Systems describing signal and noise separation

  Claerbout 1995 provides a geophysical linear inverse structure of  

$$\sv 0 \approx \st W(\st L\sv m-\sv d)$$ (43)

 

$$\sv 0 \approx \epsilon \st A \sv m,$$ (44)

where $\st W$, $\st L$, and $\st A$ are linear operators, $\sv m$ and $\sv d$ correspond to a model and to the data, and $\epsilon$ is a scale factor determining the relative weights of these two systems. The first regression involves how well the model fits the data. The second regression involves limitations on what the model is expected to be. The matrix $\st A$ often enforces a smoothness on the model.

These are systems of regressions, rather than systems of equations. It is not expected that either $\st W(\st L\sv m-\sv d)$ or $\epsilon \st A \sv m$ should ever become exactly zero, but it is expected that some minimum of these expressions can be found by adjusting the model $\sv m$.Alternatively, the regressions in () and () can be expressed as the system of equations

$$\sv e_1 = \st W(\st L\sv m-\sv d)$$ (45)

  

$$\sv e_2 = \epsilon \st A \sv m,$$ (46)

where $\sv e_1$ and $\sv e_2$ are to be minimized.

Paralleling the arguments in Claerbout 1995, if $\st W$ is replaced with a filtering operator $\st S$ that annihilates any signal s on which it operates so that $\st S \sv s \approx \sv 0$,$\st L$ is replaced by the identity matrix $\st I$,$\st A$ is replaced by a filtering operator $\st N$ that annihilates any noise $\sv n$ on which it operates, so that $\st N \sv n \approx \sv 0$,and $\sv m$ is replaced with the noise $\sv n$ that is to be calculated and removed from the data $\sv d$,equations () and () become  

$$0 \approx \st S(\sv n-\sv d)$$ (47)

 

$$0 \approx \epsilon \st N \sv n.$$ (48)

An alternative method of getting equations () and () would be to start from the definitions of the signal annihilation filter and the noise annihilation filter:  

$$0 \approx \st S\sv s$$ (49)

 

$$0 \approx \st N\sv n,$$ (50)

that is, the signal annihilation filter $\st S$ applied to the signal $\sv s$produces something that is almost zero, and the noise annihilation filter $\st N$ applied to the noise $\sv n$produces something that is almost zero. Since the data $\sv d$ is defined as the sum of the signal $\sv s$ and the noise $\sv n$,that is, $\sv d=\sv s+\sv n$, $\sv s$ in equation () may be replaced with $\sv d-\sv n$.Making this substitution and allowing for a scale factor $\epsilon$once more gives equations () and ().

Combining the systems shown in () and () into a single system gives  

$$\sv 0 \approx \left( \begin{array} {cc} \st S & -\st S \\ ... ... \left( \begin{array} {c} \sv n \\ \sv d\end{array} \right).$$ (51)

Moving the expressions that depend on the data to the left-hand side and keeping those that depend on the unknown noise on the right-hand side gives  

$$\left( \begin{array} {c} \st S \sv d \\ 0 \end{array} \rig... ...rray} {c} \st S \\ \epsilon \st N \end{array} \right) \sv n.$$ (52)

This system may be solved by either minimizing the error as in Claerbout 1995 or by substituting directly into the solution for the normal equations. In either case the solution for $\sv n$ is  

$$\sv n=\left( \st S^{\dagger}\st S + \epsilon \st N^{\dagger} \st N \right) ^{-1} \st S^{\dagger} \st S \sv d.$$ (53)

Once again, $\dagger$ indicates the conjugate-transpose, or adjoint, which is simply the transpose when all values are real.

Since the signal $\sv s$ may be expressed as $\sv d-\sv n$,the solution for the signal is  

$$\sv s =\left( \st S^{\dagger}\st S + \epsilon \st N^{\dagger} \st N \right) ^{-1} \epsilon \st N^{\dagger} \st N \sv d.$$ (54)