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

2D curved model

We start with a synthetic example (Figure 3a) created by Raymond Abma, which was originally used for testing nonstationary interpolation (Liu and Fomel, 2011). The number of time samples is 401 and the number of space samples is 240. Figure 3b is the data with uniformly-distributed random noise added. We compare $t$ -$x$ APF with $f$ -$x$ deconvolution (Gulunay, 1986), $t$ -$x$ PF (Abma and Claerbout, 1995), and $f$ -$x$ RNA (Liu et al., 2012) and test their ability for random noise attenuation. Figure 4 shows the denoised results by using stationary methods. Both the $f$ -$x$ deconvolution and the $t$ -$x$ PF fail in handling nonstationary curved events. The data was divided into 5 patches with 40% overlap along space axis. $f$ -$x$ deconvolution eliminates both signal and noise (Figure 4a and 4b) and creates some artificial events with weak energy, which are parallel with the curved event. $t$ -$x$ PF preserves signal better and introduces artifacts fewer than $f$ -$x$ deconvolution (Figure 4c), but the difference (Figure 4d) between Figure 3b and 4c also shows obvious signal.

Another approach is to apply nonstationary filters. The denoised results by using $f$ -$x$ RNA and $t$ -$x$ APF are shown in Figure 5a and 5c, respectively. The filter length of $f$ -$x$ RNA is 8 and it has a 10-sample (frequency) and 20-sample (space) smoothing radius. $f$ -$x$ RNA (Figure 5a) has a better result than stationary methods, e.g., $f$ -$x$ deconvolution (Figure 4a) and $t$ -$x$ PF (Figure 4c), however, there is still signal trend in the noise section (Figure 4b) and artificial events appear that are similar to those from $f$ -$x$ deconvolution. For the $t$ -$x$ APF, the choice of the filter length in space is similar to that in $f$ -$x$ RNA. We tend to use a 12-sample filter in space, and the filter length in time for the $t$ -$x$ APF is selected to five samples. As the time-length of the $t$ -$x$ APF increases, the $t$ -$x$ APF passes more random noise. We use the shaping regularization with a 60-sample (time) and 20-sample (space) smoothing radius to constrain the APF coefficient space. The denoised result and removed noise are shown in Figure 5c and 5d, respectively. $t$ -$x$ APF also introduces a few artifacts, but the artifacts show a random-trend distribution (Figure 5c). Meanwhile, the $t$ -$x$ APF, shown in Figure 5d, preserves signal better than the $f$ -$x$ RNA.

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Figure 3. Curved model (a) and noisy data (b).

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Figure 4. Comparison of stationary methods. The denoised result by $f$ -$x$ deconvolution (a), the noise removed by $f$ -$x$ deconvolution (b), the denoised result by $t$ -$x$ PF (c), and the noise removed by $t$ -$x$ PF (d).

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Figure 5. Comparison of nonstationary methods. The denoised result by $f$ -$x$ RNA (a), the noise removed by $f$ -$x$ RNA (b), the denoised result by $t$ -$x$ APF (c), and the noise removed by $t$ -$x$ APF (d).

For further discussion, we added extra spike noise to Figure 3b, the new noisy model with a wiggle display is shown in Figure 6a. When comparing with the $t$ -$x$ PF with patching (Figure 6b) and the $f$ -$x$ RNA (Figure 6c), the $t$ -$x$ APF shows better signal-protection ability, however, the quality of the denoised result gets worse than Figure 5c because of the spikes (Figure 6d). Larger smoothing radius can reduce the artifacts at the cost of attenuating part of the signals.

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Figure 6. Tests of hybrid noise model by using different methods. Data with hybrid noise (a), $t$ -$x$ PF (b), $f$ -$x$ RNA (c), and $t$ -$x$ APF (d).

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