Random noise attenuation by $f$ -$x$ empirical mode decomposition predictive filtering

1D EMD

The aim of empirical mode decomposition (EMD) is to empirically decompose a non-stationary signal into a finite set of subsignals, which are termed intrinsic mode functions (IMF) and are considered to be stable. 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).

Provided that $s(t)$ , $c_n(t)$ , $r(t)$ , and $N$ denote the original non-stationary signal, the separated IMFs, the residual, and the number of IMFs, respectively, the mathematical principle of EMD can be expressed as:

$$s(t)=\sum_{n=1}^{N}c_n(t)+r(t).$$ (6)

For a non-stationary signal $s(t)$ , using equation 6, we get a finite set of subsignals $c_n(t)$ ,( $n=1,2,\cdots,N$ ).

A special property of EMD is that the IMFs represent different oscillations embedded in the data, where the oscillating frequency for each subsignal $c_n(t)$ decreases as the sequence number of the IMF becomes larger (we call it a frequency decreasing property in the following context). This property results from the sifting algorithm used to implement the decomposition. Appendix A gives a detailed instruction about the sifting process, which can be summarized as a process in which low-frequency components are gradually removed to generate a more local-constant-frequency mode, which is followed by the generation of the next mode.

Figure 2 gives a demonstration for a synthetic signal. The original synthetic signal is generated through $d(t)=\sin(0.2\pi t)+\sin(0.4\pi t)+\sin(0.8\pi t)$ ; in other words, it is constructed from three individual frequency components corresponding to 0.1 Hz, 0.2 Hz and 0.4 Hz, respectively. From Figure 2, we can see that, except for small edge imprecision and negligible residual, EMD successfully decompose this signal into three components with a frequency ratio of approximately 4:2:1.

Because of the frequency decreasing property, EMD has been used outside geophysics for noise attenuation (Kopsinis, 2009; Mao and Que, 2007). Since random noise represents mainly the high-frequency components, by removing the IMFs with the highest frequency, we can attenuate this type of noise. However, in exploration geophysics, applying EMD to time traces is not effective because of the mode mixing problem. Kopecky (2010) defined mode mixing as any IMF consisting of frequencies of dramatically disparate scales. When mode mixing exists, the first one or two IMFs contain a lot of useful reflection energy. Extensions to EMD, such as ensemble empirical mode decomposition (EEMD) (Wu and Huang, 2009) and complete ensemble empirical mode decomposition (CEEMD) (Torres et al., 2011) have been proposed to solve the mode-mixing problem in signal processing and have been used in geophysics to analyze time-frequency properties, but have not been used for t-x domain seismic noise attenuation.

Random noise attenuation by $f$ -$x$ empirical mode decomposition predictive filtering