Evaluation of Multilooking Size on Single-Baseline PolInSAR Forest Height Inversion

Abstract

In polarimetric interferometric SAR (PolInSAR) technology, the random volume over ground (RVoG) model establishes the mapping relationship between polarimetric complex coherence and forest biophysical parameters (e.g., forest height). However, due to speckle noise and the finite multilooking effect, the real observed coherence region in the complex unitary circle (CUC) is an ellipse, which is biased by the ideal noise-free coherence region represented as a straight line by the RVoG model. Multilooking processing can reduce speckle noise at the cost of resolution loss. Therefore, this paper analyzes the influence of different multilooking sizes on forest height inversion. Experimental results show that the accuracy of forest height inversion first increases and then decreases with the increase in multilooking size, which means there exists an optimal size for PolInSAR forest estimation. From statistical analysis of the forest height estimation error, inversion accuracy mainly depends on estimation bias rather than estimation variance. This is mainly because, in a homogeneous forest area, a large multilooking size helps to reduce the statistical bias effect; in the textured area, the inversion accuracy benefits from a small multilooking size for avoiding the mixing of multiple types of ground targets.

1. Introduction

Forests play a vital role in the global carbon and oxygen cycle and contain many complex parameters; among them, forest height is a top priority for forest monitoring [1]. Polarimetric interferometric synthetic aperture radar (PolInSAR) technology is one of the most important methods for forest height inversion and obtains both polarimetric characteristics and interferometric information [2,3] over forest areas. For forest areas, the polarimetric characteristics are sensitive to the dielectric properties, shape distribution, and directionality of the volume scatterers, while the interferometric information is sensitive to the vertical distribution and height changes of vegetation scatterers. Therefore, PolInSAR technology has become the most commonly used remote sensing method for estimating forest height [4,5].

Widely used technology based on SAR data for forest height inversion includes random volume over ground (RVoG)-based inversion [6,7], polarization coherence tomography (PCT) [8,9], SAR tomography (TomoSAR) [10,11], and so on. Among them, the RVoG model is represented as the most extensive polarimetric coherent scattering model in PolInSAR technology. Considering different factors, many improved algorithms based on the RVoG model have been developed consecutively, such as temporal decorrelation (RVoG+TD) [12,13], volume temporal decorrelation (RVoG+VTD) [14,15], random motion over ground (RMoG) [16,17], slope (S-RVoG) [18], and varying extinction (VERVoG) [19]. Essentially, the RVoG model abstracts the forest area into both a forest layer and a surface layer and establishes the functional relationship between the polarimetric complex coherence observations and forest biophysical parameters.

In order to estimate forest height, many algorithms based on the RVoG model have been proposed. Papathanassiou et al. considered three polarimetric complex coherences with Maximum Coherence Difference (MCD) as the input observations and performed a nonlinear optimization to estimate forest height [20]. However, to avoid the rank deficiency problem in the above six-dimensional nonlinear optimization processing, Cloude et al. proposed the three-stage algorithm [21]. This algorithm considers the geometric expression of RVoG in the complex unitary circle (CUC) and uses least square line fitting to estimate the ground phase. To further reduce the number of unknown parameters, the ground-to-volume scattering ratio can be supposed to be 0, and a two-dimensional lookup table is used to calculate forest height and the extinction coefficient. Dubois-Fernandez et al. [22] and Garestier et al. [23] proposed an improved three-stage algorithm based on long-wavelength SAR, fixed to the extinction coefficient instead of the ground-to-volume scattering ratio because the scene contained considerable ground scattering contribution. Compared to six-dimensional nonlinear iterations, both three-stage algorithms take into account the geometric properties of the RVoG model, and the reliability and practicality have been improved. In addition, Zhu et al. extended the least squares adjustment from the real domain to the complex domain and proposed the complex least squares algorithm and obtained more accurate forest height results [24].

