EPS 236 is a project-oriented, hands-on course that provides a general introduction to modeling and data analysis suitable for students from many fields, with applications drawn from the atmospheric science, environmental engineering, and climate science. It is suitable for graduate students and advanced undergraduates in Earth and Planetary Sciences and Engineering Sciences/Environmental Science and Engineering, and allied natural science departments (e.g. Organismic and Evolutionary Biology, Chemistry and Chemical Biology, Physics; at MIT, students from EAPS and Civil &. Environmental have often enrolled).
There are two main topic areas for the course:
Professor Steven C. Wofsy: Geo Museum Room 453/Pierce Hall Room 100E. Telephone: 617 495-4566; email: firstname.lastname@example.org
Professor Daniel J. Jacob; Pierce Hall Room 110C. Telephone: 617 495-1794; email: email@example.com
Teaching fellow Daniel Cusworth: Pierce Hall Email: firstname.lastname@example.org .
Linear models provide the basic conceptual framework for modeling noisy data, tracer transport in the atmosphere or oceans, biogeochemical cycle, biogeochemical cycles of nutrients, and many other natural phenomena. Analogous model frameworks are used in many disciplines in natural and social sciences. Linear systems are examined to illustrate the behavior of mass-conserving and non-conserving systems affecting the distributions of chemical species in the environment, and we examine inverse and adjoint models in this basic framework.
Topics include the following: Setting up the conceptual model-how do we structure the model and obtain estimates for the magnitudes of the parameters (the simplest “inverse modeling”)? Solving the model-eigenvalues and eigenvectors, the importance of non-orthogonality, the time-evolution operator, transient and steady-state behavior. Applying the model-how do we use these models as tools to improve our understanding?
Students will receive training to use R, which will be utilized in problems focused on applications to global chemical cycles, urban atmospheric structure and chemistry, etc. (Students already proficient in Matlab, Python, or similar application may use one of those, but R will provide the course refernce material for Part Ib, data visualization).
Part Ib: Regressions, Confidence Intervals, Time Series and Images (Steve Wofsy)
Environmental data often consist of a large number disparate observations directed towards understand a particular phenomenon or set of phenomena. The data are often strictly incomparable in that they sample different spatial and/or temporal scales and different processes and attributes of the physical system. Examples include atmospheric trace gases measured from an aircraft, fluxes of these gases observed at points on the surface, long-term data acquired are remote stations on a weekly basis, and winds and temperatures obtained from radiosondes. We will use case studies to learn about data visualization and statistical inference in analysis of real data sets. The skills learned in this section will be essential for the inverse modeling and data assimilation lectures to follow.
Part Ia lectures focus on the basics of analyzing environmental data, including fitting to a curve, non-parametric methods, confidence intervals, and bootstrap/MCMC, Part Ib looks at time series, including how to assess serial correlation
Basic concepts in linear modeling of environmental systems (e.g., box models)
Introduction to regressions, curve fitting
Confidence intervals, bootstrap error estimates, non-parametric assessment of data
Time series data
Problem sets include:
Global Distributions of reactive and
greenhouse gases as observed from aircraft and surface networks.
Global data for greenhouse gases and reactive species will be examined in linear model framework and used to compare species with different reactivity and similar (or different) source regions. What does a comprehensive set of tracer data allow us to determine about sources for different gases, and to distinguish the effects of different emission locations from effects of transport?
Part 1b topics: Distributions and t tests; parametric and non-parametric regression. Analysis of data: linear regression, regressions with errors in dependent and independent variables, transformations of data; time series analysis, autocorrelated time series; error estimation: bootstrapping, correlated errors, bias, conditional sampling. Visualization of data: time series, scatter plots, missing data; smoothing and filling data using basic and advanced methods (interpolation, weighted least squares, the Savistky-Golay filter, Haar and Gaussian wavelets).
Part 2a: Chemical Transport Models (Daniel Jacob)
This first set of lectures focus on the construction of chemical transport models (CTMs). Topics will include the mass continuity equation, Eulerian and Lagrangian model frameworks, numerical solution of the advection equation and of chemical mechanisms, simulation of turbulence, simulation of aerosol dynamics, and surface-atmosphere exchange.
Text: Chemical Transport Models and Lectures on Inverse Modeling, by D.J. Jacob, online at http://acmg.seas.harvard.edu/education.html.
Requirements: weekly homework, mini-project.
1. THE CONTINUITY EQUATION
2. 1.1 Formulation
3. 1.2 Discretization of the continuity equation
4. 1.2.1 Discretization in space
5. 1.2.2 Operator splitting
6. THE TRANSPORT OPERATOR
7. 2.1 Mean and turbulent components of transport
8. 2.2 Parameterizations of turbulence
9. 2.2.1 Eddy diffusion
10. 2.2.2 Wet convective transport
11. 2.3 Numerical solution of the advection equation
12. 2.3.1 Classic schemes
13. 2.3.2 Volume schemes
14. 2.3.3 Semi-Lagrangian algorithm
15. THE CHEMISTRY OPERATOR
16. 3.1 Characteristic time scales in atmospheric chemistry mechanisms
17. 3.2 Implicit finite difference solvers
2. CONTINUITY EQUATION FOR AEROSOLS
3. DEPOSITION PROCESSES
5.1 Dry deposition
5.1.1 One-way deposition
5.1.2 Two-way exchange
5.2 Wet deposition
5.2.1 Scavenging in wet convective updrafts
5.2.2 Scavenging by large-scale precipitation
Part 2b: Inverse modeling (Daniel Jacob)
This second set of lectures focus on the construction of inverse models and data assimilation with very general applications. Topics will include Bayes’ theorem, simple inverse problem for scalars, vector-matrix tools for inverse modeling, inverse problem for vectors, Kalman filters, and adjoint methods. We will learn how to combine observations, physical models, and external information into optimal estimation of the state of a complex system.
6. INVERSE MODELING AND DATA ASSIMILATION
6.2 BAYES’ THEOREM
6.3 INVERSE PROBLEM FOR SCALARS
6.4 VECTOR-MATRIX TOOLS FOR INVERSE MODELING
6.4.1 Error covariance matrices
6.4.2 Gaussian probability distribution functions for vectors
6.4.3 Jacobian matrix
6.4.4 Model adjoint
6.5 INVERSE PROBLEM FOR VECTORS
6.5.1 Analytical maximum a posteriori (MAP) solution
6.5.2 Averaging kernel matrix
6.5.3 Pieces of information in an observing system
6.5.4 Example application
6.5.5 Sequential updating
6.6 KALMAN FILTER (“3-D Var”)
6.7 ADJOINT APPROACH (“4-D Var”)
6.8 OBSERVING SYSTEM SIMULATION EXPERIMENTS