# Uncertainties and Perspectives on Forest Height Estimates by Sentinel-1 Interferometry
## 3 Results and Discussion
### Minimizing \(\sigma_{dh}\) through Simulated Scenarios  
According to Equation (12), \(\sigma_{d\Delta\varphi}\) depends on \(B\) and \(h_{v}\). To explore its dependency, \(B\) and \(h_{v}\) were changed progressively from 5 m to 1000 m and from 5 m to 50 m, respectively, testing their effects on \(\sigma_{dh}\) (Figure 5a). Figure 3: (**a**) \(\sigma_{\omega}\) versus \(B\) and \(\theta\). (**b**) relative weights affecting \(\omega\) variance involved in VPL. Note that \(w_{\text{B}}\) is exaggerated by a factor 200. Simulations in Figure 5a show that \(h_{v}\) value \(>15\) m (i.e., the majority of forests), and \(\sigma_{dh}\) presents a minimum with respect to \(B\). It is worth noting that \(\sigma_{dh}\) values are certainly underestimated. In fact, \(\gamma_{thermal}\), \(\gamma_{Doppler}\), \(\gamma_{temp}\) are factors that cannot be properly modelled, being independent from the features of interferometer. Considering \(B\) values minimize \(\sigma_{dh}\) for the different \(h_{v}\) values, a model directly relating the \"optimal\" B value with the expected \(h_{v}\) was defined (Equation (13) and Figure 5b). A simple power model can be calibrated (Equation (13)), as reported in Figure 5b (see figure for model \(a\), \(b\) and R\({}^{2}\) coefficients). It is worth noting that the optimal \(B\) value occurs within the critical baseline (for S1, about 5 km) supporting the hypothesis that, over vegetation, the accuracy of interferometric-derived heights does not increase by using large baselines. The operational utility of this model can be easily exemplified using a case study. Suppose we investigate tree heights in a forest having an expected value of 25 m. The model of Figure 5b makes possible to obtain an optimal baseline value of 150 m. Similarly, it can be said that the optimal \(B\) value in forests with tree heights ranging between 15 m and 30 m (the majority of forest in temperate zones) should range between 250 m and 100 m, respectively. Since pixel size (NL/SGRP) has been proved to significantly impact \(\sigma_{\Delta\rho}\), and consequently \(\sigma_{\Delta\rho}\) (Equation (9)), some simulations were performed to describe SGRP dependency from \(\gamma\) (Figure 6). Equations (14) and (15) were applied repeatedly assuming a reference \(\sigma_{dh}=3\) m, and setting the following values for the involved parameters: \(H=693\) km, \(\theta=35^{\circ}\), and \(\delta_{az}=20\) m. This made it possible to describe SGRP variation against \(\gamma\) and \(B\). Figure 6 shows that SGRP is inversely proportional to \(\gamma\). Since vegetation usually presents medium-low coherence values expected in the range \(0.2<\gamma<0.6\)[17], correspondent expected SGRP falls in the range 50-300 m if \(B>50\) m. This SGRP (geometric resolution) is certainly not appropriate for a fine scale height retrieval; nevertheless, it appears to be proper for small-scale analysis (wide areas), generating a granularity that is consistent with the ones from traditional forest surveys. These are, in fact, characterized by a low-density of ground sampling (plots) [53]. Once the optimal \(B\) value was found and, consequently, the correspondent SGRP was found, \(\sigma_{dh}\) can be estimated as a function of \(\gamma_{FP}\) and expected \(dh\). To investigate these dependencies, a simulation was run assuming baseline values in the range of 50-200 m using Equation (5) (Figure 7). During simulations, the following operational parameters were used: \(NL=100\), \(\theta=35^{\circ}\), and \(H=693\) km. Figure 5: (a) Several scenarios of \(dh\) uncertainty according to \(h_{v}\) and \(B\). (b) Power model relating \(h_{v}\) and \(B\) at \(dh\) uncertainty is minimized. Figure 7 shows that \(dh\) values consistent with forest height in the range of 10-65 m generate \(\sigma_{dh}\) values varying between 1 m and 70 m, depending on \(\gamma_{FP}\) (coherence of the forest area) and \(B\) values. The most favourable conditions occur for high \(\gamma_{FP}\) and larger baselines.