# Uncertainties and Perspectives on Forest Height Estimates by Sentinel-1 Interferometry
## 2 Materials and Methods
### Interferometric Phase Modelling  
SAR interferometry relies on image pairs processing acquired from different positions separated by a proper distance known as a baseline (\(B\)). This condition makes it possible to recover \"stable\" object height through a simple geometric transformation [38; 39; 40]. Unfortunately, this technique is known to suffer from many limitations in vegetated areas, mainly due to the noise induced by canopy volume and interferometric signal decorrelation.  
\begin{table}
\begin{tabular}{c c c} \hline \hline _Feature_ & _Values_ & _Units_ \\ \hline Frequency (\(\lambda\)) & 5.54 & cm \\ Nominal Satellite Altitude (H) & 693 & km \\ Look Angle (\(\theta\)) & 30–45 & \({}^{\circ}\) \\ Attitude accuracy (\(\sigma_{\theta}\)) & 0.01 & \({}^{\circ}\) \\ Maximum Noise Equivalent Sigma Zero (NESZ) & \(-\)22 & dB \\ Spatial resolution range (\(\delta_{rg}\)) & 5 & m \\ Spatial resolution azimuth (\(\delta_{az}\)) & 20 & m \\ Satellite position accuracy POD & 5 & cm \\ Bandwidth (\(Bw\)) & 42–56 & MHz \\ Antenna real length (\(L\)) & 12 & m \\ \hline \hline \end{tabular}
\end{table}
Table 1: S1 mission nominal features.  
Figure 1: Workflow adopted in this work. \(B\) is the baseline; _NL_ is the interferogram multilooking factor; \(h_{0}\) is the expected average forest height; \(dh\) is the estimated forest height according to proposed model; \(\Delta\varphi^{FP}\) and \(\Delta\varphi^{RP}\) are the interferometric phases of a forest point and a reference (ground) point, respectively; \(\omega\) is the gain factor that allows the conversion of an interferometric phase difference (\(d\Delta\varphi\)) into height difference.  
Assuming a flattened interferogram with no significant ground shifts affecting the area, the contribution to the interferometric phase (\(\Delta\varphi\)) given by local topography can be defined according to Equation (1):  
\[\frac{d\Delta\varphi}{dh}=\frac{4\pi\;B}{\lambda\;R\;\tan\theta}=k_{z}=\frac{1}{\omega} \tag{1}\]  
where \(\Delta\varphi\) is the interferometric phase; \(h\) is the target point elevation; \(\lambda\) is the wavelength of the radar signal; \(R\) is the sensor-target slant range; \(\theta\) is the antenna off-nadilar angle (look angle); \(B\) is the baseline; and \(\omega\) is the gain factor needed to convert back a phase difference into the correspondent height difference.  
Inverting Equation (1), the height difference (\(dh\)) between the two compared points can be obtained by Equation (2). This approach appears to be similar to an ordinary topographic levelling problem [41]. Subsequently, forest point absolute elevation can be obtained by adding \(dh\) to at least one reference point (representing the ground) having known elevation:  
\[dh=h^{FP}-h^{RP}=\omega\;\left(\Delta\varphi^{FP}-\Delta\varphi^{RP}\right) \tag{2}\]  
where \(h^{FP}\) and \(\Delta\varphi^{FP}\) are elevation and interferometric phase of the forest point, respectively, and \(h^{RP}\) and \(\Delta\varphi^{RP}\) are elevation and interferometric phase of the reference point, respectively. It is worth to noting that Equation (2) can be retained phase unwrapping independent if the expected forest height is maintained lower than the so called height of ambiguity (HOA) [19,30].  
This condition is satisfied if no significant elevation difference exists between forest and reference ground levels. This condition can be a priori-verified using an external DEM (e.g., global SRTM or ASTER DEMs). The condition must be tested for each interferogram pixel to remove inadequate ones. HOA can be obtained multiplying \(2\pi\) to \(\omega\), and it is possible to model the relationship between HOA and \(B\).