Inverse Theory (P00784)

? Credit Points : 10 ? SCQF Level : 11 ? Acronym : GEO-P-RSIT

This course addresses a class of mathematical problems which occur in various branches of Earth science and elsewhere. The distinguishing feature of these problems is that they involve the estimation of an underlying continuous function from a finite number of measurements. This is a fundamentally difficult task as the measurements can never supply the infinite number of pieces of information which a continuous function could represent. The measurements do, however, supply some information on the underlying function, so what we can reasonably hope to do is to obtain an estimate of the function and an understanding of how good that estimate is. By far the commonest application of these ideas is the estimation, from remote sensing measurements, of atmospheric properties which vary with height. A problem of this type is used as an example throughout this course. The concepts presented also have applications in seismology, eomagnetism and oceanography.

Entry Requirements

? Pre-requisites : Available only to postgraduate students Radiative Transfer Fundamentals for Remote Sensing

Subject Areas

Home subject area

Postgraduate Courses (School of GeoSciences), (School of GeoSciences, Schedule N)

Delivery Information

? Normal year taken : Postgraduate

? Delivery Period : Semester 1 (Blocks 1-2)

? Contact Teaching Time : 4 hour(s) per week for 5 weeks

First Class Information

Date	Start	End	Room	Area	Additional Information
21/09/2006	09:00	11:00	Room 301, Crew Building	KB

All of the following classes

Type	Day	Start	End	Area
Lecture	Thursday	09:00	10:50	KB

Summary of Intended Learning Outcomes

On completion of this module, we expect students to be able to:
1. Explain the mathematical nature of the atmosphere remote-sensing problem.
2. Demonstrate competence in the mathematical techniques required to tackle the problem, specifically:
a) Solve simultaneous equations (including under and over-determined examples)
b) Calculate means, standard deviations and covariance matrices
c) Find the eigenvalues and eigenvectors of symmetric matrices
3. Describe some of the methods used to solve inverse problems, set out their mathematical formulation
and show clear understanding of their theoretical underpinnings. The methods to be covered are:
a) naive inversion, and why it usually doesn't work,
b) the MAP formula, its derivation and the nature of the solution,
c) the Twomey-Tikhonov formula, and the circumstances in which it is appropriate,
d) The extra difficulties of a non-linear problem, and how one can solve it.
4. Write computer programs to implement these methods, applying them to am atmospheric sounding
example

Assessment Information

-Learning outcomes 1 - 3 above will be assessed by in-class tests. These will contribute 40% of the final
mark: 10% for each test. There will be one test in each week (apart from week 1). Each test will be 30
minutes long and will cover the material from the previous week. Testing is done in this progressive
manner as material in the later weeks relies heavily on a good understanding of material in the earlier
weeks.
-Learning outcome 4 (and to some extent the other learning outcomes) will be assessed by an extended
practical exercise to be handed out a short time into the course and handed in at the end of week 6. The
practical exercise will contribute 60% of the final mark.