A normal-mode formula for the derivative of a waveguide pressure field with respect to an arbitrary three-dimensional sound speed perturbation
Semi-analytic expressions are derived for the first order derivative of a pressure field in a laterally homogeneous depth waveguide, with respect to an arbitrary three-dimensional refractive index perturbation in either the water column or ocean bottom. These expressions for the environmental derivative, derived using an adjoint method, require a three-dimensional spatial correlation between two Greens functions, weighted by an environmental parameter basis function, with the Greens functions expressed in terms of normal modes. When a particular set of orthogonal spatial basis functions is chosen, the three-dimensional spatial integral can be converted into a set of one-dimensional integrations over depth and azimuth. The use of the orthogonal basis permits environmental derivatives to be computed for any arbitrary sound-speed perturbation. To illustrate the formulas, a sensitivity study is presented that explores the impact of three-dimensional plane wave and cylindrical perturbations on the environmental derivative. Under certain circumstances it is found that perturbation components outside the vertical plane connecting the source and receiver have non-negligible effects on the pressure derivative. Potential applications of these formulas include benchmarking three-dimensional propagation codes, computing Cramer-Rao bounds for three-dimensional environmental parameter estimates, and potentially inverting for small three-dimensional refractive index distributions.