31627 - Inverted Problem in Geophysics

Learning outcomes

At the end of the course, the student will have basic knowledge about inverse problem theory and about its application to the solution of geophysical problems. The student will be able to set and solve linear inverse problems using probabilistic linear least-squares methods; he/she will understand how to approach and solve non-linear problems; he/she will know the basic features of three-dimensional earth structure as determined by seismic tomography.

Course contents

Description of inverse problems in Geophysics. Elements of Bayesian statistics. An introduction to a probabilistic approach to the solution of inverse problems. Model space, data space, parameter space. States of information and codification of information deriving from physical theories, from measurements, and from a priori knowledge. Combination of states of information. General solution to the inverse problem and special cases. Characterization of the solution. Inverse problems with Gaussian distributions. Linear problems. Data outliers and L1 minimisation. Generalized Gaussian. Non-linear inverse problems, physical interpretation. Solution to non-linear inverse problems with iterative optimization techniques. Method by Backus and Gilbert: concept of resolution. Classical solutions to least-squares problems for over-determined and under-determined cases, damped least squares. Singular value decomposition (SVD). Fully non-linear methods: Monte-Carlo, simulated annealing, neighbourhood. Application to the determination of three-dimensional earth structure by seismic tomography.

Readings/Bibliography

Edwin T. Jaynes: Probability Theory as Extended Logic, Cambridge University Press, 2003.  
Albert Tarantola: Inverse Problem Theory and Methods for Model Parameter Estimation, SIAM, 2005.  
Keiiti Aki and Paul G. Richards, Quantitative Seismology: Theory and Methods, W. H. Freeman and Co., First Edition, 1981  

Teaching methods

Theoretical lectures with presentation of practical examples and problems solved by means of computer programs.

Assessment methods

Oral exam

Teaching tools

Sample problems and computer exercises.

Office hours

See the website of Andrea Morelli