In PolInSAR technology, the coherence set corresponds to the coherence values of all polarimetric channels, which shows a continuous region in the CUC, defined as the coherence region. Usually, the coherence region is applied to analyze the geometrical distribution of all possible coherence values in the CUC. The coherence region expressed by the RVoG model is a line segment in the complex plane, defined as the coherent line segment in this paper. However, polarimetric complex coherence is easily affected by inherent speckle noise in the coherent imaging system [25,26], and hence the coherent line segment degenerates into an ellipse distributed on both sides of the coherent line segment, defined as the coherent ellipse in this paper. In practice, due to the finite multilooking effect or practical scene heterogeneity, this coherent ellipse is further biased by the coherent line segment in the complex plane, and it then results in a loss of image resolution and detailed information. Therefore, it is necessary to analyze the influence of different multilooking sizes on forest height inversion accuracy. Based on the P-band PolInSAR data provided by the AfriSAR2016 campaign, this paper evaluates and analyzes multiple Boxcar filtering sizes ranging from 5 × 5 to 25 × 25, and employs the fixed extinction coefficient algorithm to estimate forest height. Additionally, forest height results after Boxcar filtering are compared to those after refined Lee filtering [27] and intensity driven adaptive neighborhood (IDAN) [28] filtering.

The rest of this paper is organized as follows. Section 2 explains the RVoG model and fixed extinction coefficient-based forest height inversion. In Section 3, there is an overview of the study area and data sets. Results and analysis are presented in Section 4. Finally, the major conclusions of this work are given in Section 5.

2. Methodology

2.1. PolInSAR Matrix Generation

The scattering matrix $\mathit{S}$ can usually be used for describing the full polarimetric information of a single target. In the case of linear horizontal and vertical polarization bases, the Sinclair scattering matrix $\mathit{S}$ is expressed as follows [29]:

$$\mathit{S}=\left[\begin{array}{cc}{S}_{HH}& S_{HV}\\ S_{VH}& S_{VV}\end{array}\right]$$

(1)

For a reciprocal target matrix, in the monostatic backscattering case, the reciprocity constrains the matrix $\mathit{S}$ to being symmetrical, that is, $S_{HV}=S_{VH}$. The corresponding complex Pauli basis vector is [29]:

$$k={\left[\begin{array}{ccc}{S}_{HH+VV}& S_{HH-VV}& 2S_{HV}\end{array}\right]}^T/\sqrt{2}$$

(2)

where ${}^T$ denotes the transpose. The complex Pauli basis scattering vectors of two images construct the six-element complex scattering target vector $k_6={\left[\begin{array}{cc}{k}_1^T& k_2^T\end{array}\right]}^T$. Therefore, the PolInSAR data is formulated into a 6 × 6 interferometric coherency matrix by $k_6$ and its conjugate transpose [29]:

$$T_6=\langle k_6{k}_6^{\ast \mathrm{T}}\rangle =\left[\begin{array}{cc}\langle k_1{k}_1^{\ast \mathrm{T}}\rangle & \langle k_1{k}_2^{\ast \mathrm{T}}\rangle \\ \langle k_2{k}_1^{\ast \mathrm{T}}\rangle & \langle k_2{k}_2^{\ast \mathrm{T}}\rangle \end{array}\right]=\left[\begin{array}{cc}{T}_{11}& {\Omega }_{12}\\ {\Omega }_{12}^{\ast \mathrm{T}}& T_{22}\end{array}\right]$$

(3)

where $\langle \cdot \rangle$ denotes the spatial multilook average and ${}^{\ast }$ is the complex conjugate. $T_{11}$ and $T_{22}$ are master and slave polarimetric coherency matrixes that contain full polarimetric information, respectively. ${\Omega }_{12}$ denotes the interferometric cross correlation matrix. In order to acquire the interferogram of a polarization channel between two observations, a normalized complex projection vector $\mathsf{\omega }$ is introduced. For full-polarimetric SAR images, $\mathsf{\omega }$ is parameterized as follows [29]:

$$\mathsf{\omega }=\left[\begin{array}{c}\cos \alpha \\ \sin \alpha \cos \beta e^{j\delta }\\ \sin \alpha \sin \beta e^{j\psi }\end{array}\right],\{\begin{array}{l}0\le \alpha \le \pi /2\\ 0\le \beta \le \pi /2\\ -\pi \le \delta \le \pi \\ -\pi \le \psi \le \pi \end{array}$$

