Retrieval of Boreal Forest Heights Using an Improved Random Volume over Ground (RVoG) Model Based on Repeat-Pass Spaceborne Polarimetric SAR Interferometry: The Case Study of Saihanba, China

Abstract

Spaceborne polarimetric synthetic aperture radar interferometry (PolInSAR) has the potential to deal with large-scale forest height inversion. However, the inversion is influenced by strong temporal decorrelation interference resulting from a large temporal baseline. Additionally, the forest canopy induces phase errors, while the smaller vertical wavenumber ($k_z$) enhances the sensitivity of the inversion to temporal decorrelation, which limits the efficiency in forest height inversion. This research is based on the random volume over ground (RVoG) model and follows the assumptions of the three-stage inversion method, to quantify the impact of repeat-pass spaceborne PolInSAR temporal decorrelation on the relative error of retrieval height, and develop a semi-empirical improved inversion model, using ground data to eliminate the interference of coherence and phase error caused by temporal decorrelation. Forest height inversion for temperate forest in northern China was conducted using repeat-pass spaceborne L-band ALOS2 PALSAR data, and was further verified using ground measurement data. The correction of temporal decorrelation using the improved model provided robust inversion for mixed conifer-broad forest height retrieval as it addressed the over-sensitivity to temporal decorrelation resulting from the inappropriate $k_z$ value. The method performed height inversion using interferometric data with temporal baselines ranging from 14 to 70 days and vertical wavenumbers ranging from 0.015 to 0.021 rad/m. The R² and RMSE reached 0.8126 and 2.3125 m, respectively.

1. Introduction

Forest ecosystems are the main components of terrestrial ecosystems [1]. Estimating the distribution and change of biomass and carbon storage in forest ecosystems can help to understand the relationship between carbon sources and carbon sinks, and the changing trends in terrestrial ecosystems [2,3,4]. Forest height is an essential parameter for representing the vertical structure of the forest. It provides a significant reference value in estimating forest carbon storage and plays a key role in evaluating forest stand quality and climate impact [5,6,7]. Remote sensing is an essential forest monitoring method that has allowed the development of various forest height retrieval technologies. Microwave remote sensing has attracted much attention due to its intense penetration into the atmosphere and forest canopy, and independence from weather conditions [8,9,10,11,12]. Meanwhile, spaceborne synthetic aperture radar (SAR) is widely used to observe forest heights of various forest types due to its advantages of the all-weather and all-season observation [4]. The use of spaceborne SAR data are of great significance in the study of forest parameters of northern temperate forests, and could play a role in achieving the upcoming global forest biomass and carbon cycle detection mission [13,14].

The SAR interferometry is used to calculate the vertical height of the ground using the phase difference between two sensors. This method is particularly important in terrain survey and evaluating terrain deformation [15,16,17]. For a complex distributed scattering unit such as a forest, it is often difficult to accurately separate the canopy phase and the ground phase using interferometric information alone (even when a high-frequency imaging system, such as C and X band, is used) [4,11]. PolInSAR has strong penetrability in the low frequency band (such as L-band and P-band), and can effectively retrieve the distribution of forest height through the canopy [18,19]. The difference in sensitivity of polarization to different components of the forest combined with the interferometric technique [20] allows inversion of forest height by distinguishing canopy scattering centers from ground scattering centers through the difference in ground scattering contribution ratio.

However, the position of scattering center is not entirely located on top of the canopy and the ground, making it difficult to extract forest height information [11]. A variety of interferometric coherence models such as the interferometric water cloud model (IWCM) [21,22,23], third-order Fourier-Legendre (FL) polynomial inversion mode [24], random volume over ground model (RVoG) [25,26,27], and the two-level method (TLM) [28] have been used to extract forest parameters. At present, the RVoG is the widely accepted model due to its simplicity and high accuracy [27]. Additionally, it is a relatively robust inversion model [29,30] whose limitations and potential errors in terms of temporal decorrelation and terrain interference can be addressed. As a result, several improved models and methods have been developed [9,31,32,33] to improve its applicability [34]. Meanwhile, the most commonly used method is the three-stage inversion method [35]. This method significantly improves the efficiency of parameter inversion through geometric analysis and strengthens the control of errors in the inversion process. The three-stage inversion method has been successfully applied in parameter inversion of different wavebands and different forest types [36]. Recent studies have shown that there are still more improved models for the RVoG model, which are necessary to make it more adaptable to various inversion conditions and increase the inversion accuracy. After comparing a variety of forest height retrieval methods, Chen et al. found that there was a significant increase in the retrieval accuracy when the S-RVoG model was used to retrieve forest heights by ALOS PALSAR data after introducing the normalized vegetation index [37]. Shi et al. improved the RVoG+VTD model using dual baselines by P-band E-SAR data, which consequently improved forest height inversion accuracy [38]. Xing et al. added a temporal-decorrelated adaptive estimation process based on the expectation maximum (EM) algorithm to the RVoG+VTD model and converted the Euclidean distance to a generalized distance to extract the magnitude features more efficiently. The accuracy of forest height estimated by this method was significantly improved compared with the original model and was closer to the Lidar data [39].

Nevertheless, the RVoG model considers the effect of forest height and volume decorrelation, but other decorrelation caused by any other factors introduces errors in the inversion process [9,40,41,42]. Therefore, it is still affected by several limiting factors, with the temporal decorrelation being the most significant factor [43,44,45,46]. To correct for temporal decorrelation, several improved models based on RVoG have been proposed. The most widely used models include RVoG+VTD model [35] and RMoG model [32,43]. Although these models have shown some effectiveness in the inversion process, they still have some limitations. The RVoG+VTD model fixes the extinction coefficient parameter in the model and introduces a parameter representing temporal decorrelation to correct for the effect of temporal decorrelation during the inversion process. However, this means that the role of the extinction coefficient in the inversion is neglected, and the choice of the initial value of this parameter will have an impact on the model accuracy. The RMoG model considers the interaction between temporal decorrelation and volume decorrelation, and inverts both simultaneously in the model. The model is more advanced and effective, but the model introduces more positional parameters, which increases the difficulty of the model and the requirement of observation quantity. In addition, these models were validated only on airborne data, but did not take into account the case of repeat-pass spaceborne data. The repeat-pass spaceborne SAR data has a larger temporal baseline than airborne SAR, and the impact of decorrelation is more difficult to ignore [23,47]. When using these models to invert forest heights on repeat-pass spaceborne PolInSAR, forest heights in the study area are seriously overestimated. Based on the characteristics of repeat-pass spaceborne InSAR, Lei et al. proposed a nonlinear iterative model based on the RMoG model to invert the forest height [41,48]. Although the model yields high accuracy results in repeat-pass spaceborne SAR, it uses coherence for inversion and neglects the effect of temporal decorrelation on the interferometric phase. However, due to the range of effective vertical wavenumbers, the interferometric phase of the repeat-pass spaceborne SAR is more significantly affected by temporal decorrelation. The effective vertical wavenumber ($k_z$) plays an essential role in linking the forest height to interferometry [40]. However, the range of vertical wavenumber is faced with limitations when inverting the forest height through the RVoG model [49]. For instance, $k_z$ with an excessively large or excessively small value will increase the interference of decorrelation and cause considerable deviations in the inversion results [50]. The $k_z$ value of the repeat-pass spaceborne PolInSAR data are often lower than the scope of inversion, making it difficult to invert the forest height using the RVoG model.

The present research is based on the RVoG model and the three-stage inversion method. It aimed at analyzing the effects of coherence and phase errors of temporal decorrelation on repeat-pass spaceborne PolInSAR inversion performance using theoretical and experimental approaches. This study further recommends a method improved by the RVoG model and suitable for repeat-pass spaceborne PolInSAR data. The research theoretically analyzed the inversion accuracy of the improved model, and used the world’s only repeat-pass spaceborne L-band ALOS-2 PolInSAR data to test and verify the temperate forests of Hebei Province, China. Finally, a semi-empirical improved model based on RVoG model was established by using ground data as prior data to correct for the interference caused by temporal decorrelation to the forest height inversion obtained from repeat-pass spaceborne PolInSAR data.

