Inversion of vegetation height from PolInSAR using complex least ...

SCIENCE CHINA Earth Sciences • RESEARCH PAPER •

June 2015 Vol.58 No.6: 1018–1031 doi: 10.1007/s11430-015-5070-1

Inversion of vegetation height from PolInSAR using complex least squares adjustment method FU HaiQiang, WANG ChangCheng*, ZHU JianJun, XIE QingHua & ZHAO Rong School of Geosciences and Info-physics, Central South University, Changsha 410083, China Received June 15, 2014; accepted December 4, 2014; published online April 14, 2015

In this paper, we propose the novel method of complex least squares adjustment (CLSA) to invert vegetation height accurately using single-baseline polarimetric synthetic aperture radar interferometry (PolInSAR) data. CLSA basically estimates both volume-only coherence and ground phase directly without assuming that the ground-to-volume amplitude radio of a particular polarization channel (e.g., HV) is less than 10 dB, as in the three-stage method. In addition, CLSA can effectively limit errors in interferometric complex coherence, which may translate directly into erroneous ground-phase and volume-only coherence estimations. The proposed CLSA method is validated with BioSAR2008 P-band E-SAR and L-band SIR-C PolInSAR data. Its results are then compared with those of the traditional three-stage method and with external data. It implies that the CLSA method is much more robust than the three-stage method. polarimetric SAR interferometry (PolInSAR), complex least squares adjustment, random volume over ground (RVoG), vegetation height inversion, truncated singular value decomposition (T-SVD) Citation:

Fu H Q, Wang C C, Zhu J J, et al. 2015. Inversion of vegetation height from PolInSAR using complex least squares adjustment method. Science China: Earth Sciences, 58: 1018–1031, doi: 10.1007/s11430-015-5070-1

Vegetation height is vital in quantifying the terrestrial carbon cycle effectively (Balzter et al., 2007; Gama et al., 2010) and it plays an important role in estimating biomass stored in vegetation, and promoting the development of meticulous forestry management (Luo et al., 2010; Neumann et al., 2010). Therefore, accurately extracting the vegetation height on a large scale is an indispensable work and becomes a research hotspot in current. Given the fact that the polarimetric synthetic aperture radar interferometry (PolInSAR) combines the characterizes of the polarimetric synthetic aperture radar (PolSAR) and the synthetic aperture radar interferometry (InSAR) technologies, which can separate the scattering power of a single resolution cell into the contributions of surface, double-bounce and volume scattering (Neumann et al., 2010; Cloude and Papathanassiou, 1998; *Corresponding author (email: [email protected])

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Papathanassiou and Cloude, 2001). Therefore, the PolInSAR methodology provides us a great promise for accurately extracting the vegetation height. In various PolInSAR campaigns, the two-layer random volume over ground (RVoG) model (Papathanassiou and Cloude, 2001; Treuhaft and Siqueira, 2000) has been widely used to invert vegetation height (Luo et al., 2010; Papathanassiou and Cloude, 2001; Parks et al., 2007; Li et al., 2002). It describes vegetation as a homogeneous layer over impenetrable ground (Papathanassiou and Cloude, 2001). Based on the RVoG model, Papathanassiou then proposed the six-dimensional non-linear optimization method in 2001 (Papathanassiou and Cloude, 2001). This method has been successfully evaluated using different types of PolInSAR data (Parks et al., 2007; Li et al., 2002). However, the accuracy of the solution depends on the selected starting value given its nonlinear optimization feature. A poor choice of starting value results in ambiguous and/or unstable parameearth.scichina.com

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ter estimation (Li et al., 2002; Cloude and Papathanassiou, 2003). Moreover, the iterative procedure in this method is time-consuming and is unsuitable for large-scale inversions. Therefore, Cloude separated the six-dimensional non-linear parameter optimization process into three stages in 2003. This technique is known as the three-stage method (Cloude and Papathanassiou, 2003), and it has been widely used because it is time-saving, simple, and universal (Parks et al., 2007; Chen et al., 2007). However, it assumes that the effective ground-to-volume amplitude radio of a particular polarization channel (e.g., HV) is less than 10 dB (Cloude and Papathanassiou, 2003), which is unlikely in many cases. For instance, it is unsuitable for sparse forested areas or for low-frequency PolInSAR data, which are similar to L and P bands. In addition, both the six-dimensional non-linear optimization and three-stage methods do not consider the prior statistical errors of coherence observations, which may translate directly into erroneous ground-phase and volume-only coherence estimations. These methods also disregard problems related to poor geometrical dispersion, which may disrupt the estimation of vegetation height. To overcome the limitations described above, this paper proposes a complex least squares adjustment (CLSA) approach that can not only limit interferometric coherence error but also accurately determine volume-only coherence that is compensated by the effective ground-to-volume amplitude ratio. We can then enhance the accuracy of vegetation height inversion.

