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 Claerbout (1986); Fomel (1994)  

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