Smoothing by empirical mode decomposition

Empirical mode decomposition (EMD) is used to empirically decompose a non-stationary signal into a finite set of subsignals, which are termed as intrinsic mode functions (IMF) and are considered to be locally stationary. The IMFs satisfy two conditions: (1) in the whole data set, the number of extrema and the number of zero crossings must either equal or differ at most by one; and (2) at any point, the mean value of the envelope defined by the local maxima and the envelope defined by the local minima is zero (Huang et al., 1998). Let $s(t)$, $c_n(t)$, $r(t)$ and $N$ denote the original non-stationary signal, the separated IMF, the residual and the number of IMFs, respectively. The EMD can be expressed as:

$$s(t)=\sum_{n=1}^{N}c_n(t)+r(t).$$ (1)

For a non-stationary signal $s(t)$, using equation 1, we get a finite set of subsignals $c_n(t)$,( $n=1,2,\cdots,N$).

We can straightforwardly use the EMD to denoise each seismic trace because of the special frequency-decreasing property of IMF. IMFs represent different oscillations embedded in the data, and the oscillating frequency for each subsignal $c_n(t)$ decreases as the sequence number of IMF gets larger. This property results from the sifting algorithm to implement the decomposition. The sifting process can be summarized as a process in which low-frequency component is gradually removed out to generate a more local-constant-frequency mode, which is followed by generating the next mode.

Chen et al. (2014b) provided an overall introduction of the applications of EMD in random noise attenuation of seismic data. According to Chen et al. (2014b), EMD can be used to denoise each 1-D signal from the 2-D seismic profile in time-space ($t$-$x$) domain either along the time direction or space direction. However, because of the mode-mixing problem, $t$-$x$ domain EMD along the time direction will cause some damage to useful seismic signal. A better way utilize EMD to remove noise is to apply EMD along the space direction and to remove the highly oscillating components. The frequency-space ($f$-$x$) domain EMD can help to obtain faster implementation and even better performances. $f-x$ EMD refers to applying EMD on each frequency slice in the $f-x$ domain, and removing the first IMF, which mainly represent the highest wavenumber components. The methodology can be summarized as (Chen et al., 2015a):

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

where $\hat{s}(m,t)$ and $d(m,t)$ denote the estimated signal and observed 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. In order to optimize the effectiveness, the seismic events should better be flattened first.

It is worth to be mentioned here that the $f-x$ EMD does not require exactly flattened events. Unlike 1D mean and median filters, $f-x$ EMD can also preserve useful reflections with small dip angle. Compared with Karhunen-loeve (KL) or singular value decomposition (SVD) filtering, the $f-x$ EMD can be much more adaptive, because the horizontal events mainly lay in the first 1-3 IMFs after EMD in the $f-x$ domain, however, the number of singular values corresponding to the useful horizontal reflections for KL or SVD transforms have a large range and thus is not convenient to chose. The performance for preserving horizontal events of $f-x$ EMD is also much better than that of KL or SVD transforms (Chen et al., 2015a).