1 Scattering model The RVoG model constrains a physical model that associates vegetation structure parameters with multiple polarimetric SAR coherence observables. Using the RVoG model,

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we can express the complex interferometric coherence after range spectral filtering as below (Papathanassiou and Cloude, 2001; Treuhaft and Siqueira, 2000):

 ( w )  e i

0

 v   (w) , 1   (w)

(1)

where w is the unitary polarization vector that defines the choice of scattering mechanism; w is the complex interferometric coherence of polarization vector w, which is polarization-dependent and is known;  is the ground surface phase, which is an unknown real value and is polarizationindependent; wis a polarization-dependent, unknown real value that represents the effective ground-to-volume amplitude ratio accounting for attenuation through volume; and v is the polarization-independent, volume-only coherence, which can be expressed as (Papathanassiou and Cloude., 2001)

v 

2 (e(2 hv / cos  ik z hv )  1) , (2  ik z cos  )(e 2 hv / cos  1)

(2)

where  is the mean extinction coefficient of the electromagnetic wave of the vegetation layer and is a function of the scattering density in its volume and dielectric constant.  and hv are radar incident angle and vegetation height, respectively. kz is the number of the effective vertical interferometric wave. According to eq. (1), the theoretical loci of coherence sets lie along a straight line in the complex unit circle, as shown in Figure 1(a) (Papathanassiou and Cloude, 2001; Cloude and Papathanassiou, 2003). This figure also depicts the schematic of this line. However, different complex coherence observations cannot lie along a common coherence straight line because of errors (low signal-to-noise ratio and/ or temporal decorrelation) in the estimation of interferometric

Figure 1 Complex coherence sets represented by a complex plain. (a) Theoretical data; (b) simulated data using PolSARproSim. Williams (2006) provides detailed information regarding this simulator.

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coherence sets (Papathanassiou and Cloude, 2001; Cloude and Papathanassiou, 2003), as indicated in Figure 1(b). Thus, the coherence line (the black line in Figure 1(b)) is reconstructed using least squares fit (Cloude and Papathanassiou, 2003). The length of the visible line is measured according to the geometry of the SAR baseline, the radar frequency on the system side, and the forest density on the target side (Papathanassiou and Cloude, 2001). However, the least squares fit may be unstable if the ellipticity of the coherence sets is high or if geometrical dispersion is poor because of ill-posed problem. w is polarization-dependent. By using different polarization vectors w as in eq. (1), we can determine the coherences of different scattering mechanisms. In other words, if the number of polarization vectors w is M, then we can establish M equations similar to eq. (1). Unknown parameters are estimated when more than three polarization vectors are exploited. Thus, we can estimate RVoG parameters through least squares adjustment. This method can preserve many of the original attributes of the interferometric coherence observations while reducing the disturbance in the observational errors.

2 CLSA inversion model 2.1

Complex least squares

Reference (Johansson et al., 2002) mentions a complex least squares method. We can determine the residual sum of squares of the module minimum of the complex as follows (| ( ( w ))  (ˆ ( w )) |)2  min .

(3)

If the prior statistical errors of w are known, we conduct the following analysis for the criterion 2 P ( w )  Re  ( w )   Re ˆ ( w )   2   Im  ( w )   Im ˆ ( w )     min V  Re  ( w )   Re ˆ ( w )    Re , VIm  Im  ( w )   Im ˆ ( w ) 

VRe  V    subject to V T PV  min, VIm 

(4)

where Re () and Im () represent the real and imaginary parts of w. P is the weight matrix that is to be discussed in the subsequent section. In fact, this criterion is equivalent to the joint adjustment of complex real and imaginary parts. 2.2

Mathematical model

Eq. (1) should be linearized first because this equation is nonlinear. We define va+bi, and transform ei into ei cosisin. We can then rewrite eq. (1) and linearize it using the Taylor series formula

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v1w  

(aˆ  uˆw )sin ˆ0  bˆ cos ˆ0 cos ˆ0 dˆ0  daˆ 1  uˆw 1  uˆw

sin ˆ0 ˆ (1  aˆ ) cos ˆ0  bˆ sin ˆ0 db  dˆ w  l1w , 1  uˆw (1  ˆ w ) 2 (aˆ  uˆw ) cos ˆ0  bˆ sin ˆ0 sin ˆ0 cos ˆ0 ˆ dˆ0  daˆ  db  1  uˆw 1  uˆw 1  uˆw 

v2 w



(1  aˆ ) sin ˆ0  bˆ sin ˆ0 dˆ w  l2 w , (1  ˆ w ) 2

(5)

with l1w  Re( ( w )) 

