Combining ray-tracing techniques and finite-element modelling in deformable media

We propose a numerical ray-tracing algorithm for isotropic media discretized by finite elements. This algorithm, which is based on tetrahedral elements, combines the flexibility of finite-element modelling with the calculation of rays through a three-dimensional (3D) speed field, given a direction of a ray at its initial point. We implement this algorithm and investigate the sensitivity of rays (namely, their shape and the signal travel time along them) to (i) deformations of the medium and subsequent changes of its speed field and (ii) perturbations of initial direction in the presence of first- and second-order discontinuities. We consider several scenarios analytically, and use them to verify the finite-element scheme. The ray-tracing scheme accounts for Snell's law at interfaces of velocity discontinuities and intrinsically handles discontinuous velocity gradients. It accommodates objects of arbitrary 3D shape and arbitrary speed fields. Furthermore, this finite-element technique accounts for deformations of a discretized medium on the level of each node by moving the node location and associated value of velocity. For several numerical experiments, we deform different models to observe deviations of rays from their initial shape. This study shows how the computation of rays can be combined with finite-element calculations of static elastic deformations, with implications to seismic and optical studies.