# Uncertainties and Perspectives on Forest Height Estimates by Sentinel-1 Interferometry
## 3 Results and Discussion
### Theoretical Uncertainty of \(\omega\)  
\(\omega\) is the parameter that converts the interferometric phase difference (radians) into the correspondent height difference (meters, \(dh\)). \(\omega\) uncertainty (\(\sigma_{\omega}\)) depends both on the \(B\), \(R\), \(\theta\) factors value and on their accuracy (Equation (6)).  
Concerning \(\sigma_{B}\), it was obtained considering the S1-declared positional accuracy of the ESA provided POD, i.e., 5 cm [35, 36]. A reference value for \(\sigma_{B}\) was found to be equal to 12 cm. \(\sigma_{\theta}\) was assumed equal to 0.01\({}^{\circ}\), corresponding to the nominal antenna attitude accuracy for S1. \(\sigma_{R}\) was determined according to Equation (7). \(\sigma_{tropo}\) and \(\sigma_{iono}\) were set to 4 m and 1 m, respectively, according to [52]. This was obviously an approximation, since atmospheric conditions can locally change. The same paper [52] reported 0.4 m as a reference value for \(\sigma_{proc}\). \(\sigma_{TOF}\) was set equal to 3.3 m, assuming that S1 \(Bw\) (see Table 1) was 46 MHz.  
According to the above-mentioned values, \(\sigma_{R}\) was finally computed (Equation (7)) resulting in 5.3 m.  
To investigate \(\sigma_{\omega}\) dependency from involved parameters, one has to take care that \(R\) and \(\theta\) change along the scene, while \(B\) and \(H\) could be assumed constant. Measuring errors affecting these system parameters (Equation (1)) necessarily affect \(\sigma_{\omega}\). VPL can be used to explore how \(B\) and \(\theta\) affect the theoretical value of \(\sigma_{\omega}\) (Equation (6)). Figure 3a shows that for \(B\) values greater than 50 m, \(\sigma_{\omega}\) was lower than 10 m and increased while \(\theta\) values increased (i.e., tending to the swath far range). These findings raise some doubts about Equation (3), which does not take into account the error contributions of the system parameters. Additionally, \(\sigma_{\omega}\) appeared to be highly significant. A better estimate of \(\sigma_{dh}\) could come from Equation (4) where, differently from Equation (3), \(\sigma_{\omega}\) is considered.  
In order to explore the importance of system parameters on \(\sigma_{\omega}\), correspondent weights (\(w_{i}\)) were computed according to equations reported in Table 2. Results are reported in Figure 3b where it can be noted that 64%, 11% and 24% of the total variance (\(\sigma_{\omega}^{2}\)) were due to the look angle, \(R\) and the mixed term accounting for \(R\) and \(\theta\) correlation, respectively. \(B\) uncertainty participated for less than 1%.  
From an operational point of view, a rough sensitivity analysis can help to interpret results. For example, a look angle difference of 1\({}^{\circ}\) determines a \(\Delta\omega\) of about +3.7%; a satellite altitude difference of 10 km determines a \(\Delta\omega\) of about +1.4%; a slant range difference of 10 km determines a \(\Delta\omega\) of about +1.2%. \(\theta\) appears to be the most conditioning factor for \(\omega\) computation.  
Figure 2: Maximum detectable \(dh\) (HOA), avoiding phase unwrapping, versus \(B\) and \(\theta\) (simulation performed with reference to S1 nominal features).  
Given these values, \(\omega\) can be reasonably computed directly using the coarse estimates of \(\theta\), \(R\), \(H\), and \(B\), as reported in the SAR metadata information or satellite-approximated state vectors.  
While using S1 nominal features in Equation (6), it can be easily derived that \(\sigma_{\omega}\) varies between 1 m to 10 m, thus demonstrating to significantly contribute to the interferometric-derived \(dh\).