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Preconditioning the filtering method

There is a simple trick that modifies the fitting goals in equation ([*]). We can pose the following preconditioning transformations:
372#372
where 348#348 and 347#347 are new variables. Now we can derive a new system of fitting goals as follows:
   373#373
This system is almost equivalent to what I introduced in (), except for the regularization that I omitted. With 374#374 the noise-modeling operator and 375#375 the signal-modeling operator, the least-squares inverse of equations ([*]) without the regularization terms is then given by  
 376#376   
with
377#377 (148)
I showed in () that 378#378 and 379#379 can also be interpreted in term of projection filters.

The estimated noise and signal are then computed as follows
355#355 (149)
Because of the relationship that exists between the filtering and subtraction methods, the estimated noise or signal should be equivalent for both. This has been observed in a multiple attenuation problem by ().


next up previous print clean
Next: Discussion Up: From the filtering to Previous: The filtering method
Stanford Exploration Project
6/7/2002