A Fast and Global Two Point Low StorageOptimization Technique for Tracing

We present the problem of tracing rays in 2D and 3D heterogeneous isotropic media as a set of optimization problems. Each optimization problem is obtained by applying Fermat's principle to an approximation of the travel time equation from a xed source to a xed receiver. We assume a piecewise linear raypath which simpliies the computations of the problem, in the same way Mao and Stuart suggested in a very recent paper. Here, instead, the reeectors geometry and the velocity function are computed by using nonuniformly biharmonic splines. This biharmonic spline is easy to apply to interpolation problems in three and more dimensions and it also has minimum curvature. On the other hand, to solve the optimization problem we use the Global Spectral Gradient method. This recent developed optimization scheme is a low storage optimization technique that requires very few oating point operations. It only requires the gradient of the travel time function, and it does not require a close initial raypath. These three properties of the optimization method and the assumption of piecewise linear rays make this ray tracing scheme a very fast and eeective method when estimating velocities via tomography. Besides, in a 2D or 3D stratiied homogeneous medium, we show that this ray tracing approach converges to the global minimum, if the optimization problem has a solution. We present some numerical results that indicate that this scheme outperforms some classical optimization methods recently used for solving the same problem.