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Selected chapters from inverse problems - NDGF019

Title:	Vybrané partie z obrácených úloh
Guaranteed by:	Department of Geophysics (32-KG)
Faculty:	Faculty of Mathematics and Physics
Actual:	from 2007
Semester:	summer
E-Credits:	3
Hours per week, examination:	summer s.:2/0, Ex [HT]
Capacity:	unlimited
Min. number of students:	unlimited
4EU+:	no
Virtual mobility / capacity:	no
State of the course:	taught
Language:	Czech, English
Teaching methods:	full-time
Teaching methods:	full-time

Guarantor:	doc. RNDr. Oldřich Novotný, CSc. RNDr. Bohuslav Růžek, CSc.

Opinion survey results Examination dates SS schedule Noticeboard

Annotation -

Last update: JUDr. Dana Macharová (27.03.2008)

Notion of forward and inverse problems. Classification of the inverse problems solved in geophysics. Linear algebra, matrix operations. Least squares method and minimum norm method. Matrix regularisation. Inverse matrix, generalised inversion. Linear inverse problem. Resolution matrix. Methods of non-linear inversion and non-linear optimisation. Examples of inverse problems in geophysics: seismic tomography and seismic kinematic inversion; inversion of waveforms; inversion of magneto-telluric data; inversion of surface-wave dispersion curves.

Aim of the course -

Last update: T_KG (11.04.2008)

The lecture explains and compares various approaches to solving inverse problems, and presents examples from several branches of geophysics.

Course completion requirements -

Last update: prof. RNDr. František Gallovič, Ph.D. (10.06.2019)

Oral exam

Literature -

Last update: JUDr. Dana Macharová (27.03.2008)

R.C. Aster, B. Borchers, C.H. Thurber: Parameter Estimation and Inverse Problems. Elsevier, Amsterdam 2005.

W. Menke: Geophysical Data Analysis: Discrete Inverse Theory. (Revised Edition). Academic Press, San Diego 1989.

A. Tarantola: Inverse Problem Theory: Methods for Data Fitting and Model Parameter Estimation. Elsevier, New York 1987.

Teaching methods -

Last update: T_KG (11.04.2008)

Lecture

Syllabus -

Last update: JUDr. Dana Macharová (27.03.2008)

1. Basic notions

Importance of inverse problems in contemporary geophysics. A brief classification of inverse problems. Inverse problem versus optimisation. Deterministic versus statistical variables. Probability. Operations with random variables. Propagation of errors.

2. Linear algebra and mathematical methods of linear inversions

Matrix operations. The first and second Gauss transformations. System of linear equations with a rectangular matrix, least squares method and minimum norm method. Regularisation of matrices. Inverse matrix, generalised inversion, determinant, eigenvalues, eigenvectors. Projection matrix. Singular value decomposition. Transformation of matrices, contravariant/covariant coordinate bases.

3. Linear inverse problem

Parameter and data spaces. Covariance of parameters and data, mutual relation. Factorisation of the vector space, correlation. Null-space and range. Resolution matrix.

4. Methods of non-linear inversion and non-linear optimisation

Method of tangents/secants. Simplex method. Variable metric method. Newton-Raphson method. Monte Carlo method, Markov's chains. Genetic/evolution algorithms.

5. Examples of inverse problems in geophysics

Earthquake location, joint estimation of hypocentral and structural parameters. Seismic tomography. Inversion of waveforms. Inversion of seismic data in anisotropic models. Magneto-telluric inversion in 1D a 2D media. Inversion of surface-wave dispersion curves.