University of Florida

CWR 6536
Stochastic Subsurface Hydrology

Semester Taught - Fall

Catalog Description

Credits: 3

Stochastic modeling of subsurface flow and transport modeling including geostatisitcs, time series analysis, Kalman filtering, and physically based stochastic models.


Prerequisites for the course include calculus (through ordinary differential equations), undergraduate probability and statistics, and previous coursework in subsurface hydrology. Familiarity with computer operating systems and computer programming will be assumed.


Dr. Wendy Graham
UF Water Institute
570 Weil Hall
Phone: 352-392-5893 x2113

Material/Supply Fees


Class Materials Required

Course notes will be provided at

The following texts are useful as references but not required

  • Papoulis, A., Probability, Random Variables, and Stochastic Processes, 3rd edition, Published by McGraw Hill, 1991
  • Goovaerts, P., Geostatistics for Natural Resources Evaluation , Published by Oxford University Press, 1997.
  • Gelhar, L. Stochastic Subsurface Hydrology, Published by Prentice-Hall, 1993.

Recommended Journal Article Reading List: Stochastic Subsurface Hydrology Reading List 2012.docx

Course Outline

Two lectures/discussions each week.

  1. Review of Probability and Random Field Theory: Lectures 1-8
    (Weeks 1-5)
    1. Properties of Random Variables and Random Fields
      1. Distributional assumptions
      2. Moments
      3. Spatial and temporal correlation
      4. Stationarity & ergodicity
    2. Estimation of Properties from field data
      1. Estimating the mean and variance
      2. Estimating the covariance and variogram
      3. Estimating the cross-covariance and cross-variogram
      4. Estimating the pdf and cdf
    1. Mean, covariance and variogram models
      1. stationary & nonstationary mean models
      2. hole and non‑hole covariance & variogram models
      3. isotropic & anisotropic covariances & variograms

Case Study: Estimation of pdfs, sample means, covariances, cross covarainces, variograms and cross variograms from mystery random fields.

  1. Physically-Based Stochastic Modeling Methods: Lectures 9-19 (Weeks 6-12)
    1. Why Stochastic Modeling?
      1. Effective "macro‑scale" model parameters
      2. Model prediction uncertainty analysis
      3. Data assimilation/conditioning with model-derived covariances and cross-covariances
      4. Monitoring network design
    2. Theoretical Approaches to Stochastic Modeling
      1. Exact Analytic Solutions
      2. Monte Carlo Methods
      3. Approximate Analytic Solutions
      4. Approximate Numerical Soltuions
    1. Applications of Stochastic Modeling to Multidimensional Groundwater Flow and Transport Problems
      1. Monte Carlo Methods (Freeze 1975; Delhomme, 1979; Smith & Freeze 1979; Graham & McLauglin, 1989a)
      2. First-order Spectral methods (Mizell et al 1982, Bakr et al 1978, Gelhar 1993; Vomvoris, 1990)
      3. First-order state space techniques (Hoeksema & Kitanidis1985, McLaughlin & Wood 1988; James and Graham, 1998; Graham and McLaughlin 1989a,b, 1991
      4.  First‑order Lagrangian techniques  (Dagan, Neuman, Destouni, Cvetkovic&Dagan)

Case Study: Stochastic Model Development using 1-D Environmental Fate and Transport Model

  1. Optimal Estimation of Hydrogeochemical Parameters: Lectures 20-23 (Weeks 13-14)
    1. Theory of Kriging
      1. Simple Kriging (known mean)
      2. Ordinary Kriging (constant but unknown mean)
      3. Non-stationary Kriging
      4. Log–Kriging, Block-Kriging, Indicator-Kriging
    1. Theory of Co-Kriging
      1. Development of Equations
      2. Simple Co-Kriging (known mean)
      3. Ordinary Co-Kriging (constant but unknown mean)
    1. Kalman Filtering
    1. Generalized Likelihood Uncertainty Estimation
    1. Particle Filtering

Case Study: Geostatistical Analysis of Upper Floridan Aquifer Data

  1. Student Presentations of Current Literature in Data Assimilation (e.g. Kalman Filtering, Particle Filtering,  Generalized Likelihood Uncertainty Estimation, Variational Methods (Weeks 13-17 )

Select paper and submit for approval by instructor deadline. All students responsible for reading all papers and participating in discussions.


Grading Method Percentage
Term projects (3 @25% each) 75%
Written and Oral presentation

Academic Honesty

As a student at the University of Florida, you have committed yourself to uphold the Honor Code, which includes the following pledge: “We, the members of the University of Florida community, pledge to hold ourselves and our peers to the highest standards of honesty and integrity.”  You are expected to exhibit behavior consistent with this commitment to the UF academic community, and on all work submitted for credit at the University of Florida, the following pledge is either required or implied: "On my honor, I have neither given nor received unauthorized aid in doing this assignment." 
It is assumed that you will complete all work independently in each course unless the instructor provides explicit permission for you to collaborate on course tasks (e.g. assignments, papers, quizzes, exams). Furthermore, as part of your obligation to uphold the Honor Code, you should report any condition that facilitates academic misconduct to appropriate personnel. It is your individual responsibility to know and comply with all university policies and procedures regarding academic integrity and the Student Honor Code.  Violations of the Honor Code at the University of Florida will not be tolerated. Violations will be reported to the Dean of Students Office for consideration of disciplinary action. For more information regarding the Student Honor Code, please see:  

UF Counseling Services

Resources are available on-campus for students having personal problems or lacking clear career and academic goals which interfere with their academic performance. These resources include:

  1. University Counseling Center, 301 Peabody Hall, 392-1575, personal and career counseling;
  2. Student Mental Health, Student Health Care Center, 392-1171, personal counseling;
  3. Center for Sexual Assault/Abuse Recovery and Education (CARE), Student Health Care Center, 392-1161, sexual assault counseling;
  4. Career Resource Center, Reitz Union, 392-1601, career development assistance and counseling.