Ε4201 - MATHEMATICAL METHODS IN GEOSCIENCES

INSTRUCTORS

COURSE KEY ELEMENTS

COURSE CONTENT:

Combination of theoretical introductions (lectures) and practical training with scientific computing engines (MATLAB or OCTAVE) and their associated signal, modeling and statistical analysis toolboxes.

• 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.

LEARNING ACTIVITIES - TEACHING METHODS:

PLANNED LEARNING ACTIVITIES:

ASSESSMENT METHODS AND CRITERIA

Students are evaluated by a formative assessment process in Greek. Foreign students from European Union countries (attending through the Erasmus programme) may be evaluated by the same process in English.

• The final grade is the arithmetic mean of the grades of all reports prepared and submitted as part of the practical training program

RECOMMENDED BIBLIOGRAPHY

Suggested Bibliography:

• Μαθηματικές Μέθοδοι Physicsς Τόμος Ι, Βεργάδος Ι. [Κωδ. ΕΥΔΟΞΟΣ: 230]
• Μάθετε το MATLAB 7, D. Hanselman, B. Littlefield [Κωδ. ΕΥΔΟΞΟΣ: 13789]

Additional Teaching Matterial:

• Moller, C., «Numerical computing with MATLAB», MathWorks Inc., 2004 (PDF)
• Trauth, M.H., «MATLAB® Recipes for Earth Sciences», Springer, 2007.
• Snieder, R., 1997, “A guided tour of Mathematical Physics”, Samizdat Press [ PDF]
• Αναλυτικές Σημειώσεις Διδασκόντων (άνω των 140 σελίδων) και ύλη ασκήσεων α-ναρτημένες στην η-Τάξη

Optional Literature for further Study:

• Βέργαδος, Ι., «Μαθηματικές Μέθοδοι Physicsς», Τόμος ΙΙ
• Τραχανάς, Σ., «Διαφορικές Εξισώσεις, Τόμος Ι Συνήθεις Διαφορικές Εξισώσεις»
• Τραχανάς, Σ., «Μερικές Διαφορικές Εξισώσεις»
• Βεργάδος, Ι., «Μαθηματικές Μέθοδοι Physicsς», Τόμοi Ι & II, Πενεπιστημιακές Εκ-δόσεις Κρήτης.
• Arfken, G.B and Weber, H.J., 2005. Mathematical Methods for Physicists, 6th Edition, Elsevier.
• Scales, J.A. et al., 2001. Introductory Geophysical Inverse Theory, Samizdat Press. (PDF)
• Claerbout, J., 1976. Fundamentals of Geophysical Data Processing, Samizdat Press.
• Claerbout, J., 1996, Imaging the Earth’s Interior, Samizdat Press.
• Turcotte, D.L., 1997. Fractals and Chaos in Geology and Geophysics, Cambridge University Press.

¹ V.S.: Visitor Students (e.g. ERASMUS)