The use of new sensors and autonomous observing systems has produced a wealth of high-quality data in all branches of environmental science.These data contain important information about distributions, fluxes or reaction rates of key properties in the earth system. Inverting the datasets, e.g., calculating the underlying concentrations, fluxes and rate constants from the data, is an important aspect of data analysis, and a wide range of numerical methods is available for this task. This course offers an introduction to linear inverse methods. Techniques for the solution of under- and overdetermined systems of linear equations will be covered in detail. Examples of such systems are (1) linear and non-linear regression, (2) curve fitting, (3) factor analysis, (4) diagnostic tomography, (5) remote sensing from airplanes or satellites, and (6) models of atmospheric and oceanic circulation and biogeochemistry. Contrary to square linear systems that are easy to solve, in general, under- and overdetermined linear systems exhibit complications: (1) the numbers of equations and unknowns differ, and (2) coefficients and right-hand-side of the equations usually are derived from measurements and thus contain errors. Basic techniques from numerical mathematics that solve these problems will be presented and explained extensively using examples from different fields. Error analysis will be of major concern. The examples cover different aspects of environmental research and should benefit students from the “Wahlfach” Physics of Oceans and Atmosphere, as well as students from other fields of physics and geophysics. A basic knowledge of linear algebra is required.
At the end of this lecture students will have learned:
- Techniques for the optimal solution of under- and over determined systems of linear equations
- Methods for calculating variances and covariances of the solutions.
- Concepts of resolution (in solution as well as data) and methods to calculate them
- Practical examples and applications to test data sets from atmospheric remote sensing and oceanography
- W. Menke, Geophysical Data Analysis, Academic Press, 1984
- P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences, McGraw-Hill, 1969