## System of two Hamilton-Jacobi equations
for complex-valued travel time

**Ludek Klimes**
### Summary

Since waves propagate in real space
and since the material properties are known
in real space only,
we cannot calculate complex-valued rays.
In real space, the eikonal equation
for complex-valued travel time represents
the system of two Hamilton-Jacobi equations
for the real and imaginary parts
of the complex-valued travel time.
Unfortunately, the solution of this system of
Hamilton-Jacobi equations does not propagate along rays,
and has to be solved by more global 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.
This method may be suitable for application
to wavefront tracing.

We present three simple examples of the numerical calculation
of the complex-valued travel time,
and compare their results with the analytical solutions.
### Keywords

Wave propagation, attenuation, eikonal equation,
complex-valued travel time, system of Hamilton-Jacobi equations.

### Whole paper

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In: Seismic Waves in Complex 3-D Structures, Report 19,
pp. 157-171, Dep. Geophys., Charles Univ., Prague, 2009.

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