### ASA 130th Meeting - St. Louis, MO - 1995 Nov 27 .. Dec 01

## 4pUW5. Applications of optimized rational approximations to parabolic
equation modeling.

**R. J. Cederberg
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Michael D. Collins
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*Naval Res. Lab., Washington, DC 20375
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**William L. Siegmann
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*Rensselaer Polytechnic Inst., Troy, NY 12180
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Rational approximations designed using least-squares constraints are easy
to obtain and have useful applications. The coefficients are defined in terms of
accuracy constraints on the propagating part of the spectrum and stability
constraints on the rest of the spectrum. Among the applications are a filtering
operator as well as improved approximations for the split-step Pade solution [J.
Acoust. Soc. Am. 93, 1736--1742 (1993)] and radiation boundary conditions for
outgoing wave equations [Clayton and Engquist, Geophysics 45, 895--904 (1980)].
An efficient operator filter is obtained by designing a rational approximation
that is the identity function (or a weighted function) over the propagating (or
desired) spectrum and decays to zero elsewhere. Operator approximations have
previously been applied to replace the operator in the interior of the domain
(the parabolic wave equation), generate radiation boundary conditions, solve
scattering problems, and generate initial conditions. With the improved
approximations, it is possible to use range steps of more than a hundred
wavelengths with the split-step Pade solution. By enforcing both accuracy and
stability constraints, it should be possible to overcome stability problems
[Howell and Trefethen, Geophysics 53, 593--603 (1988)] and obtain useful
higher-order radiation boundary conditions for outgoing wave equations.