**2. Radiative Model and Retrieval Method**

Nearly all gas molecules have characteristic absorption/emission spectra in the infrared (IR) spectral region, due to transitions between ro-vibrational levels. For a specific line at wavenumber *ν* with absorptivity *a*, gas transmittance is given by the Lambert– Beer law:

$$\pi\_{\mathfrak{F}}(\nu, \mathbb{C}\_{\mathfrak{F}'} T\_{\mathfrak{F}}) = e^{-a(\nu, T\_{\mathfrak{F}}) \mathbb{C}\_{\mathfrak{F}} L\_{\mathfrak{F}}} \equiv e^{-a(\nu, T\_{\mathfrak{F}}) Q\_{\mathfrak{F}}} \tag{1}$$

where *Lg* is the gas optical path, *Cg* is the concentration, *Qg* = *CgLg* is the column density, and the dependence of *a* on wavenumber and temperature has been shown explicitly. If there is more than one absorbing species, *τ*(*ν*) is just a product of terms, as in Equation (1), one for each species; if the concentration is not homogeneous, the product *aCL* is replaced by an integral. Since absorptivities are well-known parameters that can be extracted from spectroscopic databases such as HITRAN [12], a transmittance measurement over a spectral range provides, in principle, an accurate way to identify gases in a sample and to determine their concentrations.

This is the basis of IR absorption spectroscopy, a classical method of analytical chemistry. In its most straightforward laboratory implementation, a gas cell in a spectrophotometer is filled with the sample to be measured, and then with a reference gas without

absorption lines in the spectral region of interest (typically N2). Transmittance is obtained as the ratio of the two spectra.

However, the full potential of absorption spectroscopy is displayed in remote measurements. In a typical field measurement with an imaging spectrometer, a gas cloud is observed against a background, and the instrument provides a measurement of the spectral radiance incoming to each pixel. In order to relate this radiance with the gas parameters, a radiative model of the measurement configuration is needed (Figure 1).

**Figure 1.** Schematics of the radiative model.