The structure of this paper is as follows: Section 2 introduces the study area and the acquisition process of ground measurement data. The basic conditions and preprocessing of the ALOS2 PALSAR data set have been introduced as well. In Section 3, we present the theoretical background of the RVoG model and the three-stage inversion method. The effect of temporal decorrelation on forest height inversion and the corresponding theoretical model have also been explained in this section. An improved model for temporal decoherence is proposed in Section 4. The paper presents theoretical background and the inversion process of the model, and performs a theoretical error analysis of the model. Section 5 uses the Saihanba ground measurement data and ALOS2 PALSAR data to retrieve the forest height and evaluate the results. Some shortcomings of the model and its robustness under different temporal baselines are discussed in Section 6, and recommendations made for the selection of future PolInSAR spatial baselines. Finally, the conclusion is presented in Section 7.

2. Research Materials and Theoretical Models

2.1. Study Area and Sample Site Data Collection

The study was conducted in Saihanba Forest, a typical temperate forest in northern China. The area is located in the transition zone from Yanshan Mountain to Inner Mongolia in Hebei Province (117°E, 42°N), and is the largest plantation forest in the world. The forest sites in the area are rugged and mountainous, with an average elevation of 1500 to 2067 m, making ground surveys difficult (Figure 1). The complex climatic conditions and the vulnerability of conventional imagery to weather conditions in the region make it necessary to conduct microwave observation studies in the region [51].

The measured data from the plots was collected in the same study area as the PolInSAR data and randomly sampled within the study area. There are various types of forests in this area, and the main are temperate mixed conifer-broad forests. Larix principis-rupprechtii Mayr. and Betula platyphylla Suk. are the dominant species in the area. In addition, they include Picea asperata Mast., Pinus sylvestris var. mongolica Litv., and some broad-leaved species (Figure 2). During the field measurements, we conducted a careful field inspection before selecting the sample sites in order to make the samples better represent the overall condition of the forest area. Among the final collected samples, 28 plots were measured at fixed points on a kilometer grid, and the remaining 69 plots were randomly sampled, hoping to represent the real situation of the forest as much as possible. The proportion of stand types in the random sampling samples was ensured to be similar to the proportion of overall stand types in the forest area as far as possible, and the forest types in the sample sites are shown in

Table 1. Forest types and the collected random sample sizes.

Forest Types	Sample Size	Forest Types	Sample Size
Pure forest of Larix principis-rupprechtii Mayr.	36	Pure forest of Pinus tabuliformis var. mukdensis	2
Pure forest of Betula platyphylla Suk.	12	Mixed coniferous forest	4
Pure forest of Picea asperata Mast.	4	Mixed broad-leaved forest	4
Pure forest of Pinus sylvestris var. mongolica Litv.	2	Coniferous and broad-leaved mixed forest	5

. Furthermore, the forest management in Saihanba area is in good condition, with obvious differences in stand age and density. Due to tending and thinning, most of the forests gradually decrease in density as the forest age increases. In order to restore the true forest condition, at least 30% of the samples were guaranteed to be young (high-density stands) or old growth (low-density stands) during the sampling process.

When collecting samples, we avoided the forest edge and large empty windows, and chose a relatively central position in the small forest class to ensure that the samples represent the real situation of the surrounding small forests as much as possible. After determining the center point, a distance of 17.32 m was measured along the four positive directions to determine the location of the four corner points, enclosing a diamond-shaped sample with an area of 0.06 ha (Figure 2a). The trees in the sample plot were inspected for each log, and the height of each log was measured with a Vertex IV ultrasonic height gauge. In this study, the forest height is defined as the average tree height of the sample plot. This assumption takes into account the value of average tree height in forest surveys on the one hand, and on the other hand provides support for the subsequent estimation of forest biomass. According to the field measurement results, the forest height was between 0 and 30 m, and the average height was 17.98 m. The Kolmogorov–Smirnov test was performed on the forest height sample data to test for normal distribution. The results proved that the sample data obeyed the normal distribution and has good representativeness (Figure 3).

2.2. PolInSAR Data

The PolInSAR datasets from the study area were in five scenes of ALOS2 PALSAR repeat-pass fully polarized synthetic aperture radar data (Figure 4). The 1.1-level L-band SLC data were developed by Japan Aerospace Exploration Agency (JAXA). This data were used in this study because it is among the few commercial L-band segment satellite-based fully polarized SAR data that is publicly available worldwide. Additionally, the L-band is less sensitive to forest vertical heterogeneity and allows accurate inversion of forest height from empirical structure functions [52]. Therefore, it is important to study independent inversion of forest height from this data before the opening of the new low-band satellite-based SAR system.

The datasets were collected over the study area from July to September 2020, representing the growing season of the forests in northern China. This was done within the same period the field observation data of the sample site was collected. The average zenith angle was 27.8°, the range pixel spacing was 5.66 m, the azimuthal pixel spacing was 2.86 m, the general observational area was 4944.62 km², and the average height of the sensor from the Earth’s surface was 636.56 km. The spatial baseline length between different data ranged from 80.2 to 170.4 m. $k_z$ ranged from 0.012 to 0.021 rad/m, and the temporal baseline between two adjacent data were 14 days. The images acquired on different dates were combined as the interferometry primary and secondary data. Four groups of PolInSAR interferometric pairs with different temporal baselines and vertical wavenumbers were set up (

Table 2. PolInSAR interferometric datasets.

Data Sets	Date of Image 1	Date of Image 2	Average Vertical Wavenumber	Temporal Baseline/Day
0711-0725	11 July 2020	25 July 2020	0.015	14
0905-0919	5 September 2020	19 September 2020	0.018	14
0808-0919	8 August 2020	19 September 2020	0.018	42
0711-0919	11 July 2020	19 September 2020	0.021	70

). The inversion of each pair of interferometric data were done independently to compare and verify the results. The inversion process was repeated in each pair of interferometric data.

Each set of interferometric pairs was pre-processed to remove decorrelation geometrically using GAMMA software [53,54]. Meanwhile, ionosphere-induced phase drift and path delays are the main sources of error in ALOS2 repeat-pass spaceborne PolInSAR when performing interferometry [55,56]. In this study, the ionospheric effect was eliminated using the distance splitting spectroscopy method (Figure 5). Terrain correction was performed using 30 m resolution SRTM DEM data [57,58].

3. Theoretical Analysis of Forest Height Inversion

3.1. RVoG Model and Three-Stage Inversion Method

PolInSAR data combines both interferometry and polarization properties [59]. The complex interferometric coherence ${\gamma }_{Obs}(\overrightarrow{\omega })$ is obtained by combining the matrices $s_1(\overrightarrow{\omega })$ and $s_2(\overrightarrow{\omega })$ of the primary and secondary images at a particular polarization $\overrightarrow{\omega }$ and can be expressed as follows:

$$\begin{array}{c}{\gamma }_{Obs}(\overrightarrow{\omega })=\frac{\langle s_1(\overrightarrow{\omega })s_2^{*}(\overrightarrow{\omega })\rangle }{\sqrt{\langle s_1(\overrightarrow{\omega })s_1^{*}(\overrightarrow{\omega })\rangle \langle s_2(\overrightarrow{\omega })s_2^{*}(\overrightarrow{\omega })\rangle }}\end{array}$$

(1)

where * represents the conjugate of the SAR image and $\langle \rangle$ represents the expected value [28,29]. The magnitude of the complex coherence ${\gamma }_{Obs}(\overrightarrow{\omega })$ ($\left|{\gamma }_{Obs}\right|$) represents the coherence between two images (i.e., degree of similarity between two images), and its value ranges from 0 to 1.

