*2.1. Radiative Model*

The following simplifying assumptions will be made:


With these approximations, the radiance measured by the radiometer can be expressed as:

$$\mathcal{L}\_{\mathfrak{m}} = \mathcal{L}^{B}(T\_{b}) \cdot \varepsilon\_{b} \cdot \tau\_{a\_{1}} \tau\_{\mathfrak{g}} \tau\_{a\_{2}} + \mathcal{L}^{B}(T\_{\mathfrak{g}}) \cdot \left(1 - \tau\_{\mathfrak{g}}\right) \tau\_{a\_{2}} \tag{2}$$

where *τg*, *τa*<sup>1</sup> and *τa*<sup>2</sup> are, respectively, the transmittances of the gas cloud and the first and second atmospheric paths (atm 1 and atm 2 in Figure 1), L*<sup>B</sup>* stands for Planck's blackbody radiance, and *Tb* and *Tg* are, respectively, the temperatures of background and gas cloud.

To obtain a transmittance measurement, a reference spectrum must be measured without gas:

$$
\mathcal{L}\_r = \mathcal{L}^B(T\_b) \cdot \varepsilon\_b \cdot \tau\_{a\_1} \tau\_{\mathcal{G}0} \tau\_{a\_2} \tag{3}
$$

where *τg*<sup>0</sup> stands for the transmittance of the region of atmosphere that was previously occupied by gas cloud; it will be assumed that *τg*<sup>0</sup> ≈ 1.

A nominal transmittance is obtained as the ratio:

$$\tau\_{\text{nom}} \equiv \frac{\mathcal{L}\_m}{\mathcal{L}\_r} = \tau\_\% + \frac{\mathcal{L}^B(T\_\%)}{\mathcal{L}^B(T\_b)} \cdot (1 - \tau\_\%) \cdot \frac{1}{\varepsilon\_b \cdot \tau\_{a\_1}} \equiv \tau\_\% + \tau' \tag{4}$$

The positive term *<sup>τ</sup>* is negligible if *<sup>ε</sup>b*L*B*(*Tb*) >> L*B*(*Tg*), i.e., when the background is much hotter than the gas; otherwise, the equation can be solved for *τ<sup>g</sup>* if *Tg*, *Tb* and *ε<sup>b</sup>* are known (it will be generally assumed that in the spectral region considered, *τa*<sup>1</sup> ≈ 1).
