## Complex-valued eikonal-transport equation

**Ludek Klimes**
### Summary

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.
### Keywords

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

### Whole paper

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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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