Previous studies have shown that the complex coherence obtained by Equation (1) is still affected by several decorrelations even after eliminating system induced decorrelation [2,54]. The observed interferometric coherence can be modeled as a combination various contribution [54,55] and illustrated as follows:

$$\begin{array}{c}{\gamma }_{Obs}={\gamma }_{SNR}{\gamma }_{Tmp}{\gamma }_{vol}\end{array}$$

(2)

Here,${\gamma }_{SNR}$ represents the decorrelation effect from thermal noise; ${\gamma }_{Tmp}$ is temporal decorrelation; and ${\gamma }_{vol}$ is volumetric decorrelation, which is widely used to forest height inversion. ${\gamma }_{SNR}$ can be eliminated during image preprocessing. However, due to the constraints of a variety of factors, ${\gamma }_{Tmp}$ can introduce bias between ${\gamma }_{Obs}$ and ${\gamma }_{vol}$; it not only affects the phase of ${\gamma }_{Obs}$, but also further reduces the overall coherence. For northern forests with lower forest heights, ${\gamma }_{Tmp}$ may sometimes mask the influence of ${\gamma }_{Obs}$, especially for repeat-pass spaceborne PolInSAR data with a temporal baseline of several days [28].

${\gamma }_{vol}$ is included into the model to allow calculation of the forest height. The RVoG model combines the forest height with the scattering properties by treating the scattering as a volume scattering and ground scattering contributions through the assumption of the forest as a random homogeneous scatterer [28]. The model expresses the forest volume scattering complex coherence as follows:

$$\begin{array}{c}{\gamma }_{vol}=e^{i{\phi }_0}\frac{{\gamma }_v+m(\overrightarrow{\omega })}{1+m(\overrightarrow{\omega })}\end{array}$$

(3)

where $e^{i{\phi }_0}$ is the ground scattering contribution, m is the effective ground-to-volume amplitude ratio, and ${\gamma }_v$ is the volume scattering complex coherence without the ground contribution. ${\gamma }_v$ can be expressed by the mean extinction coefficient $\sigma$ and the forest height $h_v$ as follows:

$$\begin{array}{c}{\gamma }_v=\frac{2\sigma }{\cos \theta \left(e^{\frac{2\sigma h_v}{\cos \theta }}-1\right)}{{\displaystyle \int }}_0^{h_v}{e}^{-i{k}_{z}z}{e}^{-\frac{\left(2\sigma z\right)}{\cos \theta }}dz\end{array}$$

(4)

where $k_z$ is the vertical wavenumber.

$$\begin{array}{c}{k}_z=\alpha \frac{2\pi ∆\theta }{\lambda \sin {\theta }_0}\approx \alpha \frac{2\pi B_{\perp }}{\lambda R\sin {\theta }_0}\end{array}$$

(5)

${\theta }_0$ is the radar incidence angle, $∆\theta$ is the incidence angle difference between the two images induced by the spatial baseline, λ is the wavelength, $B_{\perp }$ is perpendicular component of the spatial baseline, $R$ is the slant range, and $\alpha$ is an integer constant that is equal to 2 for monostatic acquisition and 1 for bistatic acquisition.

Since the ground scattering ratio is a parameter affected by polarization, Equation (3) can be observed as a straight line in the complex plane (Figure 6). The three-stage inversion method is applicable to PolInSAR developed from this geometric property [38]. The first stage of the inversion method uses different ground contributions contained in the different PolInSAR data polarizations, which fall at different positions in the complex plane (blue shade in Figure 6), and the coherence line of Equation (3) can be fitted. In the second stage, the two intersection points of the coherence line and the unit circle of the complex plane are determined as candidate points for the ground coherence point. The actual ground coherence point is determined through screening, and its influence is eliminated. In the third stage, a 2D lookup table of extinction coefficient and forest height (red points in Figure 6) can be established according to Equation (4). Generally, it is assumed that the complex coherence point under HV polarization is mainly volume-only scattering in InSAR, and the complex coherence in PolInSAR coherence optimization represents volume scattering, which allows forest height inversion, and determination of extinction coefficient value in the lookup table.

3.2. Temporal Decorrelation

The temporal decorrelation of interferometric image often causes obvious errors to the inversion results, regardless of whether RVoG model or other interferometric inversion models are used. Temporal decorrelation reduces the coherence and causes a phase shift of the interferometric data so that the forest height inversion model is greatly affected. The error sources of temporal decorrelation have a complex structure [42] and are influenced by combination of factors that are difficult to quantify. As a result, temporal decorrelation is difficult to remove when pre-processing image data.

In general, the degree of temporal decorrelation is described by the temporal baseline, which is the time interval between primary and secondary image observations. Data with larger temporal baselines undoubtedly face greater temporal decorrelation. The temporal baseline of repeat-pass spaceborne SAR data for the same observation area tends to be more significant compared to airborne PolInSAR data. This study’s temporal baseline of ALOS2 repeat-pass spaceborne SAR varies from a few days to tens of days. Therefore, there is a larger temporal decorrelation contribution in the repeat-pass spaceborne SAR interferometric data. In addition, the sensitivity of different images to the same temporal decorrelation of the interference varies and PolInSAR data with smaller $k_z$ has a more pronounced response to temporal decorrelation. For the InSAR, $k_z$ is defined by the angular difference between the primary and secondary sensors (Equation (7)) and is a factor that indicates the sensitivity of the interferometric phase to changes in terrain (height) [49]. For the PolInSAR inversion forest height model, the smaller $k_z$ makes the forest height more sensitive to changes in the coherence and phase of the interferometric data during inversion [49]. Therefore, when using interferometric images with small $k_z$ to retrieve forest height, there may be huge errors in the inversion results even if there is a weak temporal decorrelation factor [50]. The situation is reflected in the absence of intersection between the observed complex coherence in the unit circle and the LUT when using the three-stage inversion method for height inversion [60]. Meanwhile, the $k_z$ of repeat-pass spaceborne PolInSAR data tend to be lower, making the inversion results (which already contain a large temporal decorrelation factor) less accurate. Therefore, it is necessary to adopt an effective correction for temporal decorrelation inversion model when retrieving forest height from repeat-pass spaceborne PolInSAR data.

There are several improved models for temporal decorrelation including the RVoG+VTD model [60], the RMoG model [35,43], and the semi-empirical iterative model for dielectric constant and random motion modeling using Gauss-Newton iterative optimization model [42,48].

The RVoG+VTD model demonstrates that, temporal decorrelation shifts the volume-only coherence points in the complex plane unit circle during the inversion using the three-stage method. As such, there is no intersection between the height-extinction LUT and the volume-only coherence point. Therefore, correction terms for the shifted volume-only coherence points are achieved by fixing the extinction coefficient and using the SINC function. Meanwhile, RMoG model is different from the RVoG+VTD model since the effect of temporal decorrelation on volumetric decorrelation is not considered to be a multiplicative relationship. This model quantifies the cause of temporal decorrelation as a stochastic motion parameter that varies with the forest canopy. This was the first method to attempt to model the direct extraction of forest height from the mixed effect of temporal decorrelation and volumetric decorrelation. Additionally, a ten-dimensional parametric model was developed from the observations under different polarization channels. Another study decomposed temporal decorrelation into the temporal effects of dielectric constant variation and random motion, used coherence to build an empirical model, and extracted the parameters using the Gauss-Newton iterative method [42,48]. The forest height model can use the repeat-pass spaceborne InSAR data in L-band with large temporal baseline.

4. Improved Inversion Model

In this study, a new inversion method has been proposed to enable forest height inversion by empirical iteration. Corrections were performed to degrade coherence and phase shift caused by error sources such as temporal decorrelation. In addition, the inversion accuracy of the improved model was simulated, and its geometric process was analyzed theoretically.

4.1. Theoretical Background

The interferometric phase obtained from the interferometry can be used to calculate the topographic height of the ground surface.

