PHYSICAL METHODS APPLIED TO EARTH SCIENCES
Academic Year 2026/2027 - Teacher: VITO CLAUDIO LATORAExpected Learning Outcomes
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
- knowledge of the main mathematical, physical and statistical tools needed to quantitatively describe processes and data of
geological and geophysical interest;
- ability to apply inductive and deductive reasoning to the description and interpretation of physical and geophysical phenomena;
- ability to formulate simple problems using appropriate relationships between physical quantities, of algebraic, integral or
differential type, and to understand their solution through analytical methods.
Applying knowledge and understanding
- ability to apply the acquired knowledge to the description of physical phenomena using the scientific method rigorously;
- ability to apply mathematical, physical and statistical tools to the analysis and modelling of geological and geophysical
problems;
- ability to develop simple codes for the processing, visualization and interpretation of data acquired in the fields of geology and
geophysics;
- ability to use programming/scripting tools for the analysis of scientific data and time series.
Making judgements
- ability to critically select mathematical, statistical and computational methods appropriate to the type of data, the scientific
problem and the geological-geophysical context;
- ability to critically evaluate data, results and models, recognising their limitations, uncertainties and fields of applicability,
developed through laboratory activities.
Learning skills
- ability to autonomously update one’s knowledge of mathematical, statistical and computational methods applied to Earth
Sciences, also through the consultation of scientific literature, technical manuals and software documentation, developed through
laboratory activities.
Course Structure
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.
Required Prerequisites
For a fruitful understanding of the topics covered in the course, it is important to have basic knowledge of mathematics and
general physics, with particular reference to fundamental physical quantities, functions, elements of algebra, and the graphical
representation of data and functions. Basic knowledge of computational tools for data management, processing and visualization
is also useful. These knowledge requirements are to be understood as useful cultural prerequisites and not as formal course
prerequisites.
Attendance of Lessons
Mandatory.
Detailed Course Content
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.
Textbook Information
1. Dispense
2. Trauth, M. H., (2021). MATLAB recipes for earth sciences (Fifth edition). Berlin: Springer.
3. Trauth, M. H. (2021). Signal and Noise in Geosciences. Springer International Publishing.
4. CM Grinstead, JL Snell, Introduction to probability (Second revised Edition, American Mathematical Society 1997)
Course Planning
| Subjects | Text References | |
|---|---|---|
| 1 | Alignment of basic mathematical and computational knowledge | Lecture Notes |
| 2 | Differential and integral calculus | Notes |
| 3 | Series | Notes |
| 4 | Probability | Notes |
| 5 | Basics of Matlab | Notes |
| 6 | Univariate statistics | MATLAB recipes for earth sciences (Fifth edition), capitolo 3 |
| 7 | Bivariate statistics | MATLAB recipes for earth sciences (Fifth edition), capitolo 4 |
| 8 | Time series analysis | MATLAB recipes for earth sciences (Fifth edition), capitolo 5. Signal and Noise in Geosciences, capitolo 7 |
Learning Assessment
Learning Assessment Procedures
he 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).
Examples of frequently asked questions and / or exercises
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?