Random noise attenuation by a selective hybrid approach using f-x empirical mode decomposition

Synthetic example

The first synthetic example is composed of three horizontal and one dipping events. In this test, 40 Hz Ricker wavelet is used. The sampling interval is 2 ms. The clean and noise sections are shown in Figure 6. After using $f-x$ EMD, $f-x$ SSA and the proposed selective hybrid approach, the denoised results and their corresponding noise sections are shown in Figure 7. Even though $f-x$ EMD can get a very clean denoised section, the dipping event is totally removed, as shown in the noise section Figure 7d. $f-x$ SSA preserves the useful dipping event effectively, however, the noise level in the denoised section is still very strong. Compared with $f-x$ EMD and $f-x$ SSA, the selective hybrid $f-x$ EMD & SSA seems to obtain a perfect denoised section, with all the useful energy preserved and most of the noise removed. The frame box in Figure 7d is the selective hybrid processing window. We also compare the amplitude difference for the 25th trace of the linear example among the clean data, noisy data, $f-x$ EMD denoised data, $f-x$ SSA denoised data, hybrid approach denoised data and show them in Figure 8. As we can see from the comparison, although the noise level is very strong, the proposed hybrid approach does the best job in preserving the amplitude of useful signals.

The second synthetic example is composed of three hyperbolic events, with one very steep hyperbolic event crossing the other one. In this case, we use 40 Hz Ricker wavelet and 2 ms sampling interval. The clean and noise sections are shown in Figure 9. After using $f-x$ EMD, $f-x$ SSA and the proposed selective hybrid approach, the denoised results and their corresponding noise sections are shown in Figure 10. The hyperbolic shape makes it difficult for both $f-x$ EMD and $f-x$ SSA. In this example, the two conventional approaches cannot get acceptable results because of a heavy loss of useful signals. The proposed hybrid approach, however, does a nearly perfect job in removing noise and preserving useful signals. We use two selective hybrid processing windows to retrieve the dipping signals and get a clean profile without any loss of useful signal. The two selective hybrid processing windows are shown in Figure 10d. The amplitude difference for the 15th trace of the hyperbolic example among the clean data, noisy data, $f-x$ EMD denoised data, $f-x$ SSA denoised data and hybrid approach denoised data is shown in Figure 11. It's very obvious that the proposed hybrid approach preserves the amplitude of useful signals best.

In order to numerically compare the denoising performances of different approaches, we utilize a measurement used previously (Hennenfent and Herrmann, 2006),

$$SNR=10\log_{10}\frac{\Arrowvert \mathbf{s} \Arrowvert_2^2}{\Arrowvert \mathbf{s}-\hat{\mathbf{s}}\Arrowvert_2^2},$$ (7)

where $\mathbf{s}$ is the noise-free signal and $\hat{\mathbf{s}}$ is the denoised signal. The SNR comparison is listed in table 1.

Table 1: Comparison of SNR using different approaches.

Model	linear	hyperbolic
Original (dB)	-4.725	3.439
$f-x$ EMD (dB)	1.174	3.136
$f-x$ SSA (dB)	-1.217	3.899
$f-x$ EMD & SSA (dB)	3.044	5.334

According to table 1, for the linear example, the SNR of the proposed approach is obviously much higher than $f-x$ SSA. Even though the SNR of the proposed approach is similar to that of $f-x$ EMD or even lower, according to the signal preservation effect for the proposed approach, we can conclude that the selective hybrid approach works better than the other two approaches.

Random noise attenuation by a selective hybrid approach using f-x empirical mode decomposition