Jacob Roginsky
Acoust. Div., Naval Res. Lab., Overlook Ave., S.W., Washington, DC
G. W. Stewart
Univ. of Maryland, College Park, MD
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.