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.**

**Attenuation:**

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).