Theoretical foundations of ray methods

Ludek Klimes

NGEO097, summer term, 45 hours
(An alternative to lecture NGEO032: Ray methods in seismology)

Viscoelastodynamic equations:
Linear constitutive equations for viscoelastic media, relaxation functions. Anisotropic viscoelastodynamic equation in the time domain. Anisotropic viscoelastodynamic equation in the frequency domain. Isotropic viscoelastic medium. Dispersion and attenuation.

Comparison of ray methods and other methods.

Ray theory for elastic media:
Standard ray series. Christoffel equation. Polarisation. Eikonal equation. Transport equation. Polarisation of S waves in isotropic media. Coupling ray theory for S waves. Examples of synthetic seismograms in weakly anisotropic media.

Hamiltonian functions for elastic media:
Isotropic elastic medium. Anisotropic elastic medium. Phase-space derivatives of the eigenvalues of the Christoffel matrix.

Theory of the solution of the Hamilton-Jacobi equation:
Difference between the viscosity solution and the Hamiltonian solution. Geometrical interpretation. Phase-space derivatives. Hamilton's equations of rays. Ray coordinates. Hamiltonian equations of geodesic deviation. Propagator matrix of geodesic deviation. Second-order derivatives of travel time.

Theory of travel-time perturbations:
Perturbation parameters and perturbation Hamiltonian function. Perturbation derivatives. Perturbation expansion of travel time. Linear perturbation Hamiltonian function. Equations for the third-order and higher-order spatial derivatives and for the perturbation derivatives of travel time.

Transformation of the spatial and perturbation derivatives of travel time at an interface:
Travel time at a smooth interface. Transformation of the first-order derivatives of travel time. Transformation of the second-order derivatives of travel time.

Transformation of paraxial matrices at an interface:
Transformation of the matrix Q of geometrical spreading at an interface. Transformation of the matrix P of paraxial slowness vectors at an interface. Transformation of both paraxial matrices at an interface. Transformation of the propagator matrix of geodesic deviation at an interface. Transformation of the non-eikonal paraxial vector at an interface.

Transport equation:
Solution of the transport equation. Phase shift due to caustics. Examples of phase shifts.

Reflection and transmission coefficients for the amplitude at an interface.

Complex-valued Hamiltonian function. Reference Hamiltonian function. Perturbation Hamiltonian function. Reference rays. Reference travel time. Hamiltonian equations of geodesic deviation. First-order perturbation derivative of travel time. First-order perturbation derivative of travel-time gradient. Second-order perturbation derivative of travel time.

Paraxial approximation and Gaussian beams and packets:
Paraxial approximation. Gaussian beams. Gaussian packets. Summation of Gaussian beams. Summation of Gaussian packets.

Systems of rays and calculation of travel times:
Ray parameters and the continuity of multi-valued travel time. Velocity model. Smooth velocity model. Block velocity model. Elementary waves. Ray histories. Controlled initial-value ray tracing. Two-point ray tracing. Other applications of controlled initial-value ray tracing. Wavefront tracing. Interpolation within ray cells.

Green tensor:
Representation theorem. Born approximation. Born correction of an approximate wavefield.

Ray-theory Green tensor:
Paraxial matrices for the amplitude of the ray-theory Green tensor. Ray-theory Green tensor in a homogeneous medium. Elementary ray-theory Green tensor in a heterogeneous medium.

Seismic sources.

Synthetic seismograms.


Cerveny, V. (2001): Seismic Ray Theory. Cambridge Univ. Press, Cambridge (viii + 713 pages).