# A TSVD-Based Method for Forest Height Inversion from Single-Baseline PolInSAR Data
## 1 Introduction  
It is well known that vegetation height plays an important role in quantifying the terrestrial carbon cycle [1; 2]. Moreover, vegetation height is an essential factor for the estimation of the biomass stored in vegetation [3; 4]. Therefore, accurately extracting vegetation height at a large scale is an important task. Given the fact that polarimetric interferometric synthetic aperture radar (PolInSAR) can separate the scattering power of a single resolution cell into the contributions of surface, double-bounce, and volume scattering, it can be considered to be a viable remote sensing technique for estimating vegetation height in large-scale areas [4; 5; 6; 7; 8; 9]. PolInSAR can be used to extract vegetation height through its sensitivity to the vegetation vertical structure [10; 11; 12; 13; 14]. The complex interferometric coherence of the observed PolInSAR data has been related to the vertical distribution of the vegetation scattering [10; 11; 12]. In a number of PolInSAR campaigns, the random volume over ground (RVoG) model [6; 10] has been used to extract vegetation height from the complex interferometric coherence [15; 16; 17]. The RVoG model is a physical PolInSAR model that integrates the complex coherence and the biophysical parameters. Based on the RVoG model, [PERSON] [6] proposed six-dimensional nonlinear optimization method, which has beensuccessfully evaluated with different types of PolInSAR data [6; 15; 16]. However, for this method, the accuracy of the solution is greatly dependent on the selected initial value, and a poor choice of initial value can result in unstable parameter estimation. Furthermore, the iterative procedure used in this method consumes too much time and is unsuitable for the inversion of large-scale areas. To cope with this problem, [PERSON] [18] separated the six-dimensional nonlinear parameter optimization process into three stages. This solution is known as the \"three-stage method\", and has been widely used in vegetation height extraction for its simple, universal, and time-saving properties [15; 19]. However, the three-stage method assumes that there is at least one polarization channel without a ground scattering contribution (the null ground-to-volume ratio (GVR) assumption), and thus it is difficult to determine the polarization because the volume and ground scattering contributions are always mixed in all of the polarization channels due to the diverse penetration depths [20]. As a result, the pure volume coherence estimated by the three-stage method is often inaccurate. Furthermore, with the three-stage method, it is not possible to adjust the contributions of the interferometric coherence observations in the estimation of the RVoG model parameters. The aim of this work is to address the limitations described above and extract accurate vegetation heights from single-baseline PolInSAR data. Based on the RVoG model, linear observation equations are first developed from the complex interferometric coherence observations by Taylor expansion, in order to combine all the available interferometric coherence observations. Next, since the coefficient matrix derived from the linear equations is ill-conditioned, in order to overcome this ill-posed problem, the proposed truncated singular value decomposition (TSVD)-based method is used to estimate the pure volume coherence. Furthermore, due to the fact that the ordinary truncation method for TSVD is not suitable for this issue, a more adaptive truncation method is proposed so as to improve the accuracy of the estimated pure volume coherence. Finally, the forest height is extracted from the estimated pure volume coherence by the use of a least-squares method. This paper is structured as follows. The principle of the RVoG model is introduced and discussed in Section 2. Section 3 presents the TSVD-based method for the estimation of vegetation height from complex interferometric coherence observations. The validation of the experiments are presented in Section 4. The discussions are presented in Section 5.