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.
