# Uncertainties and Perspectives on Forest Height Estimates by Sentinel-1 Interferometry
## 2 Materials and Methods
### Minimizing \(\sigma_{dh}\) through Simulated Scenarios  
As demonstrated in the previous sections, \(\sigma_{dh}\) depends on both \(\omega\) and \(\sigma_{\Delta\varphi}\). While trying to minimize it, the only parameters a user can manage are image pair selection (spatial and temporal baseline) and the multilooking factor (_NL_). These can be optimized to guarantee the lowest \(\sigma_{dh}\). Concerning the temporal baseline selection, since forests are continuously changing, the longer the temporal baseline, the lower the associated coherence (temporal decorrelation) [17, 51]. Therefore, short temporal baseline and winter acquisitions are more suitable for forest height retrieval as discussed by [19, 31, 51]; however, currently, no formalized model exists to support this evidence. As far as baseline optimization (i.e., \(B\) value minimizing \(\sigma_{dh}\)) is concerned, this must be managed going back to Equations (3), (11) and (12). These make it possible to generate different scenarios depending on varying baseline and \(h_{v}\) values (i.e., expected average forest height). With reference to the scenarios, the optimal \(B\) value can be retrieved once an expected forest \(h_{v}\) is set. To make the estimate more immediate, a power model (Equation (13)) was calibrated directly relating \(h_{v}\) with the optimal B value:  
\[B_{opt}=a\cdot(h_{v})^{b}. \tag{13}\]  
where \(a\) and \(b\) are model parameters. Concerning _NL_ value optimization, one has to consider Equation (5) that relates _NL_ with \(\sigma_{\Delta\rho}\): the higher _NL_, the lower the interferometric phase uncertainty. _NL_ can therefore be optimized by fixing an expected value for \(\sigma_{dh}\) once \(\omega\) and local coherence are known (Equation (14)). \[NL=0.5\left(\frac{1}{\gamma^{2}}-1\right)\left(\frac{\sigma_{dh}}{\omega} \right)^{2} \tag{14}\]  
According to Equation (14), after setting an expected \(\sigma_{dh}\), the correspondent _NL_ value can be obtained. To make this information more operational, _NL_ can be easily translated into the correspondent squared pixel size (SGRP) by Equation (15). This works under the condition that the ratio between azimuth and range pixel size is the one of S1, i.e., 1:4 (see Table 1). \[SGRP\left(m\right)=\frac{\delta_{az}\cdot\sqrt{NL}}{2} \tag{15}\]  
Considering that the accuracy of forest height by ground survey is lower than 3 m, in this work some simulations retrieving SGRP at \(\sigma_{dh}=3\) m were performed by changing the \(\gamma\) value between 0.1 and 1 in Equations (14) and (15). This makes it possible to find an optimal value of _NL_ suitable to guarantee the expected accuracy of forest estimates. Once optimal \(B\) and _NL_ (or SGRP) values corresponding to an expected \(h_{v}\) were found, that part of \(\sigma_{dh}\), depending on settable operational parameters, can be finally minimized. Unfortunately, a significant part of \(\sigma_{dh}\) additionally depends on systematic errors such as orbital-related ones, flattening residuals and atmospheric delays. These cannot be directly accounted for, because they are difficult to model. Nevertheless, one can try to remove/minimize them by considering height differences between neighbour targets in place of absolute height measures. Measure differencing is expected to reasonably remove these errors, assuming that they occur similarly for close points [42]. Conversely, remaining error components depending on targets attributes, i.e., temporal decorrelation and random noise, cannot be further removed or minimized. Their joint effect on the final uncertainty of _dh_, described in Equation (5), can be evaluated through simulations based on a sensitivity analysis approach. These were achieved by differencing a reference ground point, external to the forest, and showing a very high coherence value (\(\geq\)0.8) with one representing the forest itself.