The Calibration Method of Multi-Channel Spatially Varying Amplitude-Phase Inconsistency Errors in Airborne Array TomoSAR

Abstract

Airborne array tomographic synthetic aperture radar (TomoSAR) can acquire three-dimensional (3D) information of the observed scene in a single pass. In the process of airborne array TomoSAR data imaging, due to the disturbance of factors such as inconsistent antenna patterns and baseline errors, there are spatially varying amplitude-phase inconsistency errors in the multi-channel Single-Look-Complex (SLC) images. The existence of the errors degrades the quality of the 3D imaging results, which suffer from positioning errors, stray points, and spurious targets. In this paper, a new calibration method based on multiple prominent points is proposed to calibrate the errors of amplitude-phase inconsistency. Firstly, the prominent points are selected from the multi-channel SLC data. Then, the subspace decomposition method and maximum interference spectrum method are used to extract the multi-channel amplitude-phase inconsistency information at each point. The last step is to fit the varying curve and to compensate for the errors. The performance of the method is verified using actual data. The experimental results show that compared with the traditional fixed amplitude-phase inconsistency calibration method, the proposed method can effectively calibrate spatially varying amplitude-phase inconsistency errors, thus improving on the accuracy of 3D reconstruction results for large-scale scenes.

1. Introduction

Conventional SAR can only achieve 2D imaging [1]. In order to meet with a wider range of needs, 3D SAR imaging technology has gradually emerged. The InSAR realizes the reconstruction of terrain height using the dual-channel data [2], and can generate a Digital Elevation Model (DEM), which has been widely used [3,4,5]. However, InSAR does not have the ability to distinguish layovers. Compared with InSAR, TomoSAR uses more channels to realize the reconstruction of layovers [6]. The airborne array TomoSAR technique utilizes array antenna and Multi-Input Multi-Output (MIMO) technology to acquire multi-channel coherent information, which can generate the 3D tomographic imaging results [7]. Because of its advantages of short data acquisition time, little loss of time decorrelation, and a flexible operation mode, it has shown broad application prospects in the field of mountain mapping [8], urban mapping [9,10], resource exploration [11,12], and other applications [13,14,15].

Limited by the array antenna’s length, the elevation resolution of array TomoSAR is poor [16]. Therefore, super-resolution algorithms are usually used to enhance the elevation resolution in the 3D imaging of array TomoSAR [17,18]. However, the performance of super-resolution algorithms usually requires precise multi-channel amplitude-phase consistency [19]. In practice, because of various factors, such as the different patterns of each transceiver antenna, and the internal circuits and baseline errors, there will be amplitude-phase inconsistency errors in multi-channel SLC images after 2D imaging [20]. The existence of the errors degrades the quality of the 3D imaging results, which suffer from positioning errors, stray points, and spurious targets [21], especially in a large-scale scene. So, the calibration of the multi-channel amplitude-phase inconsistency errors is necessary to ensure high-quality 3D reconstruction results. In addition, due to the diversity and variability of the sources of the errors, such as the array antenna and the carrier aircraft motion errors, etc., the errors vary spatially. Additionally, the antenna pattern measured in the microwave anechoic chamber is quite different from the actual result, due to the effect of the aircraft structure. Therefore, effective calibration can only be achieved by analyzing the actual measured data acquired during the flight.

