Sample Objective Functions
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Several test
objective functions were implemented as benchmarks for the
library. They range in complexity from analytic multidimensional
quadratics to highly nonconvex, numerically determined functions,
such as the stacking power.
Some are classic test functions in optimization, such as
Rosenbrock's function, and others are taken from our current work
in seismic inversion.
Here we briefly describe each of them.

Generalized quadratic function:
This dimensional
function is useful as a benchmark since the convergence
properties of most of the algorithms is known for quadratics.

Rosenbrock's function:
Rosenbrock's function, a standard test function in optimization theory,
is
Due to a long narrow valley present in this function,
gradientbased techniques may
require
a large number of iterations before the minimizer is found.

1D seismogram inversion:
This objective function measures the rootmeansquare
(RMS) misfit between the impulse responses
(i.e., seismograms deconvolved with the (assumed) known wavelet)
of the first model supplied (taken to be the observed ``data''),
and subsequent
trial models. The mathematical expression of this function is given by:
where
is the timesampled reflectivity series,
is the observed impulse response, and
is the impulse response for the model .
By minimizing this expression the true reflection coefficient series can
be estimated.

Stacking Power:
Residual statics cause
time shifts observed in reflection seismic data
and are attributed to
localized heterogeneities in the near surface. These time shifts result
in a decrease of the signal quality and can therefore be estimated by
optimizing some measure of signal quality
such as the stacking power function.
We do a summation of
the seismic traces within all CMP records of a given seismic line
as shown below:
where is the parameter vector of source and receiver statics.
The function is extremely complicated. A
picture of a twodimensional slice through
such a beast is shown in the paper ``Global search
and genetic algorithms'' by Smith et al. in the January 1992 issue
of The Leading Edge.
A working version of the stackpower demo is provided with the COOOL
release and includes a toy data set.

Refraction Statics:
To test the linear solvers, we provide as a separate ftpable
data set, a large sparse matrix associated with
firstbreak delay times.
This matrix, generated from a synthetic data set created by
Paul Docherty, has over 60 000 equations. By taking advantage
of the sparse matrix facilities in COOOL, it is quickly
solvable on a small workstation or PC.
Next: An Example
Up: The CWP ObjectOriented Optimization
Previous: Local optimization methods
Sun Feb 25 12:08:00 MST 1996