(aˆ  ˆ w ) cos ˆ0  bˆ sin ˆ0 , 1  ˆ w

l2 w  Im( ( w )) 

(aˆ   w ) sin ˆ0  bˆ cos ˆ0 , 1  ˆ w

(6)

where the two subscripts 1 and 2 indicate the real and imaginary parts, respectively; aˆ , bˆ , ˆ w , and ˆ0 are approximations of a, b, uw , and , respectively; and daˆ , dbˆ ,

duˆw , and dˆ0 are corrections to these approximations. The truncation error must first be discussed because the first-order Taylor series formula is used to approximate the RVoG function. A simulation experiment is conducted with the parameters (kz=0.3, =°, hv=20 m, rad, (w)=, and = 0.2 dB/m) with six other group of approximations with hv ranging from 5 to 35m, (w) ranging from 0.05 to 0.35,  ranging from 0.05 to 0.35 rad and ranging from 0.05 to 0.35 dB/m. These parameters are used to calculate six approximations with Taylor series formula. The results are presented in Figure 2. The values obtained with the second-order Taylor series formula and with RVoG are indicated simultaneously. Given that the (w) value is complex, we express it by applying the corresponding module (Figure 2(a)) and phase (Figure 2(b)). The lines of the firstand second-order formulas are close enough that the differences between them can be ignored. This result suggests that the first-order Taylor series formula is adequately accurate without considering the truncation error. In addition, model errors are reduced when initial values are accurate. We can select linear-basis polarization, Pauli-basis polarization, unconstrained optimization, constrained optimization, and maximum coherence separation polarization as candidate observations (Seymour and Cumming, 1994). To describe our method clearly, we choose five polarization channels, two linear-basis polarizations (HH and VV), and three Pauli-basis polarizations (HH+VV, HHVV, and HV). Thus, we obtain 10 observations (each (w) has two observations for the real and imaginary parts), 8 unknowns (, v=a+bi, , , , , ), and 2 redundant observations. The observation equations can be written as V  B X L,

101

108 81

101

(7)

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where V  v1HH

v2 HH Λ v1HH VV

cos ˆ0 1  uˆ HH



101

 (aˆ  uˆ ) sin ˆ  b cos ˆ HH 0 0 0   1  uˆ HH    (aˆ  uˆ HH ) cos ˆ0  b sin ˆ0  1   HH  B  Μ 108  (aˆ  uˆ ˆ0  b0 cos ˆ0 )sin HH VV  1  uˆ HH VV   (aˆ  uˆ ˆ ˆ HH VV ) cos  0  b0 sin  0  ˆ  1  uHH VV 

sin ˆ0 1  uˆ HH

v2 HH VV  , T

(8)

(1  aˆ ) cos ˆ0  bˆ sin ˆ0 (1  ˆ HH )2

0 0 0 0 0 0

sin ˆ0 1  uˆ HH

cos ˆ0 1  uˆ HH

(1  aˆ ) sin ˆ0  bˆ sin ˆ0 (1  ˆ HH )2

Μ cos ˆ0 1  uˆ HH VV

Μ sin ˆ0  1  uˆ HH VV

Μ

Μ

0

0 0 0

sin ˆ0 1  uˆ HH VV

cos ˆ0 1  uˆ HH VV

0

0 0 0

Μ

     0   Μ ,  ˆ (1  aˆ ) cos ˆ0  b sin ˆ0   (1  ˆ HH VV )2  (1  aˆ )sin ˆ0  bˆ sin ˆ0   (1  ˆ HH VV )2  0

Μ

(9) X  dˆ0 daˆ dbˆ duˆHH duˆ HV duˆVV duˆ HH VV duˆHH VV  ,

81

(10) L  l1HH

101

l2 HH VV  , T

l2 HH Λ l1HH VV

(11)

where N is the number of independent samples used to estimate the coherence. If a std value is minimal, then it is regarded as the mean square error of unit weight. Moreover, the weights of the other observations are expressed as P (w) 

2.3

min  2

Stochastic model

Different polarization mechanisms display distinct interferometric coherences that may be corrupted by differential level noise because the ground is strongly polarization-dependent. Equalizing the weights of all coherence observations is unreasonable in the first stage of three-stage method (Cloude and Papathanassiou, 2003). Thus, we apply CramerRao bounds (Cloude and Papathanassiou, 2003; Seymour and Cumming, 1994) to restrict the contributions of the different observations and to solve this problem. The std of the coherence amplitude can then be approximately related to the coherence given the following CramerRao bounds (Cloude and Papathanassiou, 2003; Seymour and Cumming, 1994):

  (w) 

Figure 2

1   (w) 2N

2

,

Error analysis of Taylor series formula.

