NGEO052, summer term, 30 hours

(extension of lecture
Theoretical foundations of ray methods)

** Viscoelastodynamic equations. **

Linear constitutive equations for viscoelastic medium, relaxation functions.
Anisotropic and isotropic viscoelastodynamic equations.
Dispersion and attenuation.

** Seismic model of the medium (macro model). **

Coordinate systems and metric tensors. Computer representation of the model.

** Green tensor. **

** Travel times. **

Kinds of travel times.
Isotropic and anisotropic eikonal equations.
First-arrival travel times.
Ray-theory travel times, elementary waves.

** Initial-value ray tracing. **

Hamiltonian ray tracing, rays as geodesics, wave-propagation metric tensor.
Kinematic and dynamic ray tracing.
Second and higher partial travel-time derivatives.

** Calculation of ray-theory travel times **

Two-point ray tracing, shooting methods, bending methods.
Computation of travel times on regular rectangular grids.
Ray cells, weighting of paraxial ray approximations.
Wavefront tracing.

** Calculation of first-arrival travel times **

Network shortest-path ray tracing, grid travel-time tracing.

** Ray-theory synthetic seismograms.**

Isotropic and anisotropic ray theories. Weak anisotropy.
Complex-valued rays and travel times. Space-time ray theory.
Gaussian beams and packets, Chapman-Maslov asymptotic theory.

** Accuracy and validity conditions of asymptotic ray methods. **

Kirchhoff integrals, Fresnel zones, representation theorems,
Fresnel volumes.

** Full-wave finite differences in 3-D. **

Accuracy of various finite-difference schemes, grid dispersion.
Waves as structural interfaces.
Fast calculation of the first-arrival waveforms.

** Ray method for surface waves in Cartesian and curvilinear coordinates. **