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Improving the signal-prediction filter

In the previous discussion, it was assumed that the signal filter $\st S$ completely annihilates the signal, that is $\st S \sv s =\sv 0$.In reality, imperfect filters are derived from noisy data. For prediction filtering, the filters are derived from the least-squares solutions to the expression $\st S\sv d = \sv 0$.Since the data $\sv d$ contains noise, rather than getting an $\st S$ where $\st S \sv s =\sv 0$, we must contend with an imperfect $\st S$ such that $\st S \sv s \approx \sv 0$.This section shows how a better $\st S$ may be calculated by reducing the influence of the noise.

The presence of noise in the estimation of the signal annihilation filter $\st S$affects the calculation of the estimated signal in two ways. First, spurious events may be generated. These events may be widely separated in f-x prediction or may be seen as distortions of an event's wavelet. The cause of these distortions is discussed in chapter [*]. Second, the amplitudes of the reflectors in the calculated signal are reduced due to the imperfect prediction. As the strength of the noise increases, the more corrupted the filter becomes and the more the reflectors are attenuated.

To improve the calculation of the filter $\st S$,$\st S$ should be derived from the signal $\sv s$ instead of the data $\sv d$.Since the actual signal is unavailable, I use the inversion prediction result from equation ([*]) to get an estimate of the signal. Although the signal estimate is not perfect because $\st S$ is imperfect, this signal estimate can be used to create a new $\st S$ that is less affected by the noise. The process of calculating the signal, then getting a new signal annihilation filter, may be iterated as often as desired.

At this point, you might wonder why we should bother with the inversion when a cleaned-up signal may be obtained from prediction filtering. The inversion is more expensive than prediction filtering and might be avoided until a more perfect filter $\st S$ is available. Unfortunately, the signal annihilation filter calculated from the signal derived from prediction filtering will be exactly the same as the original filter calculated from the data. The residual $\sv r$ in the filter calculation expression $\sv r=\st S\sv d$becomes zero when the data $\sv d$ is replace by the signal estimated from prediction filtering. This is because all the noise calculated in prediction filtering is orthogonal to $\st S$, but everything in the estimated signal fits $\st S$ perfectly.

Once an improved signal filter $\st S$ is calculated from the estimated signal, this new filter may be used either to produce an improved prediction-filtering result, or it may be used to derive another inversion prediction result. If the response of the filter to the noise is assumed to be small, the improved prediction-filtering result might be the final result, but generally, if the noise is large enough to corrupt the filter, the response of the filter to the noise should be removed with inversion prediction.

Figures [*] and [*] in the next section show that iterating the calculation of the signal annihilation filter has the desired effect of preserving the amplitudes of the calculated signal and reducing the wavelet distortion in cases of small signal-to-noise ratios. Both effects are the result of removing some of the noise from the data used in the filter calculations. The amplitude improvement is a straightforward result of having a filter that predicts the signal well, rather than having a filter that predicts the signal poorly. The reduction of the generated spurious events results from the filter not being forced by the noise to use events parallel to the predicted events to improve the predictionsAbma (1994).

In the examples shown in the next section, three iterations of estimating the signal annihilation filter $\st S$ were used. I have found that one or two iterations do not allow the amplitudes of the reflections to be restored properly and more iterations seem to weaken the reflections. More work needs to be done to find how the number of iterations affects weak events that do not line up with the strongest events in a section. It is possible that iterating tends to eliminate weak events not lined up with the strongest reflections, since a preliminary filter might attenuate a weak event which then would not be recovered in the following passes.

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