Tangent linear and second-order models from the Born and Rytov approximations

Abstract

Adjoint and higher-order methods for the solution of inverse problems in underwater acoustics rely on tangent linear and higher-order models of the forward propagation problem, with respect to the underlying parameters of interest. The use of Born and Rytov approximations is examined for the representation of perturbations of the Green’s function and perturbations of the phase, respectively, caused by range-dependent variations of the sound-speed distribution about a range-independent background environment. Using the normal-mode representation for the background Green’s function and applying the stationary-phase approach in combination with analytic evaluation at singular points, closed-form expressions are derived for the first- and second-order perturbations of the Green’s function, generalizing previous perturbation results derived under the assumption of range independence. The resulting expressions for the phase show the second-order character of the phase (and the associated travel times) in the case of range-dependent, zero-mean sound-speed perturbations.