In repeat-pass interferometry, the same sensor makes two recordings of the same ground target at a certain time interval to form an interferometric data pair (Figure 7). The first recording is referred to as the primary image (marked as $s_1$), and the second recording is referred to as the secondary image (marked as $s_2$).

$$\begin{array}{c}{s}_1=a_1{e}^{i{\phi }_1}\\ s_2=a_2{e}^{i{\phi }_2}\end{array}$$

(6)

The primary and secondary images are conjugated and multiplied to obtain the interferometric phase values.

$$\begin{array}{c}\varphi =\arctan \left(s_1{s}_2{}^{*}\right)={\phi }_1-{\phi }_2\end{array}$$

(7)

When phase ambiguity is not considered, the interferometric phase can be expressed as follows:

$$\begin{array}{c}\varphi =-\frac{4\pi }{\lambda }∆R\end{array}$$

(8)

where $∆R$ is the difference between $s_1$ and $s_2$ radar wave propagation distances.

The interfering phase $\varphi$ consists of the following five main components:

• Flat Earth phase ${\varphi }_{flat}$ due to reference ellipsoid.
• Topographic phase ${\varphi }_{topo}$ due to terrain undulation.
• The deformation phase caused by the deformation of the ground surface during the two imaging sessions.
• The phase difference caused by atmospheric disturbances.
• The phase difference due to noise.

In this study, the ground surface was considered undeformed. The atmospheric and noise disturbances were ignored.

After eliminating the above interference factors, only the flat earth phase and the topographic phase remained in the interferometric phase. The geometric principle of interferometry is explained using a previous example where the topographic height h was measured at the surface target point P in Figure 7 [16]. Point P in the figure is the target point of interferometry. $P_0$ is the point on the reference ellipsoid, and it allows equal distance from $s_1$ to P and $P_0$.

When the sensor is far enough from the ground target, the component $B_{\parallel }$ of the spatial baseline (between $s_1$ and $s_2$) that is parallel to the line $R_1$ (between $s_1$ and P) can be approximately equal to $∆R$ ($B_{\parallel }\approx ∆R$). The topographic phase at point P is expressed as follows:

$$\begin{array}{c}{\varphi }_{topo}=-\frac{4\pi }{\lambda }{B}_{\parallel }=-\frac{4\pi }{\lambda }B\sin \left(\theta -\alpha \right)\end{array}$$

(9)

B is the spatial baseline between $s_1$ and $s_2$, $\theta$ is the angle between the line connecting $s_1$ to the ground target point P and the vertical direction, and $\alpha$ is the angle between the baseline B and the horizontal direction. The flat earth phase at point $P_0$ is as follows:

$$\begin{array}{c}{\varphi }_{flat}=-\frac{4\pi }{\lambda }{B}_{\parallel }^0=-\frac{4\pi }{\lambda }B\sin \left({\theta }_0-\alpha \right)\end{array}$$

(10)

$B_{\parallel }^0$ is the component of the spatial baseline B that is parallel to the line $R_0$ between $s_1$ and $P_0$, ${\theta }_0$ is the angle between lines $s_1$ and $P_0$ and the vertical direction, and ${\varphi }_{flat}$ is the flat earth phase corresponding to the point P. Meanwhile, the phase difference $∆\varphi$ between P and $P_0$ can be expressed as follows:

$$\begin{array}{c}∆\varphi ={\varphi }_{topo}-{\varphi }_{flat}=-\frac{4\pi }{\lambda }B\left(\sin \left(\theta -\alpha \right)-\sin \left({\theta }_0-\alpha \right)\right)\end{array}$$

(11)

The angle $∆\theta$ between ${\theta }_0$ and $\theta$ is very small, due to the long distance between the sensor and the ground target. Thus, Equation (11) can be simplified as follows:

$$\begin{array}{c}∆\varphi =-\frac{4\pi }{\lambda }B\cos \left({\theta }_0-\alpha \right)∆\theta =-\frac{4\pi }{\lambda }{B}_{\perp }^0∆\theta \end{array}$$

(12)

$B_{\perp }^0$ is the component of the spatial baseline B, which is perpendicular to $R_0$. In the geometric relationship illustrated in Figure 7, the value of the height of point P is computed as shown below:

$$\begin{array}{c}\begin{array}{c}h=H-R_1\cos \theta =R_1∆\theta \sin \theta -∆R_1\cos \theta \\ ∆\theta =\frac{h+∆R_1\cos \theta }{R_1\sin \theta }\end{array}\end{array}$$

(13)

h is the height of the ground target point P from the horizontal plane, H is the height of $s_1$ from the horizontal plane, $R_1$ is denoted as the radar wave propagation distance of $s_1$, and $∆R_1$ is the difference between $R_1$ and $R_0$. Therefore, the topographic phase and height can be expressed as shown in the equation below:

$$\begin{array}{c}∆\varphi =-\frac{4\mathsf{\pi }{B}_{\perp }^0}{\lambda R_1\sin \theta }·\left(h+∆R_1\cos \theta \right)=-k_z·h-\frac{4\mathsf{\pi }{B}_{\perp }^0}{\lambda R_1\tan \theta }·∆R_1\end{array}$$

(14)

The first term on the right side of the equation is the terrain phase given the terrain height, while the second term is the flatland phase considering the zero change in elevation. After removing the flat earth phase, we obtain a linear relationship between height and terrain phase, and it is linked by $k_z$.

The above equation showed terrain height measurement by interferometric phase. However, there is a significant difference in measurements when using PolInSAR to measure forest height. Meanwhile, the interferometric phase between the top point of the canopy and the underlying surface point should be included in the calculation of forest height. However, the observed phase at the top of the canopy had two-phase contributions in addition to the five components mentioned in Equation (8). The first contribution is the shift of the phase center caused by the penetration of low-frequency SAR into the forest canopy [53]. The second contribution is the phase shift caused by random movement of the canopy during the imaging of the primary and secondary images [43]. Both of these contributions interfere with the forest height measurement and should be eliminated during the inversion process.

The geometry of the forest height measurement by interferometric phase is illustrated in Figure 8. $P_1$ is the phase center at the top of the canopy. However, the phase shift caused by the random motion of the canopy causes the phase center of the canopy to shift to $P_2$ when being observed. The shift in phase center that is caused by low-frequency SAR penetration makes it possible for the observed phase to lie anywhere between $P_2$ and $P_2^{\prime }$. $P_0$ is the corresponding point on the reference ellipsoid. The interferometric phase and the height from the horizontal plane at $P_2$ are then described as follows:

$$\begin{array}{c}∆{\varphi }_1=-\frac{4\pi }{\lambda }{B}_{\perp }^0∆{\theta }_1\end{array}$$

(15)

$$\begin{array}{c}{h}_1=H-R_1^{\prime }\cos \theta \end{array}$$

(16)

The relationship between interferometric phase and height at point $P_2$ is as follows:

$$\begin{array}{c}∆{\varphi }_1=-\frac{4\pi B_{\perp }^0}{\lambda R_1^{\prime }\sin \theta }·\left(h_1+∆R_1^{\prime }\cos \theta \right)=-k_z·\left(h+∆h\right)-\frac{4\pi B_{\perp }^0}{\lambda R_1^{\prime }\tan \theta }·∆R_1^{\prime }\end{array}$$

(17)

where $∆R_1^{\prime }$ is the difference between $R_1^{\prime }$ and $R_0^{\prime }$. $∆h$ is the height difference between $P_1$ and $P_2$, and the offset of the measured height of the canopy. The second term on the right-hand side of the equation is the flat-earth phase of $P_2$. The above two interference factors influence the height deviation, so it is necessary to quantify the height error of the two effects. Previous studies have described random motion as Gaussian function that varies with height and uniformly when in the vertical direction [35,43]. There was neither deformation on the surface nor phase difference caused by random motion on the surface. When only the offsets produced by the random motion on the canopy are taken into account, the height offset can be considered as a linear function of the canopy height and can be expressed as follows:

$$\begin{array}{c}∆h={\delta }_r\left(h\right)=\frac{{\delta }_r}{h_r}h\end{array}$$

(18)

where ${\delta }_r$ denotes the standard deviation of the motion at a certain reference height $h_r$, and ${\epsilon }_0$ represents the variation of this offset from the height.

$$\begin{array}{c}∆h={\epsilon }_0·h\end{array}$$

(19)

Previous studies have focused on the modeling of random motion and understanding the relationship between interferometric coherence and random motion. However, the effect phase shift errors on the height inversion results may be more pronounced. Therefore, this study attempts to correct this error and improve accuracy in the height inversion results.

