PHYSICAL METHODS APPLIED TO EARTH SCIENCES

To provide knowledge and skills in the field of physical, mathematical and computational methods applied to Earth Sciences, as tools for the processing, analysis and modelling of geological and geophysical data and problems. The course also provides basic programming knowledge for the development of simple codes aimed at analysing data acquired in the fields of geology and geophysics.

Knowledge and understanding

The course is delivered through lectures, laboratory exercises and seminars. Lectures are aimed at providing the theoretical and methodological knowledge related to physical, mathematical, statistical and computational methods applied to Earth Sciences. Laboratory exercises allow students to apply this knowledge to the formulation and solution of simple quantitative problems, the analysis of geological and geophysical data, the graphical representation of results, and the development of simple calculation procedures and analysis codes.

If the course is delivered in blended or remote mode, appropriate adjustments may be made to the above, in order to ensure consistency with the syllabus.

1. Alignment of basic mathematical and computational knowledge: Real-valued functions of a real variable. Limits and continuity. Derivatives and differentiation rules. Riemann integral. Fundamental theorem of calculus. Common integrals. Improper integrals. Review of signals and time series: amplitude, period, frequency, and phase. Continuous and discrete signals: sampling frequency, sampling interval, Nyquist frequency, aliasing, and temporal and spectral resolution. Signals in the time domain and in the frequency domain.
2. Differential and integral calculus: Functions of several variables. Limits and continuity, partial derivatives, differential and differentiable functions, higher order derivatives and Schwartz's lemma. Taylor series. Extremes. Integral calculus for functions of one variable. Riemann integral. Fundamental theorem of integral calculus. Remarkable integrals. Improper integrals. Integral calculus for functions of several variables. Vector differential calculus. Differential operators: gradient, divergence, curl and Laplacian. 
3. Numerical series and series of functions, Fourier analysis: Number series. Convergence and absolute convergence. General theorems on numerical series. Convergence test of series with positive terms, series with alternating signs. Harmonic series, geometric series. Function series. Punctual and uniform convergence. Power series. Taylor and MacLaurin series. Fourier series and their convergence. Examples and applications: square and triangular wave. Review of complex numbers. Fourier transforms. Spectral analysis of a signal. 
4. Probability: Samplings. Binomial coefficients. Conditional probability. Independent and mutually independent events. Bayes theorem. Random variables. Probability mass function. Expectation values and variances of Bernoulli, binomial, geometric and Poisson. Rare events and radioactive decay. Two or more random variables. Joint and marginal distribution. Independent variables. Covariance and correlation coefficient. Conditional expectation values. Applications to the random walk. Continuous random variables. Cumulative distribution and probability density. Uniform, exponential and normal distributions. Central limit theorem.
5. Application 1: univariate and bivariate statistics: Fundamentals of Matlab programming: variables, indexing, operators, matrix operations, scripts, functions, 2D plot, 3D plot, flow control (if statement, for loop, while loop). Empirical distributions, measure of central tendency and dispersion, correlation coefficients, linear regression, estimation of regression coefficients by bootstrap, jackknife and cross-validation.
6. Application 2: time series analysis: Creating signals in time domain, spectral analysis, spectral analysis of non-stationary signals (Short Time Fourier Transform, Wavelet Power Spectrum), comparison between signals via cross wavelet spectrum and wavelet coherence, interpolation in one dimension.

The exam consists of an oral interview lasting approximately 30 minutes, aimed at assessing the student’s knowledge of the theoretical and methodological topics covered in the course and their ability to apply mathematical, physical, statistical and computational tools to the description and analysis of problems of geological and geophysical interest. Students may start the exam by presenting a topic of their choice.

The assessment will take into account the correctness and completeness of the answers, the ability to formulate and discuss simple quantitative problems, the ability to connect different topics, the appropriate use of technical and scientific language, and the ability to discuss procedures, data, graphs and results derived from laboratory exercises.

Learning assessment may also be carried out on-line, should the conditions require it.

To ensure equal opportunities and in compliance with current laws, interested students may request a personal interview in order to plan any compensatory and/or dispensatory measures based on educational objectives and specific needs. Students can also contact the CInAP (Centro per l’integrazione Attiva e Partecipata — Servizi per le Disabilità e/o i DSA) referring teacher within their department (https://www.cinap.unict.it/content/referenti).

What is the differential of a function? 

What are the critical points for a function of two variables. 

Give the definition of integral according to Riemann 

Calculate the primitive of a given function 

What is an improper integral 

What is the geometric series? How can we prove its convergence? And to what value does it converge? 

Talk about Fourier series and Taylor series. 

How is a Fourier transform defined and what is it used for? 

Give the definition of conditional probability. 

Prove Bayes' theorem. 

Calculate expectations and variances for Bernoulli, binomial, geometric and Poisson distributions.

What is meant by empirical distribution?

What are mean, median, mode and standard deviation and how are they calculated?

Talk about the main correlation coefficients.

Talk about spectral analysis.

What are the main differences between Short-Time Fourier Transform and Wavelet Power Spectrum?