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Next: REFERENCES Up: Clapp and Clapp: Steering Previous: 2-D construction

3-D construction

Up to this stage, everything that has been shown is very similar to the filter construction in previous works Clapp et al. (1997); Fomel (2000). In 3-D, we have a fundamental shift in philosophies. In previous works, Clapp (2000, 2001) and Fomel (2001) had the goal to create a filter whose response was a plane oriented at a given direction. Instead we are going to construct a tube. There is a definite disadvantage to this approach. The resulting filter will not be appropriate for doing interpolation because it is going to be spreading information in a limited area.

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 $\cos(\theta - \theta_i)^2 \cos(\phi - \phi_i)^2$,where $\phi$ is our desired azimuth and $\phi_i$ 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.

 
draw2
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.
draw2
view

 
response-3d
response-3d
Figure 5
Impulse response of the 3-D filter at several different angles and azimuths.
[*] view burn build edit restore

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),
\begin{displaymath}
\bf d\approx \bf J\bf A^{-1}\bf p.\end{displaymath} (6)
In this case $\bf d$ is the known values interpolated on to our model space (bottom-left panel of Figure [*]), $\bf J$is a diagonal matrix of ones (where we have known data points) and zeros where we do not. $\bf A^{-1}$ is polynomial division with a steering filter operator constructed using the above methodology. Our preconditioned variable $\bf p$ is related to our model through $\bf p=\bf A\bf m$.The bottom right panel of Figure [*] shows the interpolated result. Even with this very sparse dataset we have done a good interpolation of the field.

 
examp-2d
examp-2d
Figure 6
A 2-D interpolation example. The top-left panel is the model. Top-right panel is the dip field used to construct the filters. The bottom-left panel is our initial data. The bottom-right panel is our interpolation result. Generally the result is quite accurate.
[*] view burn build edit restore

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.

 
known-3d
known-3d
Figure 7
The correct model for 3-D interpolation problem.
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initial-3d
initial-3d
Figure 8
Our starting set of traces.
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final-3d
final-3d
Figure 9
The final interpolation result. Note how we have formed tubes around our known traces.
view burn build edit restore

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.


next up previous print clean
Next: REFERENCES Up: Clapp and Clapp: Steering Previous: 2-D construction
Stanford Exploration Project
7/8/2003