(4)

where $\alpha$, $\beta$, $\delta$, and $\psi$ are the polarimetric covariance matrix written in terms of the so-called polarimetric intercorrelation parameters [30]. The complex Pauli basis scattering vectors $k_1$ and $k_2$ can be projected onto the given complex projection vectors ${\mathsf{\omega }}_1$ and ${\mathsf{\omega }}_2$, and the corresponding polarimetric image $I_1$ and $I_2$ can be obtained, i.e., $I_1={\mathsf{\omega }}_1^{\ast \mathrm{T}}{k}_1$ and $I_2={\mathsf{\omega }}_2^{\ast \mathrm{T}}{k}_2$. Thus, for both polarimetric channels ${\mathsf{\omega }}_1$ and ${\mathsf{\omega }}_2$, the complex polarimetric interferometric coherence $\gamma \left({\mathsf{\omega }}_1,{\mathsf{\omega }}_2\right)$ can be formed [29]:

$$\gamma \left({\mathsf{\omega }}_1,{\mathsf{\omega }}_2\right)=\frac{\langle I_1{I}_2^{\ast }\rangle }{\sqrt{\langle I_1{I}_1^{\ast }\rangle \langle I_2{I}_2^{\ast }\rangle }}=\frac{\langle {\mathsf{\omega }}_1^{\ast \mathrm{T}}{\Omega }_{12}{\mathsf{\omega }}_2\rangle }{\sqrt{\langle {\mathsf{\omega }}_1^{\ast \mathrm{T}}{T}_{11}{\mathsf{\omega }}_1\rangle \langle {\mathsf{\omega }}_2^{\ast \mathrm{T}}{T}_{22}{\mathsf{\omega }}_2\rangle }}$$

(5)

2.2. RVoG Model

To describe the scattering scene of the two-layer structure in a forest area, the RVoG model establishes the mapping function between the polarimetric complex coherence and the forest biophysical parameters (as shown in Figure 1) [21]:

$$\gamma (\mathsf{\omega })=e^{i{\varphi }_0}\frac{{\gamma }_v+\mu (\mathsf{\omega })}{1+\mu (\mathsf{\omega })}$$

(6)

where $\gamma (\mathsf{\omega })$ denotes the volume coherence in a polarimetric channel $\mathsf{\omega }$, which includes contributions from both the volume and ground layers, ${\varphi }_0$ is the underlying ground phase, $\mu (\mathsf{\omega })$ denotes the ground-to-volume amplitude ratio, and ${\gamma }_v$ represents the volume-only coherence. Considering the impact of terrain slope on forest height estimation, ${\gamma }_v$ can be expressed in terms of [18]:

$${\gamma }_v=\frac{2\sigma \left(e^{\left(2\sigma h_v\cos \alpha /\cos \theta +i{k}_z{h}_v\cos \alpha \right)}-1\right)}{\left(2\sigma +i{k}_z\cos \theta \right)\left(e^{2\sigma h_v\cos \alpha /\cos \theta }-1\right)}$$

(7)

where $\sigma$ represents the extinction coefficient, $h_v$ denotes forest height, $\alpha$ is the local range terrain slope, and $\theta$ is the local incidence angle calculated by the nominal incidence angle ${\theta }^{\prime }$ (i.e., $\theta ={\theta }^{\prime }-\alpha$). $k_z$ is the local effective vertical wavenumber given as follows [21]:

$$k_z=\frac{4\pi B_{\perp }}{\lambda R\sin \theta }$$

(8)

where $\lambda$ represents the radar wavelength, $B_{\perp }$ denotes the perpendicular baseline, and $R$ is slant range distance.

