- Docente: Andrea Morelli
- Credits: 4
- SSD: GEO/10
- Language: Italian
- Teaching Mode: Traditional lectures
- Campus: Bologna
- Corso: Second cycle degree programme (LM) in Physics (cod. 8025)
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.
Fully non-linear methods: Monte-Carlo, simulated annealing,
neighbourhood.
Method of 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.
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
Ralph A. Wiggins, 1972, General linear inverse problem --
implication of surface waves and free oscillations for earth
structure, Reviews of Geophysics and Space Physics, 10,
251-285.
Teaching methods
Lectures with overheads and problems.
Assessment methods
Oral exam
Teaching tools
Sample problems and computer exercises.
Office hours
See the website of Andrea Morelli