*5.3. Retrieval of Q and T by Polynomial Fitting of Principal Components*

One appealing aspect of the approach developed here is that the temperature and column density of the pollutant gas can be retrieved even without the ability to perform the complex process of spectrum simulation explained in Section 2.3. Rather, for a specific measurement conditions, with known *Tb* and expected ranges of *Tg* and *Qg*, the user can be provided with the mean spectrum and the first *p* eigenvectors of the relevant simulated spectra datacube. Then, the components on the PC base of the experimental spectra can be written by subtracting the mean spectrum and projecting onto the eigenvectors.

In the previous section, *Tg* and *Qg* for a pixel were obtained by an exhaustive search in the simulated PC datacube, to find the best agreement with those components. However, this can be further simplified for the user if explicit functions can be found, *Tg* = *Tg*(*PC*1, ... *PCp*) and *Q* = *Q*(*PC*1, ... *PCp*), that fit the dependence of *Tg* and *Qg* from the PCs, as defined in the simulated PC datacube.

This has been perfomed for the three gases under study in this work, using the function package polyfitn available for use in MATLAB. It has been found that seconddegree polynomial functions can provide values for *Tg* and *Qg* as functions of (*PC*1, *PC*2), with very small errors. As an example (Figure 7), the error of the *Qg* values furnished by the polynomial function is smaller than ±0.7 ppm·m for CH4, ±1.7 ppm·m for N2O, and ±5.5 ppm·m for C3H8 for most of the (T, Q) values of the pre-calculated datacube.

**Figure 7.** Absolute errors in the Q values obtained as second-degree polynomial functions of *Tg* (horizontal axis) and *Qg* (vertical axis) for each of the gases studied. Errors are very small except for the cases when (T, Q) values are either very large or very small (for CH4 and N2O) and only for the very small values of Q (for C3H8).
