Least-squares inversion

In order to approximate the inverse of seismic wave propagation, we can use the migration operator in a least-squares inversion problem (, ). The seismic wave propagation is a linear operator (29#29) which gives us the seismic data (30#30)from the ideal subsurface model (31#31) as seen by:

32#32 (15)

Once again, a migration operator such as downward-continuation migration is the adjoint of the wave propagation (33#33). A direct least-squares solution for the model from this equation is

34#34 (16)

However, the size of the seismic imaging problem makes it unreasonable to invert 35#35 directly. Fortunately, we can approximate the inverse by using 33#33 and 29#29 in an iterative conjugate-gradient scheme. This amounts to minimizing the objective function:

 

 36#36 (17)

I choose to write this equation in a more intuitive form known as a fitting goal:

 

 37#37 (18)

Given this formulation of the imaging problem, it is interesting to consider its relationship to the illumination problems discussed in the previous chapter. Essentially, the shadow zones are related to the null space of the inversion. We are missing elements of the data (30#30)that correspond to certain reflection angles at certain subsurface locations. Model locations that correspond to the null space in the data can fill with noise, possibly causing the inversion to diverge.