REGULARIZATION USING STEERING FILTERS

We can write our fitting goals for a linear problem as

$$\bold d \approx \bold T \bold m .$$ (64)

Where $\bold d$ is the data, $\bold{T}$ is the linearized tomographic operator, and $\bold m$ is our model. Often the problem is under-determined, so we need to add some type of regularization  

$$\bold 0 \approx \bold A \bold m .$$ (65)

Ideally, $\bold{A}$ should be the inverse of the model covariance matrix (). Unfortunately, we are estimating the model so we don't have the covariance matrix. The lack of knowledge about the model often leads to the Laplacian or some other isotropic operator being used for $\bold{A}$. As a result, we fill the null-space of the model with isotropic features, that, while explaining the data, may be unreasonable when judging the results with geologic criteria.

Fortunately, we often do have other sources of information, such as a geologist's model for the region or reflector dip from well logs, that can be used to better constrain our inversion. For example, we can the find the general dip direction by interpreting an early migration result and use this information to construct a space variant filtering operator that annihilates dips with the given direction. (Figure ), that can be used as $\bold{A}$ in equation (). The inverse of the dip-annihilation operator is really a first-order approximation for the model covariance matrix. We know that in general we have some isotropic smoothness in our velocity function, therefore adding some isotropic smoothness to our regularization operator is appropriate. This creates a space variant filter direction and produces an anisotropic blob oriented in the dip direction (Figure ).

 

filters
Figure 1 Steering filter directions as a function of geologic dip.

 

sweep
Figure 2 Preconditioning operator impulse response when oriented at -40, -20, 20, and 40 degrees from horizontal.

By forming our operators in helix-space Claerbout , we can find a stable inverse for our steering filters ($\bold{A}$) and change our regularization problem into a preconditioning problem. By substituting:

$$\bold m = \bold A^{-1} \bold p$$ (66)

we get

$$\begin{eqnarray} \bold d &\approx& \bold T \bold A^{-1} \bold p \nonumber \\ \bold 0 &\approx& \epsilon \bold I \bold p .\end{eqnarray}$$ (67)

The inverse operator ($\bold A^{-1}$) spreads information long distances at every iteration, quickly filling the null-space with reasonable values.

In general, tomography problems are not as straightforward as the one presented above. First and foremost, the tomography problem is non-linear. Perturbations in the slowness model change the raypaths, making the tomography problem non-linear. We can get around this by imposing an outer, non-linear raytracing loop over a linearized back projection operation that assumes stationary raypaths. In addition, we must deal with the inherent velocity-depth coupling problem: any changes in traveltimes can be caused by either reflector movement or slowness model changes. Therefore, we must take into account reflector movements when evaluating the linearized tomographic operator ():

$$\bold{\Delta t_m^i } \approx \bold{L_s^i\Delta s} + \bold{G^i\Delta R^i} = (\bold{L_s^i} + \bold{G^iH^i})\bold{\Delta s},$$ (68)

where

$\bold{\Delta t_m^i }$

is the difference between the modeled and the correct travel times,

$\bold{L_s^i}$

is the back-projection operator along our modeled ray paths,

$\bold{G^i}$

maps changes in reflector position to changes in traveltimes,

$\bold{\Delta R^i}$

is the change in reflector position,

$\bold{H^i}$

maps slowness changes to reflector movement,

$\bold{\Delta s}$

is -ur change in slowness.

Finally, we add in our preconditioning operator to obtain our final set of tomography goals,

$$\begin{eqnarray} \bold{\Delta t_m^i} &\approx& (\bold{L_s^i} + \bold{G^iH^i}) \b... ...-1}} \bold{p} \nonumber \\ \bold{0} &\approx& \epsilon \bold{Ip} .\end{eqnarray}$$ (69)