ABSTRACT Seismic tomography methods that use waves originating outside the volume being studied are subject to bias caused by unknown structure outside this volume. The bias is of the same mathematical order and similar magnitude as the local-structure effects being studied; failure to account for it can significantly corrupt derived structural models. This bias can be eliminated by adding to the inverse problem three unknown parameters specifying the direction and time for each incident wave, a procedure analogous to solving for event locations in local-earthquake and whole-mantle tomography. The forward problem is particularly simple: The first-order change in the arrival time at an observation point resulting from a perturbation to the incident-wave direction and time equals the change in the time of the perturbed incident wave at the point where the unperturbed ray entered the study volume. This consequence of Fermat’s principle apparently has not previously been recognized. Published teleseismic tomography models probably contain significant artifacts and need to be recomputed using the more complete theory.