Abstract
The eikonal equation in an attenuating medium has the form of a complex–valued Hamilton–Jacobi equation and must be solved in terms of the complex–valued travel time (complex–valued action function). The solution of the complex–valued Hamilton– Jacobi equation for complex–valued travel time by Hamilton’s equations of rays would require complex–valued rays (complex–valued geodesics).
Since the material properties are known in real space only, we cannot calculate complex–valued rays. A very suitable approximate method for calculating the complex–valued travel time right in real space is represented by the perturbation from the reference travel time calculated along real– valued reference rays to the complex–valued travel time defined by the complex–valued Hamilton–Jacobi equation.
In: Seismic Waves in Complex 3-D Structures, Report 20, pp. 193-205, Dep. Geophys., Charles Univ., Prague, 2010.