When collecting seismic data, we are injecting seismic energy into the earth which propagates to some point in the subsurface and reflects back to the surface. To create a model of the subsurface, we want to reverse this process. There are two different ways to go through this reversal: with an adjoint operation or with an inverse operation. The downward-continuation migration described in the previous chapter is an adjoint to seismic wave propagation, not an inverse. () explains that an adjoint is a matrix transpose. This means that it is an approximation that is not able to accurately ``undo'' the wave propagation, especially in complex areas. In such areas, the adjoint operator will be unable to account for seismic energy that leaves the survey area or becomes evanescent. In the case of the downward-continuation migration used in this thesis, overturned rays are not properly handled. If we use the inverse operation, which is more computationally expensive than an adjoint operation, it will do a better job of reversing the propagation. However, applying the true inverse is computationally infeasible for the huge matrices we face in seismic imaging. Fortunately, we can make the adjoint more like an inverse through a least-squares scheme, which will not be as computationally intensive as a true inverse.

10/31/2005