Denoising by spectral decomposition

The non-stationary decomposition model for a complex signal $d(t)$ is:

$\displaystyle d(t)=\sum_{n=1}^{N}d_n(t)=\sum_{n=1}^{N}\hat{A}_n(t)e^{i\Phi_n(t)}+r(t),$ (3)

where $d_n(t)$, $\hat{A}_n(t)$, $\Phi_n(t)$, $r(t)$ stand for decomposed signal, local amplitude, local phase and residual respectively. $\hat{A}_n(t)$ can be found by following equation 4 using regularized non-stationary regression (RNR):

$\displaystyle \min\parallel d(t)-\sum_{n=1}^{N}\hat{A}_n(t)e^{i\Phi_n(t)}\parallel_2^2,$ (4)

and $\Phi_n(t)$ can be found by time integration of the instantaneous frequency $f_n(t)$ following equation 5:

$\displaystyle \Phi_n(t)=2\pi\int_0^t f_n(\tau)d\tau.$ (5)

The instantaneous frequency $f_n(t)$ in equation 5 can be determined directly from the phase of different complex roots $\hat Z_n(t)$ of the polynomial function (shown in equation 7) following equation 6:

$\displaystyle f_n(t)=-Re\left[ arg\left(\frac{\hat{Z}_n(t)}{2\pi\Delta t}\right)\right],$ (6)

$\displaystyle F(Z)=(1-Z/Z_1)(1-Z/Z_2)\cdots(1-Z/Z_N)$     (7)
$\displaystyle \quad=1+a_1Z+a_2Z^2+\cdots+a_nZ^N.$      

In non-stationary case, the filter coefficients $a_n$ becomes a smoothly varying functions of time $a_n(t)$ (shown in equation 2), which makes the filter shown in equation 7 adapt to non-stationary changes in the input data. The complex roots $\hat{Z}_n(t)$ can be found using a eigenvalue-based algorithm (Fomel, 2013).

The final denoised data can be got by:

$\displaystyle \hat{d}(t)=\sum_{n=1}^{N}\hat{A}_n(t)e^{i\Phi_n(t)}.$ (8)

The decomposition is similar to empirical mode decomposition (EMD), but differs in that it has a mathematical formulation for controlling the decomposition. Figure 1 gives a comparison between SDRNAR and EMD in decomposing a synthetic signal that has two oscillating frequency components. As can be seen from the demonstration, both SDRNAR and EMD successfully decompose the combined signal into individual monotonic component. The residuals using both methods are very close to zero. Figure 2 shows a comparison between SDRNAR and EMD in decomposing a noise free synthetic seismic trace.

A special property of the proposed approach is that, apart from the denoised data, we can also get other valuable information from the data, such as the instantaneous frequency $f_n(t)$ and amplitude of the instantaneous frequency $\hat{A}_n(t)$ of the $n$th component. The spectral information can be valuable in interpretation jobs like identifying the oil&gas traps, which has already been shown by Fomel (2013).

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Figure 1.
Signal Separation using SDRNAR and EMD. (a) Original signal. (b) Frequency component 1 using SDRNAR. (c) Frequency component 1 using EMD. (d) Frequency component 2 using SDRNAR. (d) Frequency component 2 using EMD. (f) Residual using SDRNAR. (g) Residual using EMD.
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Figure 2.
Trace decomposition using SDRNAR and EMD. (a) Original noise free seismic trace. (b) First component using SDRNAR. (c) First component using EMD. (d) Second component using SDRNAR. (e) Second component using EMD. (f) Third component using SDRNAR.
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