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Interpolation in frequency and space

Interpolation in the frequency domain requires a different approach than in the time domain. Spitz (1991) shows that 2D plane waves can be predicted for a single frequency by using a 1D spatial prediction filter, because a single plane wave at each frequency appears as a complex sinusoid in space. The wavenumber of this sinusoid increases as a function of frequency, so each frequency requires a unique 1D PEF. Data containing a combination of plane waves appear as a summation of complex sinusoids at each frequency, which can still be predicted by a reasonably short 1D PEF. In three dimensions, this filter would be a 2D filter, in four dimensions a 3D filter, and so on, making the frequency-based approach a series of smaller problems whereas the same region in time would be solved as a single larger problem.

Nonstationarity is addressed in a slightly different manner in $ f$-$ x$. 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 ($ h$-$ s$) 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.


next up previous [pdf]

Next: Sigsbee data example Up: Interpolating with nonstationary filters Previous: Interpolation in time and

2009-04-13