# A TSVD-Based Method for Forest Height Inversion from Single-Baseline PolInSAR Data
## 2 RVoG Model
### Extraction of Vegetation Height from the Pure Volume Coherence  
Since the pure volume coherence is parameterized by \(\gamma_{v}=a+bi\), the pure volume coherence is estimated as \(\hat{\gamma}_{v}=\hat{a}+bi\) by the proposed TSVD-based method. Using the estimated parameters \(\hat{a}\) and \(\hat{b}\), the pure volume coherence which links to the vegetation height can be expressed as:  
\[\hat{a}+\hat{b}i=\frac{2\sigma\left(e^{(2\sigma h_{v}/\cos\theta+ik_{z}h_{v})}- 1\right)}{(2\sigma+ik_{z}\cos\theta)\left(e^{(2\sigma h_{v}/\cos\theta)}-1 \right)}. \tag{25}\]  
The equation can then be separated into a real part and an imaginary part:  
\[\left\{\begin{array}{l}\hat{a}=\mathrm{Re}(\gamma_{v}),\\ \hat{b}=\mathrm{Im}(\gamma_{v}).\end{array}\right. \tag{26}\]  
From Equation (25), it can be seen that \(\theta\) and \(k_{z}\) are the known parameters, and the unknown parameters are \(\sigma\) and \(h_{v}\). Therefore, \(\sigma\) and \(h_{v}\) can be estimated by the least-squares estimation from Equation (26). The Taylor series is used to convert the nonlinear function into a linear function. The residual functions for the least-squares criterion can then be expressed as [23]:  
\[\left\{\begin{array}{l}V_{\text{Re}}^{\text{P}}=\frac{\partial\text{Re}(\gamma _{\text{r}})}{\partial\sigma}\text{d}\sigma+\frac{\partial\text{Re}(\gamma_{ \text{r}})}{\partial\mu_{\text{p}}}\text{d}h_{\text{v}}-l_{\text{Re}}^{\text{ P}},\\ V_{\text{Im}}^{\text{P}}=\frac{\partial\text{Im}(\gamma_{\text{r}})}{\partial \sigma}\text{d}\sigma+\frac{\partial\text{Im}(\gamma_{\text{r}})}{\partial \mu_{\text{p}}}\text{d}h_{\text{v}}-l_{\text{Im}}^{\text{P}},\end{array}\right. \tag{27}\]  
where \(V_{\text{Re}}^{\text{P}}\) denotes the residual of the real part, and \(V_{\text{Im}}^{\text{P}}\) denotes the residual of the imaginary part, \(\text{d}\sigma\) and \(\text{d}h_{\text{v}}\) denote the corrections of the approximations of the unknown parameters, and \(l_{\text{Re}}^{\text{P}}\) and \(l_{\text{Im}}^{\text{P}}\) are the real and imaginary differences between the estimated pure volume coherence and the predicted initial value. Using the vector \(V_{\text{P}}\) to denote the residuals, \(\mathbf{X}_{\text{P}}\) denotes the unknown corrections, \(\mathbf{A}_{\text{P}}\) denotes the coefficient matrix, and \(\mathbf{L}_{\text{P}}\) denotes the real and imaginary differences. The residual functions for the least-squares criterion can then be expressed as [23; 24; 29]:  
\[V_{\text{P}}=\mathbf{A}_{\text{P}}\mathbf{X}_{\text{P}}-\mathbf{L}_{\text{P}}. \tag{28}\]  
Since the number of residual equations is the same as that of the unknowns, the least-squares criterion can be used to estimate the unknowns. The estimation is given by:  
\[\mathbf{\hat{X}}_{\text{P}}=\left(\mathbf{A}_{\text{P}}^{\text{T}}\mathbf{A}_{\text{P}} \right)^{-1}\mathbf{A}_{\text{P}}^{\text{T}}\mathbf{L}_{\text{P}}, \tag{29}\]  
where \(\mathbf{\hat{X}}_{\text{P}}\) denotes the estimation of the unknowns. The vegetation height can then be obtained from the estimation of \(\text{d}h_{\text{v}}\) and the initial value of \(h_{\text{v}}\) by:  
\[\hat{h}_{\text{v}}=h_{\text{v}0}+\text{d}h_{\text{v}}, \tag{30}\]  
where \(\hat{h}_{\text{v}}\) denotes the estimated vegetation height, and \(h_{\text{v}0}\) denotes the initial value of \(h_{\text{v}}\).