Inverse Problems in Geophysics

Author

Thomas Günther (TU Bergakademie Freiberg)

Prerequisites

Here are the lecture notes for the module Inverse Problems in Geophysics., tailored for Master’s students of Geophysics or Geoinformatics at the Institute of Geophysics and Geoinformatics, TU Bergakademie Freiberg. You can access all the materials on the GitHub repository https://github.com/halbmy/InversionGeophysics.

Enjoy studying!

Background

While this course was around already since 2000 in Freiberg (when I did my PhD), it did never meet the general title before my comeback. Striving to do so, I found no text book about learning the background of the simulation methods being increasingly used in geophysics, particularly solving partial differential equations by own codes and understanding the physics by doing so. We aim at learning to solve all common PDE types from elliptic, parabolic to hyperbolic, going from 1D to 2D and 3D, from Finite Differences to Finite Elements and Finite Volumes.

Code

The website’s course material includes sections with embedded Python code. You can easily copy the code and execute it in a compatible Python runtime environment. For an optimal experience, we recommend installing Python via miniforge. You mainly need the numerical base library NumPy for doing the computations, and Matplotlib for visualization purposes. For higher-dimensional problems, we use the pyGIMLi package to generate meshes and matrices, but stay with its core and equation levels for didactic reasons.

Moreover, you have the freedom to explore your own concepts and delve deeper into the course content by creating your personalized Jupyter notebooks. You can utilize use either Visual Studio Code or Jupyterlab Working with Jupyter Notebooks in Visual Studio Code is exceptionally straightforward.

The material was created by Quarto.

Self study

There are individual small tasks for self-study that are scattered throughout the lecture.

NoneSelf study

You can easily identify these callouts.

Introduction

  • Content and dates
  • Literature and Links
  • definition of data and model
  • error and noise
  • linear vs. non-linear
  • linear inversion: simple matrix problem

Subject and Objectives

ImportantInversion (main task in applied geophysics)

Determine a model describing the subsurface that can explain our data!

NoteAim of the lecture
  • understand the process of imaging/inversion
  • solve inverse problems by simple scripts
  • actively control the inversion process
  • use tools to get a feeling for the value of models

Content

  1. Introduction, data, models
  2. Simple matrix problems, linear regression
  3. Method of least squares
  4. Resolution and Singular Value Decomposition
  5. Regularization methods
  6. Ray tomography
  7. Non-linear minimization
  8. Time-lapse inversion
  9. Global optimization methods

What should you know already?

  • Higher mathematics: differential equations, algebra (1.-2. BGIP)
  • Experimental and theoretical physics: governing equations
  • Numerics for engineers (2. BGIP)
  • Programming (1. BGIP), Software development (3. BGIP)
  • Geophysics: feeling for physical fields & methods (2.-4. BGIP)
  • MSc level: Scientific programming, HPC, seismic imaging

Literature

License

This material is licensed under the Creative Commons License CC-by 2.0.