Subtraction method

In contrast to the Wiener-like method which filters the noise, the following method aims to model the noise and then subtract it from the input data. In this section, we show that the formalism used by Nemeth (1996) can help to better separate correlated noise and signal. But first, we detail the similarities and differences between the Wiener-like and the subtraction method.

In equation (2), the noise and signal PEFs filter the data components. Alternatively, building on Nemeth (1996), the noise and signal nonstationary PEF can predict the data components via a deconvolution as follows:  

$${\bf d} = {\bf N^{-1}m_n} + {\bf S^{-1}m_s}.$$ (4)

We call ${\bf m_s}$ the signal model component and ${\bf m_n}$ the noise model component (not to be confused with the noise model that we use to compute the noise PEF). Clearly, ${\bf N^{-1}m_n}$ models the noise vector n and ${\bf S^{-1}m_s}$ the signal vector s. Because we use PEFs in equation (4), this approach is pattern-based in essence. With ${\bf L_n}={\bf N^{-1}}$ and ${\bf L_s}={\bf S^{-1}}$,using linear algebra, we can prove that the least-squares solution of ${\bf m_s}$ and ${\bf m_n}$ is  

$$\left( \begin{array} {c} \hat{{\bf m_n}} \\ \hat{{\bf m_s... ..._n}L_s})^{-1}{\bf L_s'\overline{R_n}}\end{array}\right){\bf d},$$ (5)

with

$$\begin{eqnarray} {\bf \overline{R_s}}&=&{\bf I}-{\bf L_s}({\bf L_s'L_s})^{-1}{\b... ...line{{\bf R_n}}&=&{\bf I}-{\bf L_n}({\bf L_n'L_n})^{-1}{\bf L_n'}.\end{eqnarray}$$ (6)

The operators ${\bf \overline{R_s}}$ and ${\bf \overline{R_n}}$ can be seen as signal and noise filters respectively since ${\bf L_s}({\bf L_s'L_s})^{-1}{\bf L_s'}$ and ${\bf L_n}({\bf L_n'L_n})^{-1}{\bf L_n'}$ are the data resolution operators for the signal and the noise, respectively. In the appendix B, we give a geometrical interpretation for both ${\bf \overline{R_s}}$ and ${\bf \overline{R_n}}$.

The degree of orthogonality between the noise operator ${\bf L_n}$ and the signal operator ${\bf L_s}$ restricts the existence of $\hat{{\bf m_n}}$ and $\hat{{\bf m_s}}$ in equation (5). If the two operators overlap completely, the Hessians ${\bf L_n'\overline{R_s}L_n}$ and ${\bf L_s'\overline{R_n}L_s}$ are not invertible. If the two operators overlap only partially, Nemeth (1996) proves that the separability of the signal and noise can be improved if we introduce a regularization term. If we use a model space regularization Fomel (1997), we have  

$$\left( \begin{array} {c} \hat{{\bf m_n}} \\ \hat{{\bf m_s... ...'C_s})^{-1} {\bf L_s'\overline{R_n}}\end{array}\right){\bf d},$$ (7)

with ${\bf C_n}$ and ${\bf C_s}$ the regularization operators for the noise model ${\bf m_n}$ and the signal model ${\bf m_s}$.A data space regularization can also improve the separation but will not be considered here.

In equation (4), the outcome of the inversion is ${\bf m_n}$ and ${\bf m_s}$. The estimated signal ${\bf \tilde{s}}$ is then easily derived as follows:

$${\bf \tilde{s}} = {\bf d}-{\bf N^{-1}m_n}.$$ (8)

We call this new method the subtraction method. In the next two sections, we compare the Wiener-like approach and the subtraction method for ground-roll and multiple attenuation.