Modelling seismic waves in 3-D

Ludek Klimes

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.