# 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
Using the trace operation, the sum of the variances of the estimations can be given by:  
\[\text{Vars}_{\hat{X}}=\text{Trace}(\text{Cov}_{\hat{X}})=\sigma_{0}^{2}\left( \sum_{i=1}^{n}\frac{1}{\lambda_{i}^{2}}\right), \tag{16}\]  
where \(\text{Vars}_{\hat{X}}\) represents the sum of the variances of the estimations, and Trace denotes the trace operation. If the equation is ill-posed, the singular values gradually decrease to zero, and \(\lambda_{1}\) is much larger than \(\lambda_{n}\) (\(\lambda_{n}\) is close to zero). Equation (16) shows that small singular values greatly magnify the estimation variances. Thus, the least-squares estimation becomes highly unreliable and is unable to obtain accurate estimations of the parameters. In order to overcome the ill-posedness of the problem, truncated singular value decomposition (TSVD) [26; 27; 28] is adopted. TSVD truncates the small singular values that greatly enlarge the variances to improve the least-squares estimation. Using the TSVD-based method, the estimation of the unknown parameters is given by:  
\[\hat{\mathbf{X}}_{\mathrm{t}}=\mathbf{\textit{G}}\mathbf{\textit{S}}_{\mathrm{ t}}^{\mathrm{T}}\mathbf{\textit{U}}^{\mathrm{T}}\mathbf{\textit{L}}, \tag{17}\]  
\[\mathbf{\textit{S}}_{\mathrm{t}}^{\mathrm{T}}=\left[ \begin{array}{cccc} \mathbf{\textit{S}}_{\mathrm{P}}^{-1}&0&\cdots&0\\ 0&\mathbf{\textit{S}}_{\mathrm{T}}=\mathbf{\textit{\textbf{0}}}&\cdots&0\\ \end{array} \right], \tag{18}\]  
where \(\hat{\mathbf{X}}_{\mathrm{t}}\) denotes the improved estimation by TSVD, \(\mathbf{\textit{S}}_{\mathrm{t}}\) denotes the inverse singular value matrix which truncates the small singular values, \(\mathbf{\textit{S}}_{\mathrm{P}}\) is the singular value matrix which represents the preserved large singular values, and \(\mathbf{\textit{S}}_{\mathrm{T}}\) is a zero matrix that represents the truncated small singular values. We can then obtain the variance of the TSVD estimations as follows:  
\[\mathrm{Vars}_{\hat{\mathbf{X}}_{\mathrm{t}}}=\mathrm{Trace}\Big{(}\mathbf{ \textit{G}}\mathbf{\textit{S}}_{\mathrm{t}}^{\mathrm{T}}\mathbf{\textit{U}}^{ \mathrm{T}}\mathbf{\textit{\sigma}}_{0}^{2}\mathbf{\textit{U}}\mathbf{\textit{S }}_{\mathrm{t}}\mathbf{\textit{G}}^{\mathrm{T}}\Big{)}=\mathbf{\textit{\sigma }}_{0}^{2}\Bigg{(}\sum_{i=1}^{k}\frac{1}{\lambda_{i}^{2}}\Bigg{)}, \tag{19}\]  
where \(\mathrm{Vars}_{\hat{\mathbf{X}}_{\mathrm{t}}}\) represents the variance of the TSVD estimations. \(k\) denotes the number of preserved large singular values. Equation (19) shows that TSVD greatly reduces the variance of the least-squares estimations through truncating the small singular values. However, it is well known that TSVD results in a biased estimation. Truncating the small singular values not only reduces the variance but also introduces bias into the estimation. TSVD improves the estimation by reducing the mean-square error (MSE) of the least-squares estimation. The MSE is expressed as [26]:  
\[\mathrm{Mse}_{\hat{\mathbf{X}}_{\mathrm{t}}}=\mathrm{Vars}_{\hat{\mathbf{X}}_ {\mathrm{t}}}+\textit{Bias}_{\hat{\mathbf{X}}_{\mathrm{t}}}^{\mathrm{T}} \mathrm{Bias}_{\hat{\mathbf{X}}_{\mathrm{t}}}, \tag{20}\]  
where \(\mathrm{Mse}_{\hat{\mathbf{X}}_{\mathrm{t}}}\) represents the MSE of the TSVD estimation, and \(\mathrm{Bias}_{\hat{\mathbf{X}}_{\mathrm{t}}}\) represents the bias introduced by TSVD. The sum of squares of \(\mathrm{Bias}_{\hat{\mathbf{X}}_{\mathrm{t}}}\) is given by:  
\[\textit{Bias}_{\hat{\mathbf{X}}_{\mathrm{t}}}^{\mathrm{T}}\mathrm{Bias}_{\hat{ \mathbf{X}}_{\mathrm{t}}}=\sum_{i=k+1}^{n}\mathbf{\textit{X}}^{\mathrm{T}} \mathbf{\textit{G}}_{i}\mathbf{\textit{G}}_{i}^{\mathrm{T}}\mathbf{\textit{X}}, \tag{21}\]  
where \(\mathbf{\textit{X}}\) represents the true values of the unknown parameters, and \(\mathbf{\textit{G}}_{i}\) denotes the \(i\)-th right singular vector that corresponds to the \(i\)-th singular value. Equation (21) shows that the bias is introduced by truncating the small singular values. The more singular values that are truncated, the more bias is introduced.