Seismic signal de-noising using time–frequency peak filtering based on empirical wavelet transform

Abstract

Seismic noise suppression plays an important role in seismic data processing and interpretation. The time–frequency peak filtering (TFPF) is a classical method for seismic noise attenuation defined in the time–frequency domain. Nevertheless, we obtain serious attenuation for the seismic signal amplitude when choosing a wide window of TFPF. It is an unsolved issue for TFPF to select a suitable window width for attenuating seismic noise effectively and preserving valid signal amplitude effectively. To overcome the disadvantage of TFPF, we introduce the empirical wavelet transform (EWT) to improve the filtered results produced by TFPF. We name the proposed seismic de-noising workflow as the TFPF based on EWT (TFPF-EWT). We first introduce EWT to decompose a non-stationary seismic trace into a couple of intrinsic mode functions (IMFs) with different dominant frequencies. Then, we apply TFPF to the chosen IMFs for noise attenuation, which are selected by using a defined reference formula. At last, we add the filtered IMFs and the unprocessed ones to obtain the filtered seismic signal. Synthetic data and 3D field data examples prove the validity and effectiveness of the TFPF-EWT for both attenuating random noise and preserving valid seismic amplitude.

Appendix: Empirical wavelet transform

To represent EWT, we first define a non-stationary signal $s_{n} (t)$ as

$$ s_{n} (t) = \sum\limits_{k = 1}^{K} {c_{k} (t)} + r(t), $$

(8)

where $c_{k} (t)$ and $r(t)$ are the IMFs and residual, respectively. Based on Eq. (8), we decompose the non-stationary signal $s_{n} (t)$ into a set of IMFs $c_{k} (t),k = 1,2, \ldots ,K$ and a residual $r(t)$.

EWT builds adaptive wavelets capable for extracting IMFs using the analyzed signal (Gilles 2013). Each of IMFs has a compact support spectrum. EWT aims to separate different portions of the Fourier spectrum, which corresponds to different IMFs. After segmenting the Fourier support $[0,\pi ]$ into $N$ contiguous segments, we denote $\omega_{k}$ to be the limits between each segment, where $\omega_{0} = 0$ and $\omega_{K} = \pi$. Furthermore, each segment is denoted as $\Lambda_{k} = [\omega_{k - 1} ,\omega_{k} ]$. Then, EWT is performed in the following steps easily:

• (1) We first apply the Fourier transform (FT) to $s(t)$ and obtain the Fourier spectrum. We then assume that there are maxima $\left\{ {M_{k} } \right\},k = 1,2, \ldots ,K - 1$ of the Fourier spectrum.

• (2) After obtaining the segments of the Fourier spectrum and the set of the boundaries, the boundaries $\omega_{k}$ of each segmentation are defined as

  $$ \omega_{k} = \frac{{M_{k} + M_{k + 1} }}{2},\quad k = 1,2, \ldots ,K - 1, $$

  (9)

  where $\omega_{0} = 0$ and $\omega_{K} = \pi$.

• (3) We define the empirical scaling functions $\phi_{k}$ and empirical wavelets $\psi_{k}$ in Eqs. (10) and (11):

  $$ \phi_{k} = \left\{ {\begin{array}{*{20}l} {1,} \hfill & {\omega \le (1 - \gamma )\omega_{k} ,} \hfill \\ {\cos \left[ {\frac{\pi }{2}\alpha (\gamma ,\omega_{k} )} \right],} \hfill & {(1 - \gamma )\omega_{k} < |\omega | \le (1 + \gamma )\omega_{k} ,} \hfill \\ {0,} \hfill & {{\text{otherwise}},} \hfill \\ \end{array} } \right. $$

  (10)
  $$ \psi_{k} = \left\{ {\begin{array}{*{20}l} {1,} \hfill & {(1 + \gamma )\omega_{k} \le |\omega | \le (1 - \gamma )\omega_{k + 1} ,} \hfill \\ {\cos \left[ {\frac{\pi }{2}\alpha (\gamma ,\omega_{k + 1} )} \right],} \hfill & {(1 - \gamma )\omega_{k + 1} < |\omega | \le (1 + \gamma )\omega_{k + 1} ,} \hfill \\ {\sin \left[ {\frac{\pi }{2}\alpha (\gamma ,\omega_{k} )} \right],} \hfill & {(1 - \gamma )\omega_{k} < |\omega | \le (1 + \gamma )\omega_{k} ,} \hfill \\ {0,} \hfill & {{\text{otherwise}},} \hfill \\ \end{array} } \right. $$

  (11)

  where $\alpha (\gamma ,\omega_{k} ) = \beta \left( {\frac{1}{{2\gamma \omega_{k} }}\left[ {|\omega | - (1 - \gamma )\omega_{k} } \right]} \right)$. $0 < \gamma < 1$ is a parameter for ensuring no overlap between consecutive transition phases and $\beta (x)$ is a function defined in Eq. (12):

  $$ \beta (x) = \left\{ {\begin{array}{*{20}c} {0,} & {x \le 0,} \\ {1,} & {x \ge 1,} \\ \end{array} } \right.\quad {\text{and}}\quad \beta (x) + \beta (1 - x) = 1,\quad \forall x \in [0,1]. $$

  (12)

• (4) We calculate EWT $W_{s}^{\varepsilon } (i,t)$ by the inner product of $s(t)$ with the empirical wavelets $\psi_{k}$ as

  $$ W_{s}^{\varepsilon } (n,t) = < s,\psi_{k} > = \int {s(\tau )\overline{\psi }_{k} (\tau - t)d\tau } . $$

  (13)

  In addition, we calculate the approximation coefficients $W_{s}^{\varepsilon } (0,t)$ as

  $$ W_{s}^{\varepsilon } (0,t) = < s,\phi_{1} > = \int {s(\tau )\overline{\phi }_{1} (\tau - t)d\tau } , $$

  (14)

  where $\phi_{1}$ and $\psi_{k}$ are defined in Eqs. (10) and (11).

• (5) At last, we calculate IMF $c_{k} (t)$

  $$ \begin{gathered} c_{0} (t) = W_{s}^{\varepsilon } (0,t) * \phi_{1} (t), \hfill \\ c_{k} (t) = W_{s}^{\varepsilon } (n,t) * \psi_{k} (t). \hfill \\ \end{gathered} $$

  (15)

Moreover, the original signal is reconstructed by

$$ s(t) = W_{s}^{\varepsilon } (0,t) * \phi_{1} (t) + \sum\limits_{k = 1}^{K} {W_{s}^{\varepsilon } (n,t) * \psi_{k} (t)} . $$

(16)

Using Eq. (16), EWT decomposes a non-stationary seismic signal into a couple of band-limited IMFs with different dominant frequencies. Then, we could use TFPF-based algorithms to the selected IMFs to suppress seismic noise in this study.