Inverse Theory

Course Aims

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, geomagnetism and oceanography.

For more details, see the full course specification.

Course notes are available. Please do not regard this as a good reason for not attending lectures.

• Week 1 Course notes, In-class problems
• Week 2 homework
• Week 3 Course notes
• Week 4 Course notes I Course notes II
• Week 5 Course notes
• Week 6-7 Course notes
• Week 7 Course notes
• Week 8 Course notes
• Week 9 Course notes

Here are

• the instructions for the first practical exercise
• the instructions for the second practical exercise

For the first practical exercise, the instructions tell you where to find R code to read in the data files and that code includes the full path names of the data files. So if you are running R on the School of GeoSciences Linux machines, the code should Just Work (tm). If you are trying to do the practical on your own machine, you will need to download the data files. And you may prefer to download the read code instead of copying and pasting it. The three data files are here:

• The temperature profiles tprofs.dat
• The influence functions etc nadout.dat
• The measurements newrads.dat for which you do not know the true profile.

... abd the three code snippets are here:

• To read in the temperature profiles readprofs.R
• To read in the influence functions readks.R
• To read in the radiances for which you don't know the true profile: readrads.R

For the code snippets to work, you will need to edit them to change the filename to reflect where you put the data files. For example, if you put them in the same directory as your R programs and from where you start R up, you will need to change
hdr <- scan(file="/home/hcp/wrk/invth_pract/tprofs.dat",nlines=1)
to
hdr <- scan(file="tprofs.dat",nlines=1)
and to make similar changes wherever a data file name appears.

Please note that I expect the figures in your report to be printed clearly. This Open Office 2 document and this Microsoft Word document (which says the same thing) explain some of the issues. The hand-in date for your report will be near the end of the semester. If you prefer to use LaTeX, there are some instructions and examples linked from the introductory computing course