The calibration method can be divided into two kinds of internal calibration and external calibration, distinguished via implementation. Internal calibration is used to calculate the amplitude-phase inconsistency of different channels by constructing a data path inside the radar. In 2001, M. Werner et al. calibrated the phase errors between SRTM interference channels using the technique of internal calibration [22]. In 2010, M. Schwerdt et al. designed a special internal calibration network for TerraSAR-X to estimate the amplitude-phase errors during data acquisition [23]. Nevertheless, the internal calibration method can only measure the errors generated before the antenna radiation, and the errors generated in the antenna radiation and data processing are ignored. The external calibration method is used to correct the errors by analyzing the target in the imaging scene and combining a certain estimation algorithm. According to the different reference targets, the external calibration method can also be divided into two kinds. One is to place some active or passive calibration equipment or external sources on the ground during radar operation, and to calibrate the multi-channel amplitude-phase errors through these manually arranged calibrators. In 2018, Bu et al. used multiple corner reflectors placed on the ground to calibrate the array TomoSAR multi-channel amplitude-phase imbalances and the antenna phase center position simultaneously by combining the maximum likelihood estimation and the least squares method [19]. However, this method cannot be applied when the imbalances is spatially varying. At the same time, the manual layout of a large number of ground calibrators is time-consuming and labor-intensive. Additionally, in some inaccessible mountainous areas, such methods will fail when the calibrators cannot be deployed. Therefore, another kind of external calibration method is proposed, which uses the target or terrain in the natural scene to calibrate. In 2015, Tebaldini et al. proposed a phase center double localization (PCDL) strategy to compensate for the phase errors without any reference targets or prior information [24]. But this method may not handle situations with low correlations between the SAR images. In 2018, an approach based on phase derivative constrained optimization (PDCO) was proposed by Aghababaee et al., which aims to solve the problem of uncontrollable vertical shifts in the tomographic focusing [25]. In 2020, Z. Jiao et al. used strong scatterers in 2D SAR images to perform amplitude-phase imbalance calibration between channels by iteratively updating through two steps of calibration and reconstruction [26]. However, this method also does not consider the spatially varying amplitude-phase errors. In 2019, D. Feng et al. proposed a residual phase errors calibration method based on phase gradient autofocus (PGA) to compensate for the multi-channel phase errors caused by baseline errors in holographic SAR (HoloSAR) [27]. Whereas, the PGA method is only suitable for small areas, and it cannot be directly applied to the compensation of phase errors in a large area. Moreover, the PGA method will also cause an unwanted vertical shift in the retrieved tomograms. Aiming at this problem, in 2021, H. Lu et al. proposed a NC-PGA method based on network construction and PGA [28]. This method divides the test area into multiple sub-regions, and uses PGA to estimate the phase error of each region. In 2022, H. Lu et al. continued to propose the BBN-PGA method on the basis of NC-PGA [29]. The block-structured network (BBN) method is used to replace the NC method to estimate the elevation phase errors in urban areas, and it is combined with the PGA method to compensate for the errors.

In the above methods, either the amplitude-phase inconsistency errors are not considered at the same time, or the spatial variability of amplitude-phase errors along the range caused by antenna pattern inconsistency and baseline errors has not been considered. In view of the above problems, this paper proposes a multi-channel spatially varying amplitude-phase inconsistency errors calibration method based on the widely distributed prominent points in natural scenes, combined with signal noise subspace decomposition and the maximum interference spectrum method. Using prominent points with high signal-to-noise ratios, the amplitude inconsistency information is extracted after the subspace decomposition, and the phase inconsistency information is extracted using the maximum interference spectrum method. Because of the wide distribution of the prominent points, the varying amplitude-phase inconsistency information in the scene can be extracted. After fitting the changing curve of spatially varying amplitude-phase inconsistency, the errors in multi-channel SLC data can be compensated. This method can effectively calibrate the spatially varying multi-channel amplitude-phase inconsistency errors, and then improve the quality of airborne array TomoSAR 3D reconstruction results.

This paper is organized as follows. In Section 2, the relevant theory of the proposed method is introduced. In Section 3, the method for calibrating the spatially varying multi-channel amplitude-phase inconsistency errors is proposed. Section 4 presents the experiment results. Discussions are given in Section 5. Section 6 presents the conclusions of the paper.

2. Theory

In this section, the airborne array TomoSAR imaging Geometric model is first introduced. Then, the amplitude-phase inconsistency errors in multi-channel SLC data after 2D imaging are analyzed.