From (6), when the radar response is dominated by ground contribution, the scattering phase center is located closer to the ground, i.e., $\mu (\mathsf{\omega })\to \infty$, and complex coherence is equal to the underlying ground phase: $\gamma (\mathsf{\omega })=e^{i{\varphi }_0}$. At the other extreme, when the radar response is dominated by volume scattering, without the presence of the ground ($\mu (\mathsf{\omega })=0$), the complex coherence is equal to the volume-only coherence with the underlying ground phase: $\gamma (\mathsf{\omega })=e^{i{\varphi }_0}{\gamma }_v$. Therefore, from the perspective of the geometric interpretation, the RVoG model can be expressed as a linear function of both real and imaginary parts of the polarimetric complex coherence set and corresponds to a line segment in the CUC.

2.3. Fixed Extinction Coefficient-Based Forest Height Inversion

Due to the large amount of penetration into the vegetation volume at the P-band, the fixed extinction coefficient algorithm is an inversion better suited to the forest scene and data exploited in this study [22,23]. The inversion process of the fixed extinction coefficient algorithm is:

(1)

Line fitting. Common polarimetric complex coherences (i.e., ${\gamma }_{HH}$, ${\gamma }_{HV}$, ${\gamma }_{VV}$, ${\gamma }_{HH+VV}$, ${\gamma }_{HH-VV}$) and complex coherences ${\gamma }_{PDHigh}$ and ${\gamma }_{PDLow}$ based on the phase diversity (PD) algorithm [31] are selected, and the total least square line fitting is then used for obtaining the coherent line segment;

(2)

Ground phase estimation. There are two intersections between the line segment and CUC, one of which is the ground phase. Thus, to determine the ground phase, Kugler et al. proposed a method-based $k_z$ [32]:

$$\{\begin{array}{l}For{k}_z>0\\ if\arg \left(\exp \left(i{\varphi }_1\right)\exp {\left(i{\varphi }_2\right)}^{\ast }\right)<0then{\varphi }_0={\varphi }_1\\ if\arg \left(\exp \left(i{\varphi }_1\right)\exp {\left(i{\varphi }_2\right)}^{\ast }\right)>0then{\varphi }_0={\varphi }_2\\ For{k}_z<0\\ if\arg \left(\exp \left(i{\varphi }_1\right)\exp {\left(i{\varphi }_2\right)}^{\ast }\right)<0then{\varphi }_0={\varphi }_1\\ if\arg \left(\exp \left(i{\varphi }_1\right)\exp {\left(i{\varphi }_2\right)}^{\ast }\right)>0then{\varphi }_0={\varphi }_2\end{array}$$

(9)

where ${\varphi }_1$ and ${\varphi }_2$ are the two intersections between the line segment and CUC, respectively;

(3)

Forest height estimation. In order to avoid the influence of ground scattering contribution, the ${\gamma }_{PDHigh}$ is used as the observation, and then a fixed extinction coefficient $\sigma ={\sigma }_0$ is used. Finally, the two-dimensional lookup table of forest height $h_v$ and ground-to-volume scattering ratio $\mu (\mathsf{\omega })$ is established to estimate forest height [23]:

$$\underset{h_v,\mu }{\min }\Vert {\gamma }_{PDHigh}-e^{i{\varphi }_0}\frac{{\gamma }_v\left(h_v,\sigma ={\sigma }_0\right)+\mu }{1+\mu }\Vert$$

(10)

Compared to the traditional three-stage algorithm, the fixed extinction coefficient-based improved algorithm considers that the long-wavelength SAR scene contains considerable ground scattering contribution. This method is more suitable for the selected P-band SAR data in this paper, and effectively guarantees the accuracy of forest height inversion.

3. Study Areas and Data Sets

This paper selects two study areas, which are located in Mabounie (00°43′ S, 10°31′ E) and Mondah (00°35′ N, 9°18′ E) in Gabon, Africa. Mabounie covers an area of about 4.45 × 10⁴ km² and is dominated by hilly landforms with relatively gentle fluctuations and the average forest height is about 30 m. Mondah covers an area of about 6.62 × 10⁴ km² and is a relatively young forest with high variability in density and forest height is roughly between 5 m and 55 m.

