Robust reflection tomography in the time domain

The convergence of conventional reflection tomography is often uncer-tain when the starting velocity function is too far from the correct one. The time-domain reflection tomography that we present in this paper is more robust than conventional depth-domain reflection tomography. Our new tomographic method avoids some of the instabilities of con-ventional depth-domain tomography, by solving for a velocity model and reflector geometry defined in vertical-traveltime domain. These instabilities are often caused by the coupling between the velocity function and the depth-mapping of reflectors. Time-domain tomog-raphy keeps the distinction, which is typical of time processing, be-tween the velocity function that best focuses the data and the velocity function that correctly maps the reflectors in depth. Time-domain reflection tomography is based on a new eikonal equa-tion that is derived by a transformation of the conventional eikonal equation from depth coordinates (z; x) into vertical-traveltime coor-dinates (; ). The transformed eikonal enables the computation of reflections traveltimes independent of depth-mapping. This separation allows the focusing and mapping steps to be performed sequentially even in the presence of complex velocity functions, that would other-wise require “depth ” migration. We compute the solutions of the transformed eikonal equation by solv-ing the associated ray tracing equations. The application of Fermat’s principle leads to the expression of linear relationships between pertur-bations in traveltimes and perturbations in focusing velocity. We use this linearization, in conjunction with ray tracing, for the time-domain tomographic estimation of focusing velocity