3.4.1 Formulation

In this section we continue to restrict our attention to constant depth oceans with completely reflecting seabottoms. However, in the present example we consider the case where there is a seamount on the ocean floor. We wish to reconstruct the seamount using far-field data. To this end we generate an acoustic field using a point source at a given location, say . The acoustic pressure then satisfies

and the outgoing radiation condition. Here we assume that , and k ² is not an eigenvalue of the exterior boundary problem.

D represents the seamount, and is the surface of the seamount, which has a parameterization

here a is some positive constant where we assume that the seabottom is flat for r > a.

For a constant depth ocean without a seamount, the solution to (3.95) (3.98) is the Helmholte-Green function in , which has the form

where the g(z) are the point sources

The solution of problem (3.95) (3.98) can be represented as

for ; here p _sc( ) is the unique solution of the integral equation

and

The inverse problem is the following: Given p( ) for all ? ?, := {( r, ?, z) : z = d = constant} and ₀, determine the seamount M ^[8].

3.4.2 Uniqueness of the Seamount Problem

We assume that both T ₁ (the receiving plane) and T ₂ (the source location plane)...