non-stationary least-squares filtering

Even though the target-oriented Hessian has a smaller number of rows and columns in equation (9), its condition number could be high, making the solution of the non-stationary least-squares filtering problem in equation (5) unstable. One solution is adding a smoothing regularization operator to equation (5):

$$\begin{eqnarray} {\bf H}\hat{{\bf m}}-{\bf m}_{mig}&\approx&0, \nonumber\\ \epsilon{\bf I}\hat{{\bf m}}&\approx&0, \end{eqnarray}$$ (11)

where the choice of the identity operator (${\bf I}$) as regularization operator is arbitrary. Changing the $\epsilon$ parameter for such a simple regularization operator is equivalent to stopping the conjugate gradient solver after a different number of iterations.

A more sophisticated regularization scheme could involve applying an smoothing operator in the angle (or ray parameter) dimension Kuehl and Sacchi (2001); Prucha et al. (2000). More research need to be done regarding that subject, which is of extreme importance to obtain stable and meaningful results in real case scenarios.