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Scale-invariance introduces more fitting equations

The fitting goals (3) and (4) have about double the usual number of fitting equations. Scale-invariance introduces extra equations. If the range of scale-invariance is wide, there will be more equations. Now we begin to see the view to the end of the tunnel:
1.
Refining a computational mesh improves accuracy.
2.
Refining a data mesh makes empty bins.
3.
Empty bins spoil analysis.
4.
If there are not too many empty bins we can find a PEF.
5.
With a PEF we can fill the empty bins.
6.
To get the PEF and to fill bins we need enough equations.
7.
Scale-invariance introduces more equations.
An example of these concepts is shown in Figure 2.

 
mshole90
mshole90
Figure 2
Overcoming aliasing with multiscale fitting.


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Additionally, when we have a PEF, often we still cannot find missing data because conjugate-direction iterations do not converge fast enough (to fill large holes). Multiscale convolutions should converge quicker because they are like mesh-refinement, which is quick. An example of these concepts is shown in Figure 3.

 
msiter90
Figure 3
Large holes are filled faster with multiscale operators.

msiter90
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next up previous print clean
Next: Coding the multiscale filter Up: MULTISCALE, SELF-SIMILAR FITTING Previous: Examples of scale-invariant filtering
Stanford Exploration Project
2/27/1998