Signal and noise separation in prestack seismic data using velocity-dependent seislet transform

Validation of slope estimation and random noise elimination

A simple synthetic example is shown in Figure 1a. The synthetic data were generated by applying inverse NMO with time-varying velocities and represent perfectly hyperbolic events. Figure 1b shows local event slopes measured from the data using PWD algorithm (Fomel, 2002). PWD provides an accurate slope field for noise-free data. Figure 2a and 2b show the data after adding normally distributed random noise and local slopes from PWD, respectively. Compared with Figure 1b, PWD fails in finding exact slope field because of strong random noise. Next, we calculate slopes using NMO velocities from velocity scan. Picked NMO velocities (Figure 3a) are close to the exact velocity because velocity scan is less sensitive to strong random noise. As a consequence, VD slopes calculated from equation 7 provide a more accurate result (Figure 3b).

synt,sdip
Figure 1. Synthetic data (a) and slopes calculated by PWD (b).

noise,pdip
Figure 2. Synthetic noisy data (a) and slopes calculated by PWD (b).

svsc2,vdip
Figure 3. Velocity scanning (dash line: exact velocity, solid line: picked velocity) (a) and VD slopes (b).

A direct application of the seislet transform is denoising. We apply both PWD-seislet and VD-seislet transforms on the noisy data (Figure 2a). Figure 4a and 4b show the transform coefficients of PWD-seislet and VD-seislet, respectively. The hyperbolic events are compressed in both transform domains. Notice that PWD-seislet coefficients get more concentrated at small scale than those of VD-seislet because parts of the random noise are also compressed along inaccurate PWD slopes. Meanwhile, random noise gets spread over different scales in the VD-seislet domain, while the predictable reflection information gets compressed to large coefficients at small scales, which makes signal and noise display different amplitude characteristics. Figure 4c shows a comparison between the decay of coefficients sorted from large to small in the PWD-seislet transform and the VD-seislet transform. Seislet transform can compress the seismic events with coincident wavelets, if the slopes of the reflections are correct, the sparse large coefficients only correspond to the stacked reflection events. However, when the slopes of the reflections are not accurate, the stacked amplitude values for inconsistent wavelets will create more coefficients with smaller values. VD slopes are less affected by strong random noise than PWD slopes, which results in a faster decay of the VD-seislet coefficients. A simple thresholding method can easily remove the small coefficients of random noise. Figure 5a and 5b display the denoising results by using PWD-seislet transform and VD-seislet transform, respectively. The events after PWD-seislet transform denoising show serious distortion while VD-seislet transform produces a reasonable denoising result. For numerically comparison, we use the signal-to-noise ratio (SNR) defined as $SNR=10\log_{10}\frac{\vert\vert\mathbf{s}\vert\vert _2^2} {\vert\vert\mathbf{s}-\hat{\mathbf{s}}\vert\vert _2^2}$, where $\mathbf{s}$ is the noise-free signal and $\hat{\mathbf{s}}$ is the denoised signal. The original SNR of the noisy data (Figure 2a) is -12.53 dB. The SNR of the denoised results using the PWD-seislet transform (Figure 5a) and the VD-seislet transform (Figure 5b) are 0.53 dB and 1.94 dB, respectively.

pseis,vseis,ccomp
Figure 4. PWD-seislet coefficients (a), VD-seislet coefficients (b), and transform coefficients sorted from large to small, normalized, and plotted on a decibel scale (Solid line - VD-seislet transform. Dashed line - PWD-seislet transform) (c).

pclean,vclean
Figure 5. Denoising result using different transforms. PWD-seislet transform (a) and VD-seislet transform (b).

Signal and noise separation in prestack seismic data using velocity-dependent seislet transform