$   2.45    $                                            $   5.01    $                                            $   3.22    $
        PPNM                   $   {\mathbf{0.40}}   $                                  $   {\mathbf{0.80}}   $                                  $   {\mathbf{0.93}}   $                   $   {\textcolor[rgb]{0.00,0.00,1.00}{\mathbf{0.62}}} $
  ---- ------- -------------------------------------------------------- -------------------------------------------------------- -------------------------------------------------------- --------------------------------------------------------

  : Three-endmember forest synthetic dataset. Abundance MSE ($\times 10^{-2}$) and RE ($\times 10^{-4}$) for various linear/nonlinear mixing models. \[tab:synth\_forest\_RMSE\]

Fig. \[fig:synth\_forest\_wavelength\] shows the RDs as functions of wavelength. From the RD associated with the oracle-LMM, it clearly appears that the nonlinearity effects mostly occur in the spectral range $700\textrm{nm}-1400\textrm{nm}$, especially for the $20-80\%$ and $80-20\%$ scenes. All nonlinear mixing models provide good model fits, except the FM, as already shown by the REs reported in Table \[tab:synth\_forest\_RMSE\].

![Three-endmember forest synthetic dataset. Reconstruction difference $\textrm{RD}_{\ell}$ as a function of wavelength for various linear/nonlinear mixing models: LMM (black), oracle-LMM (black, dashed line), FM (blue), oracle-FM (blue, dashed line), NM (magenta), GBM (red) and PPNM (green).[]{data-label="fig:synth_forest_wavelength"}](results_simulated_error_wavelength_new.pdf){width="0.75\columnwidth"}

In-situ measurements
--------------------

Three types of in-situ measured mixed pixels were available to test the different mixing models, i.e., tree-weed, tree-soil and tree-soil-weed mixtures (see paragraph \[sec:in situ\]). In Table \[tab:RMSE\_real\], the reconstruction error of the mixed signal and the accuracy of the estimated abundances are depicted. From the RE associated with the oracle-LMM, it appears that most nonlinearities 