Geophysical and environmental modeling

A.Y. 2020/2021
Overall hours
Learning objectives
To provide the students with a basic knowledge on some of the modeling methods used in geophysical and environmental problems. The main focus will be on providing the students with knowledge and skills for application of statistics and the theory of stochastic processes (including some techniques of data mining), for the numerical solution of partial differential equations (finite differences/finite volumes, finite elements/spectral methods), for the spectral analysis of geophysical data sets, and for model calibration via solution of inverse problems.
Expected learning outcomes
The students will be able to: 1) read and understand scientific papers and books to expand their knowledge on the topics presented during the lectures; 2) read and critically analyze technical reports where simulation models or data processing tools are described and applied to environmental and geophysical problems; 3) set up in a proper way the modeling of phenomena relevant in the environmental and geophysical fields.
Course syllabus and organization

Single session

Lesson period
First semester
Lectures will be given in synchronous telematic method, with the platforms Zoom or MS Teams and will be recorded so that the video will be available through the web site of the course unit on the web portal of the University dedicated to teaching activities.

The syllabus and the study material will not be subject to any variation.

The oral exam will be held in telematic way, through the MS Teams platform, which can be accessed by following the instructions available on the web portal of the University.
Course syllabus
1) Development and application of a model.
2) Prototypical equations: conservation principles and phenomenological laws; prototypical partial differential equations.
3) Finite differences and finite volume methods for the numerical solution of flow and transport equations. Accuracy and convergence; Von Neumann stability analysis; Courant-Friedrichs-Lewy condition); finite volume method (or integrated finite differences); staggered grids; methods of solution of large linear systems (Gaussian elimination; successive over relaxation method; preconditioned conjugate gradient); explicit, implicit and Crank-Nicolson schemes for parabolic equations; nested models.
4) Spectral analysis. Signals and systems; Fourier transform (representation of discrete-time signals and systems in the frequency domain; sampling); discrete Fourier transform (representation of periodic and finite-length discrete-time signals; power spectrum).
5) Basic properties of spectral and finite elements method for the solution of flow and transport equations: weighted residuals and variational approaches.
6) Basic statistic and theory of probability; basic theory of stochastic processes.
7) Modeling solute transport. Analytical solutions of transport equation. Lagrangian models: particle tracking method for convective transport (Runge-Kutta method for ordinary differential equations). Stochastic models (Monte Carlo methods for the advective-dispersive equation (Kolmogorov-Dmitriev theory of branching stochastic processes; discrete-time random walk).
8) Data mining and data analysis. Regression methods; multivariate analysis (Principal Component Analysis & Factor analysis); clustering algorithms; wavelets; applications of stochastic processes to geophysics and to environmental physics (Kriging; Kalman filter; Markov chains); artificial neural networks.
9) Model calibration by solution of inverse problems. General definition and properties of the discrete inverse problem; the null space; use of prior information and weighted least squares; Bayesian approach and the maximum likelihood method; gradient-based optimization algorithms (Newton's method; Levenberg-Marquardt's algorithm); global optimization methods (genetic algorithms; differential evolution; simulated annealing); sensitivity analysis (one-at-a-time approach; adjoint-state equation; first-order sensitivity).

Students' project. Each student will be asked to complete a project by developing original codes or by application of freely available software tools (e.g., Princeton Ocean Model, CALMET/CALPUFF/CALGRID package, MODFLOW/MODPATH package).

Optional short course
* A tutorial on Sobolev spaces and weak solutions of partial differential equations
- Weak derivatives and Wk,p Sobolev spaces;
- Weak solutions to elliptic partial differential equations/boundary value problems.
* The inverse problem of identifying the leading coefficient of the prototypical elliptic equation: 1. Classical results and difficulties
- Calderón's inverse conductivity problem: the state-of-the-art until 2000;
- Determining the conductivity in transport media: integration along characteristic lines.
* The inverse problem of the identification of the leading coefficient of the prototypical elliptic equation: 2. Recent results
- Results on Calderón's problem in the third millennium;
- Univalent sigma-harmonic mappings and their relevance to inverse problems.
* The inverse problem for the prototypical reduced wave equation
- The Helmholtz equation;
- Full waveform inversion.
Prerequisites for admission
Good knowledge of differential and integral calculus and basic knowledge of statistics. Good knowledge of classical physics. Basic computer skills. Knowledge of a programming language is a useful, but not mandatory, prerequisite.
Teaching methods
Standard lectures. Fast quizzes (multiple-choice questions) will be proposed during lectures, whereas autonomous activities will be proposed as homework between successive lectures; these activities will be performed with telematic synchronous and/or asynchronous instruments.
Autonomous activity by students, through the development or the application of computer codes for a modeling exercise. At the end of this work, the student will write a report to describe the work done.
In the A.Y. 2020/21 an optional short course will be given by a visiting professor, alternating frontal lectures and open discussion with students.
Teaching Resources
The lecture notes can be downloaded from the website
These notes include useful references for appropriate detailed study.
The material will be progressively moved to Moodle and will be accessible through the link
Assessment methods and Criteria
Preparation of a written report related to the modeling project developed by each student.
Oral exam (discussion of the report and questions on the topics of the lectures) to verify the acquisition of knowledge about the topics taught during lectures.
Results of quizzes (multiple-choice questions) and homework (open questions and exercises) performed during the semester will provide an additional, integrative assessment.

For the written report the assessment criteria are the ability to describe in a clear and rigorous way the results, the skill in the use of the specialistic lexicon and the ability to apply what has been taught during the lectures.
For the oral exam, the assessment criteria are the ability to organize the presentation of knowledge and the mastery of the results illustrated in the report and of the topics taught during the lectures.

The final evaluation is expressed with a mark in thirtieth and accounts for the assessment of the written report and of the oral exam; the assessment of the results of quizzes and homework can give an incremental, additional bonus for the final mark.
Lessons: 42 hours
Professor: Giudici Mauro
By phone or mail appointment
via Cicognara 7