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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

\begin{eqnarray}
\bold 0 & \approx & \bold W ( \bold L \bold m - \bold d) \\ \bold 0 & \approx & \epsilon \bold A \bold m\end{eqnarray} (9)
(10)
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 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} (11)
(12)


next up previous print clean
Next: SIGNAL-NOISE DECOMPOSITION BY DIP Up: Nonstationarity: patching Previous: Parameters for signal enhancement
Stanford Exploration Project
2/27/1998