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IMA Lectures
Ray Theory for the Elastic Wave Equation
March 4-6, 2002

Mathematics in Geosciences, September 2001 - June 2002

Robert Burridge
Earth Resources Laboratory
Massachusetts Institute of Technology (MIT)

burridge@erl.mit.edu

Slides:   pdf

Syllabus

Introduction.

Ray theory for the scalar wave equation and scalar Helmholtz equation

  • The ray theory ansatz, amplitude and phase, slowness and travel time
  • The eikonal equation
  • The transport equation(s)
  • The ray-tube-area method to obtain the amplitudes
  • Caustics

Dynamic ray theory

  • The second derivatives of the phase function
  • The first variation of the ray equations
  • The transverse components of the Hessian of the phase
  • The dynamic ray equation and the simultaneous solution of the ray and dynamic ray equations
  • The solution of the transport equation using the results of dynamic ray theory

The elastic wave equation

  • Isotropy
  • Isotropy: P and S waves
  • The transport of the S polarization
  • Anisotropic ray theory

Ray theory for a symmetric hyperbolic system

  • Group velocity: various properties


The aim for the course is to be self contained at the expense of being formal. For instance, methods of calculating coefficients in an asymptotic series will be treated without proving that the series is indeed asymptotic.


Prerequisites:
The material will be strictly classical. The audience is expected to be familiar with the solution of first order PDE's by the method of characteristics, cartesian tensors, vector and dyadic notation, and the theory of the eigenvalues and eigenvectors of symmetric matrices.

LIST OF CONFIRMED PARTICIPANTS

Name Department Affiliation
Santiago Betelu Mathematics University of North Texas
Robert Burridge Earth Resources Laboratory Massachusetts Institute of Technology
Jamylle Carter   Institute for Mathematics & its Applications
Christine Cheng Institute for Mathematics & its Applications University of Minnesota
Dacian Daescu University of Minnesota Institute for Mathematics and its Applications
Gregory S. Duane University of Minnesota Institute for Mathematics and its Applications
Michael Efroimsky University of Minnesota Institute for Mathematics and its Applications
Selim Esedoglu   Institute for Mathematics & its Applications
Daniel Kern    
Anna Mazzucato Mathematics Yale University
Aurelia Minut University of Minnesota Institute for Mathematics and its Applications
M. Yvonne Ou University of Minnesota Institute for Mathematics and its Applications
Jianliang Qian   Institute for Mathematics & its Applications
Toshio Yoshikawa University of Minnesota Institute for Mathematics and its Applications

Slides:   pdf

Mathematics in Geosciences, September 2001 - June 2002
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