Seismic waves propagating in an anelastic medium undergo phase and amplitude distortions. Although these effects may be compensated for during imaging processes, a background -model is generally required for their successful application. We have developed a new approach to the -estimation problem, which is fundamentally related to the basic physical principle of time reversal. It is based on back-propagating recorded traces to their known source location using the reverse tomographic equation. This equation is a ray approximation of viscoelastic wave propagation. It is applied assuming a known and correct velocity model. We subsequently measure consistency between spectral shapes of traces that were back-propagated using the tomographic equation. We formulate an inverse problem using this consistency as an objective function. In conventional inversion, on the contrary, the discrepancy between modeled and recorded data, or some data characteristics, is minimized. The inverse problem is solved by ant-colony optimization, a global optimization approach, to avoid local minima present in the objective function. This method does not require knowledge of the source function and uses the full spectrum rather than its parametric reduction. Through synthetic and field cross-hole examples, we illustrate its accuracy and sensitivity in inverting for complex attenuation models. In the synthetic case, we also compare reconstructed source consistency with the conventional centroid frequency shift objective function. The latter displays poor resolution when recovering complex structures. We determine that the reconstructed source-consistency approach should be used as a part of an iterative workflow, possibly yielding initial models for a joint velocity and inversion.