In order to retrieve magnetic field from intensity and polarization spectra emitted by an inhomogeneous solar atmosphere we use the radiation transfer equations for the Stoke's parameters of total intensity and for circular polarization. These equations for the bremsstrahlung emission mechanism in an inhomogeneous plane layered atmosphere (see Appendix A) are:
The solution of the inverse problem given by Eqs. (1), (2) gives the magnetic field and the electron temperature in the solar atmosphere as functions of t. We call such solutions "tomographic" models (see Bogod & Grebinskij ) because they represent the only direct information, which may be extracted from brightness spectra without additional assumptions. The "physical" models, i.e. spatial distributions of the electron number density , electron temperature and field B(h), may be retrieved from the tomographic models by solving the inversion problem Eq. (3), which requires the assumptions on the relation between and . This approach is quite different from the standard model method which starts with the information obtained from optical observations and then does forward calculations to fit microwave observations in the following steps: , (see Avrett  for review).
It was shown by Bogod & Grebinskij (), that Eq. (1) may be effectively inverted to find by the differential deconvolution method (DDM). However, that technique is not applicable to Eq. (2), which cannot be reduced to a Laplace transform. Thus, to retrieve magnetic fields from spectral observations, we should use some estimates, well known from previous studies by Bogod & Gelfreikh () in the form of (see Appendix A):
In order to obtain such estimates from observations of solar atmosphere emission in the case of inhomogeneous magnetic fields, one should answer on two questions: (i) Which layer of the atmosphere (corona or chromosphere) gives the main contribution to the polarization at a given wavelength ? (ii) What is the relation between the estimated magnetic field [ ] (Eq. 4) and its real value in the case of an inhomogeneous field B(h)?
In order to answer these questions, we use a priori information about the distribution of temperature in the solar atmosphere, which may be extracted from observed brightness spectra by Eq. (1) and modelling.
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