Ability to create of 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.

Integration of concepts of resolution and error estimation for practical problems.

Time series analysis with non-Fourier methods. Understanding of basic statistics, confidence, implications of hypothesis testing.

(LO1) Knowledge and understanding of    eigenvalue analysis and its application to data analysis; implications of model existence, uniqueness for interpretation; basic statistics, including confidence testing, central limit theory

(LO2) Interpretation of statistical results, geophysical modelling of real data set, amd understanding and application of concepts of resolution, error estimation, and quantification of model quality (and of its limitations.

(LO3) Inverting a large data set to give a geophysical model, and time series analysis from optimisation.

(LO4) Programming skills, including fluency in a unix/linux operating system, and shell programming.

(S1) Problem solving skills

(S2) Numeracy

(S3) Communication skills

(S4) IT skills

Optimisation, including Lagrange multipliers;

Mathematical methods problem session;

Eigenvalues and eigenvectors, particularly of real symmetric matrices;

Mathematical methods problem session;

Error analysis and basic probability and statistics. The Normal distribution and Central Limit Theorem;

Central limit theorem demonstration practical;

Hypothesis testing, regression, Chi-squared;

Least squares application to a geophysical problem;

Introduction to inversion through linear algebra;

The generalised matrix inverse;

Least-squares inversion worked examples;

Philosophy of inverse theory: existence, uniqueness, construction and stability;

Least-squares foundations of the generalised inverse;

Least squares inversion worked examples;

Practical solution: Ranking and winnowing, regularisation. Numerical methods (Cholesky decomposition);

Data error covariance. A priori model covariance. Solution resol ution and covariance;

Project session one - Introduction to problem and solution formation;

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;

Alternative methods of error estimation - bootstrapping;

Project session two. Solution space investigation;

Examples of hypothesis testing applied to inverse theory;

Introduction to non-linear inversion, including simulated annealing Outliers;

Project session three: Resolution, covariance, the Eigen problem;

Earthquake location - worked example;

Introduction to tomography as a non-linear inverse problem.

Splines:

Non-linear inversion example;

Practical examples of inverse theory;

Time series analysis with optimisation - splines;

Advanced topics:

Time series analysis with splines

Teaching Method 1 - Lecture
Description: Basic background and skills instruction
Attendance Recorded: Yes

Teaching Method 2 - Laboratory Work
Description: Neptune practical and spline exercise included
Attendance Recorded: Yes
Unscheduled Directed Student Hours (time spent away from the timetabled sessions but directed by the teaching staff): 24