# 2021/22 Taught Postgraduate Module Catalogue

## SOEE5675M Inverse Theory

### 10 creditsClass Size: 10

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

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2021/22

### 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

 SOEE3250 Inverse Theory SOEE5116M Computational Inverse Theory

This module is not approved as an Elective

### 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

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

 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

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Coursework
 Assessment type Notes % of formal assessment Practical Numerical Problems 20.00 Total percentage (Assessment Coursework) 20.00

All resits will be in the same format.

Exams
 Exam type Exam duration % of formal assessment Open Book exam 2 hr 80.00 Total percentage (Assessment Exams) 80.00

Re-sits will be assessed by a single 2 hour exam only (no coursework)