Fermat’s principle and acoustic tomography in a time-dependent ocean

Abstract

Fermat’s principle states that, for nondispersive waves propagating between two given points, travel time is stationary with respect to small variations of the wave propagation path around the actual path (i.e., ray) taken by the wave. This fundamental principle governing short wave kinematics was first established by seventeenth century French mathematician Pierre de Fermat. The principle is of particular importance for acoustic tomography because it allows one to neglect perturbations in ray geometry when performing a linear inversion of acoustic travel times for sound speed variations. Derivations of Fermat’s principle available in the literature does not apply to acoustic waves in media with time-dependent sound speed. Moreover, the very formulation of the principle in this case is not obvious as travel time along a trial ray ceases to be a single-valued functional of the trial ray geometry. In this paper, the Fermat’s principle is extended to media with time-dependent parameters. We consider acoustic waves in the inhomogeneous, moving or quiescent fluid. Formulations of the principle in terms of travel time and eikonal are compared. Applications of the established principle to ocean acoustic tomography and modeling wave propagation in weakly nonstationary, inhomogeneous ocean are discussed. [Work supported by ONR.]