Syllabus
1. Basic notions Importance of inverse problems in contemporary geophysics. A brief review of the historical development. 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.
3. Linear inverse problem Space of parameters and data. Covariance of parameters and data, mutual relation. Null-space and range. Resolution matrix. System identification.
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. Artificial neural networks.
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
Forward and inverse problem, linear and non-linear inversion, inversion versus optimization, model parametrization and propagation of errors, stochastic inversion, genetic and evolution algorithms