Entry requirements

Bachelor's degree in physics

Learning outcomes

After successful completion of the course, the student is expected to:

- Understand the connection between linear regression, parameter estimation and inverse theory.

- Derive the closed form solutions to a variety of linear least squares problems.

- Understand the connection between continuous models and representers.

- Derive the Singular Value Decomposition, describe the properties of the natural inverse solution and implement an algorithm for solving a simple geophysical linear inverse problem.

- Describe the principles of Tikhonov regularisation and understand the trade-off between resolution, bias and uncertainty of the Tikhonov solution.

- Understand the basic principles behind iterative methods for solving large linear systems of equations and use the conjugate gradient method to solve a simple geophysical linear inverse problem.

- Use Fourier transforms to solve the deconvolution problem by water levelling regularisation.

- Understand basic methods for solving non-linear equations.

- Solve a simple geophysical non-linear inverse problem by Occam regularisation.

- Understand Bayesian approaches to inverse solutions and use apriori information to solve a simple geophysical inverse problem.

Content

- Short review of mathematical tools

Linear algebra

Statistics

Vector algebra

- Linear regression and linear inverse problems

- Discretisation of continuous inverse problems

- The Singular Value Decomposition

- Tichonov regularisation

- Other methods of regularisation

- Fourier techniques

- Iterative methods, including the conjugate gradient method

- Non-linear regression and non-linear inverse problems including Occam's method

- Bayesian methods

Instruction

Lectures, homework, problem solving and computer solution of simple geophysical inverse problems using MATLAB.

Assessment

Oral examination. The oral examination corresponds to 7 ECTS and the compulsory part to 3 ECTS.