# Correction of Directional Effects in VEGETATION NDVI Time-Series
## 3 Methodology
### BRDF Correction
#### 3.2.5 BRDF Model Inversion and Adjustment  
Given a set of n surface reflectance observations obtained under varying illumination conditions and viewing angles, the BRDF model can be inverted to further proceed to the directional correction. Its solution involves finding the wavelength-dependent kernel weights: \(k_{0}\), \(k_{1}\) and \(k_{2}\) (see Equation (4)). Moreover, the solution of the inverse BRDF problem allows us to obtain a model of the reflectance anisotropy of the surface; that can then be used to estimate reflectances under any observing and illumination conditions. Thus, based on an ensemble of \(n\) observations obtained from the compositing phase, the model serving for the inverse problem is expressed as:  
\[\rho(\theta_{s_{j}},\theta_{v_{j}},\phi_{r_{j}})=k_{0}f_{0}+k_{1}f_{1}(\theta_ {s_{j}},\theta_{v_{j}},\phi_{r_{j}})+k_{2}f_{2}(\theta_{s_{j}},\theta_{v_{j}},\phi_{r_{j}})\qquad\qquad(j=1,..,n) \tag{5}\]  
The expression above is an equation system which can be rewritten as follows:  
\[\mathbf{Fk}=\mathbf{r} \tag{6}\]where \(\mathbf{F}\) is an \((n\times 3)\) matrix containing the Roujean kernels for the ensemble of \(n\) observations. However, \(\mathbf{k}\) is a \((3\times 1)\) vector containing the three wavelength-dependent kernel weights and \(\mathbf{r}\) is an \((n\times 1)\) vector with the observed reflectances. As described before, uncertainties associated to individual reflectance observations (angular component as defined in [31]) and their relevance within the compositing time window) are quantified by means of weighting factors (\(w_{j}\) for \(j=1,...n\)). Thus, Equation (6) can be rewritten as:  
\[\mathbf{Ak}=\mathbf{b} \tag{7}\]  
with a weighted reflectance vector defined as \(\mathbf{b}\) with elements \(b_{j}=r_{j}w_{j}\) and a design matrix \(\mathbf{A}\) with elements \(A_{ji}=F_{ji}w_{j}\). The BRDF inversion is known for being an ill-posed problem e.g., [33; 34; 35], meaning that there is no unique set of \(\mathbf{k}\) parameters that can solve Equation (7). However, significant variations of the surface BRDF shape parameters over a short timescale due to changes in surface cover are usually not expected. An approach to ensure a smooth evolution of the inverted \(\mathbf{k}\) parameters is to make use of prior information obtained from previous BRDF inversions (see Figure 2). This approach has also been implemented in previous works e.g., [31; 32; 33]. Following [PERSON] et al. [31], the equation system to solve becomes:  
\[(\mathbf{A}^{T}\mathbf{A}+\mathbf{C}_{prior}^{-1})\mathbf{k}=\mathbf{A}^{T} \mathbf{b}+\mathbf{C}_{prior}^{-1}\mathbf{k}_{prior} \tag{8}\]  
where \(\mathbf{k}_{prior}=[\mathbf{k}_{0_{prior}},\mathbf{k}_{1_{prior}},\mathbf{k}_ {2_{prior}}]\) is a vector containing the values of the \(\mathbf{k}\) parameters found in the previous BRDF inversion and the diagonal matrix \(\mathbf{C}_{prior}=diag\bigg{(}\sigma_{k_{0_{prior}}}^{2},\sigma_{k_{1_{ prior}}}^{2},\sigma_{k_{2_{prior}}}^{2}\bigg{)}\) contains the uncertainties of the uncorrelated prior information on the \(\mathbf{k}\) parameters. The previous estimates of the kernel weights (\(\mathbf{k}_{prev}\)) and their uncertainties (\(\mathbf{C}_{prev}\)) are used as follows as prior information:  
\[\mathbf{k}_{prior}=\mathbf{k}_{prev} \tag{9}\]  
\[\mathbf{C}_{prior}=\mathbf{C}_{prev}(1+\Delta)^{(t-t_{prev})} \tag{10}\]  
where \(t-t_{prev}\) is the time difference (number of days) between the current production date and the date associated with \(k_{prev}\) and \(\mathbf{C}_{prev}\). As mentioned in Section 3.2.3, in consistency with [PERSON] et al. [32], the timescale \(\tau\) is chosen as 10 days. So that the factor \(\Delta\) (defined as \(\Delta=2^{2/\tau}-1\)) when applied to the prior covariance matrix could be equivalent to run the inversion using a temporal weight function with a decreasing exponential shape and timescale of 10 days [31; 32]. This multiplicative factor applied to the prior covariance matrix reduces the confidence in the prior estimate.