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

3D field data

For the field data test, we used a 3D time migration data to evaluate the effectiveness of the SPF (Fig. 7). The data size is 700-sample (time) $\times$ 266-sample (X) $\times$ 310-sample (Y). Strong random noise from the surface conditions contaminates both the simple layers at the shallow locations and complex structure at the deeper positions. We applied the 2D $f$ -$x$ SPF with 5-sample ($x$ ) to recover the events and selected 55 ( $\lambda_{f}$ ) and 280 ( $\lambda_{x}$ ) as the regularization terms for the improved $f$ -$x$ SPF with frequency constraint. Fig. 8a and 8b show the denoised results and the removed noise at the same clip value, respectively. Both shallow plane events and deep dipping events show better lateral continuity, but the 2D SPF also removes a part of the events because it uses the local smoothness constraints without the $y$ direction. Meanwhile, information is removed near $x=0$ and $y=0$ because of the inaccurate initial filter coefficients. We also compared the 2D SPF with the 2D $f$ -$x$ EMD prediction filter Chen and Ma (2014) to test it ability for random noise attenuation. The denoised result of $f$ -$x$ EMD prediction filter is shown in Fig. 9a. The 2D $f$ -$x$ EMD prediction filter preserves signal better than the 2D SPF, but the difference (Fig. 9b) still shows obvious signal and the method is difficult to be implemented in 3D case. For comparison, we used the 3D $f$ -$x$ -$y$ RNA to attenuate random noise. The $f$ -$x$ -$y$ RNA with 5-sample (X) $\times$ 5-sample (Y) filter size outputs a smoother result and the lateral continuity is improved (Fig. 10a), where the reflection event in both shallow and deep parts becomes clearer than original field data. Fig. 10b displays that only parts of dipping events are slightly lost. The proposed $f$ -$x$ -$y$ SPF can produce more feasible results than the 2D version because the 3D SPF has extra nonstationarity along the $y$ axis (Fig. 11a), where the continuity of the events and the geological structure are enhanced. The scale parameters are selected as 90 ( $\lambda_{f}$ ), 600 ( $\lambda_{x}$ ), and 600 ( $\lambda_{y}$ ) and the same 5-sample (X) $\times$ 5-sample (Y) filter size as the 3D $f$ -$x$ -$y$ RNA. The difference (Fig. 11b) between the noisy data (Fig. 7) and the denoised result (Fig. 11a) shows more uniformly-distributed random noise than the 3D $f$ -$x$ -$y$ RNA, which demonstrates that the proposed filter is able to depict the variations in the nonstationary signals and provide an accurate estimation of complex wavefields even in the presence of strongly curved and conflicting events. Furthermore, the $f$ -$x$ -$y$ SPF takes relatively low computational time (Table 2) that highlights its efficiency, especially in higher dimensions.

realdata
Figure 7. Three-dimensional field data.

realspf2,realerrspf2
Figure 8. The denoised result by using the 2D $f$ -$x$ SPF (a) and the difference between the noisy data (Fig. 7) and the denoised result by using the 2D $f$ -$x$ SPF (Fig. 8a) (b).

realfxemd,realerrfxemd
Figure 9. The denoised result by using the $f$ -$x$ EMD prediction filter (a) and the difference between the noisy data (Fig. 7) and the denoised result by using the $f$ -$x$ EMD prediction filter (Fig. 9a) (b).

realrna,realerrrna
Figure 10. The denoised result by using the 3D $f$ -$x$ -$y$ RNA (a) and the difference between the noisy data (Fig. 7) and the denoised result by using the 3D $f$ -$x$ -$y$ RNA (Fig. 10a) (b).

realspf3,realerrspf3
Figure 11. The denoised result by using the 3D $f$ -$x$ -$y$ SPF (a) and the difference between the noisy data (Fig. 7) and the denoised result by using the 3D $f$ -$x$ -$y$ SPF (Fig. 11a) (b).

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