The following chapter provides a short introduction to inversion, a method to find an atmospheric state that is consistent with the measurement and, in case of insufficient information content, with an a priori (prior) knowledge of the state. It is tried to keep the formalism free of pecularities of the type of the measurement, but for the present purposes measured intensities and path integrals (SCDs) and derivatives of those are meant. Since the field of inversion algorithms is wide the more interested reader is refered to [Rodgers 2000] where details and further references to literature can be found.

1 Introduction
 1.1 Forward Model
 1.2 Cost Function
2 Linear Retrieval
 2.1 Example
 2.2 Averaging Kernel
3 Nonlinear Retrieval
 3.1 Gauss-Newton Method
 3.2 Levenberg-Marquardt Method
 3.3 Total State Vector
 3.4 “Secure State Vector Transformations
 3.5 LMA with a Monte Carlo Forward Model
4 Application of the Program McArtimInversion
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References