Inversion of the radiative transfer equation for polarized light
  Since the early 1970s, inversion techniques have become the most useful tool for inferring the magnetic, dynamic, and thermodynamic properties of the solar atmosphere. The intrinsic model dependence makes it necessary to formulate specific means that include the physics in a properly quantitative way. The core of this physics lies in the radiative transfer equation (RTE), where the properties of the atmosphere are assumed to be known while the unknowns are the four Stokes profiles. The solution of the (differential) RTE is known as the direct or forward problem. From an observational point of view, the problem is rather the opposite: the data are made up of the observed Stokes profiles and the unknowns are the solar physical quantities. Inverting the RTE is therefore mandatory. Indeed, the formal solution of this equation can be considered an integral equation. The solution of such an integral equation is called the inverse problem. Inversion techniques are automated codes aimed at solving the inverse problem.   The foundations of inversion techniques are critically revisited with an emphasis on making explicit the many assumptions underlying each of them. An incremental complexity procedure is advised for the implementation in practice. Coarse details of the profiles or coarsely sam- pled profiles should be reproduced first with simple model atmospheres (with, for example, a few physical quantities that are constant with optical depth). If the Stokes profiles are well sampled and differences between synthetic and observed ones are greater than the noise, then the inversion should proceed by using more complex models (that is, models where physical quantities vary with depth or, eventually, with more than one component). Significant im- provements are expected as well from the use of new inversion techniques that take the spatial degradation by the instruments into account. 
Jose Carlos del Toro Iniesta and Basilio Ruiz Cobo
astro-ph.SR