Exact reconstruction of ocean bottom velocity profiles from monochromatic scattering data

Abstract

This thesis presents the theoretical and computational underpinnings of a novel approach to the determination of the acoustic parameters of the ocean bottom using a monochromatic source. The problem is shown to be equivalent to that of the reconstruction of the potential in a Schrodinger equation from the knowledge of the plane-wave reflection coefficient as a function of vertical wavenumber, r(kz) for all real positive k z. First, the reflection coefficient is shown to decay asymptotically at least as fast as (1/kz2) for large kz and is therefore inteqrable. The Gelfand-Levitan inversion procedure is extended to include the case of basement velocity higher than the velocity of sound in water. The neglect of bound states is shown to be justified in both clayey silt and silty clay at the 220 Hz frequency of operation. Three methods for the numerical solution of the integral equation are investigated. The first one is an "Improved Born approximation" wherein the solution is given as a series expansion the first term of which is the Born approximation while the second term represents a substantial and yet easy to implement improvement over Born. The two other methods are based on a discretization of the Gelfand-Levitan integral equation, and both avoid a matrix inversion: one by employing a recursive procedure, and the other by coupling the Gelfand-Levitan equation with a partial differential equation. Bounds are obtained on errors in the solution due either to discretization or to data inaccuracy. These methods are tested on synthetic data obtained from known geoacoustic models of the ocean bottom. Results are found to be very accurate particularly at the top of the sediment layer with resolution of less than the wavelength of the acoustic source in the water. Several effects are investigated, such as sampling, attenuation, and noise. Also examined is the gradual restriction of the reflection coefficient to a finite range of vertical wave numbers and the consequent progressive deterioration of the reconstruction. The analysis shows how to reconstruct velocity profiles in the presence of density variation when the experiment is conducted at two frequencies. Our results provide a good understanding of the issues involved in conducting a monochromatic deep ocean bottom experiment and constitute a promising technique for processing the experimental data when it becomes available.