Numerical Methods in the Earth Sciences: Application to seismic wave propagation     Heiner Igel, LMU Munich Short Course Utrecht  June 7-8, 2007

Scope
Numerical solutions to many time-dependent problems in Earth sciences (e.g., wave and rupture propagation, crustal deformation) are becoming more and more standard in data modelling projects. Often this involves use of software written by others and the user is inexperienced with the fundamental concepts of numerical methods. This is dangerous as programs might be used with inappropriate parameters, leading to wrong scientific conclusions. The goal of this short course is to illustrate and derive some of the most fundamental concepts of numerical approximations that are common to almost all methodologies currently used (e.g., finite differences, finite and spectral elements, pseudospectral methods). This involves the concepts of stability and dispersion (the Courant criterion), pseudospectral accuracy and exact interpolation, strong and weak formulations, and function approximation. These concepts shall be illustrated using the simple scalar acoustic wave equation. The lectures are supported by Matlab scripts with sample programs for various numerical methods (finite differences, finite volumes, pseudospectral method).

Level
It helps to be familiar with basic partial differential equations, and complex numbers in the context of the description of plane harmonic waves with exp[ i (kx -wt)]. We also make use of the Fourier Transform, Fourier series and Taylor series.

Lectures
Note that the slides should be considered as supporting material to the lectures, not a script in itself.

• Lecture 1: Computational methods in seismology (ppt, pdf)
• Lecture 2: Approximating arbitrary functions (ppt, pdf)
• Lecture 3: Basic methods: finite differences, pseudospectral method, finite elements (ppt, pdf)
• Lecture 4: Current challenges in computational seismology (ppt, pdf)
  

Practicals
The practicals consist of a theoretical and a practical part. Follow this link to access practical exercises based on Matlab codes and accompanying  theoretical exercises. Depending on progress a selection of these exercises will be done during the practicals. Additional codes can be found here.

Literature
Several presentation and practical material on numerical methods applied to wave propagation can be found in the digital library of the SPICE project (www.spice-rtn.org).
There is a book on the finite-difference method by P. Moczo, and on ray theory by J. Brokesova. A useful book on the basic concepts of finite differences and pseudospectral methods is:
Fornberg, A practical guide to pseudospectral methods, Cambridge University Press.


Animations
Global wave propagation (Author  G. Jahnke, www.terraemotus.org)
The Grenoble basin simulation (Author M. Kaeser)

Heiner Igel, June 2007