# DEGREE REGULATIONS & PROGRAMMES OF STUDY 2015/2016

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DRPS : Course Catalogue : School of Geosciences : Postgraduate Courses (School of GeoSciences)

# Postgraduate Course: Inverse Theory (PGGE11054)

 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
 Pre-requisites Students MUST have passed: Radiative Transfer (PGGE11055) AND Fundamentals for Remote Sensing (PGGE11053) Co-requisites Prohibited Combinations Other requirements None
 Pre-requisites None
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 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
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