The P-band full polarimetric SAR data over the two study areas were acquired from the framework of the European Space Agency (ESA) AfriSAR 2016 campaign. The German Aerospace Center’s F-SAR airborne system was used in this campaign. Two fully polarimetric SAR images were selected for the interferometric process from the Mabounie area. Similarly, two SAR images were also selected from the Mondah area.

Table 1. System parameters for Mabounie and Mondah from the AfriSAR 2016 campaign.

Study Area	Image Number	Temporal Baseline	Perpendicular Baseline (m)	Image Resolution (m)	Incidence Angle (°)
Mabounie	0708	11 February 2016 11:14:48	Master	Range: 1.2 Azimuth: 0.9	25–60
	0703	0, 0, 47	−30.0
Mondah	0502	8 February 2016 10:17:59	Master	Range: 1.2 Azimuth: 0.8	10–60
	0212	3, 22, 16	−30.0

shows the detailed system parameters of the selected SAR images. In addition, the airborne light detection and ranging (LiDAR) data covering the test areas were acquired from the National Aeronautics and Space Administration (NASA)’s Land, Vegetation, and Ice Sensor (LVIS) system, which was used to verify the accuracy of the inversion results, whose LiDAR footprint on the ground is around 20 m wide [32]. The mean of master and slave Pauli basis composite RGB (PauliRGB) images and LiDAR Canopy Height Model (CHM) products in the Mabounie and Mondah areas are shown in Figure 2 and Figure 3, respectively. Figure 3c shows the optical image of the Mondah area in 2016 acquired from Google Earth.

In order to analyze the influence of different multilooking sizes on forest height inversion accuracy, the multiple Boxcar filtering sizes were designed to vary from 5 × 5 to 25 × 25, and the fixed extinction coefficient algorithm was applied to estimate forest height. The P-band data has strong penetration and a high ground scattering contribution, which is the result of a low extinction level [33]. Thus, the following experiment has fixed the extinction coefficient to be 0.1 dB/m. Finally, taking LiDAR CHM products as the reference data, we quantitatively analyze estimation accuracy of the forest height calculations.

4. Results and Analysis

We analyzed the influence of multilooking sizes or different filtering methods on forest height estimation in detail from the following four aspects: (1) numerical analysis based on backscattering intensity; (2) geometric interpretation based on coherence region; (3) error analysis based on statistical indicators; and (4) accuracy analysis of different filtering methods.

4.1. Numerical Analysis Based on Backscattering Intensity

For the detailed investigation, we selected a 200 × 200 pixels region of interest (ROI) labeled ROI-I (the black square box in Figure 2) and ROI-II (the red square box in Figure 3) from the Mabounie and Mondah areas. Figure 4a–c shows the pixel-wise forest height estimation error of ROI-I using different multilooking sizes (5 × 5, 9 × 9, 25 × 25), and the corresponding LiDAR results, PauliRGB image, and optical image is shown in Figure 4d–f. In ROI-II, the estimation errors of the 5 × 5 size, 11 × 11 size, 25 × 25 size, the corresponding LiDAR results, the PauliRGB image, and the optical image from Google Earth are shown in Figure 5.

To prove the influence of different multilooking sizes on single-baseline PolInSAR forest height inversion, this paper first analyzes the variation of backscattering intensity from different multilooking sizes. The section selects two pixels, specifically A (see Figure 4) and D (see Figure 5) from the homogeneous areas of ROI-I and ROI-II, respectively. We then use the HH channel to calculate the backscattering intensity of pixels A and D and the standard deviation of backscattering intensity centered on pixels A and D at each space, respectively, as shown in Figure 6. Note that this standard deviation is used to evaluate the spatial smoothness of intensity after filtering of different sizes. It can be seen from Figure 6 that as size increases, the backscattering intensities tend to be stable for each space, and spatial variation is also smaller and smaller.

Moreover, pixels B and C (see Figure 4) and pixels E and F (see Figure 5) belonging to the textured region are selected in ROI-I and ROI-II, respectively. We then calculated the backscattering intensity values under different multilooking sizes as shown in Figure 7. With the increase in multilooking size, backscattering intensity values for pixels B and C gradually became closer, and pixels E and F had a similar phenomenon. This explains how two pixels could be confused with each other and hence have similar values.

