INVERSION AND NOISE REMOVAL

Here we relate the basic theoretical statement of geophysical inverse theory to the basic theoretical statement of separation of signals from noises.

A common form of linearized geophysical inverse theory is inverse theory

$$\begin{eqnarray} \bold 0 & \approx & \bold W ( \bold L \bold m - \bold d) \\ \bold 0 & \approx & \epsilon \bold A \bold m\end{eqnarray}$$ (7) (8)

We choose the operator $\bold L = \bold I$ to be an identity and we rename the model $\bold m$ to be signal $\bold s$.Define noise by the decomposition of data into signal plus noise, so $\bold n = \bold d-\bold s$.Finally, let us rename the weighting (and filtering) operations $\bold W=\bold N$ on the noise and $\bold A=\bold S$ on the signal. Thus the usual model fitting becomes a fitting for signal-noise separation:

$$\begin{eqnarray} 0 & \approx & \bold N (-\bold n) = \bold N ( \bold s - \bold d) \\ 0 & \approx & \epsilon \bold S \bold s\end{eqnarray}$$ (9) (10)