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A bread-and-butter problem in seismology is building the velocity
as a function of depth (or vertical travel time)
starting from certain measurements.
The measurements are described in many books, for example
my book
BEI (Basic Earth Imaging).
They amount to measuring the integral of the velocity squared
from the surface down to the reflector,
known as the root-mean-square (RMS) velocity.
Although good quality echoes 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 travel-time depth:
(1) the integrated (from the surface) squared velocity, and
(2) a measure of the quality of the integrated velocity measurement.
Needed definitions are as follows:
-
- is a data vector in which components range over the vertical
traveltime depth
.
Its 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 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 containing the interval velocity squared
ranging over the vertical traveltime depth
.
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:
|
(31) |
In other words, any component of
measures
the integral of a material property from the Earth surface to the depth of
.
We wish to find the material property everywhere, which is
.
If we integrate it from the surface downward with causal integration
,
we should get the measurements
.
With imperfect data, our data fitting goal is to minimize the residual:
|
(32) |
where
is some weighting function,
we need to choose.
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
.
|
(33) |
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
.
Subsections
Next: Balancing good data with
Up: Preconditioning
Previous: Need for an invertible
2015-05-07