Figure 13

Figure 13 shows three plane waves recorded on five channels
and the interpolated data.
Both the original data and the interpolated data can be described
as ``beyond **alias**ing'' because on the input data
the signal shifts exceed the signal duration.
The calculation requires only a few seconds
of a ``two-stage least-squares'' method, where the first stage
estimates an inverse **covariance matrix** of the known data,
and the second uses it to estimate the missing traces.
Actually, a
**2-D prediction-error filter**

is estimated, and the inverse covariance matrix, which amounts to the PE filter times its adjoint, is not needed explicitly.

Figure 14

Let us now examine a case with minimal complexity.
Figure 14
shows two **plane wave**s recorded on three channels.
That is the minimum number of channels required to distinguish
two superposing plane waves.
Notice on the interpolated data that the original traces are noise-free,
but the new traces have acquired a low level of noise.
This will be dealt with later.

Figure 15 shows the same calculation in the presence of noise on the original data. We see that the noisy data is interpolatable just as was the noise-free data, but now we can notice the organization of the noise. It has the same slopes as the plane waves. This was also true on the earlier figures (Figure 13 and 14), as is more apparent if you look at the page from various grazing angles. To display the slopes more clearly, Figure 15 is redisplayed in a raster mode in Figure 16.

Figure 15

Figure 16

- Interpolation with spatial predictors
- Refining both t and x with a spatial predictor
- The prediction form of a two-dip filter
- The regression codes
- Zapping the null space with envelope scaling
- Narrow-band data

10/21/1998