# Using Fatiando a Terra to solve inverse problems in geophysics

Leonardo Uieda, Vanderlei C. Oliveira Jr, ValĂ©ria C. F. Barbosa

## Info

This was my second time at Scipy presenting about Fatiando a Terra (see my Scipy2013 talk). The poster is about some of the work me and Vanderlei had been doing on a general purpose inverse problems framework.

## Abstract

Inverse problems haunt the nightmares of geophysics graduate students. I'll demonstrate how to conquer them using Fatiando a Terra. The new machinery in Fatiando contains many ready-to-use components and automates as much of the process as possible. You can go from zero to regularized gravity inversion with as little as 30 lines of code. I'll walk through an example to show you how.

The inner properties of the Earth can usually only be inferred through indirect measurements of their effects. For example, density variations cause disturbances in the gravity field and seismic velocity variations affect the path of seismic waves. From a mathematical point of view, this inference is an inverse problem. To complicate things, geophysical inverse problems are usually ill-posed, meaning that a solution:

• doesn't exist;
• exists but is non-unique;
• exists and is unique but is unstable;

These problems can usually be resolved through least-squares estimation and regularization.

Research in geophysical inverse problems involves the development of: new methodologies for parametrization, different approaches to regularization, new algorithms to handle large-scale problems, combinations of existing methods, etc. All of the aforementioned developments require the creation of software, usually from scratch. Furthermore, most scientific software are not designed with reuse in mind, making remixing published methods difficult, if not impossible.

We tackled these problems by developing `fatiando.inversion`, a framework for solving inverse problems in Fatiando a Terra. The goals of `fatiando.inversion` are:

• Enable writing code that intuitively maps to the theory (equations);
• Provide a consistent interface for all solvers (similar to that adopted by scikit-learn);
• Automate the process of implementing a new inverse problem;
• Allow reuse and remixing with as little code as possible;

In this talk, I'll briefly cover the mathematics involved and the design of our new API. I'll walk through the process of implementing a new inverse problem (in about 30 lines of code) using the example of estimating the relief of a sedimentary basin from its gravity anomaly. Finally, I'll conclude by outlining how we are using this framework in our own research, what we are currently working on, and our plans for the future.

## Bonus

As a bonus, I made this gif for the Twitter hashtag #scipy2014: