Computation of the Migration Kernel

The migration kernel in equation  can be computed in one of two ways.

1. Place a source point on the surface at ${\bf {s}}$ , and solve for the field everywhere in the model by a finite-difference method to get the Green's function $g({\bf {x}},t\vert{\bf {s}},0)$ . Reciprocity says that $g({\bf {s}},t\vert {\bf {x}},0) = g({\bf {x}},t\vert{\bf {s}},0)$ , and if we replace ${\bf {s}}\rightarrow {\bf {r}}$ then this gives $g({\bf {r}},t\vert{\bf {x}},0)$ . Thus, $g({\bf {r}},t\vert{\bf {x}},0)$ can be convolved with $g({\bf {x}},t\vert{\bf {s}},0)$ to give the migration kernel $\mathcal G({\bf {r}},{\bf {s}},{\bf {x}},t)$ in equation  with the receiver at ${\bf {r}}$ and the source at ${\bf {s}}$ for all subsurface points ${\bf {x}}$ .

2. Alternatively, a point source can be placed at depth ${\bf {x}}$ and the field can be solved everywhere to get $g({\bf {r}},t\vert{\bf {x}},0)$ . This can be cost effective for target-oriented migration (or waveform inversion) so that we only need the Green's functions for point sources along the boundary of the target (Dong et al., 2009). Reciprocity says that $g({\bf {r}},t\vert {\bf {x}},0)=g({\bf {x}},t\vert{\bf {r}},0)$ , and letting ${\bf {r}}\rightarrow {\bf {s}}$ yields $g({\bf {x}},t\vert{\bf {s}},0)$ . The migration kernel at ${\bf {x}}$ can now be computed by equation . This is the method used by Liu and Wang (2008) and Dong et al. (2009) for target-oriented migration.

Zhou and Schuster (2002) demonstrated how to efficiently compute these migration operators by finite differencing along the leading portion of the wavefront.