Correcting for Ray Refraction in Velocity and Attenuation Tomography: A Perturbation Approach

In velocity and attenuation tomography, ray refraction leads to errors in time-of-arrival, as well as to errors in attenuation due to phase cancellation and lateral beam displacement. Some authors have proposed iterative techniques based on numerical ray tracing to correct for these effects. In this paper, we consider an alternative approach using a perturbation analysis of refraction. This approach requires neither iteration nor numerical integration of the ray equation. Assuming that the index of refraction deviates from its mean on the order of the small quantity ε, we derive expressions for the refracted ray path whose departure from a straight line is first order in ε. Using this result, we obtain a perturbation expansion of the path integral of the refractive index along the refracted ray and derive a time-delay correction of order ε² arising from the deviation of the refracted ray from a straight line. The expression for the first-order path is also used to obtain explicit corrections for phase-cancellation and beam-displacement errors that affect attenuation measurements when transducers of finite extent are employed. In addition, because of refraction, large aperture transducers are susceptible to an arrival-time uncertainty in a time-of-flight measurement; a first-order expression for the maximum value of this uncertainty is derived. In both two and three dimensions, the perturbation approach is much simpler computationally than numerical ray tracing methods. Computer simulated reconstructions are presented which clearly show the improvement that can be achieved with the second-order time delay correction.