(1) | ||

(2) |

draw
An initial filter shape is designed to include
the fixed coefficient (1) and several potential filter coefficients. The center
of each filter is then used to assign a dip value for each potential coefficient.
Figure 1 |

The problem with this construction is that the angular bandwidth of the filter will vary wildly. At 0,45,63,.. the inverse filter response will be basically a line while for other dips it will have significant aperture. Clapp et al. (1997) shows that by choosing a higher order interpolation scheme, the angular bandwidth problem can be reduced. We can also solve much of this problem by focusing on our goal to construct a specific type of a filter, one for velocity estimation. Velocity is generally a slow-varying function and we aren't going to have perfect knowledge of its gradient. As a result we want a filter with some angular bandwidth that we have some control over. Let us imagine we want the angular bandwidth to be some quantity .Instead of interpolating a filter for a single dip we will interpolate for a range of dips between to scaled by a function, we choose , that gives prominence to the closest to our desired dip. Our filter construction then becomes a loop over values from to range at some step. At each value we find the bounding dip locations so that

(3) | ||

(4) |

(5) |

This construction method has another advantage. Larger angles can be effectively represented. Figure shows our initial filter construction. This filter can effectively handle dips between -70 to 90 degrees. By extending the second column of the filter a wider angle range can be effectively handled.

impulse-2d
Impulse response of a 30 degree 2-D filter. The lines
represent the integration area.
Figure 2 |

Figure 3

7/8/2003