Stochastic Methods in Water Resources

Universidad Nacional de Colombia

Facultad de Ingeniería

Departamento de Ingeniería Civil y Agrícola

Bogóta


Lecturer: Luis Alejandro Morales


General information

Class schedule

HOUR MONDAY TUESDAY WEDNESDAY THURSDAY FRIDAY
7:00-8:00 MERH 01
(Salón 312)		 MERH 01
(Salón 312)
8:00-9:00 MERH 01
(Salón 312)		 MERH 01
(Salón 312)
9:00-10:00
10:00-11:00
11:00-12:00
12:00-13:00
13:00-14:00
14:00-15:00
15:00-16:00
16:00-17:00
18:00-19:00

Description

The objective of this graduate course is to provide students with a broad context of the different mathematical and computational tools and models available to analyze systems in which uncertainty exists. The key ideas underlying stochastic analysis will be presented and illustrated using various applications drawn from engineering, the physical sciences, and economics. This course aims to introduce students to the subject at an advanced level and to offer a gateway to many other high-level stochastic courses. The course introduces the use of probabilistic techniques for the characterization of hydrological processes, including random variables such as streamflow, precipitation, temperature, etc. Statistical methods for hydrological design. Analysis of extreme value problems, relationships among various hydrological variables, time series analysis, and parameter estimation.

Required prior knowledge

Students should have a clear understanding of concepts in real variables (e.g., limits, epsilon-delta arguments, etc.) and linear algebra (e.g., vector spaces, matrices, eigenvalues, eigenvectors, diagonalization). In addition, students are expected to enter the class with some basic preparation in probability (knowledge of the sample space, events, probability, conditional probability, independence, random variables, jointly distributed random variables, probability mass functions, probability density functions, expectations, the law of large numbers, the central limit theorem).

Objective

Use of probabilistic techniques for the characterization of hydrological processes, including random variables such as streamflow, precipitation, temperature, etc. Statistical methods for hydrological design. Analysis of extreme value problems, relationships among different hydrological variables, time series analysis, and parameter estimation.

General course content

1. Introduction to probability and statistics

Foundational concepts of probability and statistics, including probability laws, random variables, distributions, estimation, hypothesis testing, and regression. Emphasis is placed on developing analytical skills and applying statistical methods to real-world problems in science and engineering.

2. Hydrological statistics and extremes

Exploration of the application of statistical methods to hydrological processes, with emphasis on the analysis and modeling of extreme events such as floods and droughts. Topics include probability distributions, frequency analysis, return periods, extreme value theory, and risk assessment in water resources engineering.

3. Random functions

Introduction of the theory and applications of random functions (stochastic processes) with emphasis on their role in modeling time-dependent natural and engineered systems. Topics include stationarity, autocorrelation, spectral analysis, and applications in hydrology, physics, and engineering.

4. Time series analysis

Theory and methods for analyzing data observed over time. Topics include trend and seasonality detection, autoregressive and moving average models, spectral methods, and forecasting techniques, with applications in water resources applications.

5. Geostatistics

Introduction of statistical methods for the analysis and modeling of spatially distributed data. Topics include spatial correlation, variogram analysis, kriging techniques, and simulation methods, with applications in hydrology, geology, and environmental sciences.

6. Stochastic modelling and prediction

Formulation and application of stochastic models to represent systems influenced by randomness and uncertainty. Topics include stochastic processes, Markov chains, time series models, uncertainty quantification, and predictive methods, with applications in engineering, hydrology, and environmental sciences.

Course assessment

ASSESSMENT VALOR
Test 1 30%
Test 2 30%
Homework 40%
100%
The grading methodology will not be modified during the semester.

Homeworks

Software

For this class, an open-access GitHub repository has been created containing various programs written in R. This tool is intended to facilitate the understanding of the topics covered in class and to support the solution of practical problems. Here is the link to access the repository:https://github.com/lamhydro/sthydro

References

Recomended

  1. Bras, R.L., Rodríguez-Iturbe, I. Random Fuctions and Hydrology. Dover Publications, New-York. 1993.
  2. Kottegoda, N.T. Stochastic Water Resources Technology Macmillan, London 1980.
  3. Yevjevich, V. Stochastic Processes in Hydrology. Water Resources Publications, Fort Collins, Colorado 1972.

