Assumptions and definitions

In the following discussion, three assumptions are made to separate signal and noise from data. First, the data is defined to be a simple sum of the signal and noise; that is, $\sv d=\sv s+\sv n$, $\sv d$ being the observed data, $\sv s$ the signal, and $\sv n$ the noise. Next, there exists a filter $\st S$ that predicts the signal, $\st S \sv s \approx \sv 0$.Finally, there exists a filter $\st N$ that predicts the noise, $\st N \sv n \approx \sv 0$.The methods of getting $\st S$ and $\st N$ will be covered later.

The assumed noise filter $\st N$ requires a change in the definition of the noise from the previous chapters, where unpredictable noise was separated from a predictable signal. Although it will be shown later that unpredictable noise may be removed with the techniques to be discussed here, more emphasis is given now to coherent noise.

Three conditions are expected to be met by the final solution for the signal and noise:  

$$\sv d = \sv s + \sv n$$ (85)

 

$$\st S \sv s \approx \sv 0$$ (86)

 

$$\st N \sv n \approx \sv 0.$$ (87)

Equation () is just a definition of how signal and noise combine make the data. Equations () and () characterize the expected properties of the signal and noise. These might be considered more as levelers than as equations, since the result of either $\st S\sv s$ or $\st N\sv n$ is very unlikely to be zero. The final solution for the signal $\sv s$ and the noise $\sv n$ is expected to minimize, in the least-squares sense, both $\st S\sv s$ and $\st N\sv n$.

Using equations () to (), two systems of regressions may be generated, one to calculate the noise and one to calculate the signal. To calculate the signal, $\sv n$ replaces $\sv d-\sv s$ in equation (), which is then combined with equation () to give a single system of regressions  

$$0 \approx \left( \begin{array} {c} \st S \\ \st N\end{arr... ...\left( \begin{array} {c} \sv 0 \\ \st N\sv d\end{array}\right).$$ (88)

A similar manipulation produces a calculation of the noise  

$$\sv 0 \approx \left( \begin{array} {c} \st S \\ \st N\end{ar... ...\left( \begin{array} {c} \st S \sv d \\ \sv 0\end{array}\right)$$ (89)

Once the signal is calculated, the noise is simply $\sv n =\sv d -\sv s$.If the noise is calculated, the signal is $\sv s=\sv d-\sv n$.