عنوان مقاله [English]
In this paper some new applications of empirical mode decomposition (EMD) and Hilbert spectrum in seismic ground-roll attenuation, random noise attenuation and spectral decomposition are introduced. Hilbert spectrum is a time-frequency representation for Hilbert-Huang transform which is obtained by combination of instantaneous frequency (IF) concept and intrinsic mode function of empirical mode decomposition. This time-frequency representation method has a suitable characteristic in analyzing non-stationary data. The advantages and the performance of the spectrum in seismic random noise attenuation and ground-roll removal are tested here by applying on real and synthetic seismic data and the results were satisfactory. In attenuation of random noise the instantaneous frequency filtering operation is different from other time-frequency decomposition methods and the characteristics of this type of filtering also are discussed.
In the case of spectral decomposition we introduced a new method. We can extract constant frequency section by using empirical mode decomposition and Hilbert-Huang transform. In addition we have used instantaneous frequency separately to construct constant instantaneous frequency sections to detect low frequency shadow zone beneath the reservoir. Spectral decomposition using constant instantaneous frequency section in comparison with other conventional time-frequency decompositions methods has some advantage. Constant frequency sections which are obtained through Hilbert-Huang transform is a time consuming process while by using instantaneous frequency separately, the massive calculation process of empirical mode decomposition is omitted and the results have no difference in comparison with Hilbert-Huang transform.
Here we explain how instantaneous frequency spectrum can be obtained from intrinsic mode functions (IMF). The empirical mode decomposition method developed by Huang et al. (1998) is a powerful signal analysis technique which known to be highly suitable for analysis of the non-stationary and non-linear signals, such as seismic data. EMD decompose data into functions which is called intrinsic mode functions. But EMD has a problem which is called mode mixing. Wu and Huang (2009) proposed the ensemble empirical mode decomposition (EEMD) to solve the mode mixing problem of EMD. However, not only the EEMD is not a complete decomposition method but also it is not reversible by summing all IMFs. Torres et al. (2011) proposed the complete ensemble empirical mode decomposition (CEEMD) algorithm. CEEMD overcome the mode mixing and provides an exact reconstruction of the original signal. In this paper we used CEEMD algorithm combined with Hilbert transform and analytic signal to evaluate instantaneous frequency. There are other methods to calculate IF from signals (for more information refer to Huang etal., 2009).
Analytic signal is obtained from signal and its Hilbert transform, we can write:
Where is the Hilbert transform of and is the analytic signal then its IF can be computed from
is the instantaneous phase and is the instantaneous frequency. In addition, for any given time in a signal we can obtain instantaneous amplitude of signal x (t) using
Having time and its corresponding frequency and instantaneous amplitude we can show 3D plot of time-frequency-amplitude, which is a time-frequency representation (TFR) similar to STFT and S spectrum. This TFR representation is called instantaneous frequency spectrum or Hilbert spectrum. If we calculate instantaneous frequency from IMFs the time-frequency analysis method is called Hilbert-Huang transform.
Here we demonstrated the performances of the Hilbert spectrum in attenuating random and coherent seismic noise as well as identifying low frequency shadow zone on seismic sections. The results were acceptable with no evidences of the negative frequency or spikes which are common in conventional instantaneous frequency.