### ASA 129th Meeting - Washington, DC - 1995 May 30 .. Jun 06

## 2aUW10. Fast inversion of the ill-conditioned shallow water propagation
operator for pulsed signals.

**Jacob Roginsky
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*Acoust. Div., Naval Res. Lab., Overlook Ave., S.W., Washington, DC
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**G. W. Stewart
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*Univ. of Maryland, College Park, MD
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Time series deconvolution in underwater acoustics is a difficult problem
due to the practically singular nature of the propagation operator. The
situation is even more complicated in the presence of a multipath environment.
In particular, when the time window is shorter than the propagation time, the
process requires the solution of a large triangular Toeplitz system that is
quite ill conditioned. Satisfactory results can be obtained by truncating the
singular value decomposition (SVD), but only at the cost of ignoring the
Toeplitz structure. The object of this work was to investigate the alternative
of using Tichonov--Phillips regularization to lessen the effects of the ill
conditioning. An algorithm of Lars Elden [SIAM J. Sci. Stat. Comput. 5,
229--236 (1984)] that takes advantage of the Toeplitz structure can be adapted
to this purpose. The results are as good as those obtained from the SVD, but
the cost is O(n[sup 2]) in the matrix size as opposed to O(n[sup 3]) for the
SVD. For the problems considered here, this amounts to a speedup of two orders
of magnitude.