Others

  1. Box, G.E.P. and G.M. Jenkins, 1976. Time series analysis, forecasting and control. Revised Edition. Holden-Day, San Francisco.
  2. Christakos, G., 2000. Modern Spatiotemporal Geostatistics. Oxford University Press, Oxford.
  3. Chatfield, C., 1989. The analysis of time series. An introduction. Fourth edition. Chapman and Hall, London.
  4. Hipel, K.W. and A.I. McLeod, 1994. Time series modelling of water resources and environmental systems. Elsevier, New York.
  5. Papoulis, A., 1991. Probability, Random Variables and Stochastic processes. Third Edition. McGraw-Hill.
  6. Priestley, M.B. Spectral Analysis and Time Series Academic Press London 1981.
  7. Salas, J.D., Delleur, J.W., Yevjevich, V., Lane, W.L. Applied Modelling of Hydrologic Time Series Water Resources Publications, Fort Collins, Colorado 1997.

Program

WEEK LECTURE UNIT TOPIC LECTURE NOTES HOMEWORK
01 01 1. Introduction to probability and statistics Course introduction, definition of stochastic hydrology, other definitions Lec01
02 1. Introduction to probability and statistics Descriptive statistics, univariate and bivariate statistics. Lec02
02 03 1. Introduction to probability and statistics Definition of random variables, probability theory, probability distributions Lec03
04 1. Introduction to probability and statistics Moments, characteritics function, known probability distributions, bivariate probability distributions Lec04
03 05 2. Hydrological statistics and extremes Definitions, analysis of extreme values Lec05
06 2. Hydrological statistics and extremes Probability distributions of extremes, distribution selection Lec06
04 07 2. Hydrological statistics and extremes Statistical test, hydrological design Lec07
08 2. Hydrological statistics and extremes Markov chains, regional frequency analysis Lec08
05 09 3. Random functions Definitions, random function types, stationary random functions Lec09
10 3. Random functions Conditional random functions, spectral representation of random functions Lec10
06 11 3. Random functions Local averaging of stationary random functions Lec11
12 4. Time series analysis Introduction, stationarity and ergodicity Lec12
07 13 4. Time series analysis Time series components Lec13
14 4. Time series analysis Univariate time series: autoregresive models Lec14
08 15 4. Time series analysis Univariate time series: movil average models Lec15
16 4. Time series analysis Univariate time series: autoregressive and movil average models Lec16
09 17 4. Time series analysis Multivariate time series Lec17
18 4. Time series analysis Disaggregation models and Hurst analysis Lec18
10 19 Parcial 1
20 5. Geostatistics Introduction and descriptive spatial statistics Lec20
11 21 5. Geostatistics Spatial interpolation by Kriging Lec21
22 5. Geostatistics Spatial interpolation by Kriging Lec22
12 23 5. Geostatistics Estimating the local conditional distribution Lec23
24 5. Geostatistics Geostatistical simulation Lec24
14 25 6. Stochastic modelling and prediction Introduction, univariate explicit functions Lec25
26 6. Stochastic modelling and prediction Multivariate explicit functions Lec26
14 27 6. Stochastic modelling and prediction Stochastic diferential equations Lec27
28 6. Stochastic modelling and prediction Stochastic diferential equations Lec28
15 29 6. Stochastic modelling and prediction Introduction to Kalman filter Lec29
30 6. Stochastic modelling and prediction Kalman filter and time series Lec30
16 31 6. Stochastic modelling and prediction Kalman filter and spatial fields Lec31
32 Parcial 2

Notas:

Exams schedule

TEST DATE HOUR BUILDING ROOM
Test 1 Monday, 20-Oct-2025 7am - 9am Ed. 409 (Lab. Hidráulica) 312
Test 2 Monday, 01-Dec-2025 7am - 9am Ed. 409 (Lab. Hidráulica) 312