\abond{i}{\nopix} \abond{j}{\nopix} \Vmat{i}\odot \Vmat{j}+ \Vnoise{p}.$$ The PPNM has demonstrated a noticeable flexibility to model various nonlinearities not only for unmixing purposes [@Altmann2012ip] but also to detect nonlinear mixtures in the observed image [@Altmann2013ip]. This model has also the great advantage of having the amount of nonlinearity to be governed by a unique parameter $b_p$ in each pixel, contrary to NM or GBM. Eq. also shows PPNM includes bilinear terms $\Vmat{i}\odot \Vmat{j}$ ($j \neq i$) similar to those involved in the NM, FM and GBM, and also quadratic terms $\Vmat{i}\odot \Vmat{i}$, which may account for interactions between similar materials.

Unmixing algorithms {#subsec:algorithm}
-------------------

To evaluate the accuracy of the mixing models of interest, the pixels of the in-situ and simulated data are unmixed with respect to each model. When analyzing the pixels with the LMM, the nonlinear contribution $\boldsymbol{\mu}\left(\MATmat,\Vabond{\nopix},\Vnabond{\nopix}\right)$ is set to zero. Based on the prior knowledge of the endmember signatures $\MATmat$, the abundance vector $\Vabond{\nopix}$ associated with each pixel $\Vpix{\nopix}$ is estimating by solving the constrained minimization problem $$\label{eq:criterion_LMM}
  \hatVabond{\nopix} =
  \operatornamewithlimits{argmin}_{\Vabond{\nopix}}
  \left\|\Vpix{\nopix} - \MATmat\Vabond{\nopix}\right\|_2^2 \quad
  \text{s.t.} \quad \eqref{eq:constraints_LMM}.$$

In this work, to solve this problem, the fully constrained least square (FCLS) algorithm [@Heinz2001] is used.

Moreover, when analyzing 