|  | MU | psNMF | BPSS2 |
| --- | --- | --- | --- |
| Quality Q | 0.35±0.12 | 0.822 ±0.005 | 0.41±0.06 |
| RMSD relative error | 0.1455±2.10−6 | 0.1461±5.10−6 | 0.1468±3.10−4 |
| Computation time (s) | 13±8 | 46±9 | 413±21 |

[Table: Table 1: Results (mean and standard deviation) from 10 realizations of a toy synthetic example with NS = 5 (in agreement with next section on synthetic tests), NO = 10000, Nν = 320 and 300 CH4 spectra hidden at a level of 1 std of the noise. Quality is computed as a correlation coefficient (see Eq. 12). RMSD is computed from Eq. 8. Computation time is expressed in second.]

if the convergence is set, there is a high variability in MU results, due to the lack of regularization. On this particular example, the best is clearly psNMF algorithm. The RMSD is computed for all cases and shown in Table 1. We can observe that the value is almost equivalent, around 0.146, for all method but MU is slightly better, due to the fact that the cost function has no other term. MU algorithm is just minimizing the reconstruction. As a comparison, the RMSD expected for a perfect reconstruction of the signal (and not the noise) of this toy example is 0.16. With 5 sources (significantly more than the 3 sources defined in this toy example), noise is also encompassed within the approximated linear model, as expected. The quality Q is the only parameter to assess the quality of the algorithm to detect minor specie (here CH4). In this particular toy example, psNMF seems to be the best algorithm, providing a source correlated with groundtruth CH4 with a correlation coefficient up to 0.8. We will extensively test this performance in the next section. We also estimate the computation time on a 2.9 GHz Intel Core i7 with 16 Go DDR3 RAM as an example. All algorithms are implemented in ©Matlab using parallelized matrix computation. Results, presented in Table 1, demonstrate that MU is faster than psNMF but both are clearly less resources consuming than BPSS2. From the computation time and efficiency, we excluded BPSS2 from the next tests.
4.2. Extended synthetic tests For the first set of tests, we used the same toy model described in section 4.1, except with 100 CH4 spectra hidden at a level of 2 and 3 standard deviation of the noise (this number is called “factor above noise level”). In order to have robust results, we made 10 realizations and averaged the results. Figure 5 represents the results as a function of the number of sources NS. It presents two quality indicators of the results: the average correlation coefficient Q (see Eq. 12) and the fraction of realization with acceptable results (with Q > 0.5). We can observe that the psNMF is always better than MU on average at cost of an higher variability (higher standard deviation). Adding sources seems to always increase the detection until reaching a plateau around NS = 5. Adding more sources will not drastically increase/decrease the source | Factor above noise level of 2 Effect of Ns for noise factor 2 with 100 CH4 spectra 1 0.9 0.8 0.7 0.6 ue 0.5 VVal al ue 0.4 0.3 0.2 MU mean corr.

[Figure: figure 5]

0.1 p Ms UN M fraF c m . ce oa rn r. c >o 0rr .. 5 psNMF frac. corr. > 0 0 3 4 5 6 7 8 NbNb e ennddmmemebmerber |  | Factor above noise level of 2 Effect of Ns for noise factor 2 with 100 CH4 spectra |  |  | 1 0.9 0.8 0.7 0.6 ue 0.5 VVal al ue 0.4 0.3 0.2 0.1 .5 0 9 3 |  | Factor above noise level of 3 Effect of Ns for noise factor 3 with 100 CH4 spectra |