Nonstationary pattern-based signal-noise separation using adaptive prediction-error filter

Signal and noise separation problem

A simple assumption is that the dataset $\mathbf{d}$ can be considered as the summation of signal $\mathbf{s}$ and noise $\mathbf{n}$ :

$$\mathbf{s} + \mathbf{n} = \mathbf{d} .$$ (1)

Usually, signal $\mathbf{s}$ represents reflection events or geologic events, and noise $\mathbf{n}$ can be regarded as random noise or ground-roll noise. Let operators $\mathbf{S}$ and $\mathbf{N}$ denote the patterns of signal and noise, respectively. PEF is reasonable to be the pattern operator as it approximates the inverse energy spectrum of the corresponding component. Claerbout (2010) described the pattern operator as the absorbing operator, for instance, operator $\mathbf{N}$ may absorb noise $\mathbf{n}$ ( $\mathbf{0} \approx \mathbf{Nn}$ ). And it can destroy the corresponding noise component from data $\mathbf{d}$ and leave signal $\mathbf{s}$ :

$$\begin{aligned}\mathbf{0} & = \mathbf{N} ( \mathbf{s} + \mathbf... ...thbf{Nd} \\ & \approx \mathbf{Ns} - \mathbf{Nd}\; . \end{aligned}$$

Meanwhile, operator $\mathbf{S}$ absorbs or destroys signal component $\mathbf{s}$ :

$$\mathbf{0} \approx \mathbf{Ss} \; ,$$ (3)

which is used to restrict the shape of signal $\mathbf{s}$ . By using above relationships, the pattern-based method raises a constrained least-squares problem, and solving such a problem can separate signal $\mathbf{s}$ and noise $\mathbf{n}$ from data volume $\mathbf{d}$ .

$$\min_{\mathbf{s}} \Vert \; \mathbf{Ns} - \mathbf{Nd} \; \Vert _{2}^{2} + \epsilon^{2} \; \Vert \; \mathbf{Ss} \; \Vert _{2}^{2} \; ,$$ (4)

where $\epsilon > 0$ is the scaling factor of regularization term, which balancing the energy between estimated signal $\mathbf{\bar{s}}$ and noise $\mathbf{\bar{n}}$ .

In practice, field data d and the noise model are often available, but the clean signal is not. We here considered the noise model to be a dataset containing the properties of noise $\mathbf{n}$ , and the noise model can be obtained as a roughly separated noise section. Therefore, data pattern $\mathbf{D}$ and noise pattern $\mathbf{N}$ are easily estimated from the field data and noise model, while it cannot directly obtain signal pattern $\mathbf{S}$ . Normally, one makes a compromise by replacing $\mathbf{S}$ with $\mathbf{D}$ : with $\mathbf{D}$ :

$$\min_{\mathbf{s}} \Vert \; \mathbf{Ns} - \mathbf{Nd} \; \Vert _{2}^{2} + \epsilon^{2} \; \Vert \; \mathbf{Ds} \; \Vert _{2}^{2} \; ,$$ (5)

where $\mathbf{D}$ may be unsuitable for constraining the shape of signal $\mathbf{s}$ , and finally lead to undesirable separation result. Assuming that noise $\mathbf{n}$ and signal $\mathbf{s}$ are uncorrelated, Spitz (1999) proposed an approximation $\mathbf{S} = \mathbf{N}^{-1}\mathbf{D}$ , which is based on the relationship between the pattern operator and the energy spectrum of the corresponding component, and the regularization term becomes $\Vert\mathbf{N}^{-1}\mathbf{Ds}\Vert _{2}^{2}$ . Furthermore, one can multiply equation 5 by $\mathbf{N}$ to avoid the acquisition of $\mathbf{N}^{-1}$ :

$$\min_{\mathbf{s}} \Vert \; \mathbf{N} (\mathbf{Ns} - \mathbf{Nd})... ...on^{2} \; \Vert \; \mathbf{N} (\mathbf{Ds} / \mathbf{N}) \; \Vert _{2}^{2} \; ,$$ (6)

then the signal-noise separation problem turns into:

$$\min_{\mathbf{s}} \Vert \; \mathbf{NNs} - \mathbf{NNd} \; \Vert _{2}^{2} + \epsilon^{2} \; \Vert \; \mathbf{Ds} \; \Vert _{2}^{2} \; .$$ (7)

Equation 7 avoids the requirement for pattern operator $\mathbf{S}$ of the clean signal, and minimizing the equation leads to the expression of the estimated signal:

$$\mathbf{\bar{s}} = \left( \frac{\mathbf{N}^T \mathbf{N} \mathbf{N... ...bf{N}^T \mathbf{N} + \epsilon^2 \mathbf{D}^T \mathbf{D}} \right) \mathbf{d} \;,$$ (8)

and the formal solution of the estimated noise is:

$$\mathbf{\bar{n}} = \left( \frac{\epsilon^2 \mathbf{D}^T \mathbf{D... ...hbf{N}^T \mathbf{N} +\epsilon^2 \mathbf{D}^T \mathbf{D}} \right) \mathbf{d} \;.$$ (9)

The conjugate gradient algorithm is implemented to calculate the numerical solution of equations 8 and 9 (Appendix section). We will discuss the estimation of the nonstationary pattern operators in the next section.

Nonstationary pattern-based signal-noise separation using adaptive prediction-error filter