Ability to create geophysical models from data. Practical experience in inversion of mathematically linear problems, with knowledge of how to approach more general nonlinear problems.

Understanding of the limitations of such models, and how they should be interpreted, with particular reference to model non-uniqueness and instability. Optimisation theory, and its application to interpretation of geophysical models. Time series analysis with non-Fourier methods.

Understanding of basic statistics, confidence.

(LO1) Knowledge and understanding of mathematical fundamentals of data modelling, including eigenanalysis, statistical foundations and implications of model existence and nonuniqueness for interpretation.

(LO2) Interpretation of statistical results and Geophysical modelling of real data sets

(LO3) Ability to invert a large data set to give a geophysical model

(LO4) Programming skills, in particular the ability to work in a unix/linux environment with shell programming

(S1) Problem solving skills

(S2) Numeracy

(S3) Communication skills

(S4) IT skills

One. Optimisation, including Lagrange multipliers
Two. Mathematical methods problem session.
Three. Eigenvalues and eigenvectors, particularly of real symmetric matrices
Four. Mathematical methods problem session
Five. Error analysis and basic probability and statistics. The Normal distribution and Central Limit Theorem
Six. Central limit theorem demonstration practical
Seven. Hypothesis testing, regression, Chi-squared.
Eight. Least squares application to a geophysical problem.
Nine. Introduction to inversion through linear algebra.
Ten. The generalised matrix inverse
Eleven. Least-squares inversion worked examples (one)
Twelve. Philosophy of inverse theory: existence, uniqueness, construction and stability
Thirteen. Least-squares foundations of the generalised inverse.
Fourteen. Least squares inversion worked examples (two)
Fifteen. Practical solution: Ranking and winnowing, regularisation. Numerical methods (Cholesky decomposition).
Sixteen. Data error covariance. A priori model covariance. Solution resolution and covariance.
Seventeen. Project session one - Introduction to problem and solution formation.
Eighteen. Interpretation through framing the inverse problem as an optimisation problem - impliations for error estimates and practical bounds on results. Introduction of hypothesis testing as an appropriate solution to this problem.
Nineteen. Alternative methods of error estimation - bootstrapping.
Twenty. Project session two - Solution space investigation.
Twenty One. Examples of hypothesis testing applied to inverse theory.
Twenty Two. Introduction to non-linear inversion, including simulated annealing. Outliers.
Twenty Three. Project session three - Resolution, covariance, the eigenproblem.
Twenty Four. Earthquake location - worked example
Twenty Five. Introduction to tomography as a non-linear inverse problem. Splines.
Twenty Six. Non-linear in version example
Twenty Seven. Practical examples of inverse theory
Twenty Eight. Time series analysis with optimisation - splines
Twenty Nine. Advanced topics, including how course material leads into "Big data" and Machine learning.

Teaching Method 1 - Lecture
Description: Description and definition of methods.
Attendance Recorded: Yes

Teaching method 2 - Pre-recorded lectures
Description: Basic content covered in recorded lectures to allow more timem for interactive teaching in main lecture

Teaching Method 3 - Laboratory Work
Description: Application of methods, particularly to mathematical problems and the modelling of the magnetic field of Neptune
Attendance Recorded: Yes
Unscheduled Directed Student Hours (time spent away from the timetabled sessions but directed by the teaching staff): 18

Teaching Method 4: Online office hours
Two hours of availability each week for discussion. To be extended if there is demand,