Module 3: Geophysical inverse problems

Phạm Thành Sơn

ARC DECRA Fellow

ANU Research School of Earth Sciences

www.tsonpham.net

Introduction to Observational Seismology

VNU Hanoi University of Science, Hanoi, 21–25 April 2025

What do we do today?

In the lecture, we learn about

• Inverse problem theory
• Parameter estimation
• Bayesian sampling and ensemble solutions

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In the exercises, we do

• Self-practice exercise: Linear regression
• Self-practice exercise: Earthquake location as an inverse problem

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Inverse problem theory

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Slides taken from Andrew Valentine’s guest lectures to PHYS3070 in 2018

Studies of earthquakes and earth structures are subject to an inference process using seismic data observed on the Earth’s surface.

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Solutions to an inverse problem are often non uniqueness, meaning multiple solutions could explain data equally well.

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We all make inferences from indirect observations

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Wild mood swings

Who are the shadows of?

by Tim Noble and Sue Webster

Courtesy of Malcolm Sambridge

Inference from indirect observations

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Wild mood swings by Tim Noble and Sue Webster

Courtesy of Malcolm Sambridge

“One of the things we don’t know is the number of things that we do not know.”

Polynomial parameter estimation

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Because it’s a linear forward problem, the least squares solution could be used.

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An inverse solution could also be a (local/global) solution of the (objective) misfit function

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Bayesian framework and �ensemble solutions

Before we show you an example of how to solve this problem from a Bayesian sampling perspective, let’s switch to a slightly different mindset:

1. Instead of getting a result as a single “best-fit” model, it’s worthwhile to obtain an ensemble of models.
2. How to express such an ensemble of models? It’s uncertain where the true model is, but given a) the data and b) some prior knowledge about the model, we can express it as a probability distribution, where p(model) is the probability at which the model is true.
3. How to estimate this distribution then? There are various ways, and sampling is a typical one of them.

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Source: https://shorturl.at/Z2rlq

The Bayes’ theorem of conditional probability

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Stuart & Ord (1994)

The Bayesian inference approach views the unknown model and observed data as stochastic variables.

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“Parameter Estimation and Inverse Problems” by Rich Aster et al. (available in Resources)

The next three slides define the prior, likelihood for Bayesian sampling of the polynomial parameter estimation problem.

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“Monte Carlo methods are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results.” (Wikipedia)

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https://en.wikipedia.org/wiki/Monte_Carlo_method

“In statistics, Markov chain Monte Carlo (MCMC) is a class of algorithms used to draw samples from a probability distribution.” (Wikipedia)

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https://chi-feng.github.io/mcmc-demo/app.html?algorithm=RandomWalkMH&target=standard

In our examples, we use the affine-invariant ensemble sampler, an advanced McMC sampling method for high-dimensional problems.

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emcee, a Python implementation of the affine-invariance ensemble sampler, is used several times in the exercises.

Hu et al., 2023, GJI

Bayesian inference provides a natural means to estimate the model parameters’ uncertainty.

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https://python.arviz.org/en/stable/api/generated/arviz.plot_trace.html

https://corner.readthedocs.io/en/latest/pages/quickstart/

Under- or over-estimation of noise levels could lead to unrealistic parameter inference.

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Hierarchical Bayesian scheme let the observed data infer their reasonable noise levels by treating it a hyper-parameter in the inference.

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Bonus project 2:

Implement the hierarchical scheme for data noise inference.

Earthquake location as �an inverse problem

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Forward problem 1: S- to P- arrival time differences as seismic observations.

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Forward problem 2: Pairwise P- arrival time differences as seismic observations.

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Summaries

• Approaches for the inverse problem include optimization and ensemble sampling.
• Ensemble sampling is often useful to tackle the non-uniqueness, but it is computationally expensive.
• Examples of the approaches are demonstrated for the polynomial parameter estimation problem.

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https://forms.office.com/r/Q3T80Hsf4T

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