(12)

 2( w )

,

(13)

where P(w) is the weight matrix. We assume that the complex coherence values that lack prior static information on the real and imaginary parts are independent and have equal weights, as is practiced in the first stage of the three- stage method (Cloude and Papathanassiou, 2003). Finally, the weight matrix P, which is a diagonal matrix, is obtained. 2.4 Parameter estimation

The design matrix B is a very sparse matrix. Thus, (BT PB) is an ill-conditioned matrix, and its inverse is unstable (Wang, 2003). Furthermore, sparse or low forest areas display a similar coherence behavior, and the corresponding interferometric coherence sets exhibit poor geometrical structures in complex planes. As a result, ill-posed problems may be generated. To overcome this limitation, we adopt a

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measure called truncated singular value decomposition (T-SVD) (Wang, 2003). X  B1 L1 ,

(14)

where B1 is the Moore-Penrose inverse of B1. B1 and L1 are rewritten as B1  D 1 B L1  D 1 L,

(15)

where D is the Cholesky decomposition of P1. Prior to adjustment, we must generate initial values for B. We determine the initial values ( aˆ , bˆ , ˆ w , ˆ0 ) of unknowns (a, b, uw, ) using Eqs. (8.4) and (8.5) from Reference (Cloude, 2009). The initial values are rough. Therefore, the iterative adjustment method is used to guarantee accurate estimation. Hence, volume-only coherence and ground phase can be obtained. The look-up table (Cloude and Papathanassiou, 2003) is then used to estimate vegetation height based on eq. (2). The inversion problem takes the form





min  v  ˆv ˆ , hˆv ,  , hv

(16)

where Λ indicates the Euclidean vector norm. Figure 3 shows the flowchart of the proposed CLSA method. In short, the flowchart consists two mainly parts: (1) PolInSAR processing: Coregistration, flat earth removal, flat earth removal, multilooking operation, and coherence estimation; (2) complex adjustment processing: Based on the complex coherence obtained in the processing of (1), it is conducted as shown at the right side of Figure 3. Finally, In order to demonstrate the influence of complex coherence noise on CLSA performance, we simulate the following scenarios: kz=0.2, =°, four different hv ranging from 10 to 25 m, rad, a group of (w) are set to 0, 0.15, 0.2, 0.5 and 1, and = 0.2 dB/m . Eqs. (1) and (2) are

Figure 3

Flowchart of the proposed CLSA algorithm.

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used to calculate the values of the complex coherence that are perturbed with amplitude std. These std values range from 1% to 10%, and for every amplitude std, 100 samples are generated. As per an analysis of effective vertical interferometric wave number and vegetation height, the length of the visible line increases with vegetation height. We then calculate the vegetation height and ground phase using CLSA, the results of which are depicted in Figure 4. As amplitude std increases, the accuracy of hv and  estimation decreases. Given a 10% amplitude std, the values of root mean square error (rmse) range from 2.75 to 3.34 m. The largest estimation error corresponds to a vegetation height of 10 m. Moreover, the rmse values increase considerably as amplitude std exceeds 5%. This increase can be attributed to the fact that complex coherence noise easily destroys the geometrical structure of the visible coherence line because this line is not long enough to tolerate the noise. As a result, the ellipticity of the coherence sets is very high. In this condition, the presence of serious ill-posed problems may destabilize the least squares fit. Although a 10% amplitude std significantly perturbs the estimation, this error may be acceptable in practice overall. Fortunately, amplitude std seems to be easily controlled within 10% because the temporal and spatial baselines can be selected. Ground phase displays a similar pattern. Therefore, the phase and coherence fluctuations should be briefer than the length of the visible line. In this condition, CLSA can handle the noise effectively.

