2023/24 Undergraduate Module Catalogue

SOEE3250 Inverse Theory

10 creditsClass Size: 20

Module manager: Prof Phil Livermore
Email: P.W.Livermore@leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2023/24

Pre-requisite qualifications

Students require a solid background in university level maths (particularly matrix algebra) and working knowledge of Python.

This module is mutually exclusive with

SOEE5116M	Computational Inverse Theory
SOEE5675M	Inverse Theory

This module is not approved as a discovery module

Module summary

Given a description of a physical system, we can predict the values of any measurements we might make. This is known as a forward problem. The inverse problem is to use actual measurements to constrain the values of the parameters that characterise the system. Various approaches can be taken to solve an inverse problem depending on the linearity of the forward problem, the form of the measurement errors, the non-uniqueness of solutions and the number of model parameters and observations. This module will cover how to characterize any specific geophysical problem and choose, then implement, an appropriate approach. Students will learn the theoretical basis behind different approaches and also put them into practice using Python on a range of geophysical problems.

Objectives

To provide training in the design and solution of inverse problems, including model formulation and parametrisation, over- and under-constrained problems, linear and non-linear solution methods. To provide an understanding of how to quantify the uncertainty in a solution, based on data uncertainty and model setup.

Learning outcomes
After completing this module, students will be able to
1. Formulate inverse problems
2. Explain the difficulties inherent in inverse problems
3. Solve linear inverse problems using least-squares
4. Linearise and solve non-linear inverse problems
5. Describe and implement methods for regularization of ill-posed problems
6. Formulate inverse problems in terms of probability distributions
7. Solve inverse problems using Markov chain Monte Carlo algorithms
8. Describe and implement some machine learning algorithms.

Syllabus

Formulation of inverse problems, linear least-squares, best linear unbiased estimator (BLUE), propagation of errors, maximum likelihood solutions, linearisation of non-linear problems, Monte Carlo error propagation, ill-posed problems, resolution matrix, regularization, cross validation, Bayesian inference, Markov chain Monte Carlo algorithms, neighbourhood algorithms, machine learning.

Teaching methods

Delivery type	Number	Length hours	Student hours
Lecture	10	1.00	10.00
Practical	10	2.00	20.00
Private study hours			70.00
Total Contact hours			30.00
Total hours (100hr per 10 credits)			100.00

Private study

Completion of practical problems (10 x 2 hours).
Background reading for lectures (10 x 2 hours).
Exam preparation and revision (1 x 30 hours).

Opportunities for Formative Feedback

Continuous monitoring during practicals with immediate formative assessment and feedback. Coursework provides a mixture of summative (counts towards 20% of the final mark) and formative assessment. Weekly short answer questions will build towards a cumulative answer to a mock exam; formative feedback will be given on answers.

Methods of assessment

Coursework

Assessment type	Notes	% of formal assessment
In-course Assessment	Continuous assessment	20.00
Total percentage (Assessment Coursework)		20.00

The re-sit consists of a single written exam only.

Exams

Exam type	Exam duration	% of formal assessment
Online Time-Limited assessment	1 hr 30 mins	80.00
Total percentage (Assessment Exams)		80.00

The re-sit consists of a single written exam only.

Reading list

The reading list is available from the Library website

Last updated: 21/08/2023

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