A normal-mode expression for the sensitivity of acoustic fields to three-dimensional refractive index perturbations in a constant-depth waveguide, using an analytic adjoint approach

Adjoint or "co-state" methods have recently been applied to the ocean acoustic inverse problem using both a 2-D range-dependent PE approach and a range-independent normal mode formulation. Here the normal mode formulation is extended to yield expressions for the derivative of an acoustic field with respect to an arbitrary three-dimensional refractive index perturbation in the water or bottom, for the restricted case of a constant-depth waveguide. These results may be useful as both an independent check for 3-D propagation codes and for ocean acoustic tomography sensitivity studies. For example, inversions for sound speed perturbations in an ocean waveguide are generally conducted along a vertical plane connecting source and receiver, thus neglecting the potential effects of horizontal refraction by "out-of-plane" perturbations. A simple sensitivity analysis of the effect of out-of-plane perturbations on a propagating modal field is presented.