Meanwhile, the phase bias caused by low-frequency SAR penetration makes the observed phase lie between the top of the canopy and a half of the height [53]. Therefore, the height deviation obtained by coupling the two factors can be simplified as a linear function that changes with the vertical target height:

$$\begin{array}{c}∆h={\epsilon }_0·h-d\end{array}$$

(20)

where $d$ is the distance between the scattering center and the underlying surface after the canopy phase shift $h/2\le d\le h$. The height error was brought into the canopy phase without the flat-earth phase in order to obtain the relationship between the observed canopy phase and the true canopy height. The equation is as follows:

$$\begin{array}{c}{\varphi }_{vol}=k_z·\left(\left(1+{\epsilon }_0\right)h-d\right)=\epsilon k_{z}h-k_{z}d=\epsilon k_{z}h+{\phi }_e\end{array}$$

(21)

where $\epsilon$ is the correction term for temporal decorrelation due to random motion, and ${\phi }_e$ is the corrected phase for the phase center shift. Noteworthy, $\epsilon \ge 1$, $-\pi \le {\phi }_e\le \pi$.

Temporal decorrelation not only causes phase shift but also leads to a reduction of factors affecting interferometric coherence, such as dielectric constant with temporal baseline. This also has a more pronounced effect on the PolInSAR inversion of forest height and, therefore, measures are needed to reduce this interference. In order to address the apparent temporal decorrelation of coherence amplitude, phase interference, and canopy phase center shift suffered during the repeat-pass spaceborne PolInSAR inversion, this study proposes a new inversion method to achieve the inversion of forest height by empirical iteration. First, to address the interference of temporal decorrelation on the interferometric coherence and correct the overall coherence, this study introduces a correction term $\left|{\gamma }_e\right|$ based on the RVoG model (Equation (4)):

$$\begin{array}{c}\widehat{\gamma }=\left|{\gamma }_e\right|·\frac{2\sigma }{\cos \theta \left(e^{\frac{2\sigma h_v}{\cos \theta }}-1\right)}{{\displaystyle \int }}_0^{h_v}{e}^{-i{k}_{z}z}{e}^{-\frac{\left(2\sigma z\right)}{\cos \theta }}dz\end{array}$$

(22)

With respect to the offset phase caused by the random motion and microwave penetration factors in the temporal coherence, the offset phase value of the height modeling in Equation (21) is introduced into Equation (22) as follows:

$$\begin{array}{c}\begin{array}{c}\widehat{\gamma }=\left|{\gamma }_e\right|·\frac{2\sigma }{\cos \theta \left(e^{\frac{2\sigma h_v}{\cos \theta }}-1\right)}{{\displaystyle \int }}_0^{h_v}{e}^{-i\left(\epsilon k_{z}z+{\phi }_e\right)}{e}^{-\frac{\left(2\sigma z\right)}{\cos \theta }}dz\\ ={\gamma }_e·\frac{2\sigma }{\cos \theta \left(e^{\frac{2\sigma h_v}{\cos \theta }}-1\right)}{{\displaystyle \int }}_0^{h_v}{e}^{-i\left(\epsilon ·k_z\right)z}{e}^{-\frac{\left(2\sigma z\right)}{\cos \theta }}dz\end{array}\end{array}$$

(23)

where ${\gamma }_e=\left|{\gamma }_e\right|·e^{i{\phi }_e}$.

In previous studies, the physical model construction method [35,42] decomposed the temporal decorrelation into factors such as dielectric constant and random motion, and then modeled together with volume decorrelation to extract forest height. These models are undoubtedly advanced and effective, but often require complex iterative processes and have many model parameters, which increases the uncertainty of the inversion. This study attempts to use an empirical model to achieve fast and efficient inversion, and achieve results that are similar to previous temporal–decoherent models. Therefore, the model is built based on the three-stage inversion method, and the geometric properties of the model are used to improve efficiency of the inversion. The error factors encountered during the inversion are integrated into Equation (23) as a correction term $\epsilon$ on the phase, and a complex correction term ${\gamma }_e$. They are both brought into the iterative process to ensure that complete error sources are considered in the model.

The model is performed under the assumption that $P_2$ is in line with $s_1{P}_1$, but this does not affect the validity of the model. This study focuses on the height difference in the vertical direction between $P_2$ and $P_1$, and does not require obtaining of the specific true phase point. Therefore, even if the two are not on the same line, the inversion results in another point with the same height on the line. Moreover, since this is an empirical model, the real data are used as prior information. Therefore, forest height can be calculated based on this assumption.

4.2. Iterative Process of the Improved Model

The iterative process of the improved model follows the flow of the three-stage inversion method. First, the polarized interferometric information needs to be extracted from the pre-processed PolInSAR data. Each set of polarized interferometric data pairs contains the original complex data under full polarization, and the Pauli basis scattering vector for each set of images is as follows:

$$\begin{array}{c}\begin{array}{c}{k}_1=\frac{1}{\sqrt{2}}{\left(s_{H{H}_1}+s_{V{V}_1} s_{H{H}_1}-s_{V{V}_1} 2s_{H{V}_1}\right)}^T\\ k_2=\frac{1}{\sqrt{2}}{\left(s_{H{H}_2}+s_{V{V}_2} s_{H{H}_2}-s_{V{V}_2} 2s_{H{V}_2}\right)}^T\end{array}\end{array}$$

(24)

where $s$ represents the scattering matrix elements collected twice at different polarizations, $H$ represents horizontal polarization, and $V$ represents vertical polarization. ${\left(·\right)}^T$ represents the transpose of the matrix. With the Pauli basis vector, we can obtain the T coherence matrix and the ${\mathsf{\Omega }}_{12}$ matrix as follows.

T coherence matrix:

$$\begin{array}{c}T=\frac{1}{2}\left(T_{11}+T_{22}\right)=\frac{1}{2}\left(\langle k_1{k}_1{}^H\rangle +\langle k_2{k}_2{}^H\rangle \right)\end{array}$$

(25)

The ${\mathsf{\Omega }}_{12}$ matrix:

$$\begin{array}{c}{\mathsf{\Omega }}_{12}=\langle k_1{k}_2{}^H\rangle \end{array}$$

(26)

where $\langle ·\rangle$ denotes the mathematical expectation, and ${\left(·\right)}^H$ is the conjugate transpose.

With the T matrix and ${\mathsf{\Omega }}_{12}$ matrix, we can calculate the coherence and the phase of the primary and secondary images with different polarization and coherence optimization. This study employs the common and easily extracted complex coherence values, including three basic polarization types HH, HV, VV, four linear combinations of different polarizations (HH+VV, HH-VV, HHVV, HV+VH), three circular polarizations (LL, LR, RR), three Opt coherence optimizations (Opt1, Opt2, Opt3), and two PD coherence optimizations (PD High, PD Low). Due to the specific polarization correlation of the ground scattering contribution, the 15 complex coherences have different ground scattering ratio coherence.

The improved model extracted the volume-only scattering complex coherence method in general agreement with the three-stage inversion method. During the inversion, 15 types of complex coherences were projected into the unit circle of the complex plane. The least-squares method was used to fit the phase trunk using the different scattering ratios of the earth’s surface under different polarizations. The two intersections of the fitted coherence line and the unit circle of the complex plane allowed determination of the candidate points for ground coherence. The external DEM phase in this image was compared with the candidate points and filter to identify the true ground coherence points and remove their influence. After the removal of terrain phase, the coherent region of each pixel was obtained, and the volume-only complex coherent points in each pixel were screened by comparing the distance between the optimized complex coherent points and the ground coherent points.

Once the volume-only scattering complex coherence is scattered, a 2D look-up table of forest height-extinction coefficients is a requirement in the flow of the three-stage inversion method. The forest height and extinction coefficient values are obtained according to the intersection of the LUT and the volume-only complex coherence.

