## 2 Experimental Section
### GlobAlbedo-Derived LAI and fAPAR
#### 2.1.2 Two-Stream Model Inversion  
Inversion of the two-stream model is described in [29]. In brief, the process is a standard inversion via the minimization of a cost function \(J\left(\mathbf{X}\right)\) of a vector of state variables (parameters) \(\mathbf{X}\) that quantifies the difference between model simulated fluxes given \(\mathbf{X}\), \(M\left(\mathbf{X}\right)\), and observed fluxes, \(\mathbf{d}\), plus the deviation of \(\mathbf{X}\) from prior information on the state variables \(\mathbf{X}_{prior}\)_i.e.,_  
\[J\left(\mathbf{X}\right)=-\frac{1}{2}\left[\left(M\left(\mathbf{X}\right)-\mathbf{d} \right)^{T}\mathbf{C}_{\mathbf{d}}^{-1}\left(M\left(\mathbf{X}\right)-\mathbf{d}\right)+\left( \mathbf{X}-\mathbf{X}_{prior}\right)^{T}\mathbf{C}_{\mathbf{X}_{prior}}^{-1}\left(\mathbf{X}-\mathbf{ X}_{prior}\right)\right] \tag{2}\]  
The parameters \(\mathbf{X}\) required by the TIP are the spectrally invariant LAI and the leaf single-scattering albedo \(\omega_{l}\left(\lambda\right)\), the leaf asymmetry factor \(d_{l}\left(\lambda\right)\) and the background albedo \(r_{g}\left(\lambda\right)\) and \(d_{l}\left(\lambda\right)\), for the visible and near-infrared spectral domains respectively. Note that only the canopy variables LAI, \(\omega_{l}\left(\lambda\right)\) and \(d_{l}\left(\lambda\right)\) are effective, in contrast to \(r_{g}\left(\lambda\right)\). By solving for the scattered and transmitted radiant fluxes, the absorbed component, fAPAR or \(A_{vgg}^{Coll}\left(z_{bgd},\mu_{0}\right)\), is given by considering energy closure in the case of a black (totally absorbing) background _i.e.,_\(A_{vgg}^{Coll}=1-\left(R_{vgg}^{Coll}+T_{vgg}^{Coll}\right)\), where \(T_{vgg}^{Coll}\) is simply the component transmitted by the vegetation through multiple interactions. The model inversion problem is simplified by assuming observation and model parameter probability distributions are Gaussian, and using a locally-linear approximation of \(M\left(\mathbf{X}\right)\). In this case, the posterior probability distribution of model parameters \(P\left(\mathbf{X}\right)\) can be represented as  
\[P\left(\mathbf{X}\right)\approx exp\left(-\frac{1}{2}\left(\mathbf{X}-\mathbf{X}_{post} \right)^{T}\mathbf{C}_{\mathbf{X}_{post}}^{-1}\left(\mathbf{X}-\mathbf{X}_{post}\right)\right) \tag{3}\]  
where \({}^{T}\) represents the transpose operator; \(\mathbf{X}_{post}\) and \(\mathbf{C}_{\mathbf{X}_{post}}\) are the mean and uncertainty covariance matrix of \(P\left(\mathbf{X}\right)\); \(\mathbf{C}_{\mathbf{X}_{post}}^{-1}\) is the inverse of \(\mathbf{C}_{\mathbf{X}_{post}}\). The resulting \(\mathbf{X}_{post}\) minimizes the cost function \(J\left(\mathbf{X}\right)\). In practice, \(J\left(\mathbf{X}\right)\) is minimized using a gradient descent method using the adjoint of \(J\left(\mathbf{X}\right)\). Uncertainty ranges \(\mathbf{C}_{\mathbf{X}_{post}}\) for \(\mathbf{X}_{post}\) are approximated through inversion of the Hessian matrix of \(J\left(\mathbf{X}\right)\). The Jacobian matrix of first derivatives of fluxes with respect to parameters is used to propagate parameter uncertainty to flux uncertainty. All derivative code is derived via automatic differentiation of the code that implements \(J\left(\mathbf{X}\right)\)[33; 34]. The authors of [29] demonstrated the utility of the TIP in retrieving LAI and fAPAR from MODIS and MISR broadband surface albedo products. The authors of [30] demonstrated the performance of the TIP in deriving surface parameters on global scales from MODIS albedo data. In particular, they showed the benefits of the TIP scheme for the generation of physically consistent information about the density and absorbing properties of the vegetation layer together with the brightness of the background underneath. LAI values followed expected trajectories of seasonal and spatial distributions. The estimated uncertainties on LAI were strongly correlated with the LAI itself and to a lesser extent to the brightness of the underlying \"soil\" background.