A Connection between Geometrical Spreading and the Adjoint Field in Travel Time Tomography

The goal of tomography is to reconstruct a spatially-varying image function s(x,m), where x is position and m is a finite-length vector of parameters. Many reconstruction methods minimize the total L2 error E ≡ eTe, where individual errors ei quantify misfit between predictions and observations, to quantify goodness of fit. So-called adjoint state methods allow the gradient ∂E/∂mi to be computed extremely efficiently from an adjoint field, facilitating image reconstruction by gradient-descent methods. We examine the structure of the differential equation for the adjoint field under the ray approximation and find that it has the same form as the transport equation, whose solution involves the well-known geometrical spreading function R Consequently, as R is routinely tabulated as part of a ray calculation, no extra work is needed to compute the adjoint field, permitting a rapid calculation of the gradient ∂E/∂mi.