Analytic derivation of adjoint nonlocal boundary conditions for a stratified ocean bottom in parabolic approximation

In underwater acoustics various types of nonlocal boundary conditions have been developed to handle the semi-infinite bottom in parabolic approximations and to efficiently reduce the computational domain. This paper proposes new exact nonlocal boundary conditions suitable for a layered ocean bottom and presents an analytic derivation of the corresponding adjoint equations. The new boundary condition has the form of a Neumann-to-Dirichlet map (NtD) that explicitly contains the geoacoustic parameters of the stratified bottom, i.e., thickness, density, sound speed, and attenuation of each layer. By means of the analytic adjoint, exact gradient information can be obtained which in turn allows a direct inversion of these parameters using a gradient-based optimization scheme