Poststack Migration to Go from Data Space to Model Space.

Zero-offset seismic traces (x,z=0,t)do not provide an accurate picture of the subsurface layers when there is a great deal of structural complexity. For example, Figure 1.18 shows that the apparent reflection point deduced from our traces does not coincide with the actual reflection point in which the reflection energy originated.

  

Figure 1.18: For a dipping layer, projecting reflection energy directly to a depth of vt below a trace defines the apparent reflection point, which is not the same as the actual reflection point. Thus the stacked section d(x,z=0,t) is not a good approximation to m(x,z)for complex structures or layers with steep dip. Here t is the 1-way reflection time.

The inability of d(x,z=0,t) to represent the seismic section gets even worse when your reflector becomes more complex as shown in Figure 1.19. For example, the faults in the data have diffraction tails which are collapsed in the migrated section. To correct for these distortions we apply the operation of migration to the zero-offset seismic data to produce an image m(x,z) of the reflectivity section in (x,z).

Formally, if $\bf L$ is the forward modeling operator so that

$${\bf d} {\bf Lm},$$ = (1.11)

then migration can be described as the first iterate of a steepest descent method:

$${\bf m} {\bf [ L^T L ]^{-1} L^T d },$$ =

$$\cong {\bf L^T d }.$$ (1.12)

The migration algorithm will be explained in detail by other speakers. In fact, migration of the prestack data (i.e., CMP traces) is now commonly used today to improve the imaging quality even more.
Question: Why is ${\bf L^T}$ such a good approximation to L^-1?
Answer: If the data ${\bf d=Lm}$ roughly resemble m, then this suggests that $\bf L$ acts almost like an identity operator in mapping model space to data space. Thus, ${\bf L^T}$ might act like an inverse operator. Also, ${\bf [L^T L]}$ is somewhat diagonally dominant so that its inverse can be roughly approximated by a weighted diagonal matrix.

  

Figure 1.19: (Top) Poststack migrated image and (bottom) stacked seismic section. Note how the faults are more clearly delineated and the diffraction frowns are collapsed in the migrated section. Data are computed for the SEG/EAGE synthetic overthrust model.