Date of Award

Spring 2014

Document Type


Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)


Mathematical Sciences

First Advisor

Eliza Zoi-Heleni Michalopoulou

Second Advisor

Sunil Kumar Dhar

Third Advisor

Hongya Ge

Fourth Advisor

David James Horntrop

Fifth Advisor

Jonathan H.C. Luke


This dissertation presents theoretical and computational approaches for estimating sound-speed in the ocean under different conditions. The first part of the dissertation discusses a fast approach for solving the inverse problem of estimating sediment sound-speed based on the Deift-Trubowitz trace formula. Under certain assumptions, this algorithm can recover the sound speed profile in the seabed using pressure field measurements in the water column at low frequencies. The inversion algorithm requires solving a non-linear integral equation. In the past, Stickler and Zhou employed a first order Born approximation for solving the integral equation. This work introduces two new methods. The first is a modified Born approximation that improves upon a standard first order approximation, yet it is easy to implement; the second one is an approximation based on interpolating the integrand. It is shown that these methods work well with synthetic data in the numerical simulations. Results are compared to those of previously developed methods and demonstrate improvement especially at sharp changes in sound speed. Although the methods are stable and effective with noise-free data, problems arise when noise is considered. Regularization methods are developed to remedy this problem. Finally, we recognize that some assumptions necessary for this algorithm to work may not be realistic; several possibilities are presented to relax these limitations.

In the second part, a method is developed for the estimation of source location and sound speed in the water column relying on linearization. The Jacobian matrix, necessary for the proposed linearization approach, includes derivatives with respect to Empirical Orthogonal Function coefficients instead of sound speed directly. First, the inversion technique is tested on synthetic arrival times, using Gaussian distributions for the errors in the considered arrival times. The approach is efficient, requiring a few iterations, and produces accurate results. Probability densities of the estimates are calculated for different levels of noise in the arrival times. Subsequently, particle filtering is employed for the estimation of arrival times from signals recorded during the Shallow Water 06 experiment. It has been shown in the past that particle filtering can be employed for the successful estimation of multipath arrival times from short-range data and, consequently, in geometry, bathymetry, and sound speed inversion. Here, probability density functions of arrival times computed via particle filtering are propagated backwards through the proposed inversion process. Inversion estimates are consistent with values reported in the literature for the same quantities. Lastly, it is shown that the results and estimates from fast simulated annealing applied to the same arrival times are very similar.

Included in

Mathematics Commons