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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

$\displaystyle \bold 0$ $\displaystyle \approx$ $\displaystyle \bold W ( \bold L \bold m - \bold d)$ (7)
$\displaystyle \bold 0$ $\displaystyle \approx$ $\displaystyle \epsilon \bold A \bold m$ (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:
0 $\displaystyle \approx$ $\displaystyle \bold N (-\bold n) = \bold N ( \bold s - \bold d)$ (9)
0 $\displaystyle \approx$ $\displaystyle \epsilon \bold S \bold s$ (10)


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Next: SIGNAL-NOISE DECOMPOSITION BY DIP Up: Nonstationarity: patching Previous: Which coefficients are really

2013-07-26