Review

Following the methodology of Clapp and Biondi (1999) we will begin by considering a regularized tomography problem. We will linearize around an initial slowness estimate and find a linear operator in the vertical traveltime domain $\bf T_{}$ between our change in slowness $\bf \Delta s$and our change in traveltimes $\bf \Delta t$.We will write a set of fitting goals,

$$\begin{eqnarray} \bf \Delta t&\approx&\bf T_{} \bf \Delta s\nonumber \\ \bf 0&\approx&\epsilon \bf A\bf \Delta s,\end{eqnarray}$$ (1)

where $\bf A$ is our steering filter operator and $\epsilon$ is a Lagrange multiplier.

However, these fitting goals don't accurately describe what we really want. Our steering filters are based on our desired slowness rather than change of slowness. With this fact in mind, we can rewrite our second fitting goal as:

$$\begin{eqnarray} \bf 0&\approx&\bf A\left( {\bf s_0} + \bf \Delta s\right) \\ -\epsilon \bf A{\bf s_0} &\approx&\epsilon \bf A\bf \Delta s.\end{eqnarray}$$ (2) (3)

Our second fitting goal can not be strictly defined as regularization but we can do a preconditioning substitution Fomel et al. (1997):

$$\begin{eqnarray} \bf \Delta t&\approx&\bf T_{} \bf A^{-1}\bf p\nonumber \\ - \epsilon \bf A{\bf s_0} &\approx&\epsilon \bf I\bf p .\end{eqnarray}$$ (4)