In this study, the process of improving the model is illustrated in Figure 9. After introducing $\epsilon$ and ${\gamma }_e$ into the RVoG model (through Equation (23)) as unknown parameters, a new set of LUTs was generated when the amplitude $\left|{\gamma }_e\right|$ and the phase ${\phi }_e$ of $\epsilon$ and ${\gamma }_e$ had different values. A triple iteration of the LUT was then performed after setting the range for the parameters. The intersection of each set of LUT with the volume-only decorrelation performed the forest height inversion under the set of parameters. Meanwhile, iteration added unknown parameters to the model. Therefore, the improved model needs the control condition of iteration, which is the true height measured on the ground. In this study, 25% of the ground-measured forest heights was randomly selected as the priori data for the iterations. The best parameter values in the iteration range and the corresponding inversion results were obtained by calculating and comparing the RMSE of each set of inversion heights with the ground data.

The improved model performs height inversion for all regions in the image, which was consistent with the original inversion method. The iterative parameters $\epsilon$ and ${\gamma }_e$ in the model are calculated at the image level, and only one parameter result is iterated for the same group of interferometric pairs. Therefore, the real ground measurement data used in the inversion need at least one true value that is a true reflection of the forest height in the image. This is particularly important in controlling the inversion error in order to understand forest height inversion from the image and create a balance in the model.

4.3. Theoretical Analysis of the Improved Model

In this study, the improved model is an empirical model. Besides the control of ground data, selecting the initial range of empirical parameters also has an important impact on the results. To discuss the feasibility of the improved inversion model, and establish whether the model satisfies the inversion conditions, this study theoretically tested the improved model using simulated data.

Previous reports have shown that the magnitude of the vertical wavenumber $k_z$ determines the sensitivity of the inversion results to temporal decorrelation interference [49,50]. The simulated data in the present study was therefore created to invert the study areas with different mean tree heights ($h_{True}$) using PolInSAR with different $k_z$. When a certain degree of temporal decorrelation interference is present, the use of improved model to compute the relative height error ($\left|h-h_{True}\right|/h_{True}\times 100\%$) can reduce the relative error to less than 15%.

The distribution of error in the inversion results is shown in Figure 10. As the control variable, the interference size of ${\gamma }_{Tmp}$ to the simulated data are fixed at $0.5\times e^{0.3i}$. The average extinction coefficient $\sigma$ and the observation Angle $\theta$ are also fixed. For each given $h_{True}$, the corresponding volume-only complex coherence ${\gamma }_v$ for different $k_z$ can be calculated using Equation (4). When ${\gamma }_v$ is multiplied with the set ${\gamma }_{Tmp}$, the relative errors of the improved model inversion results under different $k_z$ can be simulated in the actual inversion process. Based on Figure 10, it can be noted that when no improvement is made to the model (when $\epsilon$ is 1), the inversion error gradually increases with the decrease of $k_z$. When $h_{True}$= 30 m, $k_z$ value larger than 0.05 is needed to determine forest height within 15% of the inversion error. Moreover, when $h_{True}$ is smaller, the inversion can only be accurate with larger $k_z$. As $\epsilon$ increases, there exists one or more intervals of $\epsilon$ value for the compensated three-stage improvement method to accurately invert forest height within 15%, regardless of the size of $k_z$ taken from the interferometric data. Besides, when $\epsilon$ increases and reaches the next interval suitable for inversion, the range of this interval will be larger than that of the previous interval. This moderates the inversion error change, and makes the model more adaptable to the forest height change in the observed area. However, when the $\epsilon$ value selected for inversion is too large, the inversion error will be very unstable. The error of the inversion result changes rapidly and loses regularity when the change of $k_z$ and $\epsilon$ are not apparent. Inversion in this range will undoubtedly reduce the accuracy of inversion. Therefore, it should also try to avoid selecting too large ε to avoid fluctuations in the accuracy during the inversion.

Figure 11 shows the simulated errors of the inversion of the improved model for different coherent amplitude and ${\gamma }_{Tmp}$. Under the same $h_{True}$ conditions in the inversion process, the variation of ${\gamma }_{Tmp}$ coherence amplitude $\left|{\gamma }_{Tmp}\right|$ has little influence on the value of $\epsilon$ in the same $k_z$ inversion (Figure 11a). In the ${\phi }_{Tmp}$ phase, change of ${\gamma }_{Tmp}$ will affect the value of $\epsilon$. The larger the ${\gamma }_{Tmp}$, the larger the value of $\epsilon$ when taken at the same $k_z$ (Figure 11b). Therefore, it can be proved that the influence of decorrelation on the inversion accuracy is mainly due to the phase change caused by ${\gamma }_{Tmp}$. Based on our results, small $k_z$ was found to be more sensitive to phase change and highly likely to cause significant errors, which was consistent with previous research results [49,50]. In addition, the decrease in coherence amplitude caused by temporal decorrelation had an effect on the inversion results, and should not be ignored.

The most significant difference between the improved model in this study and previous models is that it is an improvement on the three-stage inversion method, which is based on the geometric properties of the model. Therefore, there is need to analyze the geometric significance of the parameters expressed in the inversion process in order to understand each parameter’s intrinsic logic in improving the accuracy of the inversion.

The correction term ε on the phase was corrected for the phase shift with height, and it ranged between 1 and 50 as shown in Figure 10. When the phase is corrected by using $\epsilon$, the sensitivity of $h_v$ to complex coherence changes in the complex plane unit circle is reduced, the phase value at the maximum height is increased, and the coherence amplitude is decreased (Figure 12). In this case, it confirmed that there is an intersection point between the height-extinction coefficient curve and the volume-only coherence point. This method achieved accurate inversion results. Meanwhile, the magnitude of the complex correction term $\left|{\gamma }_e\right|$ is the correction for the reduction of coherence due to decorrelation ($\left|{\gamma }_e\right|<1$), while the phase ${\phi }_e$ is the correction for the change of scattering center ($-\pi <{\phi }_e<\pi$). In the complex plane unit circle, $\left|{\gamma }_e\right|$ moves the curve in the lookup table closer to the center of the circle (Figure 13a), and ${\phi }_e$ causes changes in the starting phase of the lookup table curve (Figure 13b). Therefore, the constant change of parameters during the iteration process generates new inversion results.

5. Results

A total of 25% of the ground measured height was randomly selected as the prior data to control the iterative process. The inversion process follows the inversion procedure in Section 4.2. After quantifying the inversion error (Figure 10 and Figure 11), the initial range limit of $\epsilon$ was set to 1–50, the initial range limit of $\left|{\gamma }_e\right|$ was set to 0–1, and the phase ${\phi }_e$ was set to $-\pi ~\pi$. Figure 14 shows scatter plots of height results of the iterations using the improved model, with prior true data.

Table 3. Iteration parameters and inversion accuracy.

Data Sets	Parameters		Inversion Accuracy
	$\mathit{\epsilon }$	${\mathit{\gamma }}_{\mathit{e}}$	RMSE	${\mathit{R}}^2$	RSD
0711-0725	31.1	$0.60\times e^{i·0.1\pi }$	2.1836	0.8355	32.66%
0905-0919	24.0	$0.75\times e^{i·0.2\pi }$	2.4885	0.8154	30.98%
0808-0919	26.5	$0.49\times e^{-i·0.3\pi }$	2.5199	0.7712	30.47%
0711-0919	19.9	$0.48\times e^{-i·0.6\pi }$	3.3373	0.6941	30.99%

shows the iterative results and the inversion accuracy.

It can be noted that the inversion results of the improved model were all within 15% RMSE compared with the true values, and there was a good fit. The iteration parameter ε ranged from 20 to 40, indicating that the random motion of the canopy produces a more significant interference with the phase error of the repeat-pass spaceborne PolInSAR data. Moreover, the decorrelation correction term of the parameter ${\gamma }_e$ became larger as the temporal baseline increased. This indicates that the improved model has a better correction for the interference of temporal decorrelation. However, the magnitude of $\epsilon$ does not increase with the temporal baseline across the data but decreases with increasing $k_z$. This also shows that the smaller the $k_z$, the greater the sensitivity of the data to temporal decorrelation, leading to a decrease in inversion accuracy.

