EPS 104 An Introduction to Mathematical Methods in Geophysics
Instructor: Mark Jellinek (firstname.lastname@example.org, McCone 377)
Office hours: Wednesday 2-4 pm.
Text: Advanced Engineering Mathematics by Kreysig.
A Course Outline
Weeks 1 8. Solution and analysis of ordinary differential equations (ODEs) with applications.
Topics: Techniques for solving 1st order and 2nd order, as well as systems of, linear ODEs. In addition to standard solution methods we will discuss Laplace Transform methods and investigate the application of Bessel and Greens functions in geophysical problems. Other special functions (e.g. Legendre polynomials and Airy functions) may be covered, depending on the interests of students.
Weeks 9 14. Derivation, solution and analysis of partial differential equations (PDEs) encountered in geophysical problems.
Topics: Conservation laws (i.e. balance equations) in geophysics, Reynolds Transport Theorem, Divergence Theorem, Dimensional Analysis and Scaling, Fourier Series, and solution of linear PDEs by Separation of Variables. Depending on time (and student inclination) we will also explore the application of Laplace Transform, Fourier Transform or similarity solution methods to PDEs encountered in geophysics.
Weeks 15 - 16. Working with geophysical data: an introduction to time series analysis.
Topics: An introduction to spectral analysis.
Course Goals For Students
1) Improve math skills. The student will be able to classify and solve (where possible) a large variety of ODEs. The student will acquire mathematical tools and strategies useful for understanding and (where possible) solving certain partial differential equations encountered in geophysics. The student will be able to apply the basic tools of time series analysis to understand appropriate geophysical data sets.
2) Learn how to learn and understand math. The student will develop a personalized approach for constructing conceptual and technical knowledge of mathematical techniques used in geophysics.
3) Invert a mathematical representation of a geological system to determine its physical meaning, significance and limitations. The student will be able to reconstruct and understand existing mathematical representations of geological systems. The student will be able to discern and articulate verbally the strengths and weaknesses of such models.
4) Create a mathematical representation of a geological system. The student will be able to represent a hypothesis mathematically. The student will understand the strengths and limitations of the model system as well as the method of solution.
5) Discuss mathematical models and analysis techniques clearly and intelligently.
There will be a quiz every other Wednesday (7 in total). Quizzes will only address basic facts and the routine of solving math problems (i.e. pattern recognition). Students can bring 1 sheet (1 side of one page) of review notes to each quiz. Students may drop the lowest grade.
Midterm Paper / Talk (Topics due 4th week of class; Paper due: tba; Talk to be scheduled) 20%
Final Paper / Talk (Paper due on last day of class; Talk to be scheduled) 20%
Your notebooks are not just for class notes in this course. A central goal of this course is that each student develops their own strategy for constructing knowledge from the ground up about the conceptual and technical basis for mathematical techniques of interest in geophysics. In particular, understanding the conceptual basis for a given mathematical technique is enormously powerful in determining how it may be used or modified. The notebooks for this course are intended to be a record of your learning and will complement your class notes. They should include the following components:
a) Problem sets that are self-graded with your own comments.
b) Detailed summaries of the knowledge structure for each math topic we cover. What are the main concepts? How do they fit together? For what sorts of problems would you expect a given technique to be useful? There are no rules for constructing the main links between important concepts underlying a math topic. How you will do this depends on how each of you learn. Flow charts, cartoons, text are just a few ways. Be creative. Learn how you like to learn.
c) Detailed summaries of how each mathematical technique is implemented. How are calculations performed in detail (what is the recipe)? Under what conditions may such calculations be performed? Are there important limitations?
Papers and Talks
In detail most problems in the Earth sciences are intractable. Consequently, determining ways to identify and pose simpler analog problems comprises a major part of basic research in geophysics. The goals of the midterm and final papers are to give you an opportunity to think deeply about how math is applied to develop precise understandings of geophysical problems. Projects can be (but are not limited to) an analysis of an existing model of a geophysical problem of your choice, an analysis of a dataset resulting from your own research, or you may choose to construct a model of a geophysical process that you find interesting. At the end of the semester I will bind all of your papers into an anthology that will be distributed to each member of the class. The goal of the talks, which will accompany your papers, are for you to learn to communicate mathematical and physical ideas in a precise, clear and simple way.
Some guidelines for your papers
Length and structure: 6 pages text (max), 4 figures (max) and appendices for math details if necessary.
Your paper should consider the following questions
1. What is the real problem?
2. What is the analog problem?
3. How is the problem posed mathematically?
4. What is the strategy for solving the problem?
5. What are the meaning and significance of the results?
6. What are the strengths and limitations of the model (including the development of the model and the solution approach)?
7. How might the model be realistically improved?
Some guidelines for your talks
Talks: 25 minutes. Talks must be at a level suitable for 2nd year undergraduate students in any science.