EPS 236 Environmental Modeling and Analysis, Fall Term 2012

Transcription

1 EPS 236 Environmental Modeling and Analysis, Fall Term 2012 Prof. Steven C. Wofsy and Prof. Daniel J. Jacob Location: Geological Museum 105 (Daly Seminar Room) Time: Tuesday and Thursday, 6:00 to 7:30p 1st meeting Tuesday 04 September EPS 236 is a project-oriented, hands-on course 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). There are two main itopic areas for the course:: 1. Data analysis focusing on understanding the science content and quantifying sources of error in complex data sets from environmental networks, complex sensors, and environmental instrumentation (e.g. a laser spectrometers deployed in an uncontrolled). Application of basic principles to real data is emphasized over theoretical aspects. We learn to use R as a tool for vvisualization, time series analysis, Monte Carlo methods, and statistical assessment. 2. Models in environmental science emphasizing (a) linear models (mathematical principles, time evolution operator, eigenvalues and eigenvectors; application to the global carbon cycle) 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 are expected to take EPS 200 and 236. Other students may take this course as a introduction to modeling and data analysis. Student projects will use R, Matlab, or Octave software application tools. 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%) Textbook: Dalgaard, P. (2008) Introductory Statistics with R (Statistics and Computing). Instructor Room Contact information Professor Steven C. Wofsy Professor Daniel J. Jacob Geo Museum Room 453. Pierce Hall Room 110C. Telephone: swofsy@seas.harvard.edu Telephone: djacob@seas.harvard.edu TBD, teaching fellow TBD Telephone: TBD TBD Part I Introduction to Deterministic and Markovian Linear Models; (Wofsy) We examine the basic conceptual framework for modeling biogeochemical cycles in the atmosphere, soils, or oceans. 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. Clear understanding of these ultra-simple models provides a strong foundation for Part II, which deals with modern concepts and implementation of atmospheric chemical models, inverse modeling, and related topics. 1/5

2 The material is basic and the treatment will be concise, but the underlying concepts are far-reaching. Many natural systems are represented conceptually as a set of linked compartments, with transfers between compartments specified by a linear process (represented as an exchange time), or by stochastic processes (represented by a transition probability). In two lectures, we will introduce students to the analytic properties, utility, and limitations of these model systems. The global carbon cycle and/or the global distribution of atmospheric greenhouse gases will constitute a case study for linear modeling, and simple environmental systems for Markovian and other stochastic models. Topics include: 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 recieve training to use R, which will be utilized in a problem focused on the global carbon cycle. (Students already proficient in Matlab or similar application may use one of those, but R will be very much preferred in Part Ib, data visualization.) Lecures in Part I Lecture 1. Basic concepts in linear modeling of environmental systems Lecture 2-3. More advanced linear models: time evolution operators, tangent linear approximations for non-linear dynamical systems; application to the Carbon Cycle in the Ocean (What is the turnover time for CO 2 in the atmosphere-ocean system?) Reading assignment: Time Scales in Atmospheric Chemistry, M. J. Prather, 1996 Problem Set 1: Oscillating Box Due 13/9/2012. Lecture 4. Markov Processes Class Project: We will do one or more class projects. 1. Thresholds and hysteresis in geophysical systems: Floating Ice in the Arctic. One project would study instability in linear and non-linear cases; conditional instability; multiple solutions; effect of model structure, stochastic processes, and diffusivity on threshold behavior. Students will generate a basic model, then modify its parameters and structure to understand "irreversible" behavior in models, and maybe in the environment. Notz D., 2009, "The future of ice sheets and sea ice: Between reversible retreat and unstoppable loss (PDF)": Ice Energy Balance Model, prototype for class project. 2. Global Distributions of reactive and greenhouse gases as observed from aircraft transecting the atmosphere In the HIPPO program, we have obtained three cross sections of greenhouse gases and reactive species, almost from pole-to-pole, down the centr of the Pacific Ocean. This presentation gives a concept for an "Erector Set" model that might be used to compare species with different reactivity and similar (or different) source regions. We ask the qeustion: does a comprehensive set of tracer data allow us to determine sources or emission ratios for different gases, and to distinguish the effects of different emission locations from effects of transport? The idea is that species with different reactivity decay at different rates as they are transported from source regions. When the concentration of on species is plotted against another, a curvilinear plot results in which the shape depends on relative rates of reaction and transport. This curve can be contrasted with the straight line obtained from inert species, or species that decay at the same rate as one another, which is linear with a slope reflecting the emission ratios but not sensitive to transport rates. Part Ib Data Visualization and Analysis; ( Wofsy) Hands-on introduction to data visualization, time series analysis, Monte Carlo methods, and statistical assessment of data. Environmental data often consist of a large number disparate observations directed towards understand a 2/5

