The Course is designed so as to maintain a fair technical level and focus on principal data reduction and analysis techniques appropriate for a broad spectrum of Earth-scientific problems.
Introduction to MATLAB/OCTAVE with parallel review of the principles of Linear Algebra; Fourier analysis, Fourier series and the Fourier transform. Power spectra and their physical interpretation. Concepts of sampling and digitization. The z-transform. Correlation and Convolution. Fast Fourier Transforms. Examples and applications in the analysis of natural phenomena; Coordinate systems, vector spaces and metric spaces. Matrices and their properties. Metric tensors: concepts, properties and utilization. Eigenvalue/eigenvector decomposition, singular value decomposition and their physical interpretation. Applications to the analysis of matrices and images; applications to geophysical and geotechnical problems – analysis of the stress, strain and impedance tensors; Solution of linear systems of equations with applications to earth-scientific problems; Simulation and modelling of data and physical processes: Linear, general and non-linear least squares. Multiple Linear Regression and applications. Non-linear least-squares inversion theory and applications; Linear Filters and Systems. Transfer functions and causality. Wavelets and wavelet transforms. Applications to the description of physical systems, time series, maps and images. Data smoothing and accentuation; application to time series, maps and images; Interpolation and extrapolation in one dimension (interpolating polynomial, linear and non-linear interpolation techniques). Interpolation in two and three dimensions with introduction to the concepts of triangulation and tessellation. Geostatistical interpolation methods (e.g. Kriging); Introduction to fractals and fractal objects. Fractal distributions and fractal clustering. Dynamic systems and self-organized criticality – introduction to the non-extensive statistical mechanics. Examples from the Earth Sciences (terrain, drainage systems, coastlines, fragmentation and porosity, faulting and tectonics, seismicity and seismogenesis, etc.); Simple differential equations: concept and solutions. Examples and applications (radioactive decay, remanent magnetization, geothermal gradient); Non-linear differential equations: basic concepts and applications; Partial differential equations (Laplace, diffusion, wave): Concepts and solution. Examples and applications (e.g. static potentials, hear transfer, wave diffusion and propagation); Numerical solution of partial differential equations – the finite difference approach with examples and applications.