# Uncertainties and Perspectives on Forest Height Estimates by Sentinel-1 Interferometry
## 2 Materials and Methods
### Modelling \(dh\) Uncertainty
#### 2.3.2 Theoretical Uncertainty of \(d\Delta\varphi\)  
The theoretical uncertainty of \(d\Delta\varphi\) (\(\sigma_{d\Delta\varphi}\)) can be computed by Equation (8):  
\[\sigma_{d\Delta\varphi}=\sqrt{\sigma_{\Delta\varphi FP}^{2}+\sigma_{\Delta \varphi RP}^{2}} \tag{8}\]  
where interferometric phase uncertainties, \(\sigma_{\Delta\varphi FP}\), and \(\sigma_{\Delta\varphi RP}\) were proved [18,44,45] to be strictly correlated to the local coherence magnitude, \(|\gamma|\), according to Equation (9):  
\[\sigma_{\Delta\varphi}=\frac{1}{\sqrt{2 NL}}\frac{\sqrt{1-|\gamma|^{2}}}{| \gamma|} \tag{9}\]  
where \(NL\) is the multilooking factor (i.e., number of pixels used to compute the complex multilooked interferogram). These considerations drive to admit that \(\sigma_{dh}=f\big{(}\omega,\,\sigma_{\Delta\varphi}\big{)}\) and, consequently, they depend on the operational parameters \(B\) and \(NL\). Equations (1) and (3) show that \(dh\) accuracy is higher for long baselines. This is what is ordinarily reported in the literature. Nevertheless, this condition appears to fail over \begin{table}
\begin{tabular}{c c} \hline \hline _Parameter_ & \(w_{i}\)_Formula_ \\ \hline _Baseline (B)_ & \(w_{B}=\frac{\left(\frac{\omega}{\partial R}\right)^{2}\cdot\sigma_{B}^{2}}{ \sigma_{\omega}^{2}}\) \\ _Slant range (R)_ & \(w_{R}=\frac{\left(\frac{\omega}{\partial R}\right)^{2}\cdot\sigma_{B}^{2}}{ \sigma_{\omega}^{2}}\) \\ _Lok angle (\(\theta\))_ & \(w_{\theta}=\frac{\left(\frac{\omega}{\partial R}\right)^{2}\cdot\sigma_{B}^{2}}{ \sigma_{\omega}^{2}}\) \\ _Mixed term (R, \(\theta\))_ & \(w_{corr(R,\theta)}=\frac{\left(\frac{\omega}{\partial R}\right)\rho_{(R, \theta)}\sigma_{R}\sigma_{\theta}}{\sigma_{\omega}^{2}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: “Weights” defining the relative importance of factors to determine \(\sigma_{\omega}^{2}\). vegetation. In fact, longer baselines decrease coherence, thus negatively affecting \(\sigma_{\Delta\varphi}\) and, consequently, \(\sigma_{\Delta\varphi}\). This effect is more evident while analysing the decorrelation model of Equation (10), as proposed by [44, 46]. This model defines the coherence magnitude as the product of several contributions related to decorrelation agents. \[\gamma_{obs}=\gamma_{thermal}\cdot\gamma_{Doppler}\cdot\gamma_{temp}\cdot \gamma_{geom}\cdot\gamma_{vol} \tag{10}\]  
where \(\gamma_{obs}\) is the observed coherence magnitude; \(\gamma_{thermal}\) is the signal decorrelation due to additive thermal noise [47]; \(\gamma_{Doppler}\) accounts for non-perfect overlap of the azimuth spectra for the master and slave SAR acquisitions and misregistration error [48]; \(\gamma_{temp}\) is associated to changes in the dielectric and structural proprieties of the target [49]; \(\gamma_{geom}\) depends on geometric relationships between the two compared SAR acquisitions; and \(\gamma_{vol}\) is the decorrelation component due to canopy scattering from multiple heights within each scattering cells. Because \(\gamma_{thermal}\), \(\gamma_{Doppler}\gamma_{temp}\) are independent from any operational parameter that can be controlled by users, these were not taken into consideration in the optimization process addressed by authors in [50]. Differently, \(\gamma_{geom}\) is dependent from baseline according to Equation (11):  
\[\gamma_{geom}=1-\frac{2\ B\ \cos^{2}\theta\ \delta_{rg}}{\lambda R} \tag{11}\]  
where \(\delta_{rg}\) is the range resolution of the radar. Similarly, \(\gamma_{vol}\) has to be taken into consideration, since this is caused by the volumetric decorrelation related to the vegetation canopy volume (distributed scatters). Assuming a uniform volume showing an exponential extinction of absorption and scattering, \(\gamma_{vol}\) can be modelled by Equation (12) ([18, 49, 50]). It states that SAR signal decorrelation depends on vegetation volume depth \(h_{v}\) (i.e., tree height) and from the system parameter \(k_{z}\) (see Equation (1)).