Accurate tree height is of great value for forest resource inventory and global carbon assessment since the height is regarded as one of the most critical parameters in forest aboveground biomass retrieval. It is particularly true when a tree is coniferous [1]. However, timely tree height surveying in forested areas at a large spatial extent and with high accuracy is a huge challenge currently. Synthetic aperture radar (SAR) has irreplaceable advantages in forest parameter inversion due to its unique imaging characteristics [2]. Utilizing the advantages of the interferometric SAR (InSAR) and polarimetric SAR (PolSAR) techniques, Cloude and Papathanassiou studied the polarimetric interferometric SAR (PolInSAR) technique for the inversion [3]. With the random volume over the ground (RVoG) model [4] and to streamline the inversion procedure, Cloude et al. next proposed the three-stage algorithm to estimate a tree height [5]. In recent years, successful airborne PolInSAR studies have led to significant development in forest parameter estimation [6], including height estimation at variable stand densities [7]. Since the radar wave penetration into tree canopies or through the canopies depends on its wavelength, the PolInSAR data with variable wavelengths are studied. The X-band PolInSAR data were successfully applied to estimating tree heights in boreal forest stands [8] or forest stands with low stand densities [9]. Satisfactory results were obtained in temperate forests using C-band [10,11] and L-band [12,13] PolInSAR data. The P-band PolInSAR technique and its data are widely considered when estimating tree heights in tropical rainforests with thicker and denser canopies [14,15].

Four primary prerequisites must be met in tree height inversion with the PolInSAR technique and data. 1) A short time baseline is required for the PolSAR data pair to have good temporal coherence. 2) The 2π height of ambiguity (HoA) should be much larger than a tree height for reliable height estimation. Thus, a good spatial baseline is required. 3) A SAR sensor with adequate canopy penetration depth is preferred. Thus, the radar wave can reach the ground surface for sure. 4) At least two PolSAR data are available. The prerequisites, unfortunately, limit the PolInSAR application. It is especially true when the global application is desired since there is no global coverage of the PolSAR data yet. For example, the advanced land observation satellite (ALOS) and phased array type L-band SAR (PALSAR) observed earth between 2006 and 2011. The SAR data are downloadable for free, but the global PolSAR data coverage does not exist (https://www.eorc.jaxa.jp/ALOS/en/alos/a1_about_e.htm). Currently, the PALSAR-2 (https://www.eorc.jaxa.jp/ALOS-2/en/about/palsar2.htm) and SAOCOM (https://earth.esa.int/eogateway/missions/saocom) are operational. Unfortunately, the acquired datasets are unavailable for free download, hindering further development of the algorithm.

To overcome the unavailability of PolInSAR data, researchers have alternatively and successively proposed PolInSAR algorithms using dual-pol SAR data [16-18], laying a solid foundation for this study. Nevertheless, after examining the retrieved tree heights using the L-band dual-pol InSAR data, the heights were significantly off the in situ values [17,18]. One possible cause is attributed to the missing PolSAR data. Then, simulating the PolSAR data with the dual-pol SAR data was explored [19-21], and the estimated tree heights were somewhat promising.

Sentinel-1A or 1B has a revisit cycle of 12 days (https://sentinel.esa.int/web/sentinel/missions/sentinel-1). The cycle can be shortened to 6 days if observations of Sentinel-1A and 1B are combined (before the malfunction of the Sentinel-1B SAR). Therefore, good temporal coherence can be produced using a pair of Sentinel-1 data in the InSAR analysis. The first prerequisite is met. Due to the interferometric intention of the Sentinel-1 mission, the spatial baseline for revisit orbits is short. Then, HoA is large enough for the tree height estimation, or the second prerequisite two is satisfied.