The inversion accuracy was found to be suitable for height inversion. However, all the relative standard deviations (RSDs) of the models were around 30% (Table 3), indicating that the inverse performance of the models still exhibits some volatility and randomness on a few samples. This phenomenon may be because the effects arising from temporal decorrelation were not completely eliminated by semi-empirical iterations. The calibration of temporal coherence could improve the deviation of coherence, but cannot solve the increase of phase line fitting variance caused by low ${\gamma }_{vol}$ coherence [49]. In addition, since the conditions of forest height and coherence between different elements in the same image are not the same, the size of the correction term is subject to various constraints. As such, the use of the same correction term may inevitably lead to inconsistent parameter effects of different pixel values. Nevertheless, the errors in the inversion results are still within reasonable limits and do not affect the forest height distribution or hinder further studies.

To test the robustness of the improved model, the iterative model was validated using an additional 75% of the ground truth data. Performance of the model was further evaluated by introducing the nonlinear least squares inverse model based on repeat-pass spaceborne InSAR [42] for comparison. In this model, the coupling coherence of volume decorrelation and temporal decorrelation is taken as the image observation, and the random motion of forest and the change of dielectric constant are taken into account under a certain temporal baseline. The model can be expressed as follows:

$$\begin{array}{c}\begin{array}{c}\left|{\gamma }_{v+t}\right|=S_{scene}·exp\left(-\frac{1}{2}{\left(\frac{4\pi {\delta }_r\alpha }{\lambda h_r}\right)}^2{h}_v{}^2\right)\\ \approx S_{scene}·sinc\left(\frac{h_v}{C_{scene}}\right), h_v<\pi ·C_{scene}\end{array}\end{array}$$

(27)

where $S_{scene}$ is a non-negative real value less than or equal to 1 and $C_{scene}$ represents the random motion level of the volume scatterers. Consistent with the original study, HV polarization was used in this study as the polarization to represent the volume scatterer. The same modeling data as the improved model in this study were used for training, and the model parameters were obtained by nonlinear least squares iteration. The scatter plots of the height inversions using models and the validation data are shown in Figure 15. The validation accuracy of the four data sets is also shown in

Table 4. The validation accuracy of the four data sets.

Data Sets	Validation Accuracy of the Improved Model			Validation Accuracy of the Nonlinear Least Squares Model
	RMSE	${\mathit{R}}^2$	RSD	RMSE	${\mathit{R}}^2$	RSD
0711-0725	2.7305	0.7401	29.06%	3.2597	0.6342	29.94%
0905-0919	2.3125	0.8126	30.58%	3.3024	0.6782	32.51%
0808-0919	3.1490	0.6871	32.33%	3.8472	0.6007	33.32%
0711-0919	4.1016	0.5978	34.51%	4.1194	0.5522	34.69%

.

Based on Figure 15, it can be noted that the inversion accuracy of the improved model is relatively close for different interferometric pairs. Therefore, the model can still perform forest height inversion within 15% accuracy even when the temporal baseline is different from $k_z$. This indicates that the inversion of the improved model is more robust and can be applied to repeat-pass spaceborne PolInSAR data with larger temporal baselines and smaller $k_z$. In contrast, the nonlinear least squares model had lower inversion accuracy and a larger randomness due to the lack of correction phase. The accuracy of the model decreased more obviously when the data with large temporal baseline was inversed. Moreover, the lack of polarization information may be another reason for the decrease in accuracy.

6. Discussion

By improving the model inversion of forest height, the significant errors caused by repeat-pass spaceborne PolInSAR temporal decorrelation are reduced, increasing the scope of application of this data in forest height inversion. Based on the results of the improved model in Section 4 under different interferometric data, the following conclusions can be drawn:

• The correction of temporal decorrelation can improve the robustness and accuracy of the inversion and meet the needs of remote sensing for forest height inversion.
• A more accurate forest height inversion of common SAR data can be performed using the improved model, but there may still be a small degree of error in the inversion results.
• Data with large temporal baselines should be carefully selected when using models for height inversion.

6.1. Inversion Performance of the Model

Temporal decorrelation causes abrupt changes in the interferometric phase on the one hand and decreases in coherence on the other. In Figure 16, the coherence of the larger temporal baseline is lower than that of the smaller temporal baseline, which also proves that for repeat-pass spaceborne PolInSAR data, temporal decorrelation is the main source of inversion errors [61]. Besides, the coherence of the 0711-0919 data are slightly lower than that of the 0808-0919 dataset, so it can be shown that the increase of the temporal baseline will not cause more loss of decorrelation when the loss of coherence reaches a certain level. Combined with the empirical parameters obtained by inversion of the improved model (Table 3), it can be seen that the continuous increase of the temporal baseline causes more changes in the phase of the observed complex coherence, compared to decreases in the coherence. The changes in phase are also affected by the sensitivity of $k_z$ to temporal decorrelation in the inversion process, and correction for this error can be better achieved using an improved model. In addition, the magnitude of coherence varies at the same temporal baseline, which can be caused by various factors, such as wind-induced motion, weather-induced changes in dielectric constant, etc. [9,26,43,62]. The coherence is not uniformly distributed in the same image, resulting from the joint action of temporal decorrelation and volume decorrelation. The validated improved inversion model summarizes the temporal decorrelation in the same data set as a complex parameter, which can also control the error of inversion results within a relatively small range.

Figure 17 shows the forest height inversion results of the improved model with different interference datasets. The inversion results show that the improved inversion method is robust for the forest height inversion under different $k_z$ and temporal baselines. Although there was some biasness between the data sets, this error may have been influenced by abrupt changes in the ground phase in addition to the reasons mentioned in Section 5. Moreover, the inversion effect does not decrease significantly for the data sets 0808-0919 and 0711-0919 with larger temporal baselines. The temporal baseline of the 0711-0919 data set was found to be larger, and it showed a better inversion effect than 0808-0919. This could be because the difference in the coherence between the two sets of data were little. Meanwhile, the data are less sensitive to temporal decorrelation interference due to the large original $k_z$ of 0711-0919 data, resulting in the preference in the overall inversion result. Therefore, for repeat-pass spaceborne PolInSAR, the improved model can be relatively robust to forest height inversion.

6.2. Error Analysis of Inversion Results

Errors in the inversion results and the effect of residual temporal decorrelation on the model are the common factors that affect the accuracy of the inversion results. However, other factors outside the model may also cause the problem of accuracy degradation. This study was based on the assumption of the traditional three-stage method [38]. Besides the temporal decorrelation source, the inversion results are still faced with two aspects of errors: residual ground contribution in the volume scattering complex coherence and the estimated shift of the real ground phase [63]. The inversion process usually assumes that the ground scattering contribution of volume-only scattering complex coherence is 0. However, when there is a strong interaction at the dihedral angle of the earth, the polarization channels with low ground contribution will still be affected by the ground scattering center, increasing the variance of the inversion results. Thus, the accuracy of the fitting is reduced [38].

Previous studies have attempted to optimize this phenomenon and can be categorized into three groups based on their principles: separation of the contribution of volume scattering based on polarization decomposition [8,64], optimization by coherence to minimize the effect of ground scattering contributions [39,45,65], and methods that introduce other data sources, such as different baselines [66], and different frequency bands [67,68]. At present, these methods can only reduce the error of ground contribution to a certain extent. Based on the repeat-pass spaceborne PolInSAR data with low coherence, the improved model introduces partial coherence optimization to help in the model calculation. This method is very valuable in research. However, removal of ground contribution to the inversion effect deserves further in-depth study.

