The Sobolev gradient steepest descent provides a numerically efficient iterative bending algorithm for two-point ray tracing. This new algorithm has advantages in its relative ease of numerical implementation, its ability to provide accurate solutions in discontinuous velocity fields (including head wave solutions), and its ability, when smooth velocity fields are used, to provide accurate solutions while using relatively few ray segments. The algorithm owes its advantages to its implicit nature, which it inherits from the Sobolev gradient solution method. For most models, the algorithm can be implemented such that at convergence, the norm of the gradient approaches zero. Thus, convergence implies a stationary point of the traveltime integral, guaranteeing that the rays found satisfy Fermat’s principle.

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