Beyond ray tomography: Wavepaths and Fresnel volumes

Abstract

Two techniques that account for the band‐limited nature of seismic data are incorporated into tomographic traveltime inversion schemes. The first technique, the wavepath algorithm, is based upon the wave equation, the Born approximation, and an adjoint method for computing Frechet derivatives. Computation of a single wavepath requires the forward propagation of the seismic wavefield, as well as the reverse propagation of a residual wavefield. The second technique, the Fresnel volume approach, is based upon the paraxial ray approximation. The Fresnel volume algorithm requires little more computation than does conventional ray tracing and an order of magnitude less computer time than our calculation of wavepaths. When the Fresnel volume sensitivity functions are normalized by the area of the Fresnel ellipse perpendicular to the ray, the sensitivity estimates are very similar to the wavepaths. In particular, there is heightened sensitivity to velocity structure near the source and receiver locations. The normalization by the Fresnel ellipse area is necessary to ensure ray theoretical results in the limit of infinite frequency. Tomographic inversion based upon wavepaths or Fresnel volumes is more appropriate when considering the arrival time of the peak of the initial pulse rather than the first‐arrival time. Furthermore, using the traveltime of the peak instead of the first‐arrival time reduces the bias of tomograms to high velocity anomalies. The raypath, wavepath, and Fresnel volume techniques were applied to a set of cross‐borehole traveltime observations gathered at the Grimsel Rock Laboratory. All methods imaged a low velocity fracture zone in the granitic site, in agreement with independent well information. Estimates of model parameter resolution are similar for the wavepath and Fresnel volume schemes. The source‐receiver regions are the most well resolved areas. However, the model parameter resolution computed using a conventional ray‐based formalism is more evenly distributed over the cross‐borehole area.