On numerical solving the complex eikonal equation using ray tracing methods

1. On numerical solving the complex eikonal equation using ray tracing methods Václav Vavryčuk Institute of Geophysics, Czech Acad. Sci., Prague

2. Wave propagation in dissipative media

3. Description of dissipative media Tensor of viscoelastic parameters – elastic parameters – viscous parameters Quality matrix (Q-matrix) all quantities and equations are complex Christoffel tensor and the eikonal equation

4. Ray tracing in isotropic elastic media Eikonal equation non-linear partial differential equation G (x, p) is the eigenvalue of the Christoffel tensor c, x, p, τare real Ray tracing equations system of linear ordinary differential equations

5. Ray tracing in isotropic viscoelastic media Eikonal equation non-linear partial differential equation G (x, p) is the eigenvalue of the Christoffel tensor c, x, p, τ are complex Ray tracing equations system of linear partial differential equations Solution of complex ray tracing equations are complex rays !

6. Troubles of complex ray theory Rays are curves in complex space 3D real space 6D complex space Physical meaning of rays is more involved Rays are not parallel to energy flux Model of the medium must be defined in complex space For simple models: analytic continuation It is not clear how to apply the complex ray theory to inhomogeneous media with interfaces Analytic continuation is not possible

7. Real ray tracing in dissipative media

8. Approximate ray tracing equations Approximate approaches: 1. Perturbation approach of elastic ray theory (Gajewski & Pšenčík, 1992) • rays are calculated in an elastic reference medium • dissipation effects are included by perturbations • weak attenuation 2. Real ray approach (Vavryčuk, 2008, 2010, 2012) • complex rays are approximated by real rays • no perturbations are used • includes strong attenuation

9. Approximate ray tracing equations Ray tracing equations same equations as for elastic medium Perturbation approach Real ray approach

10. Numerical examples

11. Model of the medium Isotropic medium with a constant gradient of c-2: real part of c-2 imaginary part of c-2 imaginary part of c real part of c

12. Complex travel time: exact solution real part of τ imaginary part of τ

13. Complex rays: exact solution propagation rays (curves of the gradient of the real travel time) attenuation rays (curves of the gradient of the imaginary travel time)

14. Errors of approximate travel time Perturbation approach errors of real travel time errors of imaginary travel time (%) (%) Real ray approach errors of real travel time errors of imaginary travel time (%) (%)

15. Errors in geometry of approximate rays Perturbation approach propagation rays attenuation rays Real ray approach propagation rays attenuation rays exact approximate

16. Conclusions • The real ray approach deals with real rays and works in isotropic as well as anisotropic dissipative media • The real ray approach is approximate but highly accurate • The real ray approach is applicable to a broad family of models including 3-D heterogeneous structures with interfaces • Computational costs of the real ray approach are the same as for perturbationsapproach