2.1. The Airborne Array TomoSAR Imaging Model

The airborne array TomoSAR obtains elevation resolution from the cross-heading array antenna, and then combines the 2D imaging of the traditional SAR to achieve the 3D resolution of the target scene. The array antenna generally adopts MIMO mode to form multiple virtual antenna phase centers [30]. The imaging Geometric model of the airborne array TomoSAR is shown in Figure 1, which has N data acquisition channels. There are two different coordinate systems in the figure. The geodetic coordinate system $(x,y,z)$ represent azimuth, ground, and height.The radar coordinate system $(x,r,s)$ represents azimuth, range, and elevation. The two coordinate systems follow the principle of the right-hand coordinate system.

The transmitted signal is a linear frequency modulation pulse signal, which is expressed as

$$s\left(t\right)=rect\left(\frac{t}{T_p}\right)exp\left\{j2\pi f_{0}t+j\pi K_r{t}^2\right\}$$

(1)

where $t,T_p,f_0$, and $K_r$ denote the range fast time, the pulse width of the transmitted signal, the center frequency, and the linear frequency modulation ratio.The delay signal reflected by the nth channel’s transmitting signal encountering the target is expressed as

$$s_n\left(t\right)=\gamma ·rect\left(\frac{t-2R_n/\phantom{2R_{n}c}c}{T_p}\right)exp\left\{-j4\pi f_0\frac{R_n}{c}\right\}exp\left\{j\pi K_r{\left(t-\frac{2R_n}{c}\right)}^2\right\}$$

(2)

where $R_n$ represents the slant range from the nth channel’s phase center to the target, and $\gamma$ represents a complex value related to the target backscattering coefficient. After the process of 2D imaging, the observed value in the azimuth-range unit can be expressed as

$$I_n\left(x_0,r_o\right)=\int \int dxdrf\left(x_0-y,r_0-r\right)\int ds\gamma \left(x,r,s\right)exp\left[-j\frac{4\pi }{\lambda }{R}_n\left(r,s\right)\right]$$

(3)

where $\gamma \left(x,r,s\right)$ represents the complex scattering value at $(x,r,s)$, $R_n\left(r,s\right)$ denotes the slant range value at $(r,s)$, $\lambda$ is the wave length of the signal, and $f\left(x_0-y,r_0-r\right)$ is a two-dimensional point target expansion function. Based on the assumption of the point target, it can be considered as a two-dimensional Dirac function. So, Equation (3) can be written as

$$I_n\left(x_0,r_0\right)=\int ds\gamma \left(x_0,r_0,s\right)exp\left[-j\frac{4\pi }{\lambda }{R}_n\left(r_0,s\right)\right]$$

(4)

2.2. Airborne Array TomoSAR Multi-Channel Amplitude-Phase Errors Analysis

In order to obtain a higher elevation resolution, the airborne array TomoSAR baseline is generally sparse [31], and so the coupling between the antennas is weak. However, due to the different transmitting and receiving antennas of each channel, the equivalent antenna pattern of each channel will also be different. Figure 2 is the antenna pattern schematic diagram.

Assuming that there are two data receiving channels, ch1 and ch2, the corresponding equivalent antenna patterns are $G_1\left(\theta \right)$ and $G_2\left(\theta \right)$. According to the radar equation [32],

$$P_{r,n}=\frac{P_t{G_n}^2\left(\theta \right){\lambda }^2\sigma }{{\left(4\pi \right)}^3{R}^4}$$

(5)

where $P_t,\lambda ,\sigma ,R$ denote transmitted power, wave length, radar cross-section, and slant range. These parameters are same at all channels. However, the equivalent antenna patterns $G_1\left(\theta \right),G_2\left(\theta \right)$ are different. $P_{r,n}$ is the received power related to amplitude. Thus, the amplitude inconsistency of different channels is

$$k=\sqrt{\frac{P_{r,1}}{P_{r,2}}}=\frac{G_1\left(\theta \right)}{G_2\left(\theta \right)}$$

(6)

where k will vary with the angle of view $\theta$ at different slant range units. The antenna patterns are difficult to measure, resulting in amplitude inconsistency errors varying with slant range in multi-channel SLC data.

