Transformations for dynamic ray tracing in anisotropic media

Summary

Six-dimensional dynamic ray tracing in (phase-space) Cartesian coordinates was introduced by Cerveny [Geophys.J.R.astr.Soc. 29(1972), 1-13]. Hanyga [Tectonophysics 90(1982), 243-251] showed that it reduces to 4-dimensional dynamic ray tracing in (phase-space) ray-centred coordinates. This paper concentrates on the explicit transformation equations of dynamic ray tracing between Cartesian and ray-centred coordinates. Many of the transformation equations have not been published before even for isotropic medium. Also proposed is an efficient way of reducing the number of equations being solved when numerically evaluating the paraxial-ray propagator matrices, both in Cartesian and ray-centred coordinates.

Keywords

Anisotropy, paraxial rays, dynamic ray tracing, ray-centred coordinates.

Contents

1. Introduction

2. Anisotropic ray theory

3. Phase-space coordinates

4. Ray tracing

5. Isotropic medium

6. Dynamic ray tracing

7. Ray-centred coordinates

8. Transformation

8.1 Transformation of the paraxial-ray phase-space coordinates

8.2 The second Hamiltonian derivatives in the local Cartesian coordinates

8.3 The second Hamiltonian derivatives in the ray-centred coordinates

9. Particular ray-centred coordinates

10. Trivial dynamic ray tracing solutions

11. Paraxial-ray propagator matrices

12. Numerical computation of paraxial-ray propagator matrices

Appendix A: Cartesian coordinates

Appendix B: Model coordinates

Acknowledgements

References

Whole paper

The scanned reprint of the paper is available in PDF 300dpi (436 kB).

The manuscript is available in PDF (117 kB). The manuscript does not contain missprints in equations (45) and (62).

Note

For the generalization of this paper from a homogeneous Hamiltonian of the second degree towards a homogeneous Hamiltonian of an arbitrary degree, refer to
Klimes, L. (2002): Transformations for dynamic ray tracing in anisotropic media with a homogeneous Hamiltonian of an arbitrary degree. In: Seismic Waves in Complex 3-D Structures, Report 12, pp. 67-78, Dep. Geophys., Charles Univ., Prague.

SW3D