One of the applications of refraction-traveltime tomography is to provide an initial model for waveform inversion and Kirchhoff prestack migration. For such applications, we need a refraction-traveltime tomography method that is robust for complicated and high-velocity-contrast models. Of the many refraction-traveltime tomography methods available, we believe wave-based algorithms to be best suited for dealing with complicated models.
We developed a new wave-based, refraction-tomography algorithm using a damped wave equation and a waveform-inversion back-propagation technique. The imaginary part of a complex angular frequency, which is generally introduced in frequency-domain wave modeling, acts as a damping factor. By choosing an optimal damping factor from the numerical-dispersion relation, we can suppress the wavetrains following the first arrival. The objective function of our algorithm consists of residuals between the respective phases of first arrivals in field data and in forward-modeled data. The model-response, first-arrival phases can be obtained by taking the natural logarithm of damped wavefields at a single frequency low enough to yield unwrapped phases, whereas field-data phases are generated by multiplying picked first-arrival traveltimes by the same angular frequency used to compute model-response phases.
To compute the steepest-descent direction, we apply a waveform-inversion back-propagation algorithm based on the symmetry of the Green's function for the wave equation (i.e., the adjoint state of the wave equation), allowing us to avoid directly computing and saving sensitivities (Fréchet derivatives). From numerical examples of a block-anomaly model and the Marmousi-2 model, we confirm that traveltimes computed from a damped monochromatic wavefield are compatible with those picked from synthetic data, and our refraction-tomography method can provide initial models for Kirchhoff prestack depth migration.