Adaptive prediction filtering in $t$ -$x$ -$y$ domain for random noise attenuation using regularized nonstationary autoregression

Field data test

For the field data test, we use a time-migrated seismic image from Liu and Chen (2013). The input is shown in Figure 11. The three sections in Figure 11 show the time slice at time position of 0.34 s (top section), X line section at Y space position of 7.62 km (bottom left section), and Y line section at X space position of 1.41 km (bottom right section). The datacube displays simple plane layers above 1.0 s and complex structure below 1.0 s. Noise is mainly strong random noise caused by the surface conditions in this area. For comparison, we apply $f$ -$x$ -$y$ NRNA to remove the random noise. We use a total of 24 neighboring traces around each output trace after applying a Fourier transform along time axis. The denoised result is shown in Figure 12, which gives a much clearer lateral continuity than original field data. $f$ -$x$ -$y$ NRNA improves the both shallow plane events and deep dipping events. Figure 14 shows the difference section, in which the processed data using $f$ -$x$ -$y$ NRNA have been subtracted from the original data. Some horizontal events are shown in Figure 14, especially at locations from 0.3 s to 1.0 s in Y line section (bottom right section). Figure 13 shows that the proposed $t$ -$x$ -$y$ APF method also produces reasonable result, where continuity of events and geology structure are enhanced, and there is little noise left. The $t$ -$x$ -$y$ APF parameters correspond to 5 (time) $\times$ 4 (X) $\times$ 4 (Y) coefficients for each sample ($M$ =2, $N$ =2, and $L$ =2 in Equation 6 and 7) and 20-sample (time), 10-sample (X), and 10-sample (Y) smoothing radius for regularization operator $\mathbf{R}$ . After carefully comparing with the filtering result of 3D $f$ -$x$ -$y$ NRNA (Figure 12), 3D $t$ -$x$ -$y$ APF (Figure 13) can be seen to preserve more detailed structure because $t$ -$x$ -$y$ APF has extra nonstationarity along time axis while $f$ -$x$ -$y$ NRNA only consider nonstationarity along the two space axis. Splitting into windows can partly help $f$ -$x$ -$y$ NRNA to improve the result, but it still cannot provide a naturally nonstationary domain. Comparing with Figure 14, the difference (Figure 15) between Figure 11 and Figure 13 shows no obvious horizontal events and random noise is more uniformly-distributed.

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Figure 11. 3D field data.

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Figure 12. The denoised result by using 3D $f$ -$x$ -$y$ NRNA.

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Figure 13. The denoised result by using 3D $t$ -$x$ -$y$ APF.

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Figure 14. The difference between the noisy data (Figure 11) and the denoised result by using 3D $f$ -$x$ -$y$ NRNA (Figure 12).

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Figure 15. The difference between the noisy data (Figure 11) and the denoised result by using 3D $t$ -$x$ -$y$ APF (Figure 13).

Adaptive prediction filtering in $t$ -$x$ -$y$ domain for random noise attenuation using regularized nonstationary autoregression