3 Experimental results and discussion 3.1

P-band E-SAR observations

In October 2008, DLR (German Aerospace Center) and FOI (Swedish Defence Research Agency) evaluated the biomass estimation potential of PolSAR in boreal forests through

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BioSAR2008. The test site was located at Krycklan Catchment, Sweden and mainly consisted of managed coniferous forests (Scots pine, Pinus sylvestris, and Norway spruce, Picea abies) that included broad-leaf trees such as birch. Figure 5 presents this scenario (P-band Pauli image (a) and optical image (b)). We then evaluated the proposed CLSA method based on 15 forest stands with in situ measurements, as indicated in Figure 5(a). Quad-polarimetric and interferometric data at the P-band were acquired using the airborne E-SAR sensor of DLR under multi-pass and multi-band configurations. Table 1 presents the two tracks selected for this study. The image range-azimuth extension measured 1501 pixels × 6301 pixels after a multilooking operation at a scale of 1:2. We then applied the CLSA as displayed in Figure 3. For the E-SAR data, Phase diversity (PD) and C&P optimization were optimized to produce the polarizations of PDHigh, PDLow, opt1, opt2, and opt3 (Cloude, 2009). The corresponding

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complex interferometric coherence sets were then observed. Polarization PDHigh is defined as volume scattering that is dominated by the least ground contributions. The temporal decorrelation over forests can usually be neglected over a short repeat-pass time (Simard et al., 2012). Thus, we estimated vegetation parameters using the RVoG model without considering temporal decorrelation. (i) Ground phase. Figure 6 depicts the ground phase estimations generated using (a) the three-stage method and with (b) CLSA. These two are difficult to distinguish directly; for instance, bare areas alone, such as the marked regions A and B, differ significantly. This finding demonstrates that the Taylor series formula does not induce significant truncation errors in the RVoG model. To examine the differences between (a) and (b) in greater detail, we scrutinize profile A-A′ in Figure 6(b). Figure 7(a) shows the profiles of ground phases estimated using the three-stage method and CLSA. Figure 7(b) depicts digital elevation

Figure 4 Inversion performance given the simulated data. Vegetation heights (top) ranging from 10 to 25 m are estimated, and the corresponding ground phases (bottom) are also presented. For each amplitude std, 100 samples are generated.

Figure 5 Test site Krycklan Catchment. (a) P-band SAR image in the Pauli basis, including the locations of 15 forest stands that are marked with red polygons. (b) Optical image (copyright Google Earth) of the test site, including the swath of radar data.

1024 Table 1

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Parameters of the airborne interferometric pair

Track

Acquisition time

1

2008-10-28, 14:53

2

2008-10-28, 15:41

Vertical baseline

Horizontal baseline

master 1.32 m

16.67 m

model (DEM) height values that derived from Light Detection and Ranging (LiDAR), and indicates that ground elevation does not vary significantly. In fact, ground phases should not differ considerably according to the geometry of the interferometric baseline. Nonetheless, Figure 7(a) highlights three phase gradient disparity regions. Hence, the ground-phase estimations with CLSA are smoother than those of the three-stage method. This finding may be attributed to the fact that we applied T-SVD to prevent the ill-posed problem induced when the observations display poor geometrical dispersion. This application may be important in deriving DEM from forest areas. (ii) Forest height. Figure 8 displays the forest heights estimated using (a) the three-stage method and (b) CLSA. As per a comparison of both sets of results, the forest

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heights measured with the three-stage method are greater. Figure 9 presents the forest heights of the 15 evaluation stands. With CLSA, forest heights are underestimated by an average of 1.5 m. Furthermore, the average std is 2.45 m, and the rmse is 3.78 m. Under the three-stage method, forest heights are overestimated by 2.9 m. In addition, the average std is 4.00 m and the average rmse is 5.37 m. Although the forest heights of stands 2, 3, 6, and 9 as obtained with the three-stage method are more accurate than those determined using CLSA, the results of the former are more biased and overestimated than those of the latter. Specifically, Figure 6 highlights one point P1 within the vegetation area, while its geometrical interpretation of volume coherence is exhibited in Figure 10(a). The volume coherence determined using CLSA is closer to the corresponding ground point than the coherence of PDHigh. This result indicates that the phase difference between the canopy and ground phase centers can be easily reduced. Therefore, Figure 10(a) illustrates why the CLSA method can prevent the overestimation of vegetation height, which is a limitation of the three-stage method. Although the CLSA results are better than three-stage

Figure 6 Ground phase estimations. Results for the three-stage method (a) and CLSA (b). Boxes A and B in (a) are bare areas. Points P1 and P2 are analyzed in the next section.

Figure 7

Ground phase (a) and DEM height (b) profiles along A-A′ in Figure 5.