The canopy penetration of Sentinel-1 C-band (about 5.5 cm in wavelength) SAR is more shallow than that of L-band SAR, but its penetration is deeper than that of X-band (about 3.0 cm in wavelength) SAR. The X-band PolSAR data were used to estimate pine tree heights at low stand densities [9], demonstrating the possibility of estimating tree height using hort-wavelength SAR. In addition, Sentinel-1 SAR is operational. Repetitive global coverage of the dual-pol C-band Sentineal-1 SAR data is acquired, and the data are freely downloadable through the Internet (e.g., https://sentinel.esa.int/web/sentinel/missions/sentinel-1). The dual-pol SAR data have global and repetitive coverage. A pair of Sentinel-1 datasets are generally available in the winter, the most favorable season for radar wave penetration. Deciduous trees are leaf-off. Although conifers are evergreen, the needles can be low in moisture content, usually true in subtropical and temperate zones. Thus, the third prerequisite is met to some extent. The precondition of available PolSAR data remains.

Sentinel-1 SAR acquires dual-pol SAR (S_HH and S_HV) in the early stage after the launch and almost exclusively S_VV and S_VH afterward. S_HV or S_VH partially contains H- and V-polarization phase information. Thus, one may construct the missing co-pol data in Sentinel-1 dual-pol SAR data with available S_HV or S_VH data. If the construction is successful, estimating the tree heights and even above-ground biomass worldwide becomes feasible. One is better positioned to understand environmental and climate changes related to the global carbon exchange. Therefore, our primary objective is to develop an approach to estimate the tree height using PolInSAR data constructed by the Sentinel-1 dual-pol SAR data and RVoG model.

The backscattering components of each image pixel of PolSAR data can be represented as S, a 2×2 Sinclair matrix, or

$ {\bf{S}} = \left[ {\begin{array}{*{20}{c}} {{S_{{\text{HH}}}}}&{{S_{{\text{HV}}}}} \\ {{S_{{\text{VH}}}}}&{{S_{{\text{VV}}}}} \end{array}} \right] $ (1)

where S_pq is the scattering component with p polarization transmitted, q polarization received; p or q represents the horizontal (H) or vertical (V) polarization. The data type of S_pq is complex. The Pauli scattering vector of S is

$ {{\bf{k}}_p} = \frac{1}{{\sqrt 2 }}{\left[ {\begin{array}{*{20}{c}} {{S_{{\text{HH}}}} + {S_{{\text{VV}}}}}&{{S_{{\text{HH}}}} - {S_{{\text{VV}}}}}&{2{S_{{\text{HV}}}}} \end{array}} \right]^{\text{T}}} $ (2)

where superscript T represents a transpose operator. The Pauli scattering vector elements can better reflect radar targets’ scattering characteristics.

Assume that two single-look complex (SLC) PolSAR data meeting the spatiotemporal interference conditions are available for the interferometric analysis. Then, one is considered the primary, and the other is secondary. The corresponding Pauli scattering vectors are $ {\bf{k}}_p^{{\text{primary}}} $ and ${\bf{k}}_p^{{\text{secondary}}}{\text{{\mathrm{,}}}} $ respectively. Using the vectors, one obtains a six-dimensional (6D) polarization interference matrix or T₆ as

$ {{\bf{T}}_6} = \left[ {\begin{array}{*{20}{c}} {{\bf{k}}_p^{{\text{primary}}}} \\ {{\bf{k}}_p^{{\text{secondary}}}} \end{array}} \right] \times {\left[ {\begin{array}{*{20}{c}} {{\bf{k}}_p^{{\text{primary}}}} \\ {{\bf{k}}_p^{{\text{secondary}}}} \end{array}} \right]^{{\text{T}}*}} = \left[ {\begin{array}{*{20}{c}} {{{\bf{T}}_{11}}}&{{{{\boldsymbol{\mathbm{Ω}}}} _{12}}} \\ {{{{\boldsymbol{\mathbm{Ω}}}} _{21}}}&{{{\bf{T}}_{22}}} \end{array}} \right] $ (3)

where * is a conjugate operator. T₁₁ is the coherency matrix of the primary image, T₂₂ is the coherency matrix of the secondary image, Ω₁₂ is a non-Hermitian polarization interference matrix containing polarization and interference information of the primary and secondary images, Ω₂₁ is another non-Hermitian polarization interference matrix. Due to the different distances between the antenna and the resolution unit during the acquisition of the two images, the spatiotemporal decorrelation exists. Thus, $ {\bf{k}}_p^{{\text{primary}}} \ne {\bf{k}}_p^{{\text{secondary}}} $, or $ {{{\boldsymbol{\mathbm{Ω}}}} _{12}} \ne {{{\boldsymbol{\mathbm{Ω}}}} _{21}} $.

