Comparing his method with Vidale's method, Van Trier pointed out that his algorithm is vectorizable and thus more efficient. For the present purposes, there is also another reason to choose Van Trier's method. The partial differential equation (4) contains the partial derivatives of traveltimes, but not the traveltimes themselves. If we use Vidale's method, the partial derivatives of traveltimes have to be calculated by the finite difference approximations

Since the use of finite difference operators increases numerical noise, it it should be avoided whenever possible. With Van Trier's method, we do not have this problem because the partial derivatives of traveltimes are intermediate results of the algorithm. For reasons of stability, we need the partial derivatives to be smooth. Van Trier's algorithm sometimes gives rough derivative functions because the finite difference scheme depends on the signs of the partial derivatives that are discontinuous functions. Therefore I take some special measures to maintain smooth solutions.12/18/1997