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A breadandbutter problem in seismology is building the velocity
as a function of depth (or vertical travel time)
starting from certain measurements.
The measurements are described elsewhere (BEI for example).
They amount to measuring the integral of the velocity squared
from the surface down to the reflector.
It is known as the RMS (rootmeansquare) velocity.
Although good quality echos may arrive often,
they rarely arrive continuously for all depths.
Good information is interspersed unpredictably with poor information.
Luckily we can
also estimate
the data quality by the ``coherency'' or the
``stack energy''.
In summary, what we get from observations and preprocessing
are two functions of traveltime depth,
(1) the integrated (from the surface) squared velocity, and
(2) a measure of the quality of the integrated velocity measurement.
Some definitions:

 is a data vector whose components range over the vertical
traveltime depth ,and whose component values contain the scaled RMS velocity squared
where
is the index on the time axis.

 is a diagonal matrix along which we lay the given measure
of data quality. We will use it as a weighting function.

 is the matrix of causal integration, a lower triangular matrix of ones.

 is the matrix of causal differentiation, namely, .

 is a vector whose components range over the vertical
traveltime depth ,and whose component values contain the interval velocity squared
.
From these definitions,
under the assumption of a stratified earth with horizontal reflectors
(and no multiple reflections)
the theoretical (squared) interval velocities
enable us to define the theoretical (squared) RMS velocities by
 
(15) 
With imperfect data, our data fitting goal is to minimize the residual
 
(16) 
To find the interval velocity
where there is no data (where the stack power theoretically vanishes)
we have the ``model damping'' goal to minimize
the wiggliness of the squared interval velocity .
 
(17) 
We precondition these two goals
by changing the optimization variable from
interval velocity squared
to its wiggliness .Substituting gives the two goals
expressed as a function of wiggliness .
 
(18) 
 (19) 