Downward continuation is a process in which we ``push'' the data recorded
at the surface down into the earth to obtain the wavefield at each depth
in the subsurface. When we use the correct velocity to push the wavefield
deeper, energy that was reflected at some point in the subsurface gets
moved back to the reflection point ^{}. The process of
downward continuation is essentially a phase shift applied to the
wavefield recorded at the surface (*U*_{z0}) () so that it becomes
the wavefield at some depth (*U*_{z}). This is expressed
mathematically as:

16#16 | (5) |

17#17 | (6) |

Fortunately, the SSR equation can be modified to handle the source and receiver wavefields separately. This modification is called the Double Square Root (DSR) equation (equation ()):

18#18 | ||

(7) |

which is the 3-D equation
where *k*_{m<<388>>x} and *k*_{m<<389>>y} are the midpoint horizontal wavenumbers
and *k*_{h<<390>>x} and *k*_{h<<391>>y} are the offset horizontal wavenumbers.
The velocities 19#19 and 20#20 are the velocities at a
given depth that are associated with the downward-continued source and
receiver wavefields, respectively. The 19#19 and
20#20 allow us to have subsurface models with laterally varying
velocities
^{}.

When the DSR equation is used to obtain the prestack wavefield at depth (21#21) and the velocities used for the downward continuation are correct, the result is to collapse all of the energy to zero-offset. If we were only interested in the zero-offset information, we could then extract an image from the downward-continued wavefield at zero time:

22#22 | (8) | |

(9) |

However, since this image will only contain the zero offset information (left panel in Figure ), it has limited diagnostic information for velocity updating, and no information on the amplitude variation with reflection angle (AVA). If we extract the image at this point only, we are losing valuable information that exists in the data.

10/31/2005