# 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
\begin{array}{l}V_{\text{Re}}^{j}=\frac{\partial\text{Re}\left(\gamma( \omega_{j})\right)}{\partial\varphi_{0}}\text{d}\varphi_{0}+\frac{\partial \text{Re}\left(\gamma(\omega_{j})\right)}{\partial\text{a}}\text{d}a+\frac{ \partial\text{Re}\left(\gamma(\omega_{j})\right)}{\partial\text{b}}\text{d}b+ \frac{\partial\text{Re}\left(\gamma(\omega_{j})\right)}{\partial\text{f}\left( \omega_{j}\right)}\text{d}\mu\left(\omega_{j}\right)-l_{\text{Re}}^{j}\\ V_{\text{Im}}^{j}=\frac{\partial\text{Im}\left(\gamma(\omega_{j})\right)}{ \partial\varphi_{0}}\text{d}\varphi_{0}+\frac{\partial\text{Im}\left(\gamma( \omega_{j})\right)}{\partial\text{a}}\text{d}a+\frac{\partial\text{Im}\left( \gamma(\omega_{j})\right)}{\partial\text{b}}\text{d}b+\frac{\partial\text{Im} \left(\gamma(\omega_{j})\right)}{\partial\text{f}\left(\omega_{j}\right)} \text{d}\mu\left(\omega_{j}\right)-l_{\text{Im}}^{j}\end{array} \right. \tag{10}\]  
where \(\partial\) represents the partial derivative operation, \(\text{d}\varphi_{0}\), \(\text{d}a\), \(\text{d}b\), and \(\text{d}\mu\left(\omega_{j}\right)\) denote the corrections of the approximations of the unknown parameters, and \(l_{\text{Re}}^{j}\) and \(l_{\text{Im}}^{j}\) are the real and imaginary differences between the observed complex coherence and the predicted initial value. The vector \(\mathbf{V}\) denotes the residuals, \(\mathbf{X}\) denotes the unknown corrections, \(\mathbf{A}\) denotes the coefficient matrix, and \(\mathbf{L}\) denotes the real and imaginary differences. The residual functions for the least-squares criterion can then be expressed as [23; 24]:  
\[\mathbf{V}=\mathbf{A}\mathbf{X}-\mathbf{L}. \tag{11}\]  
The unitary polarization vector can be constructed by linear-basis polarization, Pauli-basis polarization, magnitude diversity optimization polarization [6], and phase diversity polarization [25]. Therefore, more than three polarization projection vectors can be obtained for the estimation of the unknown parameters. Using the least-squares criterion, an estimate of the unknown corrections can be given by:  
\[\mathbf{\hat{X}}=\left(\mathbf{A}^{\text{T}}\mathbf{A}\right)^{-1}\mathbf{A}^{\text{T}}\mathbf{L}, \tag{12}\]  
where \(\mathbf{\hat{X}}\) denotes the estimation of the unknown corrections. The estimation of the model parameters can then be obtained by combining the approximations and the corrections. However, during the computation, we find that matrix \(\mathbf{A}\) is seriously ill-conditioned. It is well known that an ill-posed problem is a great hindrance to obtaining accurate parameter estimation from observation equations. For an ill-posed equation, a small amount of noise in the observations can often bring large uncertainties to the estimation. The detrimental effect of the ill-posed problem is reflected in the large variance of the least-squares estimations. The variance-covariance matrix of the least-squares estimations can be given by [23]:  
\[\text{Cov}_{\hat{X}}=\sigma_{0}^{2}\left(\mathbf{A}^{\text{T}}\mathbf{A}\right)^{-1}, \tag{13}\]  
where \(\text{Cov}_{\hat{X}}\) represents the variance-covariance matrix of the estimations, and the diagonal elements of the matrix are the variances of the estimations, \(\sigma_{0}^{2}\) denotes the unit weight variance. We perform singular value decomposition (SVD) on matrix \(\mathbf{A}\)[24]:  
\[\mathbf{A}=\mathbf{U}\mathbf{S}\mathbf{G}^{\text{T}}, \tag{14}\]  
\[\mathbf{S}^{\text{T}}=\left[ \begin{array}{cccccc}\lambda_{1}&0&0&0&\cdots&0\\ 0&\ddots&0&0&\cdots&0\\ 0&0&\lambda_{n}&0&\cdots&0\end{array} \right], \tag{15}\]  
where \(\mathbf{U}\) is a \(2m\times 2m\) orthogonal matrix of the left singular vectors, \(\mathbf{G}\) is an \(n\times n\) orthogonal matrix of the right singular vectors, and \(n\) denotes the number of unknowns, which is equivalent to \(m+3\), \(\mathbf{S}\) is the matrix of singular values, and \(\lambda\) are non-negative real numbers that are conventionally listed in decreasing order, i.e., \(\lambda_{1}>\lambda_{2}>\cdots>\lambda_{n}\).