With two normalized complex vectors ω₁ and ω₂, one can project $ {\bf{k}}_p^{{\text{primary}}} $ and $ {\bf{k}}_p^{{\text{secondary}}} $ onto vectors ω₁ and ω₂, respectively, resulting in values in the complex data type or complex values

$ \left\{ {\begin{array}{*{20}{l}} {{\mu _1} = \omega _1^*{\mathrm{,}}\;{\bf{k}}_p^{{\text{primary}}}} \\ {{\mu _2} = \omega _2^*{\mathrm{,}}\;{\bf{k}}_p^{{\text{secondary}}}} \end{array}} \right. $ (4)

where μ₁ and μ₂ are the coefficients. Then, the observed polarization interference coherence can be expressed as

$ {\gamma _{{\text{obs}}}}\left( {{\omega _1}{\mathrm{,}}{\omega _2}} \right) = \frac{{\left\langle {{\mu _1}\mu _2^*} \right\rangle }}{{\sqrt {\left\langle {{\mu _1}\mu _1^*} \right\rangle \left\langle {{\mu _2}\mu _2^*} \right\rangle } }} = \frac{{\omega _1^*\left[ {{\Omega _{ 12}}} \right]{\omega _2}}}{{\sqrt {\left( {\omega _1^*\left[ {{T_{11}}} \right]{\omega _1}} \right)\left( {\omega _2^*\left[ {{T_{22}}} \right]{\omega _2}} \right)} }} {\mathrm{.}} $ (5)

It should be noted that when ω₁ and ω₂ are the same, (5) represents a pure interference process. Specifically, when $ {{{\boldsymbol{\mathbm{ω}}}} _1} = {{{\boldsymbol{\mathbm{ω}}}}_2} = {\left[ {\begin{array}{*{20}{c}} {{1 \mathord{\left/ {\vphantom {1 {\sqrt 2 }}} \right. } {\sqrt 2 }}}&{{1 \mathord{\left/ {\vphantom {1 {\sqrt 2 }}} \right. } {\sqrt 2 }}}&0 \end{array}} \right]^{\text{T}}} $, it corresponds to S_HH interference information. $ {{{\boldsymbol{\mathbm{ω}}}} _1} = {{{\boldsymbol{\mathbm{ω}}}}_2} = {\left[ {\begin{array}{*{20}{c}} {{1 \mathord{\left/ {\vphantom {1 {\sqrt 2 }}} \right. } {\sqrt 2 }}}&{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} {\sqrt 2 }}} \right. } {\sqrt 2 }}}&0 \end{array}} \right]^{\text{T}}} $ relates to the S_VV interference information. If $ {{{\boldsymbol{\mathbm{ω}}}} _1} = {{{\boldsymbol{\mathbm{ω}}}} _2} = {\left[ {\begin{array}{*{20}{c}} 0&0&1 \end{array}} \right]^{\text{T}}} $, it represents the S_HV interference information.

With the reciprocal rule (S_HV = S_VH), the covariance matrix of C of PolSAR data can be written as

$ {\bf{C}} = \left[ {\begin{array}{*{20}{c}} {\left\langle {{{\left| {{S_{{\text{HH}}}}} \right|}^2}} \right\rangle }&{\left\langle {{S_{{\text{HH}}}}S_{{\text{HV}}}^*} \right\rangle }&{\left\langle {{S_{{\text{HH}}}}S_{{\text{VV}}}^{\text{*}}} \right\rangle } \\ {\left\langle {{S_{{\text{HV}}}}S_{{\text{HH}}}^*} \right\rangle }&{2\left\langle {{{\left| {{S_{{\text{HV}}}}} \right|}^2}} \right\rangle }&{\left\langle {{S_{{\text{HV}}}}S_{{\text{VV}}}^*} \right\rangle } \\ {\left\langle {{S_{{\text{VV}}}}S_{{\text{HH}}}^{\text{*}}} \right\rangle }&{\left\langle {{S_{{\text{VV}}}}S_{{\text{HV}}}^*} \right\rangle }&{\left\langle {{{\left| {{S_{{\text{VV}}}}} \right|}^2}} \right\rangle } \end{array}} \right] {\mathrm{.}} $ (6)

