Narrow-band data

Spitz's published procedure is to Fourier transform time to $(\omega, x)$, where, following Canales, he computes prediction filters along x for each $\omega$.Spitz offers the insight that for a dipping event with stepout $p=k_x/\omega$,the prediction filter at trace separation $\Delta x$ at frequency $\omega_0$should be identical to the prediction filter at trace separation $\Delta x/2$ at frequency $2\omega_0$.There is trouble unless both $\omega_0$ and $2\omega_0$ have reasonable signal-to-noise ratio. So a spectral band of good-quality data is required. It is not obvious that the same limitation applies to the interlacing procedure that I have been promoting, but I am certainly suspicious, and the possibility deserves inspection. Figure 20 shows a narrow-banded signal that is properly interpolated, giving an impressive result. It is doubtful that an observant human could have done as well. I found, however, that adding 10% noise caused the interpolation to fail.

 

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Figure 20 Narrow-banded signal (left) with interpolation (right).

On further study of Figure 20 I realized that it was not a stringent enough test. The signals obviously contain zero frequency, so they are not narrow-band in the sense of containing less than an octave. Much seismic data is narrow-band.

I have noticed that aspects of these programs are fragile. Allowing filters to be larger than they need to be to fit the waves at hand (i.e., allowing excess channels) can cause failure. We could continue to study the limitations of these programs. Instead, I will embark on an approach similar to the 1-D missif() program. That program is fundamentally nonlinear and so somewhat risky, but it offers us the opportunity to drop the idea of interlacing the filter. Interlacing is probably the origin of the requirement for good signal-to-noise ratio over a wide spectral band. Associated with interlacing is also a nagging doubt about plane waves that are imperfectly predictable from one channel to the next. When such data is interlaced, the PE filter really should change to account for the interlacing. Interlacing the PE filter is too simple a model. We can think of interlacing as merely the first guess in a nonlinear problem.