Ray tracing over smooth elastic shells of arbitrary shape

An efficient numerical scheme based on ray theory is developed for the analysis of elastic waves traveling over a fluid-loaded smooth elastic shell of arbitrary shape. The shell’s surface is first discretized into a number of small patches. The local geometry of each patch is then approximated in a parametric form using bi-cubic spline functions. A local curvilinear coordinate frame is defined on each patch. The ray trajectories and ray-tube areas are obtained by solving a set of ordinary differential equations, the ray and transport equations, within each patch. Several numerical tests of the accuracy and efficiency of the scheme were carried out on spherical and ellipsoidal elastic shells. The numerical results for the spherical shell agree well with analytical solutions. The ray trajectories and the ray-tube areas over an ellipsoidal shell with three distinct semiaxes clearly illustrate the inhomogeneous and anisotropic effects due to the variable curvature on the shell’s surface. It is also observed numerically that the magnitude of the ray-tube area along a ray is directly correlated with the stability of its trajectory.