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

Ludek Klimes


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.


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

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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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