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Short course on geophysical inversion at RWTH Aachen University.


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Let's build a geophysical inversion with Python

Instructor: Leonardo Uieda1

1 Department of Earth, Ocean and Ecological Sciences, School of Environmental Sciences, University of Liverpool, UK

Short course on geophysical inversion at RWTH Aachen University graduate school IRTG-2379 Modern Inverse Problems.

When 15 December / 13:00 - 16:30 CEST (UTC+1)
Where Online via Zoom
YouTube TBD


Since there is large component of live coding, participants will have to set up their computers before the workshop. It's extremely important that everyone has a working Python environment ahead of time as there will not be enough time to sort out individual problems during the workshop.

  1. Download and install the Anaconda Python Distribution. Please follow the instructions here:
  2. Make sure your installation works by opening JupyterLab through the Anaconda Navigator app (on Windows) or by running jupyter lab in a terminal (Mac/Linux). You browser should open with JupyterLab.
  3. Brush up on your coding skills with Software Carpentry's Introduction to Python lesson.

If you encounter any problems please open an "Issue" in this repository (it will require a free GitHub account).

We will be using Jupyter Notebooks to run our Python code and the libraries numpy, scipy, and matplotlib. Anaconda already comes with all of these installed.

In case of emergency (can't get things installed locally), you can use the link below to lauch Jupyter online on Binder:

Launch Binder

Beware that Binder will not store your notebooks so if the connection drops, you will lose all your code. I recommend periodically downloading your notebook to your computer so you don't have to start over. Alternatively, all the code I'll be writing is provided in gravity-inversion.ipynb.


Inverse problems abound in geophysics. It is the primary way in which we investigate the subsurface of the Earth, which is largely inaccessible to us beyond the first dozen or so kilometers. From measurements acquired on land, sea, air, and from space, geophysicists tease out the inner structure of the Earth - from a few meters to thousands of kilometers deep in the inner core. Observations of disturbances in the Earth's gravity field are one of the key elements used by geophysicists to investigate the crust-mantle interface, the large-scale structure of sedimentary basins (which are reservoirs for water and hydrocarbons), and even the mass balance of the world's ice sheets. However, the gravity inverse problem is particularly challenging due to the physics of potential fields. Unique solutions are difficult to come by and only exist under strict assumptions, which often don't hold for real world scenarios. For these problems, regularization plays a critical role and has been the focus of much research in the past 20 years.

In this tutorial, we will work together to solve a 2D gravity inverse problem in Python. Our code will estimate the shape of a sedimentary basin from gravity observations. This non-linear inverse problem will allow us to visually explore the effects of different types of regularization from a geometric perspective (smoothness, equality constraints, and more). We will discuss the challenges involved in real world applications and the difficulties of quantifying the uncertainty in the solutions. The main goal of this tutorial is to impart theoretical and practical skills that can be easily transfered to other domains.

Learning objectives

This course is designed to empower you to:

  • Learn/revise the mathematics of non-linear inverse problems
  • Translate mathematical knowledge into code
  • Apply non-linear inversion theory to a real geophysical problem
  • Analyze the effects of regularization on geophysical models


I will assume that you:

  • Are comfortable with linear algebra (matrix and vector operations, norms, inverses, linear systems, etc)
  • Have an understanding of basic calculus (partial derivatives, gradients, Taylor series expansions)
  • Are able to program a computer to build and manipulate matrices and vectors, solve linear systems, and make graphs/plots (in any language but Python or Matlab would be best)

Format and schedule

The course will be a mix of short presentations mixed with live coding (I will lead and learners will follow on their own computers). Participants will also be assigned short challenges to complete.

The following is a tentative schedule:

Session 1 13:00 - 14:30
Introduction 10 min
Maths: geophysical inverse problem formulation 10 min
Exercise: what is the gradient of the misfit function? 10 min
Maths: solving non-linear inverse problems 30 min
Code: implementing a non-linear inverse problem 30 min

Break: 30 minutes

Session 2 15:00 - 16:30
Exercise: add random noise to the data 10 min
Maths: smoothness regularization 20 min
Code: add smoothness to the inversion 20 min
Exercise: sharp vs smooth models (what about faults?) 10 min
Wrap up 10 min
Questions and extra time for discussion 20 min

Workshop material

All of the code and notes for this workshop are (or will be) uploaded to this repository. In here, you'll find:


All Python source code in this repository is free software: you can redistribute it and/or modify it under the terms of the MIT License. A copy of this license is provided in LICENSE.txt.

All other materials, including text and images, are distributed under the CC-BY 4.0 license (except where otherwise noted).


Short course on geophysical inversion at RWTH Aachen University.



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