Entry requirements

120 credits including 80 credits in physics and mathematics.

Learning outcomes

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

• Explain the connection between linear regression, parameter estimation and inverse theory.

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

• Explain 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 critically interpret the trade-off between resolution, bias and uncertainty of the Tikhonov solution.

• Explain 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.

• Give an account of and apply the basic methods for solving non-linear equations.

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

• Explain 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, and 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 (7 ECTS) and compulsory part (3 ECTS).