In addition, the phase center position errors of each channel antenna, caused by the baseline position measurement errors when the carrier is moving, will result in the spatially varying phase errors in the SLC image after the 2D imaging of each channel. The side-looking plane geometry diagram is shown in Figure 3.

$A_n$ is the phase center measurement of the nth channel and ${A_n}^{\prime }$ is the actual position. There is a deviation of $\left(\Delta y,\Delta z\right)$ between $A_n$ and ${A_n}^{\prime }$. $b_{\Vert n}$ and $b_{\perp n}$ are the parallel and vertical components of the position deviation in the line of sight direction. $R_n$ is the slant range from the measured to target, $\theta$ is the downwards angle of visibility, and ${R_n}^{\prime }$ is the actual slant range. The slant range error caused by phase center deviation can be expressed as

$$\Delta R_n={R_n}^{\prime }-R_n=\sqrt{{\left(R_n-b_{\Vert n}\right)}^2+{b_{\perp n}}^2}-R_n$$

(7)

In the far field, Equation (7) can be expressed as

$$\Delta R_n\approx -b_{\Vert n}+\frac{b_{\perp n}}{2R_n}$$

(8)

where

$$\left\{\begin{array}{c}{b}_{\perp n}=\Delta ycos\theta +\Delta zsin\theta \\ b_{\Vert n}=\Delta ysin\theta -\Delta zcos\theta \end{array}\right.$$

(9)

The phase error caused by the slant range error is

$$\Delta {\phi }_n=-\frac{4\pi \Delta R_n}{\lambda }$$

(10)

It can be seen from the above formula that the phase errors of each channel caused by the phase center errors change with the downwards angle of visibility $\theta$. Additionally, because the phase center deviation of each channel is different, there is spatial variability in multi-channel phase inconsistency.

In summary, the multi-channel amplitude-phase inconsistency errors caused by the antenna pattern and the phase center position errors have the property of changing with the slant range. Therefore, the traditional fixed multi-channel amplitude-phase inconsistency errors calibration method cannot be applied to the actual flight data process.

3. Method

In this section, a multi-channel, spatially varying, amplitude-phase inconsistency errors calibration method for airborne array TomoSAR, based on multi-prominent points is proposed. Firstly, the prominent points with high amplitude intensity and scattering stability are selected in the whole scene. Then, the multi-channel amplitude inconsistency at each point is estimated by the amplitude of the point after subspace decomposition. The maximum interference spectrum method is used to estimate the multi-channel phase inconsistency at each point. The next step is to fit the amplitude-phase inconsistency curve of each channel. Finally, we complete the calibration by compensating for the SLC data using fitted curves. The details of the whole process are as follows:

• Registering multi-channel SLC images of the airborne array TomoSAR system;
• Screening the prominent points with obvious amplitude characteristics and scattering stability from the multi-channel SLC images, and recording the position and complex value information of these points in each channel;
• Extracting the multi-channel amplitude-phase inconsistency information at all prominent points by using subspace decomposition and the maximum interference spectrum method;
• Drawing the amplitude-phase inconsistency of each channel with the range and deleting the outliers;
• Fitting the curve of multi-channel amplitude-phase inconsistency varying with range via the polynomial fitting method;
• Calibrate the multi-channel amplitude-phase inconsistency errors in SLC data via the fitting curve.

The overall flow chart of the proposed is shown in Figure 4.

3.1. Prominent Points Selection

The selection of the prominent points needs to combine the characteristics of scattering stability and obvious scattering intensity. The scattering stability can ensure that its backscattering properties in each channel change little with the angle. Additionally, the significant scattering intensity can ensure that it has a higher signal-to-noise ratio, which is more conducive to ensuring the accuracy of amplitude-phase inconsistency estimation. Scattering stability can be calculated according to amplitude dispersion information [33], which is defined as:

$$\mu =\sqrt{\frac{1}{N}\sum _{n=1}^N{\left|I_n\right|}^2-{\left(\frac{1}{N}\sum _{n=1}^N\left|I_n\right|\right)}^2}/\phantom{\sqrt{\frac{1}{N}\sum _{n=1}^N{\left|I_n\right|}^2-{\left(\frac{1}{N}\sum _{n=1}^N\left|I_n\right|\right)}^2}\frac{1}{N}\sum _{n=1}^N\left|I_n\right|}\frac{1}{N}\sum _{n=1}^N\left|I_n\right|$$

