2-D signal and noise estimation examples

Figure  shows a shot gather with strong coherent noise at a velocity of about 1800 feet per second. This noise train is relatively strong and narrow so it is easy to separate spatially. For this file, the noise filter was calculated from the data in a window 0.4 seconds long with a starting velocity of 1800 feet per second. This window can be seen from the distribution of the background noise seen around the ground roll in Figure . The signal filter was calculated from the data with the previous window zeroed out, as well as having a start time mute of 3900 feet per second.

The most expensive part of this process is calculating the noise filter $\st F_n$, since it is a fairly large filter. Using a smaller filter reduced the effectiveness of the process, probably because the noise train is more complicated than it appears. If the noise is consistent from shot to shot, the noise filter might be reused to reduce the cost. Calculating an effective noise filter is likely to be a problem on many land lines, since the noise will vary from shot to shot, and since the coupling of the receivers to the ground is variable. This variability of the coupling will show up as unpredictable parts of the ground roll. Since the ground roll is very strong when compared to the signal, the unpredictable part of the ground roll is likely to have significant energy. One method of correcting for the variable coupling was presented by Berlioux and Lumley1994. If the variations in the coupling are not corrected for, the results of this process are likely to be unsatisfactory. The shot file shown in Figure  appears to have an unusually uniform receiver coupling.

 

WFshot
Figure 9 A shot gather showing strong ground roll.

Once the signal and noise filters were calculated, system () was inverted for the noise and the missing data. As mentioned above, the noise was initialized with the noise window filtered by the signal annihilation filter. If the inversion was attempted with the noise initialized to zero, many iterations were required to get a reasonable result. Even with many iterations, the result was not as good as using a few iterations with the noise initialized with a good estimate. For the results shown here, only ten iterations were used. Little improvement was found when using more iterations.

Figure  shows the noise estimated by the inversion. Notice that there is little energy outside the zone where the noise dominated. There is no obvious signal showing in the noise section. When the noise in Figure  is subtracted from the original data in Figure , the resulting signal is seen in Figure . Almost all of the coherent noise was removed.

 

condif
Figure 10 The noise estimated from the inversion of system () using the data from the previous figure.

 

cohinv
Figure 11 The estimated signal obtained by subtracting the noise from the original data.

There is a small change in the data character of Figure  between the zone where the noise originally dominated and the area outside the zone. This was the result of the treatment of noise not predicted by either the signal filter $\st F_s$ or the noise filter $\st F_n$,especially random noise. The distribution of unpredicted noise is controlled by the weighting functions $\st W_s$ and $\st W_n$, as shown in chapter . Since the value of $\st W_n$ was small in the noise zone, most of the unpredictable noise fell there. Outside the noise zone, the value of $\st W_n$ was large, keeping the unpredictable noise out of the noise section and putting it into the signal section. If this change of character affects analysis of the data, removing random noise from the entire file using the techniques of chapter  will eliminate the effect.

The values of the weighting functions $\st W_s$ and $\st W_n$ will also control the distribution of events that are predictable with both $\st F_s$ and $\st F_n$,as shown in chapter . If an event is equally predictable with either $\st F_s$ or $\st F_n$,the event will fall into the noise zone because of the small values of $\st W_n$ there, while outside the noise zone, equally predictable events will tend to fall into the signal. Generally, events are unlikely to be well predicted by both $\st F_s$ and $\st F_n$because of the different shapes of the filters.

The results of the inversion were relatively insensitive to the exact values used in $\st W_s$ and $\st W_n$. As long as the values outside the noise zone were between 8 and 40 times the values inside the noise zone, the results appeared basically the same. This insensitivity simplifies the calculation of $\st W_s$ and $\st W_n$ considerably, since the signal can be corrected for spherical spreading before the process, while the noise, even though it will not have its amplitude perfectly represented in $\st W_n$, can be removed effectively.

As a comparison, an F-K filter was run on the previous data, muting out the noise in the F-K domain. The results, as shown in Figure , appear similar to the inversion results, although some of the ground roll is left in the shallow section in the F-K plot. The inversion result shows more character change than the F-K result in the area where the ground roll originally dominated, as discussed above. The F-K filter shows some artifacts of the mute, which is expected since the mute will produce a long impulse response in the time-space domain. Even when the inversion method uses a long filter, the response of this filter is removed from the output. In cases where the noise is aliased over a significant part of its bandwidth, the inversion may be able to characterize and remove noise more effectively than simple F-K muting.

 

fk
Figure 12 An F-K filter of the previous example.