A derivation of three-dimensional ray equations in ellipsoidal coordinates

Abstract

Conventional three-dimensional (3-D) ray equations ignore the effect of Earth flattening, whereas this effect, as has been shown by Munk et al. [J. Phys. Oceanogr. 18, 1876–1898 (1988)], cannot be ignored for long-range transmissions. To take into account the earth flattening, new 3-D ray equations are derived, in this paper, in terms of the ellipsoidal coordinates: Geographic latitude, longitude, and depth. The new 3-D equations account for both Earth curvature and Earth flattening, and thus are more accurate than those conventional 3-D ray equations. It is shown that under certain circumstances, the new equations reduce to the horizontal ray equations of Munk et al., the Aki-Richards 3-D spherical ray equations, and the conventional two-dimensional ray equations, respectively. The advantages of using the new equations in ocean acoustics are discussed. Numerical examples are presented.