To prepare for 3D anisotropic applications of traveltime tomography I found it necessary to reformulate the estimation of raypaths and the parameterization of velocities. Raypaths should be efficient to calculate and store in memory without recalculation. Velocities should be able to change with angle when necessary.
Ray relaxation can optimize ray paths through an anisotropic medium described only by group velocities. Three parameters adequately describe transversely isotropic group velocities with a vertical axis of symmetry. One parameter changes most arbitrarily, one changes only along the vertical axis of symmetry, and one remains a constant.
3D raypaths are described as sums of smooth curves with a small number of coefficients. A generic Gauss-Newton algorithm perturbs these coefficients to minimize traveltimes between endpoints. A small number of coefficients are saved in memory to describe paths efficiently.
Diving wave tomography can use conventional traveltime tomographic algorithms to model and invert the traveltimes of direct arrivals. Early synthetic tests minimized the complexity of velocity anomalies necessary to explain the data. Raypaths robustly converge to their appropriate distribution.