(11)

If $\mu$ is less than the given threshold $T_{\mu }$ (the typical value is 0.25), the point is considered to have scattering stability. The complete method of selecting the prominent points is presented in Algorithm 1.

Algorithm 1: Prominent Points Selection.

Input: N channels SLC images Initialization: Preliminary registration of multi-channel images. Perform the following steps at each channel, respectively:      Step 1: Select the first k strongest points at each range bin;      Step 2: Judge whether the point in the $M\times M$ range is the strongest point, or otherwise delete it; Perform the following steps on the points selected above:      Step 3: Screen out the points which can be selected in all channels;      Step 4: Select the prominent points with stable scattering properties according to Formula (9), and the threshold $T_{\mu }$ is set to 0.25;      Step 5: Record the position information of all the prominent points at each channel, and read the complex values of these points via interpolation. Output: Output the position and complex value information of the selected prominent points at each channel.

3.2. Amplitude-Phase Inconsistency Errors Extraction

Subspace decomposition is usually used for the direction of arrival (DOA) estimation problem [34]. The elevation resolution of array TomoSAR is a kind of DOA estimation problem. For the (i,j)th pixel unit in the registered multi-channel 2D SAR images, the first channel is used as the reference channel, and the amplitude and phase of each channel at this point are recorded as ${\Gamma }_n\left(i,j\right)={\rho }_n\left(i,j\right)e^{j{\phi }_n\left(i,j\right)}$. The phase related to the slant range is ${\alpha }_n=exp\left\{-j\frac{4\pi }{\lambda }{R}_n\left(i,j\right)\right\}$. The data vector received by the cross-track array antenna can be written as

$$\mathbf{I}=\left[\begin{array}{c}{I}_1\hfill \\ I_2\hfill \\ ⋮\hfill \\ I_N\hfill \end{array}\right]=\left[\begin{array}{c}1·{\alpha }_1\hfill \\ {\Gamma }_2\left(i,j\right)·{\alpha }_2\hfill \\ ⋮\hfill \\ {\Gamma }_N\left(i,j\right)·{\alpha }_N\hfill \end{array}\right]·{{\gamma }_i}_j+\mathbf{E}$$

(12)

where $I_n$ is the complex value of the $(i,j)$th pixel at the nth channel, $I_n$ is the slant range of the $(i,j)$th pixel at the nth channel, ${{\gamma }_i}_j$ represents the complex value related to the signal power and target backscattering coefficient, and $\mathbf{E}$ is the observation noise. The covariance matrix of the receiving data can be expressed as

$$\mathbf{R}=E\left\{\mathbf{I}{\mathbf{I}}^H\right\}={\left|{{\gamma }_i}_j\right|}^2\Gamma \mathbf{A}\left(\theta \right){\mathbf{A}}^H\left(\theta \right){\Gamma }^H+{\sigma }^2\mathbf{I}$$

(13)

where $\Gamma =\left[1,{\Gamma }_2\left(i,j\right),\dots ,{\Gamma }_N\left(i,j\right)\right]$, $\mathbf{A}\left(\theta \right)={\left[{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_N\right]}^T$ and ${\sigma }^2$ represents the noise power. By performing eigenvalue decomposition of the covariance matrix $\mathbf{R}$, we can obtain the eigenvector corresponding to the maximum eigenvalue, written as $\mathbf{e}=\left[e_1,e_2,\dots ,e_N\right]$. There is the following relationship

$$\Gamma \mathbf{A}\left(\theta \right)=k\mathbf{e}$$

(14)

where k is a unknown constant, and the amplitude inconsistency parameter can be calculated by

$${\rho }_n\left(i,j\right)=\frac{\left|e_n\right|}{\left|e_1\right|}$$

(15)

The subspace decomposition method can effectively reduce the interference of noise and ensure the accuracy of the amplitude extraction at the prominent points. At the same time, the amplitude inconsistency of each channel at the point can be obtained by comparing the amplitude of each channel. The specific steps of the amplitude-phase inconsistency errors extraction method based on the subspace decomposition and the maximum interference spectrum method are presented in Algorithm 2.