Then, under the assumption that trees in a forest stand are azimuthally symmetric, C is simplified as

$ {\bf{C}} = \left[ {\begin{array}{*{20}{c}} {\left\langle {{{\left| {{S_{{\text{HH}}}}} \right|}^2}} \right\rangle }&0&{\left\langle {{S_{{\text{HH}}}}S_{{\text{VV}}}^*} \right\rangle } \\ 0&{2\left\langle {{{\left| {{S_{{\text{HV}}}}} \right|}^2}} \right\rangle }&0 \\ {\left\langle {{S_{{\text{VV}}}}S_{{\text{HH}}}^*} \right\rangle }&0&{\left\langle {{{\left| {{S_{{\text{VV}}}}} \right|}^2}} \right\rangle } \end{array}} \right] {\mathrm{.}} $ (7)

There are the single-bounced scattering from the ground surface, the double-bounced scattering from the ground-trunk interactions, and the volumetric scattering from the tree canopy in a forest stand. When tree branches are modeled as long-thin cylinders that are randomly distributed in azimuth and zenith directions, the volumetric scattering component is [22,23]

$ {{\bf{C}}_{{\text{vol}}}} = \left[ {\begin{array}{*{20}{c}} {\dfrac{1}{3}}&0&0 \\ 0&{\dfrac{1}{3}}&0 \\ 0&0&{\dfrac{1}{3}} \end{array}} \right] {\mathrm{.}} $ (8)

According to (7) and (8), the S_HH or S_VV intensity is twice the S_VH or S_HV intensity in canopy volume scattering. S_VH or S_HV simultaneously contains the phase information of H and V polarizations. The H- or V-polarization phase will be uniformly distributed between 0 and 2π for the volume scattering target modeled as a long-thin cylinder. The phase information of H-polarization in S_HH and S_VH data should be somewhat similar. Additionally, the phase information of V-polarization in S_VV and S_HV data should be alike to some extent. Thus, the S_VH or S_HV phase can be used to replace the phase information of the missing S_HH or S_VV data. Then, the missing S_HH data within the available dual-polarization (S_VH and S_VV) data can be constructed by S_VH or S_HH = $ \sqrt 2 {S_{{\text{VH}}}} $. One rewrites the Pauli scattering vector as

$ {{\bf{k}}_{p\_{\text{construct}}}} = \frac{1}{{\sqrt 2 }}{\left[ {\sqrt 2 {S_{{\text{VH}}}} + {S_{{\text{VV}}}}} \qquad {\sqrt 2 {S_{{\text{VH}}}} - {S_{{\text{VV}}}}} {2{S_{{\text{VH}}}}} \right]^{\text{T}}} {\mathrm{.}} $ (9)

Then, the constructed PolInSAR data matrix ${{\bf{T}}_{{\text{6-construct}}}} $ is

$ {{\bf{T}}_{{\text{6-construct}}}} = \left[ {\begin{array}{*{20}{c}} {{\bf{k}}_{p{\text{\_construct}}}^{{\text{primary}}}} \\ {{\bf{k}}_{p{\text{\_construct}}}^{{\text{secondary}}}} \end{array}} \right] \times {\left[ {\begin{array}{*{20}{c}} {{\bf{k}}_{p{\text{\_construct}}}^{{\text{primary}}}} \\ {{\bf{k}}_{p{\text{\_construct}}}^{{\text{secondary}}}} \end{array}} \right]^{{\text{T}}*}} $ (10)

where superscripts primary and secondary indicate the Pauli vectors of the primary and secondary images, respectively. For simplicity, the PolInSAR data matrix is called PolInSAR data onwards.

