# A TSVD-Based Method for Forest Height Inversion from Single-Baseline PolInSAR Data
## 2 RVoG Model  
The RVoG model is a physical model that associates vegetation vertical structure parameters with multiple complex interferometric coherence observables [10]. It is a basic and popular model for describing vegetation scenarios. The model depicts the vegetation layer as a volume with randomly oriented particles over an impenetrable ground surface. Without considering the temporal decorrelation, the complex interferometric coherence \(\gamma(\omega)\) is expressed as [6]:  
\[\gamma(\omega)=e^{i\varphi_{0}}\frac{\gamma_{\text{$\upsilon$}}+\mu(\omega)}{1 +\mu(\omega)}, \tag{1}\]  
where \(\omega\) is the unitary polarization vector that defines the choice of scattering mechanism, \(\varphi_{0}\) denotes the ground surface phase, \(\mu(\omega)\) represents the ground-to-volume ratio (GVR) accounting for the polarization diversity, which varies with \(\omega\), and \(\gamma_{\text{$\upsilon$}}\) denotes the pure volume coherence, which is linked to the vegetation height and is expressed as [6]:  
\[\gamma_{\text{$\upsilon$}}=\frac{2\sigma\Big{(}e^{(2\sigma h_{\text{$ \upsilon$}}/\cos\theta+ik_{z}h_{\text{$\upsilon$}})}-1\Big{)}}{(2\sigma+ik_{z} \cos\theta)(e^{(2\sigma h_{\text{$\upsilon$}}/\cos\theta)}-1)}, \tag{2}\]  
where \(\sigma\) denotes the mean extinction coefficient, \(h_{\text{$\upsilon$}}\) denotes the vegetation height, \(\theta\) is the incidence angle, and \(k_{z}\) is the vertical wave number, which depends on the imaging geometry and wavelength [21]. \(k_{z}\) is given as:\[k_{z}=\frac{4\pi\Delta\theta}{\lambda\sin\theta}, \tag{3}\]  
where \(\lambda\) denotes the wavelength, and \(\Delta\theta\) represents the incidence angle difference between the master and slave images. According to Equation (1), the theoretical loci of the complex coherence sets of different polarizations follow a straight line in the complex plane, as shown in Figure 1 [18]. The figure also depicts the determination of the ground surface phase, which is applied in the three-stage method. However, the complex coherence observations no longer follows a common straight line in practice, due to the coherence fluctuations caused by all the possible decorrelations. In order to reconstruct the straight line accurately from noisy coherence sets, a line-fit approach based on least squares is proposed by the use of the following regression model [10; 18]:  
\[\text{Im}(\gamma(\omega))=c\text{Re}(\gamma(\omega))+d, \tag{4}\]  
where Re() and Im() denote the real and imaginary operations, respectively, \(c\) is the slope of the coherence line, and \(d\) is the intersection point to the imaginary axis. Once more than two complex coherence observations are provided, the coherence line can be determined by the line-fit approach. The ground phase \(\varphi_{0}\) can then be identified by the two intersections (\(\varphi_{1}\) and \(\varphi_{2}\)) of the coherence line and the unit circle [18]. However, the least-squares criterion may be unstable if the ellipticity of the coherence sets is high or if the complex coherence observations are too discrete, due to the noise of the polarizations. In addition, the three-stage method assumes that there is a polarization channel without ground scattering contribution, i.e., \(\mu(\omega)=0\), and then uses this assumption to further estimate the pure volume coherence. However, the volume and ground scattering contributions are always mixed in the polarization channels due to the diverse penetration depths. Therefore, it is difficult to find a polarization that can fit this hypothesis. As a result, when relying on this assumption, the estimated pure volume coherence will bias the extracted forest height. As shown in Equation (1), \(\gamma(\omega)\) is polarization-dependent. The complex coherence observations of the different scattering mechanisms can be obtained by the use of different unitary polarization vectors \(\omega\). In other words, if the number of polarization vectors is \(m\), then \(m\) equations like Equation (1) can be established.