# A TSVD-Based Method for Forest Height Inversion from Single-Baseline PolInSAR Data
## 2 RVoG Model
### Estimation of Pure Volume Coherence from the Complex Interferometric Coherence Observations  
From Equations (1) and (2), it can be seen that \(\gamma(\omega)\) and \(\gamma_{v}\) are complex numbers. Therefore, the equations can be separated into two parts: a real part and an imaginary part. In order to simplify the nonlinear equations, we parameterize the pure volume coherence by \(\gamma_{v}=a+bi\), and then the complex interferometric coherence can be given by:  
\[\gamma(\omega)=e^{i\varphi_{0}}\frac{a+bi+\mu(\omega)}{1+\mu(\omega)}, \tag{5}\]  
and the observational function for the least-squares criterion can be formulated as:  
\[\gamma\big{(}\omega_{j}\big{)}=f\big{(}\varphi_{0},a,b,\mu\big{(}\omega_{j} \big{)}\big{)}j=1,2,\ldots,m, \tag{6}\]  
where \(\omega_{j}\) is the \(j\)-th unitary polarization vector, \(f\) is the RVoG model function, as described in Equation (5), \(\mu\big{(}\omega_{j}\big{)}\) is the \(j\)-th GVR associated with vector \(\omega_{j}\), and \(m\) represents the number of polarization projection vectors. It is clear that there are four unknown model parameters in each function, and \(m+3\) unknown parameters in \(m\) functions. To estimate the unknown model parameters, least-squares estimation is adopted to adjust the contributions of the complex coherence observations and suppress noise [20; 22; 23]. The corresponding least-squares criterion can be formulated as:  
\[\sum_{j=1}^{m}\left|\hat{\gamma}\big{(}\omega_{j}\big{)}-\gamma\big{(}\omega_{ j}\big{)}\right|^{2}=\text{min}, \tag{7}\]  
where \(\sum\) represents the summation operation, \(\left|\cdot\right|\) represents the modulus operation, and \(\hat{\gamma}\big{(}\omega_{j}\big{)}\) is the estimation of \(\gamma\big{(}\omega_{j}\big{)}\). Since the complex coherence can be separated into a real part and an imaginary part, Equation (7) can be equally converted to:  
\[\sum_{j=1}^{m}\Big{(}\left|\text{Re}\big{(}\hat{\gamma}\big{(}\omega_{j} \big{)}\big{)}-\text{Re}\big{(}\gamma\big{(}\omega_{j}\big{)}\big{)}\right|^{ 2}+\left|\text{Im}\big{(}\hat{\gamma}\big{(}\omega_{j}\big{)}\big{)}-\text{Im }\big{(}\gamma\big{(}\omega_{j}\big{)}\big{)}\right|^{2}\Big{)}=\text{min}. \tag{8}\]  
In this way, the residual functions for the least-squares criterion are formulated as:  
\[\left\{ \begin{array}{l}V_{\text{Re}}^{j}=\text{Re}\big{(}\hat{\gamma}\big{(} \omega_{j}\big{)}\big{)}-\text{Re}\big{(}\gamma\big{(}\omega_{j}\big{)}\big{)},\\ V_{\text{Im}}^{j}=\text{Im}\big{(}\hat{\gamma}\big{(}\omega_{j}\big{)}\big{)} -\text{Im}\big{(}\gamma\big{(}\omega_{j}\big{)}\big{)},\end{array} \right. \tag{9}\]  
where \(V_{\text{Re}}^{j}\) denotes the residual of the real part corresponding to the \(j\)-th unitary polarization vector, and \(V_{\text{Im}}^{j}\) denotes the residual of the imaginary part. From Equation (9), it can be seen that, if the number of polarization projection vectors is \(m\), then \(2m\) residual equations can be established. The number of unknown parameters is \(m+3\). Therefore, the unknowns can be estimated by least-squares estimation when the number of polarization projection vectors is more than three. From Equation (5), it can be seen that the complex coherence function is highly nonlinear, which greatly affects the efficacy of the least-squares estimation. In view of this, a linearized strategy based on a Taylor series is adopted to convert the nonlinear function into a linear function [23]. In this case, the residual functions for the least-squares criterion can be rewritten as:\[\left\{