3-D extension

A 3-D dip annihilation filter can be approximated by cascading two perpendicular 2-D filters, one oriented in the x-z plane and the other oriented in the y-z plane. We can write this new filter as  

$$\bf A^{-1}_{3d} = \bf A^{-1}_x \bf A^{-1}_y ,$$ (2)

where $\bf A^{-1}_{3d}$ is our inverse 3-D steering filter operator, $\bf A^{-1}_x$ is an inverse steering filter oriented in the x-direction, and $\bf A^{-1}_y$ is an inverse y-direction steering filter.

The construction of two 2-D filters allows for a large degree of freedom in filter construction. The method can deal with a range of spatial dips with little variation in bandwidth response, Figure 1. In addition we can vary the range over which the smoother operates, both isotropically (Figure 2) and to some degree anisotropically (Figure 3).

 

angles
Figure 1 An example of filter behavior at different dip angles. Note how the bandwidth response is approximately the same in all panels.

 

iso
Figure 2 By changing what the non-zero lag coefficients sum to the area over which the filter acts can be drastically altered.

 

aniso
Figure 3 The cascading of two filters allows us to create a filter that acts over different range in the x and y direction.