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

Hybrid denoising approach

Provided that the noisy data $\mathbf{d}$ is composed of the clean data $\mathbf{s}$ and noise $\mathbf{n}$ , $f-x$ EMD can get a denoised section with all the horizontal events $\hat{\mathbf{s}}_{h}$ , while leaving the dipping events in the noise section:

$$\begin{split}\hat{\mathbf{s}}_{h} &\approx \mathbf{E}[\mathbf... ...hbf{n} &\approx \mathbf{d}- \mathbf{E}[\mathbf{d}]. \end{split}$$ (4)

Here, $\mathbf{E}$ denotes the noise attenuation operator by $f-x$ EMD, $\mathbf{s}_d$ denotes the true dipping events, and $\mathbf{n}$ denotes random noise in the original seismic section.

We can retrieve the useful dipping events by applying another denoising operator onto the noise section,

$$\hat{\mathbf{s}}_d \approx \mathbf{P}[ \mathbf{d}- \mathbf{E}[\mathbf{d}] ],$$ (5)

where $\mathbf{P}$ denotes a denoising operator which estimate the lost dipping events from the initial noise section, and $\hat{\mathbf{s}}_d$ denotes the estimated dipping events. The final denoised section $\hat{\mathbf{s}}$ is given by the summation of the horizontal and dipping signal section:

$$\hat{\mathbf{s}} = \hat{\mathbf{s}}_h + \hat{\mathbf{s}}_d \approx \mathbf{E}[\mathbf{d}] + \mathbf{P}[ \mathbf{d}- \mathbf{E}[\mathbf{d}] ].$$ (6)

The denoising operator $\mathbf{P}$ in equation 5 can be chosen as $f-x$ predictive filtering (Chen and Ma, 2014), wavelet transform (Chen et al., 2012), or curvelet transform (Dong et al., 2013). Thus, equation 5 becomes a general framework for all those $f-x$ EMD based random noise attenuation approaches. To extend its generality, we propose to use $f-x$ SSA (Oropeza and Sacchi, 2011) as $\mathbf{P}$ in this paper.

The effectiveness of the novel approach can be ascribed to the strong horizontal-preservation ability of $f-x$ EMD. When most of the useful horizontal energy is preserved after $f-x$ EMD, we turn to deal with less number of plane-wave components in the noise section, which is much easier because less signal components usually correspond to a more strict control over the random noise, e.g., smaller prediction length in $f-x$ predictive filtering (Canales, 1984) and lower rank for extracting useful components in $f-x$ SSA (Oropeza and Sacchi, 2011). For example, lower rank corresponds to less-serious rank-mixing problem, that is , less amount of random noise can be leaked into the denoised section.

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