This section reviews the discussion in chapter , where it was shown that one effect of the long filter length in time for f-x prediction is the possible generation of false events. If strong parallel events are embedded in a background of random noise, f-x prediction generates weak but significant events parallel to the original events. While both t-x and f-x techniques tend to line up noise with strong events, f-x prediction actually generates new events with its long filter length in time. These spurious events occur because parallel events in a random noise background produce an f-x filter where the prediction of one strong event is influenced by the presence of other parallel events. An event corrupted by noise can have its prediction improved by using information from a nearby parallel event if the filter length in time is long enough to take advantage of the extra information. The t-x prediction is much less likely to be affected by the influence of widely separated events because of its short filter length in time.
The strength of the spurious events depends on the ratio of the signal to the background noise and the number of parallel events that contribute to the output sample's prediction. With no noise, no spurious events are generated. If there is no signal, or a very weak signal with respect to the noise, no prediction is done and no spurious events are generated. If many events parallel to the signal exist, each event contributes little to the prediction, and the spurious events generated by each of these parallel events will be too small to notice in most practical settings.
An example of the spurious events generated by f-x prediction can be seen in Figure . In this case, two flat events are immersed in noise on the left side of the displays, and the right side of the display is left noise free to show the response of the filter. For both f-x and t-x predictions, only one design window is used. With f-x prediction, events widely spaced in time can be used to predict any output point. For the f-x prediction in Figure , the noise on the left side of the input allows the predictions to be made using input from both events, so the equivalent t-x filter has significant coefficients far in time from the output point. When this filter is applied to clean data on the right side, spurious events generated by these widely separated coefficients are seen. In the f-x prediction result, strong events are generated above and below the two original events, and weak events line up with the original events on the right side. The t-x prediction results do not show these erroneous events. While the f-x prediction may be modified to eliminate this problem by constraining the filter coefficients to be smooth in frequency, the t-x prediction does not generate spurious events since its time length is more naturally controlled.
The false events generated by the f-x prediction generally appear to be weak. To give a better idea of the amplitudes of the false events, the results of the f-x prediction seen in the Figure are displayed as an elevation plot in Figure . In real data with an even distribution of noise, these spurious events are unlikely to be mistaken for real events, since the remaining noise should hide the errors.
Figure 2 The relative amplitudes of the false events to the original events for the f-x prediction of Figure 1.
Finally, to demonstrate that the f-x prediction and the t-x prediction show the same effects when a long filter length in time is used and that the generated events are not Fourier domain wraparound, Figure shows a t-x prediction filter with a time-length of 50 samples. Notice that the results are similar.
Figure 3 The relative amplitudes of the false events to the original events for a t-x prediction with an extremely long filter length in time.
A typical f-x prediction program has an advantage over a similar t-x prediction program because the filter length in time is fixed and does not need to be specified. Therefore, this length cannot be accidentally made too short. The cost of not requiring this parameter is creating the risk of generating false events and by passing more noiseAbma and Claerbout (1993).