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Figure 8 Forest height estimation. Results for the three-stage method (a) and CLSA (b).

results, it is worth to discuss the reasons for the estimation bias. Upon analyzing the characteristics of the 15 forest stands, we ascribe the errors to the following four reasons: (1) The terrain slope affects the penetration depth as discussed in (Lu et al., 2013). After examination of the DEM, we know that the topographic slope values of stands 1–11 and 14 are larger than those of stands 12, 13 and 15. We can see the heights of stands 12, 13 and 15 are more accurate. (2) The morphology of coniferous forest disproves the hypotheses of RVoG, the two-layer (canopy layer and ground surface layer) scattering model. To demonstrate this and exclude the influence of terrain slope, we select stands 1 and 9 whose terrain slopes are similar. The mean age of

Figure 9

Figure 10

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two stands are near 65 and 122 years, respectively. According to the morphology of coniferous forest, the old coniferous forest owns relative low canopy-fill-factor (Neumann et al., 2010) and the proportion of stem layer is relatively high. We note that the stand 9 displays relatively low canopyfill-factors as exhibited in Figure 11(a). In contrast, the stand 1 displays relatively high canopy-fill-factors as exhibited in Figure 11(b), which is similar to the hypotheses of RVoG. As a result, the height of stand 1 is more accurate. (3) Poor geometrical dispersion of the coherence sets makes it difficult to estimate ground phase using coherence line. The coherence sets are often shrunk to a point of stand 9 as indicated in Figure 10(b). As a result, it introduces a poor geometrical dispersion, which complicates accurate coherence line estimation. In this process, the estimated ground phases and vegetation heights are highly erroneous. (4) The leaves of the broad-leaf trees had fallen from the trees when the PolInSAR data were acquired, which made the phase difference underestimated. There are some broad-leaf trees like birch in the stand 9 as shown in Figure 11(c). In this situation, the phase center of forest canopy move towards to ground surface. As a result, the phase difference is underestimated. In contrast, forest stand 13 is dominated by coniferous forests (80% pine and 20% spruce),

Comparison of the average estimated and in situ heights of forest stands.

Geometrical interpretation of polarimetric interferometry. (a) and (b) correspond to P1 and P2 in Figure 6, respectively.

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more accurate phase center can be obtained. 3.2

L-band SIR-C SAR Observations

This section discusses the use of spaceborne L band SIR-C SAR data to test and verify the validity of CLSA. As displayed in Figure 12, the test site is located in Kudara (52°9.9′N, 106°40.8′E), which is near Lake Baikal, Russia. The characteristics of the interferometric pair are detailed in Table 2. Specifically, the region marked with a yellow rectangle is the test site, which consists of mixed forests, clear cuts, and farmlands. Figure 13 presents the data as a color composite. The visual interpretation of the corresponding optical remote-sensing image highlights four typical regions, which are marked with red rectangles. Regions A-D represent the farmland, the dense forest, the clear cut, and the sparse forest, respectively. The region corresponding to the sparse forest is marked in purple because the contributions of surface and double-bounce scattering are dominant in this site. The topographic relief in the left region is significant, whereas that in the right region is relatively flat. The complex interferometric coherence sets of the HH, HV, VV, HHVV, and HH+VV polarimetric channels are selected as

Figure 11

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observations, unlike in the experiment on E-SAR data. (i) Geometrical interpretation of RVoG model. To determine whether the liner model of RVoG can accurately describe the vegetation scattering given L band SIR-C SAR data, we validate this model using the test region data. Figure 14 shows the typical coherence signatures of regions A–D in Figure 13, along with their least squares fit lines. The loci of the interferometric coherence values that are associated with the RVoG model fall along a straight line in the agricultural area (a). However, the region occupied by the interferometric coherence points is not long enough because of the low density (Papathanassiou and Cloude, 2001). The sparse forest area, which is denoted by Figure 14(c), shares the disadvantage of Figure 14(a). This disadvantage results in the poor geometric dispersions of complex planes, which negatively affects least squares fit (Neumann et al., 2010). By contrast, the region occupied by the interferometric coherence points of the dense forested area (denoted by Figure 14(d)) is longer than those represented in Figures 14(a) and 14(c). Thus, ground contribution can be minimized effectively. In addition, Figure 14(b) displays the typical surface scattering in region C. As predicted with Eq. (1), the straight line shrinks into a point on the complex unit

The in situ photos of partial region.

Figure 12 Map of Russia (low right) and a magnified image of the investigated area. The red box is the swath of radar data; the yellow box is the study region; and the background optical images were downloaded from Google Earth.

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Table 2

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Parameters of the spaceborne interferometric pair

Track

Acquisition time

Perpendicular baseline

Parallel baseline

1

1994-10-07

master

master

2

1994-10-09

50.0 m

12.5 m

Figure 13 L-band PolSAR image in the Pauli basis and four optical remote sensing images which were download from Google Earth.

circle. However, the loci of the coherence values lie along a radical line because of the temporal decorrelation effect. (ii) Ground phase. Figure 15 depicts the ground topographic phase values estimated using the three-stage method and CLSA. The results of the two methods generally agree, as illustrated in Figure 16. In Figure 15(a), the area marked with a red rectangle was highlighted for statistical purposes. Under the three-stage method, the mean topographic phase value and the std are 2.22 and 0.48 rad, respectively. With CLSA, the mean topographic phase value and the std are

Figure 14 scatter.