Fig. 1 schematically shows the RVoG model [4]. The location of the ground layer is at z₀, and the height of the vegetation layer is z₀ + h_v. Then, h_v is the vegetation or tree height. Ignoring the influence of noise and temporal decoherence, one can express the interference coherence coefficient of the model as

$ \gamma \left( \omega \right) = \exp \left( {{\mathrm{i}}{\varphi _0}} \right)\frac{{{\gamma _v} + m\left( \omega \right)}}{{1 + m\left( \omega \right)}} $ (11)

where γ_v is the complex coherence coefficient of canopy volume scattering, and φ₀ is the ground interferometric phase. m(ω) is proportional to the ratio of the amplitude of the effective ground backscatter divided by the amplitude of the vegetation volume backscatter. m(ω) relates to the radar wave’s polarization state (ω) and is

$ m\left( \omega \right) = \frac{1}{{{I_0}}}\frac{{{m_g}}}{{{m_v}}} $ (12)

with

$ {I_0} = \int\nolimits_{{z_0}}^{{h_v}} {\exp \left( {\frac{{2\sigma z'}}{{\cos {\theta _0}}}} \right) } {\text{d}}z' $ (13)

where θ₀ is the radar look angle of the primary image, σ is the average extinction coefficient, m_v and m_g represent the scattering amplitude per unit volume of the vegetation layer and the ground scattering amplitude, respectively. When the interference coefficient comes from the vegetation canopy entirely, m(ω) is 0. If the coefficient comes from the ground, m(ω) is infinite. Thus, one has 0 ≤ m(ω) < ∞. Then, based on the RVoG model and PolInSAR data, the three-stage algorithm [5] is used to estimate the tree height.

To verify the effectiveness of the proposed approach, we apply it to the simulated PolInSAR data (i.e., (10)) from a pine stand. Table 1 shows the simulation parameters. The SAR system frequency is 5.405 GHz (i.e., C-band) and is the same center frequency as the Sentinel-1 SAR. Fig. 2 (a) is the simulated forest scene. The gaps between tree crowns and within individual pine-canopies ensure the adequate acquisition of ground scattering information from the short wavelength PolInSAR data.

Fig. 2 (b) is the derived tree heights with the constructed PolInSAR data (i.e., (9)). Changes of colors from blue to yellow indicate tree heights from 0 to 22.0 m. The entire forest area is almost displayed in light brown to dark yellow, indicating that the estimated tree height is close to the true value of 18.0 m. In the sampling area (shown in the black box with 40×40 pixels), the average tree height is 18.1 m, with one standard deviation of 2.1 m. The proposed approach performs well. Fig. 2 (c) shows the results derived from the original PolInSAR data. Compared with Fig. 2 (b), there are fewer cyan areas where the estimated heights are lower than the true value. The height estimation using the original PolInSAR data should be better than the constructed PolInSAR data. Nevertheless, the mean value and the standard deviation of the estimated tree heights in the sampling area (Fig. 2 (c)) are 18.8 m and 2.0 m, respectively. The mean and standard deviation values in the Fig. 2 (b) and Fig. 2 (c) sample areas are similar. The estimated height from the constructed PolInSAR data is acceptable.

Furthermore, Fig. 3 is the height histogram. The green curve represents the heights using the constructed PolInSAR data, and the red one indicates those using the original PolInSAR data. Both should be similar overall. Although the modes of the estimated tree heights using the constructed and original PolInSAR data in Fig. 3 are larger than the true value of 18.0 m (the gray vertical dashed line), one can consider the off as a systematic error that can be mitigated. Thus, the proposed approach to constructing the PolInSAR data and estimating the tree height is encouraging and effective, and the estimated tree heights should be satisfactory.

Actual forest scenes are often more complex than the simulated forest environments. Thus, to validate the proposed approach further, we use it to estimate tree heights in the study area near Duke Forest primarily. The area is over 280000 km² with forested and open lands within the Durham, Orange, and Alamance counties, North Carolina (NC), USA. Duke Forest is owned and managed by Duke University and has been used for teaching and research since 1931 (https://dukeforest.duke.edu/). The forest stands consist of mainly coniferous evergreen (e.g., loblolly pine, pinus taeda L.), deciduous hardwood (oak, genus Quercus; hickory, genus Carya), and mixed forest of evergreen and hardwood trees.

