Courses:

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Advanced Calculus for Engineers (18.075) or Mathematical Methods for Engineers II (18.086)

Although the course is self-contained, students will find some undergraduate exposure to Fourier analysis and applied linear algebra, least-squares, etc. to be helpful. Useful background includes Mathematical Methods for Engineers I (18.085) and/or the material covered in Strang, G. Introduction to Applied Mathematics. Wellesley, MA: Wellesley-Cambridge Press, 1986. ISBN: 9780961408800.

The course is directed at making scientifically sensible deductions from the combination of observations with dynamics and kinematics represented, generically, as "models". There are two overlapping central themes:

• Theme 1: Linear "inverse" methods and data "assimilation" including regression, singular value decomposition, objective mapping, non-stationary models and data, Kalman filters, adjoint methods ("assimilation") etc.
• Theme 2: Standard time series analysis, including basic statistics, Fourier methods, spectra, coherence, filtering, etc.

Because of the amount of material to be covered, there is little time for specific scientific applications ("case studies"), but examples from various branches of the earth sciences will be discussed.

Theme 1 is covered by:

Wunsch, Carl.  Discrete Inverse and State Estimation Problems: With Geophysical Fluid Applications. Cambridge, UK: Cambridge University Press, 2006.  ISBN: 9780521854245. This material will be distributed as the working draft of a book. No knowledge of fluids is required.

An earlier edition of the book, with physical oceanographic applications, is:
Wunsch, Carl. The Ocean Circulation Inverse Problem. Cambridge, UK: Cambridge University Press, 1996. ISBN: 9780521480901. But there is no need to have it.

Partial alternatives are:

Menke, William. Geophysical Data Analysis: Discrete Inverse Theory. 2nd ed. New York, NY: Academic Press, 1989. ISBN: 9780124909212.

Daley, Roger. Atmospheric Data Analysis. Cambridge, UK: Cambridge University Press, 1991. ISBN: 9780521382151.

Theme 2 is discussed in:

Priestley, M. B. Spectral Analysis and Time Series. Combined ed. London, UK: Academic Press, 1981. ISBN: 9780125649018.

Other highly recommended useful books are:

Bracewell, R. N. The Fourier Transform and its Applications. 3rd ed. New York, NY: McGraw-Hill, 1999. ISBN: 9780073039381.

Percival, D. B., and A. T. Walden. Spectral Analysis for Physical Applications: Multitaper and Conventional Univariate Techniques. Cambridge, UK: Cambridge University Press, 1993, pp. 83. ISBN: 9780521355322.

There will be homework roughly every two weeks, plus a modest term paper (not to exceed 10 pages) on a theme related to the class. These are most likely to be applications of some methodology to real data, or a theoretical extension of a standard result. The homework will require some small computations as can be done in any of MATLAB®, Maple®, Mathematica®, etc.