The filter construction in 3-D is simply adding an extra dimension to the procedure described above for the 2-D case. Our potential filter coefficients are generally oriented one plane away from the fixed coefficient (Figure ) and range in the first axis in a similar manner as shown in Figure . Our weighting function becomes ,where is our desired azimuth and is the azimuth at the given filter location. Finally, our linear interpolation is 2-D rather than 1-D. Figure shows the filter response at different dip-azimuth combinations. The response is somewhat deceiving. What we are creating is a tube oriented at a given dip-azimuth combination so the response is only symmetric along the dip-azimuth line.
Figure 4 The front face of the cube shows where the fixed coefficient are located. The intersection of the As, Bs, and Cs represent the variable coefficient locations.
EXAMPLE To test the methodology we performed two different interpolation problems, one that we expect to succeed, the other to fail. The first is a simple 2-D interpolation problem, the top-left panel of Figure shows our model. We assume some a priori knowledge of the dips of the field (top-right panel of Figure ) and a sparse set of known data traces (bottom-left panel of Figure ). We apply the fitting goals for a model preconditioned estimation problem described in Claerbout (1999),
The 3-D case is a different story. Remember that we are creating tubes rather than planes. The area over which they act is limited. With a sparse set of data points we expect to see tubes form around our known data points. The results bear out this hypothesis. Figure shows the known model, Figure our initial data, and Figure our interpolation result. As anticipated what we have ended up doing is forming tubes that curve with our predefined dip and azimuths around the known traces.
CONCLUSIONS We demonstrated another way to form 3-D plane-wave anhilation filters. The filters do not suffer from the inaccuracies and/or computational expense of previous methods. The main difference from previous constructions is that they form tubes rather than planes. This fact makes them effective regularizers for problems like tomography but inappropriate for interpolation.