Gradient calculation of the traveltime cost function without ray tracing

Gradient based traveltime tomography algorithms require the computation of the gradient of the traveltime misfit cost function many times. This calculation is customarily done by ray tracing, the path length of the rays being closely related to the gradient. We propose in this work an alternative method to compute the gradient of the traveltime cost function without ray tracing. We use upwind finite difference schemes to compute the traveltime field by solving the eikonal equation. Then by adjoint state techniques we derive a closed-form expression of the gradient of the traveltime cost function. This approach allows an accurate computation of the gradient as well as the freedom to change the norm on the model space.