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Up: Curry: Interpolation with pseudoprimariesPseudoprimary
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Interpolation can be cast as a series of two inverse problems where a predictionerror
filter is estimated on known data and is then used to interpolate
missing data. A predictionerror filter (PEF) can be estimated by
minimizing the output of convolution of known data with an unknown
filter (except for the leading 1), which can be written in matrix form
as
 
(2) 
where f_{i} are unknown filter values and d_{i} are known data values.
The filters used in this paper are all multidimensional, which are
computed with the helical coordinate. In the case of
a stationary multidimensional PEF, this is an overdetermined
leastsquares problem with a unique solution.
Seismic data is nonstationary in nature, so a single stationary PEF is
not adequate for the many changing dips present. We estimate a single
spatiallyvariable nonstationary PEF and solve a global optimization
problem Guitton (2003). In that case the problem is now
underdetermined, and a regularization operator is introduced to the
leastsquares problem (in matrix notation) so that,
 

 (3) 
where represents nonstationary convolution with the data, is now a
nonstationary PEF, (a selector matrix) and (a copy of the data)
both constrain the value of the first filter coefficient to 1, is a
regularization operator (a Laplacian operating over space) and is a
tradeoff parameter for the regularization. Solving this system will
create a smoothly nonstationary PEF.
Once the PEF has been estimated, it can be used in a second least
squares problem that matches the output model to the known data while
simultaneously regularizing the model with the newly found PEF,
 

 (4) 
where is a selector matrix which is 1 where data
is present and where it is not, represents convolution with the
nonstationary PEF, is now a tradeoff parameter and is the desired
model.
interped
Figure 2 (a) Original data with near offsets (<2000 feet) missing. (b)
Original complete data. (c) Interpolation with PEF based upon complete
data. (d) Interpolation with PEF based upon pseudoprimaries.
Next: Results
Up: Curry: Interpolation with pseudoprimariesPseudoprimary
Previous: Generation of pseudoprimaries
Stanford Exploration Project
4/5/2006