Preconditioning

The proposed regularized tomography problem still has the problem of slow convergence. By reformulating the problem in helix space Claerbout (1998c), we can take advantage of 1-D theory to change our regularized problem into a preconditioned one. We start by defining a new variable $\bf p$:

$$\bf p= \bf A\bf \Delta s.$$ (6)

By applying polynomial division to our steering filters, we can create $\bf A^{-1}$ which becomes a smoothing operator. We can then rewrite our fitting goals as

$$\begin{eqnarray} \bf \Delta t&\approx&(\bf T_{z,ray} + \bf T_{z,ref}) \bf A^{-1}\bf p\nonumber \\ - \bf A\bf s_{0} &\approx&\epsilon \bf p .\end{eqnarray}$$ (7)