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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)

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

2013-07-26