interactions that can not be properly taken into account by a simple LMM [@Borel1994; @Ray1996; @Zhang1998; @Chen2006; @Fan2009; @Somers2009; @Tits2012igarss; @Somers2014]. Indeed, for these specific scenarios addressed in this paper, differences in elevation between the transparent $3$D vegetation canopies and the relatively flat soil surfaces submit photons to multipath and scattering effects. Similar interaction effects have been also encountered when analyzing urban scenes [@Huard2011whispers; @Fontanilles2011; @Meganem2014]. Therefore, various attempts have been conducted to overcome the intrinsic limitations of the LMM. A large family of nonlinear models that have been proposed to analyze vegetated areas can be described as $$\label{eq:NLMM}
  \Vpix{\nopix} = \MATmat\Vabond{\nopix} +
  \boldsymbol{\mu}\left(\MATmat,\Vabond{\nopix},\Vnabond{\nopix}\right)
  + \Vnoise{\nopix}.$$ In , the observed pixel is composed of a linear contribution similar to the LMM and an additive nonlinear term $\boldsymbol{\mu}\left(\cdot\right)$ that may depend on the endmember matrix $\MATmat$, the abundance coefficients in $\Vabond{\nopix}$ and additional nonlinearity coefficients $\Vnabond{\nopix}$ introduced to adjust the amount of nonlinearity in the pixel. This class of models includes the bilinear models [@Altmann2011whispers], the quadratic-linear model [@Meganem2014], the post-nonlinear model [@Altmann2012ip] and the bilinear-bilinear model [@Eches2014grsl] (the most commonly used will be fully described in Section \[sec:models\]).

However, to our knowledge, most of these models have been derived following 