Uncertain simulations: towards probabilistic modelling for earthquake early warning

Many scientific and practical scenarios require us to take decisions based on data that is known to be incomplete and/or subject to significant uncertainties. For example, earthquake and tsunami early warning (ETEW) requires us to predict earthquake impacts using whatever information may be available in the seconds and minutes immediately following rupture. In such settings, probabilistic methods provide a rigorous mathematical framework for representing both knowns and unknowns.

However, computational simulations of physical processes – such as the propagation of seismic waves – are typically deterministic, requiring a discrete set of inputs and return discrete outputs. As a result, incorporating physical models into probabilistic inference typically requires sampling: a representative ‘ensemble’ of many sets of input parameters is generated, simulations are performed for each, and then a probability distribution is inferred to match the ensemble of outputs. This is a computationally-inefficient and computationally-expensive process, and a major bottleneck in inference pipelines.

This project will investigate the feasibility of inherently-probabilistic physical simulation: can we develop computational models that can directly map from a probability distribution in the input space to a probability distribution in the output space? What are the limitations, challenges and opportunities? How would this be employed within a real-time inference or prediction framework? Exploration of these questions will be driven by practical issues faced in real-time analysis of terrestrial and glacial seismic events, but the work has potential impact and application across a wide range of problems.


This project will explore a range of concepts and tools drawn from machine learning and statistics, including Gaussian processes, generative models and neural operators. The project is primarily theoretical and computational, and the student will design, develop and test computer models using Python and a machine learning framework such as pytorch (or similar). The project focus will likely evolve to reflect the progress and discoveries made by the student, and the timeline below is indicative only. Work will focus initially on toy problems – such as modelling simple harmonic oscillators – to establish feasibility, and then investigate progressively more-complex scenarios in earthquake modelling.

Project Timeline

Year 1

• Background reading and familiarisation with conventional, deterministic strategies for modelling geophysical systems, and with machine learning methods;
• Identification and establishment of exemplar problems for development and testing;
• Investigation of feasibility/applicability of probabilistic modelling using existing tools;
• Exploration of generative models, physics-informed machine learning and neural operators for probabilistic simulations;
• Visit to BAS in Cambridge;
• Attend and present at national meeting e.g. BGA PGRIP meeting.

Year 2

• Development and testing of probabilistic simulation for simple geophysical problems;
• Publication on proof-of-concept and methodology;
• Development of application to rapid ground motion prediction;
• Visit to BAS in Cambridge;
• Attend and present at national meeting, e.g. BGA NAG meeting.

Year 3

• Publication on ground motion prediction
• Development of seismic source inversion by variational inference using the probabilistic forward model;
• Publication on source inversion;
• Visit to BAS in Cambridge;
• Attend and present at a major international conference, e.g. AGU/EGU.

Year 3.5

• Wrap up research;
• Finalise publications and thesis.

& Skills

The student will receive training and experience including code and algorithm development, machine learning, numerical modelling, uncertainty quantiifcation, Bayesian statistics and inverse theory. They will also have opportunities to attend relevant summer schools, and national and international research conferences.

References & further reading

Käufl, 2016. Rapid probabilistic source inversion using machine learning. PhD Thesis, Universiteit Utrecht. https://dspace.library.uu.nl/bitstream/handle/1874/321502/kaufl.pdf

Li et al., 2021. Physics-informed neural operator for learning partial differential equations. https://arxiv.org/abs/2111.03794

Smith et al., 2020. EikoNet: Solving the Eikonal equation with deep neural networks

Valentine & Sambridge, 2023. Emerging directions in geophysical inversion. In Applications of Data Assimilation and Inverse Problems in the Earth Sciences, Cambridge University Press. doi:10.1017/9781009180412.003

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