We used two Sentinel-1A SAR data acquired in the winter to ensure adequate penetration through tree canopies. The acquisition date of the primary image was 12 December 2018, and the secondary one was obtained on 24 December 2018. Both were acquired along ascending orbits. The time baseline of the InSAR pair is 12 days. The perpendicular spatial baseline is –44.45 m. HoA is 344.80 m, which is much taller than a tree. As the primary and secondary SAR data are in dual-pol, we constructed the related PolSAR data with (9) accordingly. Then, the PolInSAR data is ready for the tree height inversion. The ICESat-2 ATL08 product (https://icesat-2.gsfc.nasa.gov/science/data-products) contains tree heights and is used for verification. The ATL08 data were acquired on 19 December 2018. The acquisition time of the ICESat-2 data is close to those of the Sentinel-1A SAR data, and the tree heights can effectively validate the tree height estimation after the constructed PolInSAR data.

Fig. 4 shows the estimated tree heights. The image has 4095 rows and 3825 columns with a cell size of 15 m by 15 m. The background along four sides was coded blue. The tree heights are coded from blue (the lowest height of 0) to red (the highest value of 33.59 m). The clear-cut or open/water areas are clearly shown as blue spots scattered around Fig. 4. Variable tree heights are noted, with the tall trees in the northeast and southwest. The overall patterns of trees and non-trees should be consistent with those of the study area. Fig. 5 is the height histogram. The heights are mainly centered between 10 m and 25 m. The number of trees with heights between 5 m and 10 m or >25 m is small. According to local observations and forest descriptions, trees in the lower left and upper right areas may be overestimated. Nevertheless, the tree height ranges reasonably.

There are 23 ICESat-2 data points within the study area, shown as red dots in Fig. 4. The ICESat-2 tree height data are considered the truth. of 23 sample points, tree heights range from 20.3 m to 34.5 m. Then, the related 23 tree heights from Fig. 4 are extracted. The comparison is shown in Fig. 6. In one case, the estimated height is higher than that of the ICE-Sat-2 data. The estimated heights in the remaining 22 points are shorter than those of the ICESat-2 data. Our approach using the constructed PolInSAR data underestimates the heights. Quantitatively, the relative error of 5 points is within ±30%. Errors of 8 points change from 30% to 40%, and errors of the remaining 10 points are >40%.

As the stand density varies, the canopy density and closure change in forested environments. Thus, the stand density influences the scattering mechanisms in PolSAR data from the canopy and ground and their interactions. To further examine the impact on the accuracy of PolInSAR tree height inversion, we studied two pine stands through simulations. One stand has a density of 30000 trees/km², and the other is 50000 trees/km². Other simulation parameters are the same as those in Table 1. The estimated tree heights using the constructed and original PolInSAR datasets are shown in Fig. 7. The results at the 30000 trees/km² and 50000 trees/km² densities are shown in rows one and two, respectively. The truth tree height is 18.0 m in both stands.

At the stand of 30000 trees/km², tree heights derived from the constructed (Fig. 7 (a)) and original (Fig. 7 (b)) PolInSAR data are overall similar. However, more cyan pixels exist in Fig. 7 (b), or an underestimation occurs in the original PolInSAR data. Fig. 7 (c) shows height histograms. The mode derived by the constructed PolInSAR data is 18.0 m, and the mode from the original PolInSAR data is 19.0 m. The mean and standard deviation of estimated tree heights using the former are 17.9 m and 2.2 m, respectively, while the values are 18.2 m and 2.9 m using the latter. The two mean values are similar. However, the latter has a larger standard deviation than the former. Additionally, Fig. 7 (c) shows tree heights using the latter are more dispersed than those using the former. Thus, the proposed approach seems to achieve slightly better results using the constructed PolInSAR data than the original PolInSAR data. One may attribute the cause to the strong ground surface in the original PolSAR data pair. Since the S_HH has a stronger penetration ability than the S_HV wave, more S_HH energy reaches the ground, producing more ground surface backscatter. Thus, the relative importance of the amplitude of the effective ground backscatter related to the amplitude of the vegetation volume backscatter increases, producing an overestimation of tree heights. In contrast, less S_HV energy hits the ground, creating less ground-surface backscatter. The overestimation may not occur.