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2.30 and 0.31 rad, respectively. Thus, the trends of the CLSA results are smoother than those of the three-stage method results. Furthermore, the results of the latter undergo significant transitions (the region marked by a blue rectangle in Figure 16), unlike those of the former. (iii) Vegetation height. Figure 17 depicts the forest height maps generated using the three-stage method and CLSA. The regions with no vegetation are masked. Moreover, solutions cannot be obtained for some regions, such as regions A-C. Nonetheless, the CLSA method generates a greater forest region area than the three-stage method does because the use of T-SVD can prevent ill- posed problems to some extent when the observations display poor geometrical dispersion, as in Figure 14(c). Under the three-stage method, these problems corrupt the estimations as a consequence of the poor geometrical dispersion. Figure 18 presents two histograms of the vegetation heights displayed in Figure 17. Histogram (a) indicates the height distribution under the three-stage method; the mean value and the std are 30.8 and 10.9 m, respectively. Height histogram (b) illustrates the distribution under CLSA; the mean value and std are 33.6 and 8.6 m, respectively. These results confirm that the std value of the CLSA results is lower than that of the findings under the three-stage method. Due to the accurate ground truth data are not available, we have no method to do accurately quantitative comparison. To support this, we now conduct the following analysis that may be help us to accept the CLSA results. Figure 19(a) corresponds to the height of the vegetation along line A-A′ in Figure 17. It was estimated using the three-stage method. This figure also synchronously presents the associated effective ground-to-volume amplitude ratios

Geometrical interpretation of polarimetric interferometry. (a) Agricultural scatter. (b) Surface scatter. (c) Sparse forest scatter. (d) Dense forest

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Figure 15 Ground topographic phase values estimated using the three-stage method (a) and CLSA (b). The black dotted line B-B′ is applied in future analyses. In Figure 14(a), the area marked with a dashed box is highlighted for statistical purposes.

Figure 16

Ground phase estimations associated with the marked line B-B′ in Figure 15.

of the HV channel, which are estimated using CLSA. The effective ground-to-volume ratios of sampling points exceed 10 dB; thus, the coherence of the HV channel can no longer be considered the volume coherence because the assumption of the three-stage method is disproved (Cloude and Papathanassiou, 2003). However, the volume coherence under CLSA is compensated by an effective ground-to-volume amplitude ratio. Hence, volume coherence is accurate and the derived vegetation heights are robust. Figure 18(b) displays the profiles of the vegetation heights estimated

using the three-stage and CLSA methods along line A-A′ in Figure 17. The three-stage method underestimates vegetation height considerably. To illustrate this finding, point P from Figure 17 is examined, and the geometrical interpretation of its volume coherence sets are presented in Figure 20. The volume coherence at point P as estimated using CLSA is more distant from the corresponding ground point than the coherence of the HV channel is. This result indicates that a significant phase difference between the canopy and the ground phase centers. Figure 20 thus demonstrates why

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Figure 17 Forest height maps generated using the three-stage method (a) and CLSA (b). The three dashed boxes A-C represent the areas in which the vegetation heights cannot be estimated. The marked dashed line A-A′ and point P are used in forest height analysis.

Figure 18

Histograms of vegetation height. Results for the three-stage method (a) and CLSA (b).

CLSA can avoid underestimating vegetation height unlike the three-stage method. Figure 21 exhibits the std values of the coherence amplitudes of polarizations HH, HV, VV, HH-VV, and HH+VV, which were estimated using eq. (12). The five histograms suggest that almost all std values are less than 10% in all polarizations. Considering the simulated experiment results presented in Figure 4, we may conclude that the results of CLSA are reliable. Despite the good performance of CLSA in some regions, such as in regions A-C (Figure 17(b)), most of the vegetation heights either could not be estimated or the estimations fluctuated considerably. As per an examination of the

aforementioned characteristics, region A is an agricultural area, B is an area near the mountains, and C is an area with sparse vegetation. This study defines these findings as follows: (1) Area A is an agricultural area. We assume that the vegetation layer satisfies the RVoG model (Papathanassiou and Cloude, 2001). However, the agricultural layer is represented by the oriented volume over ground model (Lopezsanchez et al., 2012). Therefore, the RVoG model is no longer appropriate. (2) Area B is located near the mountains, where the terrain relief is evident. Shifts in the polarization orientation

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Figure 19 Parameter estimations associated with the marked line A-A′ in Figure 17. (a) Vegetation height estimated using the three-stage method and the corresponding effective ground-to-volume ratio that is estimated with CLSA and (b) comparison of vegetation height profiles estimated using the three-stage and CLSA methods.

