Methods of using a filter

As stated in the introduction, the filters calculated here will be used in two techniques: in simple filtering and in inversions.

For signal and noise separation by filtering, the noise is expected to be whatever remains after a signal-annihilation filter is applied to the data. This can be expressed as $\sv n_{\rm pef}=\st S\sv d$,where $\st S$ is the signal-annihilation filter, $\sv d$ is the data, and $\sv n_{\rm pef}$ is the prediction-error filter estimate of the noise. While the desired action of the filter is $\sv 0 \approx \st S\sv s$, where $\sv s$ is the signal, the signal $\sv s$ is not available for calculating $\st S$.In the case where the noise is considered to be unpredictable, the filter $\st S$ can be calculated by minimizing $\st S\sv d$.This makes $\st S\sv d$ the prediction error, and the result for $\sv n_{\rm pef}$is the prediction-error filtering estimate of the noise. The t-x and f-x prediction filtering discussed in chapters  and   calculates the prediction-error filtering estimate of the noise $\sv n_{\rm pef}$with the filtering $\st S\sv d$ done in two and three dimensions.

As pointed out by Soubaras1994, the noise as defined by prediction-error filtering is inconsistent with the definition that the data is the sum of the signal and noise, $\sv d=\sv s+\sv n$,even though $\sv s$ is calculated as $\sv s=\sv d-\sv n_{\rm pef}$.If the signal $\sv s$ is perfectly predicted by filter $\st S$,the signal will be completely annihilated by the filter so that $\st S \sv s =\sv 0$.When $\st S$ is applied to $\sv d=\sv s+\sv n$, the result is $\st S\sv d = \st S\sv n$,as opposed to $\st S\sv d=\sv n_{\rm pef}$, the definition of noise by prediction-error filtering. Thus, to avoid a conflict of these definitions, prediction-error filtering requires a filter that removes the signal $\sv s$ without disturbing the noise $\sv n$,which is expressed as $\st S\sv n \approx \sv n$.If the prediction-error filtering result $\st S\sv d$ or $\st S \sv n$ is not close to the actual noise $\sv n$, the accuracy of the signal calculated from $\sv d-\sv n_{\rm pef}$ will be compromised. In short, a prediction-error filter must not distort the noise. This requirement will always be violated to some extent.

If the filters are used in an inversion, the assumption that $\st S\sv n \approx \sv n$ is not required. The inversion only requires that $\sv 0 \approx \st S\sv s$.The form of $\st S \sv n$ is less important. $\st S \sv n$ may be, for example, reversed in polarity or time-shifted when compared to $\sv n$, and the inversion will still function well. This might be understood as making the phase of the filter unimportant, since the spectral power is the main concern.

One advantage of using a filter in an inversion is that more freedom is allowed for correcting the results to account for missing data, as described in the previous chapter and examined in more detail in chapter . While the application prediction-error filters can be modified to treat some missing data problems by predicting in only one direction, inversion gives a more natural method of allowing for missing data, as well as predicting and restoring the missing data.