4.2. Geometric Interpretation Based on Coherence Region

Generally, the estimation error of PolInSAR forest height mainly depends on estimation bias, which may be derived from the finite multilooking effect and multilooking average without any discrimination. In order to analyze the source of the estimation bias, we make a detailed analysis of the geometrical characteristics of the coherence region under different multilooking sizes from ROI-I and ROI-II.

To analyze the influence of the finite multilooking effect on bias for forest height inversion in detail we selected the same two pixels A (see Figure 4) and D (see Figure 5) from the homogeneous areas of ROI-I and ROI-II, respectively. The LiDAR forest heights of pixels A and D are 30.47 m and 33.14 m, respectively. It can be seen from Figure 4a–c and Figure 5a–c that the forest heights of pixels A and D are overestimated, but as the size increases, the phenomenon of overestimation weakens significantly. For pixel A, the estimation errors are 22.87 m, 15.74 m, and 9.57 m for the three sizes, respectively. Meanwhile, the estimation errors for the three sizes for pixel D are 20.85 m, 13.96 m, and 3.06 m, respectively. In the homogeneous area, bias reduction benefits from the large multilooking size. We then used a boundary sampling method to draw the contour of the coherence regions [34] for pixels A and D, obtained the farthest point pairs as the endpoints of the coherent line segment, and calculated the coherence amplitude at the HH channel, as shown in Figure 8. It can be seen from Figure 8 that, due to the existence of speckle noise, all polarimetric complex coherence sets are represented as a coherent elliptical region in the CUC. As the multilooking size increases, the coherence set shrinks and becomes flatter. The finite multilooking effect makes the observed coherence region biased due to the coherent line segment of the RVoG model. In a word, the aforementioned results have proved that a large multilooking size helps to reduce the statistical bias effect in a homogeneous forest area.

To further explain the bias effect of the multilooking average without discrimination, two pixels (B and C (see Figure 4)) in ROI-I were selected from the textured areas and the corresponding LiDAR forest heights are 2.85 m and 15.60 m, respectively. Pixel B has a low forest height which does not satisfy the requirements of the RVoG model. Thus, regardless of multilooking size, the estimation error is very large. For pixel C, as size increases, the estimation error is affected by pixel B and increases continuously due to the averaging of different scatterer types. Concretely, for the 5 × 5 size, the estimation errors of pixels B and C are 40.84 m and 3.21 m. The 9 × 9 size also has a difference in pixels B and C: 40.40 m and 6.68 m. In regards to a relatively small multilooking size, compared to pixel C, pixel B has a larger estimation error. This explains that the scattering characteristics of pixel C are not so affected by those of the texture area which pixel B belongs to. However, for the 25 × 25 size, pixels B and C were confused with each other and hence had a similar estimation error. The estimation errors of pixels B and C were 50.27 m and 48.25 m, respectively. This illustrates that with increasing size different types of forest scatterers are more likely to interact with each other, and the corresponding estimated height results are unreliable. For further analysis, Figure 9a–c shows the geometrical interpretation of pixels B and C in the 5 × 5, 9 × 9, and 25 × 25 sizes. It can be seen from Figure 9a–c that as size increases, the coherent regions, the line segment, and the coherence amplitude of pixels B and C becomes closer. This is because the scattering characteristics within the surrounding area of pixel B spread pixel C, and the corresponding estimated height is rather close to that of pixel B.

Similarly, pixels E and F were selected from ROI-II (see Figure 5) and the corresponding LiDAR forest heights are 2.55 m and 14.20 m, respectively. Numerically, for pixels E and F, the corresponding estimation errors are 52.44 m and 14.2 m for the 5 × 5 size, 48.77 m and 15.46 m for the 11 × 11 size, and 30.45 m and 25.23 m for the 25 × 25 size. From the numerical data and Figure 9d–f, we can also obtain the same conclusion as the Mabounie area. In the textured area, inversion accuracy benefits from a small multilooking size which avoids mixing multiple types of ground targets. Therefore, we can determine that large multilooking size causes the wrong selection of multiple heterogeneous pixels and the serious detail-blurring effect on the forest height image, which decreases the accuracy of forest height inversion.

