Postgraduate Course: Inverse Theory (PGGE11054)
Course Outline
School  School of Geosciences 
College  College of Science and Engineering 
Credit level (Normal year taken)  SCQF Level 11 (Postgraduate) 
Availability  Available to all students 
SCQF Credits  10 
ECTS Credits  5 
Summary  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. 
Course description 
Not entered

Information for Visiting Students
Prerequisites  None 
Course Delivery Information
Not being delivered 
Learning Outcomes
On completion of this module, we expect students to be able to:
1. Explain the mathematical nature of the atmosphere remotesensing problem.
2. Demonstrate competence in the mathematical techniques required to tackle the problem, specifically:
a) Solve simultaneous equations (including under and overdetermined 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 TwomeyTikhonov formula, and the circumstances in which it is appropriate,
d) The extra difficulties of a nonlinear problem, and how one can solve it.
4. Write computer programs to implement these methods, applying them to am atmospheric sounding
example

Contacts
Course organiser  Dr Hugh Pumphrey
Tel: (0131 6)50 6026
Email: h.c.pumphrey@ed.ac.uk 
Course secretary  Ms Caroline Keir
Tel: (0131 6)51 7192
Email: caroline.keir@ed.ac.uk 

