![]() |
![]() |
![]() |
![]() | Interpolation of near offsets using multiples and prediction-error filters | ![]() |
![]() |
Nonstationarity is addressed in a slightly different manner in
-
. Along the spatial axes, a nonstationary PEF can be used to
capture slopes that change as a function of position. Since a Fourier
transform is performed on the time axis to convert both the training
and interpolated data to the frequency domain, we implicitly assume
that the slopes of the plane waves do not vary as a function of time.
We address this problem of time nonstationarity by breaking up the
problem into overlapping time windows that we assume to be stationary.
First, perform a water-velocity normal move-out correction on both the
pseudo-primaries and the original recorded data to (roughly) flatten
them, and then break up the data into overlapping windows along the
time axis. We perform the NMO to reduce the amount of energy crossing
the boundaries between patches. From there, each time window of both
the pseudo-primaries and the recorded data is Fourier transformed
along the time axis, so that the data are sorted into source, offset,
frequency, and time window. A unique 2D nonstationary complex-valued
PEF is estimated in source and offset on each frequency of each time
window of the pseudo-primary data by solving equation
3. This nonstationary source-offset (-
) PEF
is then applied to fill the missing data in offset on the
corresponding time window and frequency of the recorded data by
solving equation 4, and this series of problems
is repeated for each frequency of each time window.
After the data are interpolated, they are first inverse-Fourier transformed back to time, then the time-windows are reassembled with appropriate weighting in overlapping regions, and finally are inverse NMO-corrected to return the data to their original form of time, source, and offset.
![]() |
![]() |
![]() |
![]() | Interpolation of near offsets using multiples and prediction-error filters | ![]() |
![]() |