*2.2. Temperature and Column Density Retrieval*

Our aim is to obtain the values of gas concentration *Cg* from experimental measurements of L*r*(*ν*) and L*m*(*ν*) but, since only the product *CL* appears in the equations (cf. Equation (1)), the result can only be the column density *Qg* ≡ *CgLg* rather than the concentration *Cg*. The amount of gas will be measured, as usual by spectroscopic remote sensing methods, in units of ppm·m (parts per million per meter).

Since absorptivity *a*(*ν*, *Tg*) is a known parameter, the most straightforward method to recover *Qg* for each gas is to solve Equation (4) for *τ<sup>g</sup>* and then use Lambert–Beer law (1) to obtain *Qg*. However, in many practical cases the gas cloud temperature *Tg* will be unknown, and therefore should also be retrieved simultaneously with *Qg* from the experimental measurements.

Thus, measurements of L*r*(*ν*) and L*m*(*ν*) over a spectral range rather than at a single *ν* will be necessary to provide a set of equations, but even so it is not possible to solve Equations (4) and (1) simultaneously for *Tg* and *Qg*, because both parameters are coupled in the Lambert–Beer expression of transmittance (1), where the absorptivity *a* depends on *Tg* in a nontrivial way. Instead, they will be determined by a fitting process: we will calculate theoretical spectra for L*r*(*ν*) and L*m*(*ν*), divide them to obtain a theoretical nominal transmittance *τth nom*(*ν*) and assign to each pixel the column density and temperature values which provide the best fit to the experimental spectra *τnom*(*ν*).

In summary, the final results of our method are a "column density image" and a "temperature image" with values of, respectively, *Qg* and *Tg* at each point in the field of view, obtained by iteratively fitting the experimental nominal transmittance spectra with theoretical spectra generated according to the radiative model of Figure 1, through the Equations (1)–(4).
