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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.
Last update: Macharová Dana, JUDr. (27.03.2008)
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The lecture explains and compares various approaches to solving inverse problems, and presents examples from several branches of geophysics. Last update: T_KG (11.04.2008)
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Oral exam Last update: Gallovič František, prof. RNDr., Ph.D. (10.06.2019)
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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.
Last update: Macharová Dana, JUDr. (27.03.2008)
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Lecture Last update: T_KG (11.04.2008)
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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.
Last update: Macharová Dana, JUDr. (27.03.2008)
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