Warping as residual migration

Warping provides a mapping between different migration velocities that is kinematically equivalent to velocity continuation for plane-wave events. () showed this directly from the zero-offset velocity continuation equation, but it is apparent intuitively if you consider the effect map-migration () has on a planar dipping events. In this context, warping bears the same relationship to residual migration as `map-migration' bears to conventional zero-offset migration. Map-migration and warping are both point-to-point operators; whereas conventional zero-offset migration and residual migration are based on a convolutional model. Warping, therefore, can be thought of as `residual map-migration'.

The relationship between warp-function and velocity change can be derived from kinematic map-migration equations. The following three equations describe migration of a zero-offset planar event at $({\bf x}_{\rm zo},t_{\rm zo})$dipping with slowness, ${\bf p}_{\rm zo}$, with velocity v:

$$\begin{eqnarray} t & = & t_{\rm zo} \sqrt{1-v^2 p_{\rm zo}^2} \\ {\bf x} & = & {... ... {\bf p} & = & \frac{{\bf p}_{\rm zo}}{\sqrt{1-v^2 p_{\rm zo}^2}} \end{eqnarray}$$ (168) (169) (170)

Differentiating with respect to v, and eliminating the zero-offset variables leads to the equations that describe residual map-migration along Fomel's velocity rays, providing a link between the warp-function and the residual velocity correction.

$$\begin{eqnarray} \Delta {\bf x} & = & -2 v t \; {\bf p} \; \Delta v \\ \Delta t & = & v t p^2 \; \Delta v \end{eqnarray}$$ (171) (172)

Using an algorithm based on map-migration may seem questionable when we are considering an amplitude-sensitive issue such as reservoir monitoring. However, for this application the shifts we apply are so small (a few sample points), that such an approach is valid.