Robust reflection tomography in the time domain

The convergence of conventional reflection tomography is often uncertain 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 conventional 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 tomography keeps the distinction, which is typical of time processing, between 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 equation that is derived by a transformation of the conventional eikonal equation from depth coordinates (z; x) into vertical-traveltime coordinates (τ,ε). 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 otherwise require “depth” migration. We compute the solutions of the transformed eikonal equation by solving the associated ray tracing equations. The application of Fermat’s principle leads to the expression of linear relationships between perturbations 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.