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Since signal and noise are uncorrelated,
the spectrum of data is the spectrum of the signal plus that of the noise.
An equation for this idea is

| |
(15) |

This says resonances in the signal
and resonances in the noise
will both be found in the data.
When we are given and it seems a simple
matter to subtract to get .Actually it can be very tricky.
We are never given and ;we must estimate them.
Further, they can be a function of frequency, wave number, or dip,
and these can be changing during measurements.
We could easily find ourselves with a negative estimate for
which would ruin any attempt to segregate signal from noise.
An idea of Simon Spitz can help here.
Let us reexpress equation (15) with prediction-error filters.

| |
(16) |

Inverting
| |
(17) |

The essential feature of a PEF is its zeros.
Where a PEF approaches zero, its inverse is large and resonating.
When we are concerned with the zeros of a mathematical function
we tend to focus on numerators and ignore denominators.
The zeros in
compound with the zeros in
to make the zeros in
.This motivates the ``Spitz Approximation.''
| |
(18) |

It usually happens that we can
find a patch of data where no signal is present.
That's a good place to estimate the noise PEF *A*_{n}.
It is usually much harder to find a patch of data where no noise is present.
This motivates the Spitz approximation which by saying
*A*_{d} = *A*_{s} *A*_{n}
tells us that the hard-to-estimate *A*_{s} is the ratio
*A*_{s} = *A*_{d} / *A*_{n}
of two easy-to-estimate PEFs.

It would be computationally convenient if
we had *A*_{s} expressed not as a ratio.
For this, form the signal
by applying the noise PEF *A*_{n} to the data .The spectral relation is

| |
(19) |

Inverting this expression
and using the Spitz approximation
we see that
a PEF estimate on is the required *A*_{s} in numerator form because
| |
(20) |

** Next:** Noise removal on Shearer's
** Up:** SIGNAL-NOISE DECOMPOSITION BY DIP
** Previous:** Signal/noise decomposition examples
Stanford Exploration Project

4/27/2004