3 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 rawinsondes. 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. Topics: 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. 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. Students learn the statistics and analysis package R ( in a laboratory setting, and apply this package in the data analysis projects. (Substitution of Matlab or the freeware version, octave, is acceptable but will make the more advance statistical treatments difficult.) Bootstrap:.RData.distribute (Data sets for class discussion). make_bootstrap.r (Code for class discussion). Smoothing: Moving averages, Savitsky-Golay and lowess filters--download code for discussion: smoothing_savgol.r R-Tutorials / Help Simple start-up: configure your computer: Powerpoint R for Matlab Users - excellent resources for those switching between R and Matlab! Installation List R Packages (not all will work) R tutorial #1 (Basic R Tutorial) Rtutorial1.pdf R basic statistics (data_t-test.zip) ttools.pdf t-test_r-code.zip R-intro.pdf (Users Manual). fitexy.zip Snow Leopard Exec Students should plan to bring a laptop computer for use in class during Part Ib of the course. Those who do not have such a computer should notify Prof. Wofsy so that one can be obtained on loan. Part Ib will have open-ended, individual or small group mini-projects, with one major project extending into reading/exam period. Reading: lmodel2 --R package for OLS/MA/SMA/RMA fitting lmodel2 --User guide for R package for OLS/MA/SMA/RMA fitting Efron/bootstrap estimation of uncertainty Combining Monte Carlo simulations and bootstrapping Efron/Elementary discussion of bootstrapping. Data Distribution Harvard Forest Data 3/5

4 Confidence Intervals: Summary MEI Index Page CRU Global Anom Data [ ] are there differences between NH and SH variability? CRU NH anomaly data CRU SH anomaly data Part II. Chemical transport models, optimization methods. (Prof. Jacob) The 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. The 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. Hopefully you will not think of opposing observations to models ever again. Text: Chemical Transport Models and Lectures on Inverse Modeling, by D.J. Jacob, on-line at Table of Contents at the bottom of this page. Requirements: weekly homeworks, mini-project. Table of contents for the on-line text material for Part II CHEMICAL TRANSPORT MODELS 1. THE CONTINUITY EQUATION chapter 1 lecture notes 1.1 Formulation 1.2 Discretization of the continuity equation Discretization in space Operator splitting 2. THE TRANSPORT OPERATOR chapter 2 lecture notes 2.1 Mean and turbulent components of transport 2.2 Parameterizations of turbulence Eddy diffusion Wet convective transport 2.3 Numerical solution of the advection equation Classic schemes Volume schemes Semi-Lagrangian algorithm 3. THE CHEMISTRY OPERATOR chapter 3 lecture notes 3.1 Characteristic time scales in atmospheric chemistry mechanisms 3.2 Implicit finite difference solvers 4/5

5 4. CONTINUITY EQUATION FOR AEROSOLS 5. DEPOSITION PROCESSES chapter 5 lecture notes 5.1 Dry deposition One-way deposition Two-way exchange 5.2 Wet deposition Scavenging in wet convective updrafts Scavenging by large-scale precipitation 6. INVERSE MODELING AND DATA ASSIMILATION On Line Text (Draft) 6.1 INTRODUCTION 6.2 BAYES THEOREM 6.3 INVERSE PROBLEM FOR SCALARS 6.4 VECTOR-MATRIX TOOLS FOR INVERSE MODELING Error covariance matrices Gaussian probability distribution functions for vectors Jacobian matrix Model adjoint 6.5 INVERSE PROBLEM FOR VECTORS Analytical maximum a posteriori (MAP) solution Averaging kernel matrix Pieces of information in an observing system Example application Sequential updating 6.6 KALMAN FILTER ( 3-D Var ) 6.7 ADJOINT APPROACH ( 4-D Var ) 6.8 OBSERVING SYSTEM SIMULATION EXPERIMENTS 5/5