**Prof. Steven C. Wofsy****Prof. Daniel J. Jacob**- Location:
Geological Museum 105 (Daly Seminar Room)

Time: Wednesday and Thursday, 2:30 to 4:00p

*
Click here to download lecture notes
*

**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:

**Data analysis**focusing on*understanding the science content*and*quantifying sources of error*in analysis of complex data sets from environmental networks, complex sensors, and environmental instrumentation (e.g. a laser spectrometer deployed in an airplane wing pod). Model concepts that underlie data analysis, and application of basic principles to real data, are emphasized.*R will be used as a tool for*and__visualization__,__time series analysis__,__Monte Carlo methods__,__statistical assessment__.**Models in environmental science**emphasizing*(a) linear models*(mathematical principles, time evolution operator, eigenvalues and eigenvectors; Markov chains) and*chemical transport models*including basic principles, numerical methods, and inverse models (Bayes’ theorem, optimal estimation, Kalman filter, adjoint methods.

- Note:
Graduate students in atmospheric chemistry or related fields usually take
EPS 200, 208, and 236.
- Students
are requested to bring laptop computers to class for use in the discussion
of the material.
*Prerequisite:*Applied Mathematics 105b or equivalent (may be taken concurrently); a course in atmospheric chemistry (EPS 133 or 200 or equivalent) is helpful, but not required; or permission of the instructors.*Requirements:*Homeworks (50%), Projects (50%); there is an oral exam in place of a written final.*Recommended Textbook:*Dalgaard, P. (2008)*Introductory Statistics with R*.- Office
hours and section times TBD
*Collaboration policy*: For assignments in this course, you are encouraged to consult with classmates as you work on problem sets. However, after discussions with peers, make sure that you can work through the problem yourself and ensure that any answers you submit are the result of your own efforts. In addition, you must cite any books, articles, websites, lectures, etc that have helped you with your work using appropriate citation practices.

Professor Steven C. Wofsy: Geo Museum Room
453/Pierce Hall Room 100E. Telephone: 617 495-4566; email:
swofsy@seas.harvard.edu

Professor Daniel J. Jacob;
Pierce Hall Room 110C. Telephone:
617 495-1794; email: djacob@seas.harvard.edu

Teaching fellow Daniel Cusworth:
Pierce Hall Email: __dcusworth@fas.harvard.edu____ __.

__Lecture topics __

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 *

**Lecture Schedule**

*Basic concepts in linear modeling of environmental systems
(e.g., box models)*

- Sep.
3 -- Linear Modeling; time evolution operator, solutions to the general
problem.
- Sep.
4 -- Linear Modeling; part II Analytical properties of linear systems;
Markov chain equiv. Mean Age and Age spectrum; time evolution operators,
tangent linear approximations for non-linear dynamical systems;
application to estimating global fluxes of atmospheric tracers (e.g. CO
_{2}, SF_{6}, CH_{4}, etc.) from atmospheric observations.

*Introduction to regressions, curve fitting*

- Sep.
10 -- Introduction to linear regressions: Fitting a line (curve) to data;
- Sep.
11 -- Regressions; correlated parameters, degrees of freedom, overfitting.
- Sep.
17 -- Introduction to the First Class Project: the 5-box model of greenhouse
gases in the atm.
- Sep.
18 -- Type II regressions, Fitexy (Chi-sq fitting)

*Confidence intervals, bootstrap error estimates,
non-parametric assessment of data*

- Sep.
24 -- Confidence intervals; t-tests and bootstrap
- Sep.
25 -- Bootstrap and t-test (workshop)
- Oct.
1 -- Data Filtering; Classifying data smoothing methods
- Oct.
2 -- Workshop: Modeling and analyzing N
_{2}O and CO_{2}data from WLEF

*Time series data*

- Oct.
8 -- Autoregressive data and modeling systems with serial correlation
- Oct.
9 -- Serial correlation
- Oct.
15 -- Filtering and interpolation of data: wavelets and image processing
- Oct.
16 -- Filtering and interpolation of data: Frequency domain; FFT and
spectral decomposition
- To
Be Scheduled: -- workshop: 5-box model results, and given problems for
working on in the session

*Problem sets include: *

- Oscillating
Box.
- Reconstructing
inputs from filtered output: sensor data;
atmospheric data.

*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.

*Lecture outline: *

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.

*Lecture outline *

6. INVERSE MODELING AND DATA ASSIMILATION

6.1 INTRODUCTION

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