imaging technique can be applied successfully. Performing wave-equation migration, though, usually requires that data exist on regularly sampled meshes before a wavefield extrapolation procedure can begin. Usually, this involves a data regularization step implemented as either pre-migration datuming or through a wavefield injection plus interpolation migration strategy.

An alternative to these standard approaches is discussed in Shragge and Sava (2005), who pose seismic imaging directly in acquisition coordinates and use Riemannian wavefield extrapolation to propagate wavefields. Initially, a conformal mapping approach was used to generate structured, locally orthogonal coordinate meshes (see top panel of figure ). However, the ensuing grid clustering and rarefaction demanded by orthogonality led to significant spatial variance in metric tensor coefficients. Importantly, this variance caused artifacts in the resulting image, which remains the main drawback of this migration approach.

 

Topography
Figure 3 Meshing example for the wave-equation migration from topography application. Top panel: Locally orthogonal coordinate system calculated by conformal mapping Shragge and Sava (2005). Note the severe amount of grid clustering indicating a need for meshing regularization. Middle panel: Blended coordinate system $\{ s^1,s^2 \}$ forming the differential gridding algorithm input. Bottom panel: Regularized mesh $\{x^1,x^2\}$ after 15 iterations.

The extension of RWE to non-orthogonal coordinate systems Shragge (2006) allows for greater flexibility in coordinate system design. The middle panel in figure  represents a blended coordinate system $\{ s^1,s^2 \}$ that forms the input to the differential mesh algorithm. Lines predominantly in the horizontal direction mimic topography in the near-surface and slowly heal to form an evenly sampled wavefield at depth. The bottom panel shows the coordinate system output $\{x^1,x^2\}$ from the gridding algorithm after 15 iterations. Grid irregularities now heal more rapidly and the mesh becomes very regular after a few extrapolation steps into the subsurface.