Streaming orthogonal prediction filter in $t$ -$x$ domain for random noise attenuation

Curved model

We use a synthetic example (Figure 1a) created by Raymond Abma (Liu and Fomel, 2011) to test the effectiveness of the proposed method in handling the nonstationarity. Figure 1b show data with uniformly distributed random noise. Figures 2a and 2b show the denoised result and the noise removed using $f$ -$x$ deconvolution, respectively. The data are divided into five patches with 40% overlap along the space axis. The $f$ -$x$ deconvolution creates artificial events and passes quite a lot of random noise. Nonstationary $f$ -$x$ RNA performs better. We set the filter length of the $f$ -$x$ RNA as 8 samples and the smoothing-radius size as 20 samples (in frequency) $\times$ 10-sample in space). Figure 3a shows that the $f$ -$x$ RNA passes less random noise than the $f$ -$x$ deconvolution. However, some artificial events still exist in the denoised result and there is signal energy leakage in the noise section (Figure 3b). Figure 4 is the result of processing by the $t$ -$x$ space-noncausal SOPF. The filter size is 7-sample (time) $\times$ 8-sample (space). The four scale parameters are 0.1 ($\lambda_t$ ), 0.08 ($\lambda_x$ ), 0.03 ($\gamma_t$ ), and 0.05 ($\gamma_x$ ), respectively. Figure 4a shows that the SOPF also introduces a few artifacts, but the artifacts follow a random distribution. Meanwhile, the difference (Figure 4b) between Figure 1b and Figure 4a indicates that the SOPF preserves signal better than the $f$ -$x$ RNA.

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Figure 1. Curved model (a) and noisy data (b).

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Figure 2. Denoised result (a) and noise removed (b) by the $f$ -$x$ deconvolution with patching.

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Figure 3. Denoised result by the nonstationary $f$ -$x$ RNA (a) and noise removed by the nonstationary $f$ -$x$ RNA (b).

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Figure 4. Denoised result by the $t$ -$x$ SOPF (a) and noise removed by the $t$ -$x$ SOPF (b).

Streaming orthogonal prediction filter in $t$ -$x$ domain for random noise attenuation