Exploring three-dimensional implicit wavefield extrapolation with the helix transform

Three-dimensional implicit velocity continuation

Velocity continuation is a process of navigating in the migration velocity space, applicable for time migration, residual migration, and migration velocity analysis (Fomel, 1996a). In the zero-offset (post-stack) case, the velocity continuation process is described by the simple partial differential equation (Fomel, 1994; Claerbout, 1986)

$${\frac{\partial^2 P}{\partial v\,\partial t}} + {v\,t\,\le... ...al x^2}} + {\frac{\partial^2 P}{\partial y^2}}\right)} = 0\;,$$ (18)

where $t$ is the vertical time coordinate of the migrated image, $x$ and $y$ are spatial (midpoint) coordinates, and $v$ is the migration velocity. Slightly different versions of two-dimensional implicit extrapolation with equation (18) have been described by Li (1986) and (Fomel, 1996a).

velcon
Figure 7. Impulse responses of the velocity continuation operator, computed by an implicit, unconditionally stable extrapolation via the helix transform. The left plot corresponds to continuation towards higher velocities (migration mode); the right plot, smaller velocities (modeling mode).

The helix approach has allowed us to modify the old code for three dimensions. Figure 7 shows impulse responses of an implicit helix-based three-dimensional velocity continuation.

qdome Figure 8. Qdome synthetic model, used for testing the 3-D velocity continuation program.

Figure 9 illustrates the velocity continuation process on the Qdome synthetic model (Claerbout, 1997b), shown in Figure 8. Continuation backward in velocity corresponds to the ``modeling'' mode, while forward continuation corresponds to the ``migration'' mode. It is possible to balance the amplitudes of the two processes so that the finite-difference velocity continuation behaves as a unitary operator (Fomel, 1996a,b).

modmig
Figure 9. Modeling (left) and migration (right) with the Qdome synthetic model, obtained by running the 3-D velocity continuation backward and forward in velocity.

Exploring three-dimensional implicit wavefield extrapolation with the helix transform