Downward continuation

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)

where k_z can be found from the dispersion relation (equation ()). Rearranging equation (), we find that:

17#17 (6)

which you may recognize as the Single Square Root (SSR) equation. This equation is only valid for constant velocity or velocity that varies only with depth. The inability to handle lateral velocity variations makes this equation not directly applicable in complex areas. To be able to handle lateral velocity variations, we first need to handle the source and receiver wavefields separately.

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 ()):

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.