Traveltimes are a line integral of the slowness along the ray path expressed as
This nonlinear problem can be linearized by subtracting a reference problem and expressing the forward problem of the traveltime deviations from the reference model as follows:
In order to solve the original nonlinear problem (), the above linearized inversion is applied iteratively with the updated back projection operator Li, which is obtained by ray tracing through the new reference slowness model
Therefore, traveltime tomography can be summarized as an iterative two-step process. First, traveltime deviations are measured by comparing picked traveltimes with expected traveltimes obtained through an assumed velocity model. Then the differences are projected back over the traced ray paths through the assumed velocity model to update the model.
In contrast to transmission traveltime tomography, in which computation of the operator L only requires a slowness model to trace rays, reflection traveltime tomography requires additional information about reflectors, such as the dip and location. Therefore, an image space after prestack migration that can provide both the reflector information and the traveltime deviation is a good choice for picking.