ance space:
T ∗(ν) = T(ν) −C(ν) (3)
Then convert the spectra into absorbance:
X(ν) = 1 −T ∗(ν) (4)
The final step is the linear mixture :
NS
�
X(ν) ≈
Si(ν).Ai (5)
i=1
with Si(ν) the source spectra and Ai the spectral abundance. In this description, the physical meaning of Si(ν) and Ai is lost but the apparent SNR 4 is dramatically increased, which is much more important for our analysis.

Nevertheless, assumptions required in eq. 1 are usually not relevant. Radiative transfer model used for precise quantification is highly non-linear. One has to consider that this unsupervised linear unmixing problem is already very difficult for machine learning. Solving non-linear model in a unsupervised way is a research area that is clearly not solved yet. In addition, we would like to focus on spectral detection, rather than quantification. Thus, we will focus on S(ν) much more than A. We will show that for linear, but also non-linear simulation and real data, meaningful S(ν) can be retrieved. Due to non-linearity, A may differ significantly from truth, but the big tendencies should be respected. After the quick-look analysis, estimating Si and A, one must go back to the real data. The most trivial strategy is to pick the spectra X out of the collection, with the highest abundance of a selected source Si. In the following, we will use the continuum estimation C(ν) using asymmetric least square (Eilers and Boelens, 2005), with parameters : νsmooth = 103 and p = 1 −10−2, 10 number of iterations.
3.2. Non negative matrix factorization For a collection of spectra, eq. 5 can be written in matrix form Xkj≈Ski.Aij, with i the source index (from 1 to NS) , j the observation index (from 1 to NO) and k the wavenumber index (from 1 to Nν). Thus, one have to estimate S and A, by minimizing the objective function:
F = ∥X −S.A∥2 (6) with ∥.∥, the Frobenius norm (usual L2 norm). Several algorithms have been proposed to solve this problem, subject to positivity (both S and A are non-negative). Such problem is called Non negative Matrix Factorization (NMF). This constraint is important to keep the physical meaning, but also to promote sparsity of S (a signal is sparse when most of the values are close to zero except several non-zero values). Let ˙S and ˙A be the estimation of those quantities.
MU. We propose to use the Multiplicative Updates (MU) of Lee and Seung (1999) accelerated by Gillis and Glineur (2012). We used the convergence parameter αMU = 1. Other alternative algorithms are possible but give equivalent results since they minimize the same cost function. This algorithm has the advantage of very fast computation time but the result may depend on initialization.
BPSS2. We propose to test another kind of algorithm: the Bayesian Prior Source Separation (Moussaoui et al., 2006; Dobigeon et al., 2009), that has been optimized (Schmidt et al., 2010), hereafter called BPSS2. This algorithm has the main advantage to account for extra constraint : the sum-to-one or sum-lowerthan-one on the abundances (�
i Aij = 1) that also promotes sparsity of S. This algorithm, based on Monte Carlo approach is much more time consuming. One approach to reduce the computation time is to select only relevant spectra out of the dataset (Moussaoui et al., 2008), but then the statistics may be biased (Schmidt et al., 2010).