Two improvements on conventional ray bending are presented in this paper. First, gradient search methods are proposed to find the locations of successive points on a ray such that the travel time between its endpoints is minimal. Since only the integration of the travel time along the whole ray is involved, such methods are inherently more stable than methods that use first and second derivatives of the ray path to solve the raytracing equations. Second, it is shown that interpolation between successive points on a ray is generally necessary to obtain sufficient precision in the quadrature, but also advantageous in terms of efficiency when Beta-splines are used.
Velocity discontinuities are easy to handle in the proposed bending algorithm. The target for the travel time precision is 1 part in 104, which can be considered necessary for many applications in seismic tomography. With the proposed method, this precision was reached for a constant gradient velocity model using the conjugate gradients algorithm on a five-point Beta-spline in single precision arithmetic. Several existing methods for parametrization and minimization failed to produce this target on this simple model even with computing times that are an order of magnitude larger. Calculations in a spherical Earth yield the required precision within feasible computation times. Finally, alternative search directions, which can be obtained from an approximate second order expansion of the travel time, accelerate the convergence in the first few iterations of the bending process.