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Subtracting coherent noise

In contrast to the filtering method which attenuates the coherent noise by approximating the noise covariance matrix with a PEF, the following method aims to model the coherent noise and then subtract it from the input data. This method is based on the addition of a coherent noise modeling operator to the fitting goal Nemeth (1996) as follows:
\begin{displaymath}
{\bf 0} \approx {\bf Hm_s}+{\bf A_n^{-1}m_n}-{\bf d},\end{displaymath} (3)
where H is the signal modeling operator, ${\bf m_s}$ denotes the model for the signal, ${\bf A_n}$ is a PEF that models the coherent noise present in the data d, and ${\bf m_n}$denotes the model for the noise. This method is based on the central assumption that the data d result from the addition of noise n and signal s. Thus, ideally, each operator H and ${\bf A_n}$ models a different part of the data. The PEF ${\bf A_n}$ and the operator H have different physical units. To balance the two operators, I use the fitting goal  
 \begin{displaymath}
{\bf 0} \approx {\bf Hm_s}+\gamma{\bf A_n^{-1}m_n}-{\bf d},\end{displaymath} (4)
where
\begin{displaymath}
\gamma = \frac{\Vert{\bf H'd}\Vert _2}{\Vert{\bf A_n^{-1'}d}\Vert _2}.\end{displaymath} (5)
The constant ${\bf \gamma}$ is computed at the first iteration and kept constant. If I set ${\bf B}=\gamma{\bf A_n^{-1}}$, I can write the fitting goal in the following form:
\begin{displaymath}
{\bf 0} \approx 
\left( \begin{array}
{cc} 
 {\bf H} & {\bf ...
 ...ray}
{c} 
 {\bf m_s} \\  {\bf m_n}\end{array}\right) - {\bf d}.\end{displaymath} (6)
With ${\bf L}=({\bf H} \;\; {\bf B})$ and ${\bf m'}=({\bf m_s} \;\; {\bf
m_n})$, the fitting goal becomes
\begin{displaymath}
{\bf 0} \approx {\bf Lm}-{\bf d},\end{displaymath} (7)
which leads to the familiar normal equations
\begin{displaymath}
{\bf L'Lm} = {\bf L'd}\end{displaymath} (8)
that can be written as  
 \begin{displaymath}
\left( \begin{array}
{cc} 
 {\bf H'H} & {\bf H'B} \\  {\bf B...
 ...in{array}
{c} 
 {\bf H'} \\  {\bf B'}\end{array}\right){\bf d}.\end{displaymath} (9)
Based on the results derived in the appendix (equation (34)), the least-squares inverse of m is  
 \begin{displaymath}
\left( \begin{array}
{c} 
 \hat{{\bf m_s}} \\  \hat{{\bf m_n...
 ...ne{R_s}B})^{-1}{\bf B'\overline{R_s}}\end{array}\right){\bf d},\end{displaymath} (10)
where $\overline{{\bf R_s}} = {\bf I}-{\bf H}({\bf H'H})^{-1}
{\bf H'}$ and $\overline{{\bf R_n}} = {\bf I}-{\bf B}({\bf B'B})^{-1}
{\bf B'}$.
${\bf H}({\bf H'H})^{-1}{\bf H'}$ is the signal resolution matrix and ${\bf B}({\bf B'B})^{-1}{\bf B'}$ is the noise resolution matrix. With ${\bf d} = {\bf s}+{\bf n}$, we have
\begin{eqnarray}
\overline{{\bf R_s}}{\bf d} &\approx& \overline{{\bf R_s}}{\bf ...
 ...\overline{{\bf R_n}}{\bf d} &\approx& \overline{{\bf R_n}}{\bf s}.\end{eqnarray} (11)
(12)
The operators $\overline{{\bf R_s}}$ and $\overline{{\bf R_n}}$perform signal and noise filtering, respectively. These operators are also called projectors and discussed in detail by Guitton et al. (2001) in this report. We can see that equation (10) for the signal model ${\bf m_s}$ resembles equation (2), except for the filtering operator $\overline{{\bf R_n}}$. In other words, ${\bf A_n'A_n}$ in equation (2) plays the role of $\overline{{\bf R_n}}$ in equation (10). The correct separation of the signal s and noise n depends on the invertibility of the Hessians ${\bf H'\overline{R_n}H}$ and ${\bf
B'\overline{R_s}B}$ in equation (10) Nemeth (1996).

The signal/noise separation is perfect if the signal and noise operators predict distinct parts of the data. The separation becomes more difficult if the two operators overlap. However, Nemeth (1996) has proven that the Hessians can be inverted if we introduce some regularization in the fitting goal. Hence, the noise prediction can proceed even if signal and noise are correlated. For instance, if we use a model space regularization Fomel (1997), we can consider
\begin{eqnarray}
{\bf 0} &\approx& {\bf Lm}-{\bf d}, \\  {\bf 0} &\approx& \epsilon {\bf Cm},\end{eqnarray} (13)
(14)
where
\begin{displaymath}
{\bf C}=
\left(\begin{array}
{cc}
 {\bf C_s} & {\bf 0} \\  {\bf 0} & {\bf C_n}
 \end{array}\right)\end{displaymath} (15)
is the covariance operator for the model m. It is important to keep in mind that $\epsilon$ does not have to be the same for the noise model covariance and the signal model covariance. As proven in the appendix (equation (39)), for $\epsilon$ constant, the least-squares solution of the regularized problem is  
 \begin{displaymath}
\left( \begin{array}
{c} 
 \hat{{\bf m_s}} \\  \hat{{\bf m_n...
 ... C_n'C_n})^{-1}{\bf B'\overline{R_s}}\end{array}\right){\bf d}.\end{displaymath} (16)
The regularization operator can also include cross-terms accounting for the correlation between signal and noise, as follows:
\begin{displaymath}
{\bf C}=
\left(\begin{array}
{cc}
 {\bf C_s} & {\bf C_{sn}} \\  {\bf C_{sn}} & {\bf C_n}
 \end{array}\right).\end{displaymath} (17)
Although I have not explored this possibility, the regularization clearly offers the possibility to better separate noise and signal when the two components are correlated.

With respect to the filtering method, I concluded that a noise model is not mandatory as long as we can derive it iteratively using a two-stage process. Accordingly, I used the following algorithm to accomplish the separation with the subtraction method:

1.
Solve ${\bf 0}\approx{\bf Hm_s} - {\bf d}$.
2.
Estimate a PEF ${\bf A_n}$ from the residual.
3.
Restart a new inverse problem for the fitting goal in equation (4).
4.
Stop when the residual has a white spectrum.
With the data in my study, I did not reestimate the PEF iteratively because I noticed that the first filter was accurate enough to predict the noise. However, for more complicated noisy events, an iterative scheme might be preferable. In the next section, I use the subtraction scheme to separate coherent noise from synthetic and real data. So far, no regularization is included in the inversion.



 
next up previous print clean
Next: Subtracting the coherent noise Up: Guitton: Coherent noise attenuation Previous: Discussion of the filtering
Stanford Exploration Project
4/29/2001