Geometric Theory of Ray Tracing

A temporal metric tensor is defined by combining the sound-speed function with the spatial metric tensor for a Riemannian space. Fermat's principle implies that spatial rays are temporal geodesics. Ray equations generalized to Riemannian spaces are shown to be temporal geodesic equations expressed in spatial terms. This geometric derivation leads to the consideration of geodesic deviation and its relation to three-dimensional spreading loss. Previous results [E. S. Eby, “Frenet Formulation of Three-Dimensional Ray Tracing,” J. Acoust. Soc. Amer. 42, 1287–1297 (1967)] are generalized to Riemannian spaces, and tensor expressions are derived for ray curvature and torsion.