# 2024/25 Undergraduate Module Catalogue

## SOEE3250 Inverse Theory

### 10 creditsClass Size: 29

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

**Taught:** Semester 1 (Sep to Jan) View Timetable

**Year running** 2024/25

### 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, optimisation 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 |

Standard exam (closed essays, MCQs etc) (S2) | 1 hr 30 mins | 80.00 |

Total percentage (Assessment Exams) | 80.00 |

A student who fails this Module may be offered a resit. The re-sit for this module will be a single unseen examination, of duration 1.5 hours. If the re-sit is granted as a new first attempt, the original examination mark will be discarded, and replaced by the re-sit examination mark even if it is lower. It will then be aggregated with the first-attempt coursework to provide a new Module mark. If the re-sit is a second and final attempt, the re-sit mark provides a new alternative mark for the whole Module and will be capped at 40%.

### Reading list

The reading list is available from the Library websiteLast updated: 25/06/2024

## Browse Other Catalogues

- Undergraduate module catalogue
- Taught Postgraduate module catalogue
- Undergraduate programme catalogue
- Taught Postgraduate programme catalogue

Errors, omissions, failed links etc should be notified to the Catalogue Team.PROD