In brief, through the analysis of geometric interpretations based on coherence region, we can determine that large multilooking size helps to reduce the statistical bias effect, and small multilooking size can prevent the accuracy of different forest height results being influenced by each other. Thus, a moderate multilooking size is suitable for PolInSAR forest height inversion.

4.3. Error Analysis Based on Statistical Indicators

We estimated the forest height results of both Mabounie and Mondah areas under the 5 × 5 to 25 × 25 multilooking sizes. Figure 10 and Figure 11 show the forest height inversion results of three multilooking sizes in Mabounie and Mondah, respectively. As the multilooking size increases, the image resolution decreases and the detail-blurring effect becomes more serious, indicating that a relatively small Boxcar filtering window size can effectively suppress imagery speckle noise while preserving the structure and textured features of different ground targets. This is due to the Boxcar filter averaging all sample pixels in the square window without any discrimination.

For a more intuitive accuracy assessment, we selected a total of 5840 forest stands and 2385 forest stands in a 50 × 50 window (100 m × 100 m) from the entire Mabounie and Mondah areas, respectively. For a forest stand $i$, the estimation error of forest height $\mathsf{\Delta }\epsilon$ can be calculated as:

$$\mathsf{\Delta }{\epsilon }_i=h_{v,i}-h_{LiDAR,i}$$

(11)

where $h_{LiDAR}$ represents the forest height results of LiDAR. As shown in the estimation error histograms of the Mabounie and Mondah areas in Figure 12, there exists an overestimation problem in filtered results. Compared to the 5 × 5 size, the error distribution in larger sizes is more concentrated and also is closer to 0, especially in the Mabounie area. The overall curves in the 9 × 9 size are closer to 0, which means the bias is smaller.

To further quantitatively analyze height estimation performance for different multilooking sizes, the following equations are used to generate statistics for estimation errors of all forest stands using three statistical indicators, i.e., Root Mean Square Error (RMSE), Mean (MEAN), and Standard Deviation (STD). The three aforementioned evaluation indicators can be calculated based on the estimation error $\mathsf{\Delta }\epsilon$:

$$\mathrm{RMSE}=\sqrt{\frac{1}{n}\sum _{i=1}^n(\mathsf{\Delta }{\epsilon }_i{)}^2}$$

(12)

$$\mathrm{MEAN}=\frac{1}{n}\sum _{i=1}^n\mathsf{\Delta }{\epsilon }_i$$

(13)

$$\mathrm{STD}=\sqrt{\frac{1}{n}\sum _{i=1}^n(\mathsf{\Delta }{\epsilon }_i-\mathrm{MEAN}{)}^2}$$

(14)

The RMSEs, MEANs, and STDs are calculated under different multilooking sizes for Mabounie (as shown in Figure 13a–c) and Mondah (as shown in Figure 13d–f). In the Mabounie area, it can be seen from Figure 13a–c that the RMSEs, MEANs, and STDs are consistent evaluation indicators in relation to changing trends. As size increases, the three statistical indicators first decrease and then increase and simultaneously achieve the smallest value in the 9 × 9 size. However, compared to the MEANs and RMSEs, the amplitude of variation for STDs is smaller, which explains how the performance of height estimation with different multilooking sizes is not so affected by STD and is seriously affected by bias. Specifically, compared with the maximum MEAN (5.40 m) and RMSE (6.75 m) corresponding to the 5 × 5 size, the MEAN (4.66 m) in the 9 × 9 size is reduced by 0.74 m, and the RMSE (6.07 m) in the 9 × 9 size is reduced by 0.68 m. Simultaneously, the difference between the maximum STD and minimum STD is only 0.2 m (from 4.10 m to 3.90 m for the 25 × 25 size and the 9 × 9 size, respectively).

