Syllabus
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.
Annotation
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.