Figure 20

Geometrical interpretation of volume coherence estimation of P marked in Figure 17.

angle disrupt the accurate estimation of the ground and volume phases, and they are caused by surfaces with nonzero azimuth slopes (Lee and Ainsworth. 2011). Thus, the slope of the terrain should be considered in the derivation of vegetation height in mountainous areas. (3) Area C is covered with relatively sparse vegetation. As mentioned in Reference (Papathanassiou and Cloude, 2001), the coherence behavior of the sparsely forested area is similar to that of Figure 14(c). Hence, the phase centers of the ground and vegetation layers cannot be separated effectively. This condition may induce serious ill-posed problems that cannot be adequately addressed by T-SVD.

4 Conclusions In this study, we established the combined adjustment mathematical, stochastic, and parameter estimation models of a CLSA method to estimate vegetation heights robustly. We developed these models by dividing the interferometric complex coherence into real and imaginary parts. We also applied the T-SVD method during parameter estimation to address the generation of ill-posed problems. With CLSA,

we can effectively reduce the observation errors that translate into estimation errors. Furthermore, this method does not hold to the assumption of the polarization channel whose effective ground-to-volume ratio is less than 10 dB, as per the analysis results of E-SAR P-band and SIR-C SAR Lband data (Cloude and Papathanassiou, 2003). The estimation performance of CLSA was evaluated with airborne E-SAR P-band data under a single baseline configuration. The ground-phase estimations obtained using this method are smoother and more accurate than those generated with the three-stage method. While CLSA underestimated forest height by an average bias of 1.5 m compared with the in situ measurements, it remains superior to the three-stage method. The RVoG model possesses a sufficient physical structure to interpret spaceborne SIR-C SAR L-band data sets. Similarly, the ground-phase estimations by CLSA are smoother than those by the three-stage method. In addition, the forest height results obtained with CLSA are better than those generated with the three-stage method. Nonetheless, poor geometrical dispersions in the sparse vegetation, mountainous, and agricultural regions complicate the accurate fitting of the corresponding coherence lines.

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Figure 21

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The standard deviation of coherence amplitude estimated by eq. (12).

Hence, complicated vegetation structures may be considered to facilitate the expression of the canopy-fill-factor (Neumann et al., 2010), density, and underlying terrain in the RVoG model. Temporal decorrelation was also disregarded in this study, which limits the investigation of spaceborne SAR data sets with long temporal baselines, such as ALOS 2-PALSAR. To account for these temporal effects, we may increase the number of repeated observations according to optimum polarization. However, the addition of a single complex coherence observation is accompanied by an unknown effective ground-to-volume amplitude ratio for the new polarization. Nonetheless, this disadvantage may be overcome by the multi-baseline PolInSAR (Neumann et al., 2010; Papathanassiou and Cloude, 2001). This work was supported by the National Basic Research Program of China (Grant No. 2013CB733303) and National Natural Science Foundation of China (Grant Nos. 41274010, 41371335). We acknowledge the provision of the E-SAR data by European Space Agency. And this paper was supported by PA-SB ESA EO Project Campaign of “Development of methods for Forest Biophysical Parameters Inversion Using POLInSAR Data” (Grant No. ID. 14655). We would also like to thank Alaska Satellite Facility for providing the SIR-C SAR data. Balzter H, Rowland C S, Saich P. 2007. Forest canopy height and carbon estimation at Monks Wood National Nature Reserve using dual-wavelength SAR interferometry. Remote Sens Environ, 108: 224–239 Cloude S R, Papathanassiou K P. 1998. Polarimetric SAR interferometry. IEEE Trans Geosci Remote Sensing, 36: 1551–1565 Cloude S R, Papathanassiou K P. 2003. Three-stage inversion process for polarimetric SAR interferometry. Iee P-Radar Son Nav, 150: 125–134 Cloude S R. 2009. Polarization: Applications in Remote Sensing. London: Oxford University Press Chen E X, Li Z Y, Pang Y, et al. 2007. Polarimetric synthetic aperture radar interferometry based mean tree height extraction technique. Sci Silvae Sin, 43: 66–70 Gama F F, Santos J R, Mura J C. 2010. Eucalyptus biomass and volume

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