# A TSVD-Based Method for Forest Height Inversion from Single-Baseline PolInSAR Data
## 4 Examples  
### Simulated Experiments  
In order to evaluate the performance of the proposed TSVD-based inversion method for the estimation of vegetation height from single-baseline PolInSAR data, we simulated single-baseline PolInSAR data through the PolSARpro tool released by European Space Agency (ESA), Rome, Italy, using the following forest scenario, as shown in Table 1: simulated broad-leaved forest with an average height of 18 m; and ground phase of 0 degrees, corresponding to a ground elevation of 0 m. The complex interferometric coherence of the linear-basis polarizations (transmitted polarization and received polarization are horizontal polarizations, HH, transmitted polarization and received polarization are vertical polarizations, VV) and the Pauli-basis polarization (transmitted polarization is horizontal polarization and received polarization is vertical polarization, HV) could then be obtained. The simulated forest area and the Pauli RGB composite image are shown in Figure 2.  
Furthermore, using Pauli-basis polarization, magnitude diversity optimization polarization [6], and phase diversity polarization [25], the complex interferometric coherence observations of HH+VV, HH-VV, opt1, opt2, opt3, phase diversity (PD)-high, and phase diversity (PD)-low could be obtained. Therefore, 20 residual equations could be established from the above 10 complex interferometric coherence observations based on Equation (10) with 13 unknown parameters. The unknown parameters were then estimated by the TSVD-based method. From the estimated parameters and the pure volume coherence model which links to the forest height, the forest height could be extracted by Equation (29). The extraction results are shown in Figure 3b. For comparison, the extraction results of the three-stage method are shown in Figure 3a.  
From Figure 3, it can be seen that the TSVD-based method performs much better than the three-stage method in forest height inversion in this test. Clearly, from Figure 3a, the three-stage method fails to invert the forest heights of the rectangular areas of the figure, but the TSVD-based method effectively improves the inversion of the rectangular areas, as can be seen in Figure 3b. The mean values of the extracted forest heights by the three-stage method and the TSVD-based method are 13.383 m and 15.4440 m. This demonstrates that the forest height estimated by TSVD is more  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Platform** & **Center** & **Incidence** & **Vertical** & **Horizontal** & **Vertical** & **Ground** & **Forest** \\
**Height** & **Frequency** & **Angle** & **Baseline** & **Baseline** & **Wave** & **Phase** & **Height** \\
**(m)** & **(Hz)** & **(D)** & **(m)** & **(m)** & **Number** & **(D)** & **(m)** \\ \hline
3000 & 1.3 G & 45 & 1 & 10 & 0.1154 & 0 & 18 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of the simulated scenario.  
Figure 2: (a) simulated forest area; (b) Pauli RGB composite image.  
accurate than that estimated by the three-stage method. For a further comparison, 16 forest stands were selected from the simulated forest area, and the root-mean-square error (RMSE) of each stand was adopted to compare the performance of the two methods. The RMSEs of each stand are shown in Figure 4.  
For the 16 forest stands, the RMSE of the TSVD-based method is consistently less than that of the three-stage method. This indicates that the inverted forest height obtained by TSVD is closer to the true height than the height obtained by the three-stage method. This further validates the performance of the TSVD-based method.