Smoothing slowness rather than change of velocity

In general, the tomography problem is under-determined and requires some type of regularization. Ideally, this regularization should be the inverse model covariance Tarantola (1987) but that is not readily available. In many cases we do have well logs, initial migration surfaces, or a geologist's model of the region that can at least indicate the trend that velocity should follow. Following the method described in Clapp, et al. 1999 we can build a space-varying operator composed of small plane wave annihilation filters that can smooth our velocity along this predetermined trend.

The problem is that our model is not slowness, but change in slowness. To a degree, we can get around this problem by following the method similar to the one described by Bevc 1994. We start by stating our goal to smooth the slowness field:

$$\bf 0\approx \bf A\bf s_{}$$ (4)

where $\bf A$ is our steering filter operator. But $\bf s_{}$ is actually $\bf s_{0} + \bf \Delta s$, so we can write a new regularization goal as

$$\begin{eqnarray} \bf 0&\approx&\bf A(\bf s_{0} + \bf \Delta s) \\ \nonumber - \bf A\bf s_{0} &\approx&\bf A\bf \Delta s.\end{eqnarray}$$ (5)

A problem with this method is where the adjoint of our modeling operator ($\bf T_{z}^{'}$) does not contribute at all to the model we can introduce artifacts. Our best solution to date for this problem is to introduce a smooth masking operator that tapers off to zero in locations unaffected by $\bf T_{z}^{'}$.