In the Mondah area, the changes in MEAN, RMSE, and STD evaluation indicators are similar to the Mabounie area (see Figure 13d–f). Specific numerical comparison of the three statistical indicators shows that the MEAN (5.63 m) of the 11 × 11 size is 0.90 m less than the MEAN (6.53 m) of the 5 × 5 size, and the RMSE (7.12 m) of the 11 × 11 size is 0.84 m less than the RMSE (7.96 m) of the 5 × 5 size. The difference between the maximum STD and minimum STD is only 0.35 m (from 4.70 m to 4.35 m). This also explains that the RMSE is mainly affected by estimation bias under multiple multilooking sizes. Therefore, from the above statistical analysis of two different areas, we can determine that the accuracy of forest height inversion mainly depends on estimation bias rather than STD, indicating that an appropriate size can effectively reduce the estimation error.

4.4. Accuracy Analysis of Different Filtering Methods

To compare forest height inversion performance using different filtering methods we selected Boxcar filtering (optimal size of 9 × 9 in Mabounie, 11 × 11 in Mondah), refined Lee filtering [27] (11 × 11 edge-aligned neighborhood), and IDAN filtering [28] (adaptive neighborhood with maximum size of 50). The fixed extinction coefficient algorithm was also used with the same parameters as the above experiment for estimating forest height. Figure 14 shows forest height accuracy evaluation results for different filtering methods in the Mabounie and Mondah areas. It can be seen from Figure 14 that the MEAN, STD, and RMSE from refined Lee filtering are all the smallest, indicating that refined Lee filtering had the best-estimated result among the three filtering methods. In Mabounie, compared with Boxcar filtering and IDAN filtering, the RMSE of refined Lee filtering is increased by 7.4% and 9.9%, respectively. In Mondah, the RMSEs of the three filtering methods have a small difference, with refined Lee filtering only 0.18 m less than Boxcar filtering and 0.35 m less than IDAN filtering. This is because refined Lee filtering selects the optimal local template adaptively based on the edge-aligned gradient windows, and better distinguishes between pixels of different types of ground targets.

5. Conclusions

In this paper, we used AfriSAR2016 SAR data sets including the Mabounie and Mondah areas to evaluate the performance of different multilooking sizes on single-baseline PolInSAR forest height inversion. Moreover, we also compared the forest height inversion performance using different filtering methods. Through this experiment, the main three contributions of our analysis work are as follows:

(1)

The pixel-wise detailed analysis based on the coherence set and scattering intensity reveals how multilooking size affects the tree height result. Polarimetric complex coherence is easily affected by the inherent speckle noise and finite multilooking effect, hence the coherent line segment degenerates into an ellipse which is biased by the ideal noise-free coherence region. Multilooking processing is an effective method which reduces speckle noise and the finite multilooking effect. Therefore, through analysis based on backscattering intensity and coherence region, we can determine that a large multilooking size helps to reduce the influence of statistical bias caused by the finite multilooking effect in homogeneous forest areas and a small multilooking size can prevent the influence of multilooking average without any discrimination. Thus, a moderate multilooking size is suitable for PolInSAR forest height inversion.

(2)

From the perspective of the overall error analysis, forest height inversion results mainly depend on estimation bias rather than estimation variance. Through statistics regarding estimation errors using three statistical indicators, we can determine that the accuracy of forest height inversion mainly depends on estimation bias rather than STD, indicating that an appropriate size can effectively reduce estimation error. Thus, the experimental result shows that the 9 × 9 size is the optimal one for the selected Mabounie area data set, and the 11 × 11 size is the optimal size in the Mondah area data set.

(3)

Three different filtering methods were compared, and refined Lee filtering is a more superior method. Comparing forest height inversion performance using different filtering methods, we found that refined Lee filtering had the best-estimated result because this filtering method selects the optimal local template adaptively based on the edge-aligned gradient windows and better distinguishes between pixels of different types of ground targets.

Thus, according to the estimated forest height results of different filtering methods, refined Lee filtering achieved the highest estimation accuracy. Therefore, this paper suggests selecting refined Lee filtering to reduce speckle noise and preserve detailed information in single-baseline PolInSAR forest height inversion processing.