With the stand density changing from 30000 trees/km² to 50000 trees/km², the crown density and closure increase. The canopy-layer attenuation increases, decreasing the radar wave’s penetration ability. There might not be enough radar energy (downward) to reach the ground, and the ground surface backscatter (upward) to the SAR sensor decreases. Then, the reverse may occur. That is, the relative importance of the amplitude of the effective ground backscatter related to the amplitude of the vegetation volume backscatter decreases. Tree heights are underestimated. Thus, when the construed PolInSAR data is used to derive tree heights, underestimation is expected, and it is indeed, as shown in Fig. 7 (d). Many cyan areas exist. On the other hand, tree heights derived from the original PolInSAR data may still be satisfactory, as shown in Fig. 7 (e). Fig. 7 (f) is histograms of estimated tree heights using the constructed (green curve) and original (red curve) PolInSAR data. The curves are the evidence for the above discussion. Quantitatively, the mean and one standard deviation of estimated tree heights using the constructed PolInSAR data are 16.8 m and 3.1 m, respectively. The mean is about 6.7% lower than the true value. In contrast, using the original PolInSAR data, one gets a mean value of 18.2 m and one standard deviation of 2.3 m. In summary, the stand density affects the tree height inversion with the PolInSAR technique and PolInSAR data. Different penetration abilities of the co-pol (S_HH or S_VV) and cross-pol (S_VH or S_HV) radar waves for the same forest stand are primarily attributed to the overestimation and underestimation. The backscattered information of S_HH is severely disturbed by ground scattering, reducing the accuracy of the height estimation.

Forests are the carbon sink and play an irreplaceable role in understanding environmental and climate changes. However, large-scale, high-precision, and timely tree height estimation is a huge challenge currently. Although one can use the PolInSAR technique and PolSAR data to estimate tree heights in a forest stand, the PolSAR data with global coverage are unfortunately unavailable. However, the dual-polarization (dual-pol) Sentinel-1 SAR data have worldwide coverage and are freely downloadable online. Thus, to extend the PolInSAR technique and PolSAR data for tree height inversion globally, we used $ \sqrt 2 {S_{{\text{HV}}}} $ to replace the unavailable co-polarization (co-pol) S_HH data in the dual-polarization (S_VH and S_VV) SAR datasets to construct the unavailable PolSAR data. Then, using the PolInSAR technique and a pair of constructed PolSAR data, we estimated tree heights. The estimated heights were verified with the simulation and the ICESat-2 height data, and the verification should be satisfactory.

Before ending, three remarks are offered. First, for the short wavelength SAR (e.g., C-band), the application of the repeat-pass spaceborne InSAR technique is usually affected by atmospheric effects. As shown in the study, trees in the lower-left and upper-right parts appear significantly taller than those in the middle. Regarding GoogleEarth^® optical images, the areas are spatially homogeneous in the tree height and species. Then, a possible cause could be the atmospheric variations using the repeat-pass method to create the InSAR data pair, creating variable interferometric phases when inverting tree heights using the PolInSAR data. The effect is typically heterogeneous spatially. Thus, errors can happen. Nevertheless, an ensemble of tree heights derived from multiple PolInSAR datasets can be considered since tree heights do not vary rapidly with time. The accuracy of tree height estimation can be improved.

Second, the forest stand density influences estimated tree heights. When the density is low, the ground surface backscatter can be strong. Then, the relative importance of the amplitude of the effective ground backscatter related to the amplitude of the vegetation volume backscatter increases, producing an overestimation of tree heights. Conversely, the underestimation may occur when the stand density is high or very high.

Third, since the C-band SAR wave has limited penetration ability through tree canopies in forested environments, the scattering/backscattering from the ground surface should be less unless trees are young and small (e.g., in Ref. [24]) or trees are sparse (e.g., in Ref. [25]) in a forest stand. However, terrain correction for estimating forest parameters (e.g., tree height) is needed, particularly using PolSAR data acquired by a long wavelength (e.g., P-band) SAR and in terrain with complex topography. The P-band wave has a much stronger penetration ability than the C-band wave. The P-band PolSAR backscatter data from forests are more susceptible to adverse effects from scattering/backscattering of the ground surface or even subsurface. With the anticipated 2024 launch of the ESA BiomassSAR (https://earth.esa.int/eogateway/missions/biomass), the terrain correction algorithm is urgently needed.

The authors declare no conflicts of interest.