EMD based smoothing

Chen et al. (2014b) provided an overall introduction of the applications of EMD in random noise attenuation of seismic data. The best way to utilize EMD to remove spatially incoherent noise is to apply EMD along the space direction in each frequency slice and to remove the first decomposed mode that contains the high-wavenumber noise.

The methodology can be summarized mathematically as:

$$\hat{s}(m,t) = \mathcal{F}^{-1}\left(\sum_{n=2}^{N}C_n(m,w)\right),$$ (2)

$$\mathcal{F} d(m,t) = \sum_{n=1}^{N} C_n(m,w),$$ (3)

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.

The detailed workflow can be summarized as

1. Transform the data from $t-x$ domain to $f-x$ domain.
2. For each frequency slice,
   1. separate real and imaginary parts in the spatial sequence,
   2. compute the first IMF, for the real signal and subtract it to obtain the filtered real signal,
   3. repeat for the imaginary part,
   4. combine to create the filtered complex signal.
3. Transform data back to the $t-x$ domain.

Smoothing via EMD has two main advantages. The first is its adaptivity. One does not need to decide any parameter before using the EMD based smoothing approach. As can be seen in equation 2, the first component after EMD is removed in each frequency slice to attenuate any spatially incoherent noise, which is very convenient to implement and has a robust performance. Another advantage is that EMD based smoothing can preserve the spatial amplitude variation details well since it does not require the exactly horizontal data structure. Smoothing via EMD can get very smooth final results. Unlike some other spatial smoothing operators such as mean filter, median filter or KL filter, the EMD based smoothing can also perform well even when the events have small curvatures. The main difference between the EMD and other decomposition algorithms is that it is empirical and thus adaptive to different input data sets, while other algorithms are based on a carefully designed mathematical model that requires a time-consuming parameter setting process Li et al. (2016); Liu et al. (2016a,b).