Abstract:
The Rytov approximation that expresses phase residuals as an explicit function of the slowness perturbations, is also related to the generalized Radon transform (GRT). Using Beylkin's formalism, we derive the corresponding inverse GRT to give the slowness model as an explicit function of the phase residuals. This expression is used to deduce the resolution limits of wave path traveltime tomograms as a function of source frequency and source-receiver geometry. Its validity is restricted to arbitrary models with smooth variations in velocity, where the scale wavelength of the velocity variations must be at least three times longer than the characteristic source wavelength. The formula explicitly gives the slowness perturbation function as a function of the product of the frequency and the traveltime gradient that can be obtained by ray tracing. It shows that the spatial resolution limits of a slowness anomaly can be estimated by calculating the available wavenumbers of the slowness perturbation function. Using this procedure, resolution limits are obtained for several types of data: controlled source data in a crosswell experiment, data from a diving-wave experiment and earthquake data using the reference velocity model of the whole Earth. This formula can also be used for diffraction-limited pixelization of velocity models or for direct inversion of traveltime data.
