Liouville partial-differential-equation methods for computing 2-D complex-valued eikonals in the multivalued sense in attenuating media

We present Liouville partial-differential-equation (PDE) based methods for computing complex-valued eikonals in the multivalued (or multiple arrival) sense in attenuating media. Since the earth is comprised of attenuating materials, seismic waves usually attenuate so that seismic data processing calls for properly treating the resulting energy losses and phase distortions of wave propagation. In the regime of high-frequency asymptotics, the complex-valued eikonal is a crucial ingredient for describing wave propagation in attenuating media, since it is a unique quantity which summarizes two wave properties into one function: its real and imaginary parts are able to capture the effects of phase dispersions and amplitude attenuations, respectively. Because the usual ordinary-differential-equation (ODE) based ray-tracing methods for computing complex-valued eikonals distribute the eikonal solution irregularly in real space, we are motivated to develop PDE based Eulerian methods for computing complex-valued eikonals on regular meshes. Therefore, we propose to solve novel paraxial Liouville PDEs in real phase space so that we can compute the real and imaginary parts of the complex-valued eikonal in the multivalued sense on regular meshes. We dub the resulting method the Liouville PDE method for complex multivalued eikonals in attenuating media. We also provide Liouville PDE formulations for computing multi-valued amplitudes. Numerical examples, including a synthetic gas-cloud model, demonstrate that the proposed methods yield highly accurate complex-valued eikonals in the multivalued sense.