Solving the frequency-dependent eikonal equation

Biondo Biondi

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ABSTRACT

The eikonal equation that is commonly used for modeling wave propagation is frequency independent. Therefore, it cannot correctly model the propagation of seismic waves when rapid variations in the velocity function cause frequency dispersion of the wavefield. I propose a method for solving a frequency-dependent eikonal equation, that can be derived from the scalar wave equation without approximations. The solution of this frequency-dependent eikonal is based on the extrapolation of the phase-slowness function downward along the frequency axis starting from infinite frequency, where the phase slowness is equal to the medium slowness. The phase slowness is extrapolated by solving a non-linear partial differential equation using an explicit integration scheme. The extrapolation equation contains terms that are functions of the phase slowness, and terms that are functions of the ray density. When the terms that are dependent on the rays are neglected, the extrapolation process progressively smooths the slowness model; this smoothing is frequency-dependent and consistent with the wave equation.

Numerical experiments of wave propagation through two simple velocity models show that the solution of the frequency-dependent eikonal computed using the proposed method matches the results of wave-equation modeling significantly better than the solution of the conventional eikonal equation.