Even after terrain correction, the influence of ground phase estimation is still inevitable [11]. The ground phase images were estimated from different interferometric data during the three-stage inversion (Figure 18). The ground phase of all interferometric pairs, as distinguished from the interferometric DEM (Figure 4c), was affected by abrupt changes resulting to incorrect ground point selection during the inversion. The phase error was particularly prominent in 0711-0725, where the $k_z$ of this data set was smaller than the other groups. Thus, the spatial baseline of this data set was shorter than the other groups during interferometry. In addition, the abrupt changes of terrain phase were more obvious in rugged terrain, which was consistent with previous findings [69]. In contrast, the abrupt ground phase error for 0711-0919 was not significant, indicating that the estimation error of the ground phase was not significantly affected by the temporal baseline when compared to spatial baseline. This also supported previous conclusions about the same [32,61].

The forest average height serves as the standard forestry table measurement parameter and plays an important role in forest inventory and subsequent biomass estimation [70]. However, it has also been suggested that forest dominance height is closer to the effective height of the remote sensing signal [63], and the difference tends to be more pronounced for forests with lower heights. In addition, differences in forest species and density may also have an impact on the inversion results. The reasons are mainly the differences arising from the canopy cover and the ratio of canopy to bare ground. In the general boreal temperate forests of China, the distribution of forest species and density is complex. They were not strictly classified and studied separately in this study. This will undoubtedly have some influence on the inversion accuracy, which is worthy of our further study. It is worth noting that although the sample size does not bias the fit results, it may still cause a reduction in correlation [52].

The model proposed in this study is only for the fuzzy inversion of forest height caused by temporal decorrelation. It does not focus on the inversion errors caused by other factors that may exist. As such, the model is only suitable for data whose main error source is temporal decorrelation. In addition, this study is a regional study, and the model has only been verified in the height inversion of forests in northern China. Therefore, the inversion of the model in other regional forests requires further studies.

6.3. Suitable Range of Spatial Baseline for Forest Height Inversion

In this study, height inversion of repeat-pass spaceborne PolInSAR data revealed that the degree of influence of temporal decorrelation on the inversion accuracy depends not only on the length of the temporal baseline, but also on the sensitivity of the data to temporal decorrelation, such as the magnitude of $k_z$. Given the upcoming large-scale forest biomass observation mission, the multifaceted parameters of PolInSAR data should be carefully considered. The spaceborne SAR sensor has a relatively fixed flight trajectory when compared to the airborne SAR, and it is relatively difficult to change its space parameters. Therefore, this study aims at providing more insights about the spatial parameter suitable for the forest height of the spaceborne SAR inversion. This will help future studies in establishing a suitable spaceborne inversion system.

Given the sensitivity of the ground target height to the interferometric phase, the phase shift caused by temporal decorrelation causes the most interference to the forest height calculation. Therefore, the correction term $\epsilon$ in the model (Equation (23)) has the most significant impact on the inversion accuracy. In Equation (23), the function of the $\epsilon$ parameter is to correct the phase deviation. However, from another perspective, the inversion accuracy can be measured by converting $\epsilon$ to the size of $k_z$. As such, the inversion accuracy can be improved by using the data suitable for $k_z$.

In this study, we used L-band ALOS2 repeat-pass spaceborne PolInSAR data to retrieve the height of temperate forest in the Saihanba region of northern China, and obtained the $\epsilon$ parameters under different $k_z$ and temporal baselines. When the $k_z$ value of the spaceborne data reached $\epsilon ·k_z$ of the corresponding height, the data were considered suitable for inversion of the forest height using the polarized interferometric model. The suitable $k_z$ for different data pairs can be obtained from Table 2 and Table 3, as shown in

Table 5. Suitable $k_z$ for different data pairs.

Data Sets	Temporal Baselines (Day)	$\mathbf{Original} {\mathit{k}}_{\mathit{z}}$	$\mathbf{Suitable} {\mathit{k}}_{\mathit{z}}$	Suitable Vertical Baseline
0711-0725	14	0.015	0.465	2391.7 m
0905-0919	14	0.018	0.432	2723.7 m
0808-0919	42	0.018	0.477	3007.4 m
0711-0919	70	0.021	0.418	2635.4 m

.

The analysis of Table 5 shows that a larger vertical baseline is more suitable for repeat-pass spaceborne PolInSAR inversion of forest height, and a similar conclusion was reached in a previous study upon analysis of airborne data [11,50]. This is because when the $k_z$is larger, the baseline decorrelation caused by the spatial baseline between the primary and secondary images is greater. The larger $k_z$ results obtained in the study indicate that the disturbance caused by baseline decorrelation can be hardly considered when compared to temporal decorrelation. Conversely, smaller $k_z$ may cause abrupt changes in the phase estimation of the underlying surface, causing a decrease in the inversion accuracy. In addition, the suitable $k_z$ of different temporal baselines slightly differed for the same area data, indicating that the sensitivity of $k_z$ to temporal decorrelation is more important than that of temporal baseline. Therefore, more attention should be paid to the use of suitable $k_z$ when using repeat-pass spaceborne SAR data interferometry to measure forest height.

However, according to previous studies, the choice of $k_z$ was negatively correlated to the forest height in the observation area [11]. The $k_z$ value could be appropriately lowered when the forest height of the study area was higher. According to the average forest height in this study, the suitable $k_z$ exceeded the fuzzy interval of $2\pi$. Since the relationship between $k_z$ of the data and the ground height has to be within the $2\pi$ fuzzy interval, there may not be a suitable value of $k_z$ for inversion if the inversion model is not improved. Therefore, the correction of the $\epsilon$ parameter for the phase is essential when the temporal decorrelation of the data has a significant effect.

7. Conclusions

The performance study of forest height through repeat-pass spaceborne PolInSAR inversion will effectively improve large-scale forest height estimation efficiency. To address the limitations of the current inversion of forest height with repeat-pass L-band spaceborne PolInSAR data, a theoretical analysis of the effects of phase deviation and temporal decorrelation on the inversion performance and an improved inversion method are proposed based on the RVoG model. In this respect, this study makes three main conclusions: (1) the correction of temporal decorrelation can improve the robustness and accuracy of the forest height inversion. (2) A more accurate forest height inversion of common SAR data can be performed using the improved model, but there may still be a small degree of error in the inversion results. (3) Data with large temporal baselines should be carefully selected when using models for height inversion.

The repeat-pass spaceborne sensors have a more extended temporal baseline than the airborne sensors, which exposes the repeat-pass spaceborne PolInSAR to more excellent decorrelation effects when inverting forest heights in the study area. Besides, the vertical wavenumber of the repeat-pass spaceborne sensor is lower than the interval suitable for inversion, making the inversion height more sensitive to decorrelation. When affected by these two factors, the spaceborne PolInSAR data often loses the ability to invert the forest height using the traditional three-stage inversion method.

To avoid the interference of both factors as much as possible, we propose a semi-empirical improvement model that controls the iterations by ground true data, based on the three-stage inversion method. Through theoretical analysis, it was found that in general cases, there is a correction term to make the error between the inversion results and the true value to converge to less than 15%. Moreover, there is more than one correction term for any reasonable range of specified forest height and $k_z$ can accurately invert the forest height under the influence of temporal decorrelation.

This method uses ALOS2 repeat-pass spaceborne L-band PolInSAR data with large temporal baseline to perform accurate forest height inversion in the mixed conifer-broad forest of the Saihanba of northern China. For different interferometric pairs, the RMSEs of inversion results were less than 15%. For the interferometric data of the larger temporal baseline, the inversion results were slightly lower than other interferometric pairs because of the low coherence caused by decorrelation. Nevertheless, this study is based on the assumptions of the traditional three-stage inversion method. Therefore, there is need for further research to improve the accuracy of this method using previous improved models, such as ground contribution and terrain impact.

The present study achieves adequate accuracy in forest height inversion by repeat-pass spaceborne PolInSAR data through the improved model. However, accuracy degradation may occur during inversion of heterogeneous forests because the results are affected by height variation. Further research is therefore needed to establish whether general patterns can be summarized to avoid the above problems by examining data from different forest height intervals.