Noniterative f-x-y streaming prediction filtering for random noise attenuation on seismic data

3D synthetic qdome model

We start with the 3D qdome model Claerbout and Fomel (2008) containing curve events and faults (Fig. 3a) to evaluate the proposed method by handling the nonstationarity problem. The model size is 200 (time samples) $\times$ 150 (X traces) $\times$ 100 (Y traces). Fig. 3b displays the data with Gaussian noise added. We compared the 3D $f$ -$x$ -$y$ SPF with the 2D $f$ -$x$ SPF and the 3D $f$ -$x$ -$y$ RNA Liu and Chen (2013) to test their ability for random noise attenuation. The filter length of the $f$ -$x$ SPF is 5-sample ($x$ ). We also selected the scale parameters with 0.008 ( $\lambda_{f}$ ) and 0.06 ( $\lambda_{x}$ ). Fig. 4a shows the denoised result obtained by using the $f$ -$x$ SPF that eliminates most of the random noise. However, there is still an obvious signal in the noise section (Fig. 4b) because the 2D $f$ -$x$ SPF has a low accuracy owing to the local similarity of filter coefficients only along the $f$ and $x$ directions. A more effective approach is to apply global smoothness. The denoised result obtained by using the 3D $f$ -$x$ -$y$ RNA is shown in Fig. 4c. The filter size of the $f$ -$x$ -$y$ RNA is 5-sample ($x$ ) $\times$ 5-sample ($y$ ). The 3D $f$ -$x$ -$y$ RNA has a better result than the 2D $f$ -$x$ SPF, and it is visually difficult to detect the signal in the difference between the noisy data (Fig. 3b) and the denoised result (Fig. 4c). The global smoothness constraints along two spatial directions can help RNA to improve the result (Fig. 4d), but it also increases the computational cost because it iteratively solves the regularized least-squares problem (Table 1). We designed a 3D $f$ -$x$ -$y$ SPF with 5-sample ($x$ ) $\times$ 5-sample ($y$ ) coefficients for each sample and the scale parameters 0.008 ( $\lambda_{f}$ ), 0.06 ( $\lambda_{x}$ ), and 0.06 ( $\lambda_{y}$ ). The proposed method produces a reasonable result (Fig. 4e), where the geological structure is improved. It is also difficult to distinguish the signal in the removed noise (Fig. 4f), which is an indication of successful signal and noise separation. The signal-to-noise ratio (SNR) and time consumption were used to analyze the performance of each method (Table 2). The SNR is defined as:

$$SNR=10\log_{10}\frac{\vert\vert\mathbf{s}\vert\vert _2^2} {\vert\vert\mathbf{s}-\hat{\mathbf{s}}\vert\vert _2^2},$$ (17)

where $\mathbf{s}$ is the noise-free signal and $\hat{\mathbf{s}}$ is the denoised signal. The computing platform uses Intel E5-2650 2.0GHz CPU and the displayed time consumption is the average of ten records. Although the denoised result obtained by using the $f$ -$x$ SPF shows a relatively high SNR, the amplitude of the curved events is partly damaged, which is shown in Fig. 4b. The $f$ -$x$ -$y$ RNA preserves a more detailed structure than the $f$ -$x$ SPF at the cost of a lower SNR and longer computational time. In general, the $f$ -$x$ -$y$ SPF provides a higher SNR while maintaining the computational cost at a feasible level.

qdmod,qdnoise
Figure 3. 3D synthetic model (a) and noisy data (b).

qdspf2,qderrspf2,qdrna,qderrrna,qdspf3,qderrspf3
Figure 4. Denoised result by the $f$ -$x$ SPF (a), noise removed by the $f$ -$x$ SPF (b), denoised result by the $f$ -$x$ -$y$ RNA (c), noise removed by the $f$ -$x$ -$y$ RNA (d), denoised result by the $f$ -$x$ -$y$ SPF (e), and noise removed by the $f$ -$x$ -$y$ SPF (f).

Noniterative f-x-y streaming prediction filtering for random noise attenuation on seismic data