Complex-valued eikonal-transport equation

Ludek Klimes


It is well known that the representation of amplitude variations in terms of complex-valued travel time increases the accuracy of the ray methods. In order to represent the amplitude variations in terms of complex-valued travel time, we supplement the eikonal equation for complex-valued travel time with the frequency-dependent term corresponding to the transport equation. We refer to this equation for the complex-valued travel time including amplitude variations as the "eikonal-transport equation".

In real space, the eikonal-transport equation for complex-valued travel time represents the system of two second-order partial differential equations for the real and imaginary parts of the complex-valued travel time. The solution of this system of equations does not propagate along rays, and has to be solved by suitable numerical methods.

We propose to consider a system of surfaces and to calculate the complex-valued travel time from one surface to the subsequent surface numerically, analogously to the system of two Hamilton-Jacobi equations for complex-valued travel time. This method may be suitable for application to wavefront tracing.

We present two simple numerical examples, including comparisons with the standard ray theory and with the Gaussian beam summation.


Wave propagation, complex-valued travel time, amplitude, caustic, eikonal equation, transport equation.

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In: Seismic Waves in Complex 3-D Structures, Report 19, pp. 173-190, Dep. Geophys., Charles Univ., Prague, 2009.
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