# 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
Considering Equation (20), it is clear that TSVD reduces the MSE by truncating the small singular values. However, the reduction of the MSE relies on the reduced variance being more than the introduced bias. Therefore, the truncation parameter k which determines the preserved large singular values and truncated small singular values is a key factor for TSVD to reduce the MSE. The method that is commonly used for the determination of the truncation parameter is related to the condition number, i.e., \(\lambda_{1}/\lambda_{i}\), as defined in Equation (15). If the condition number is larger than the given upper limit, the singular value \(\lambda_{i}\) should be truncated [28; 29; 30]. However, large-scale PolInSAR data usually consist of millions of pixels, and the singular values in ill-conditioned matrices of pixels are different from each other. Therefore, it is difficult to determine a reasonable upper limit for the condition number. In order to determine a reasonable truncation parameter for this issue, we need to develop a new approach. From Equations (19) and (21), it can be seen that if singular value \(\lambda_{i}\) is truncated, the reduced variances are \(\mathbf{\textit{\sigma}}_{0}^{2}/\lambda_{i}^{2}\) and the introduced bias is \(\mathbf{\textit{X}}^{\mathrm{T}}\mathbf{\textit{G}}_{i}\mathbf{\textit{G}}_{i}^ {\mathrm{T}}\mathbf{\textit{X}}\). Therefore, if \(\mathbf{\textit{\sigma}}_{0}^{2}/\lambda_{i}^{2}>\mathbf{\textit{X}}^{\mathrm{ T}}\mathbf{\textit{G}}_{i}\mathbf{\textit{G}}_{i}^{\mathrm{T}}\mathbf{\textit{X}}\), the singular value should be truncated, and if \(\mathbf{\textit{\sigma}}_{0}^{2}/\lambda_{i}^{2}<\mathbf{\textit{X}}^{\mathrm{ T}}\mathbf{\textit{G}}_{i}\mathbf{\textit{G}}_{i}^{\mathrm{T}}\mathbf{\textit{X}}\), the singular value should be preserved. In order to determine the small singular values, the values of \(\mathbf{\textit{\sigma}}_{0}^{2}/\lambda_{i}^{2}\) and \(\mathbf{X}^{\mathrm{T}}G_{i}\mathbf{G}_{i}^{\mathrm{T}}\mathbf{X}\) need to be calculated accurately. From the least-squares estimation, the estimation of is given by [23]:  
\[\hat{\sigma}_{0}^{2}=\frac{\mathbf{\
u}^{\mathrm{T}}\mathbf{\
u}}{2m-n}=\frac{ \left(\mathbf{U}\mathbf{U}^{\mathrm{T}}\mathbf{L}-\mathbf{L}\right)^{\mathrm{ T}}\left(\mathbf{U}\mathbf{U}^{\mathrm{T}}\mathbf{L}-\mathbf{L}\right)}{2m-n}, \tag{22}\]  
where \(\mathbf{U}\) denotes the \(2m\times n\) dimensional left singular vectors, \(\hat{\sigma}_{0}^{2}\) is the estimation of \(\sigma_{0}^{2}\)[23, 26], and \(n\) is the number of unknowns. It is clear that the small singular values have no adverse effect on the estimation of \(\sigma_{0}^{2}\). Therefore, \(\sigma_{0}^{2}/\lambda_{i}^{2}\) can be calculated by the least-squares solution if \(2m>n\)[23, 26]. In this paper, 10 polarizations are selected for the estimation of the pure volume coherence. Therefore, \(\sigma_{0}^{2}\) can be estimated. Using the SVD matrices of \(\mathbf{A}\), the estimation of \(\mathbf{G}_{i}^{\mathrm{T}}\mathbf{X}\) by least squares can be expressed as:  
\[\mathbf{G}_{i}^{\mathrm{T}}\mathbf{\hat{X}}=\lambda_{i}^{-1}\mathbf{U}_{i}^{ \mathrm{T}}\mathbf{L}. \tag{23}\]  
It can be seen from Equation (23) that the singular values greatly affect the estimation of \(\mathbf{G}_{i}^{\mathrm{T}}\mathbf{X}\). Small singular values seriously enlarge the value of \(\mathbf{G}_{i}^{\mathrm{T}}\mathbf{\hat{X}}\), whereas large singular values do not. The variance of the estimation of \(\mathbf{G}_{i}^{\mathrm{T}}\mathbf{X}\) can be given by:  
\[\mathrm{Var}_{\mathrm{g}}=\lambda_{i}^{-1}\mathbf{U}_{i}^{\mathrm{T}}\sigma_{ 0}^{2}\mathbf{U}_{i}\lambda_{i}^{-1}=\sigma_{0}^{2}\lambda_{i}^{-2}, \tag{24}\]  
where \(\mathrm{Var}_{\mathrm{g}}\) represents the variance of estimation \(\mathbf{G}_{i}^{\mathrm{T}}\mathbf{\hat{X}}\). The equation shows that the variance of \(\mathbf{G}_{i}^{\mathrm{T}}\mathbf{\hat{X}}\) is negatively correlated with the singular value. Therefore, the variance of \(\mathbf{G}_{i}^{\mathrm{T}}\mathbf{\hat{X}}\) which corresponds to a large singular value is small.