** Next:** A self-fulfilling prophecy?
** Up:** HOW TO DIVIDE NOISY
** Previous:** Deconvolution with an unknown

In all the signal analysis above there was no explicit model
for **noise**,
but implicitly the idea of noise was there.
Now we will recognize it and solve explicitly for it.
This leads to what is called ``**linear-estimation** theory."
Instead of simply *Y*=*FX*, we add noise into the defining equation:
| |
(19) |

To proceed we need to define the ``**variance**"
(described more fully in chapter ) as
| |
(20) |

and likewise the noise variance .
The general linear-estimation method minimizes something
that looks like a sum of **relative error**s:

| |
(21) |

Notice that the variances put both terms of the sum
into the same physical units.
I have not derived equation (21) but stated it as reasonable:
from it we will derive reasonable answers
which we have already seen.
The rationale for the minimization of (21)
is that we want the noise to be small,
but because we must guard against zero division in *X*=*Y*/*F*,
we ask for *X* to be small too.
Actually, by introducing equation (19),
we have abandoned the model *X*=*Y*/*F* and replaced it
with the model *X*=(*Y*-*N*)/*F*.
Thus, instead of thinking of falsifying *F*
to avoid dividing by zero in *X*=*Y*/*F*,
we now think of finding *N* so the numerator in (*Y*-*N*)/*X* vanishes
wherever the denominator does.
By introducing
(19) into
(21) we can eliminate either *N* or *X*.
Eliminating *N*,
we have

| |
(22) |

Minimizing *Q*(*X*) by setting its derivative by to zero gives
| |
(23) |

| (24) |

Equation (24)
is the same as equation (12),
except that it gives us a numerical interpretation
of the value of in equation (12).
We can find an explicit equation for the noise
in terms of the data and filter
by substituting
equation (24) into
equation (19)
and solving for *N*.

** Next:** A self-fulfilling prophecy?
** Up:** HOW TO DIVIDE NOISY
** Previous:** Deconvolution with an unknown
Stanford Exploration Project

10/21/1998