*2.3. Theoretical Spectra and Fitting Procedure*

The spectral positions and intensities of the emission/absorption lines have been obtained from the HITRAN database [12]. For methane and nitrous oxide, the HAPI [13] Python-based interface to HITRAN has been used to download the respective absorption coefficients. However, in this free-access database, there is no detailed information about propane. Absorption coefficients for it have been obtained from the absorption cross sections at an atmospheric pressure of 1 atm and three temperatures (278.15 K, 298.15 K and 323.15 K) available on the webpage of HITRAN online [14]. With this information, it is possible to calculate the absorption coefficients by multiplying the cross-section data by the number of molecules per volume unit at ambient conditions.

Theoretical spectra have been generated by summing up the standard linehapes of single absorption lines ("line-by-line method"). The dependence of *a* on temperature, due to variation of absorption cross sections with T, has been fitted by seventh-order polynomial functions with a spectral resolution of 0.01 cm−<sup>1</sup> [10]. With this parametrization it is easy to construct theoretical transmittance spectra *τth <sup>g</sup>* (*ν*) for arbitrary values of *Tg* and *Qg*, using Equation (1) and, in turn, theoretical *τth nom*(*ν*) spectra with (2)–(4).

In order to compare these spectra to the measured ones, the effect of finite instrument resolution must be accounted for. In our case, a triangular apodization was used, so that the instrumental lineshape function (ILS) is a squared *sinc* function [15].

However, when calculating the theoretical transmittance spectrum, it is not correct to simply convolve the ideal spectrum with the ILS. The reason is that the experimental nominal transmittance spectrum *τnom*(*ν*) is not measured directly, but rather as a ratio (Equation (4)) of two radiance spectra measured by our instrument, L*<sup>m</sup>* and L*r*. Therefore, the correct theoretical spectrum *τth nom*(*ν*) must be calculated as a ratio of widened radiances:

$$\pi\_{\text{nuv}}^{\text{th}}(\nu) = \frac{\int \left[ \mathcal{L}^{\text{B}}(\nu', T\_{\text{b}}) \cdot \boldsymbol{\varepsilon}\_{\text{b}} \cdot \boldsymbol{\tau}\_{\text{u}\_{1}}(\nu') \cdot \boldsymbol{\tau}\_{\text{g}}(\nu') \cdot \boldsymbol{\tau}\_{\text{u}\_{2}}(\nu') + \mathcal{L}^{\text{B}}(\nu', T\_{\text{g}}) \cdot (1 - \boldsymbol{\tau}\_{\text{g}}(\nu')) \cdot \boldsymbol{\tau}\_{\text{u}\_{2}}(\nu') \right] \cdot ILS(\nu - \nu') d\nu'}{\int \mathcal{L}^{\text{B}}(\nu', T\_{\text{b}}) \cdot \boldsymbol{\varepsilon}\_{\text{b}} \cdot \boldsymbol{\tau}\_{\text{u}\_{1}}(\nu') \cdot \boldsymbol{\tau}\_{\text{b}}(\nu') \cdot ILS(\nu - \nu') d\nu'} \tag{5}$$

where *τg*(*ν*) and *τa*<sup>1</sup> (*ν*), *τa*<sup>2</sup> (*ν*) stand for the ideal transmittance spectra of the gas cloud and first and second atmospheric paths, respectively, as provided by HITRAN. They are functions (not explicitly displayed) of the temperatures (*Tg*, *Ta*) and column densities of the gas cloud (*Qg*) and the atmospheric gases. In this work, it has been assumed that *τa*<sup>1</sup> ≈ *τa*<sup>2</sup> ≈ 1, which is a very good approximation for the measurement configuration and the spectral regions involved.

At each pixel, the fitting procedure is as follows (a single gas will be assumed; for each additional gas the procedure is the same but there is an additional unknown value of column density to be determined). We start by assuming a value for the couple (*Qg*, *Tg*). The theoretical transmittance spectrum *τth nom*(*ν*) is calculated with Equations (1) and (5) at the points of the wavenumber axis of the experimental spectra. The differences with *τnom*(*ν*) for each wavenumber are added up in quadrature to get the sum of squared errors (SSE). The Nelder–Mead minimization algorithm, as implemented in MATLAB software, is used then to find the value of (*Qg*, *Tg*) for the next iteration, until convergence is reached. This iterative process is repeated for each pixel to obtain the images of column density and temperature.
