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

$f-x$ EMD

$f-x$ EMD was proposed by Bekara and van der Baan (2009) to attenuate random noise. They applied EMD on each frequency slice in the $f-x$ domain, and remove the first IMF, which mainly represent the higher wavenumber components. The methodology can be summarized as:

$$\begin{split}\hat{s}(m,t) &= \mathcal{F}^{-1}\left(\sum_{n=2}... ..., \\ \mathcal{F} d(m,t) &= \sum_{n=1}^{N} C_n(m,w), \end{split}$$ (2)

where $\hat{s}(m,t)$ and $d(m,t)$ denote the estimated signal and acquired noisy signal, respectively. $\mathcal{F}$ and $\mathcal{F}^{-1}$ denote the forward and inverse Fourier transforms along the time axis, respectively. $C_n$ denotes the $n$ th EMD decomposed component. $w$ denotes frequency. However, a problem occurs when applying $f-x$ EMD, because the dipping events will also be removed. This problem occurs because, for many data sets, the random noise and any steeply dipping coherent energy make a significantly larger contribution to the high-wavenumber energy in the $f-x$ domain than any desired signal (Bekara and van der Baan, 2009).

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