Algorithm 2: Amplitude-phase inconsistency errors extraction method based on subspace decomposition and maximum interference spectrum.

Input: The complex vector of a prominent point at different channels $A_0=\left(a_1,a_2,\dots ,a_N\right)$, the multi-channel data after registration of the square area $S_0=\left(s_1,s_2,\dots ,s_N\right)$, selected by taking the prominent point as the center with the size of X and Y, channel m is the reference channel. Initialization: Normalized interference phase diagram ${s_n}^{\prime }=s_n.*s_m/\left|s_n.*s_m\right|$, $S=\left({s_1}^{\prime },{s_2}^{\prime },\dots ,{s_N}^{\prime }\right)$. Calculate amplitude inconsistency:      Step 1: Calculate covariance matrix $\mathbf{R}=A_0{A_0}^H$;      Step 2: Calculate the eigenvector $\mathbf{e}$ corresponding to the maximum eigenvalue via the eigenvalue decomposition of $\mathbf{R}$.;      Step 3: Calculate multi-channel amplitude inconsistency $A_m=\left|\mathbf{e}/{\mathbf{e}}_{\mathbf{m}}\right|$ according to Formula (12)∼(15); Calculate phase inconsistency:      Step 4: Make a Fourier transform of the interferograms of each channel and the reference channel $s=fftshift\left(fft2\left({s_n}^{\prime }\right)\right)$;      Step 5: Search for the point with the largest amplitude in s and record its complex value $\alpha$ and position $(x,y)$ ;      Step 6: Calculate the multi-channel phase inconsistency $A_p=\left(p_1,p_2,\dots ,p_N\right)$, where $p_n$ is calculated by: $p_n=phase\left(\alpha \right)+2\pi ·\left\{\left(X/2+1\right)/X*\left(x-X/2\right)+\left(Y/2+1\right)/Y*\left(x-Y/2\right)\right\}$. Output: Output the amplitude-phase inconsistency $A_m$, $A_p$.

4. Experiments and Results

In this section, the proposed method is applied to the experimental results of the actual airborne flight data. Firstly, the aircraft system and related parameter settings on the experimental data acquisition are introduced. Then, the experimental results, including the results of points selection and multi-channel amplitude-phase inconsistency errors extraction during the experiment, and the comparison results of 3D imaging, verify the effectiveness of the method.

4.1. Experimental System and Data Acquisition

The experiment adopts the airborne array TomoSAR system developed by the Aerospace Information Research Institute, Chinese Academy of Sciences (AIRAS) [35]. The system structure is shown in Figure 5. The array antenna and IMU are installed on the belly. In order to reduce the influence of aircraft vibration, the antenna is connected by a rigid structure. Additionally, the position of the multi-channel phase center is shown in Figure 6. The baseline lengths between ch1 and ch2 and between ch6 and ch7 are about twice those of other adjacent channels.

The key parameters of the system are shown in

Table 1. The parameters of airborne array TomoSAR system.

Parameter	Symbol	Value
Baseline Length	B	2 m
Number of transmitting antennas	N1	2
Number of receiving antennas	N2	5
Flight Height	H	5 km
Flight Velocity	v	100 m/s
Polarization	\	HH
Wave Band	\	12~18 GHz (Ku)
Band Width	$Bw$	600 MHz

below.

The array antenna works in MIMO mode. There are two transceiver antennas and five receiving antennas, forming 10 data transceiver channels. The radar works in the Ku band. The bandwidth of the transmitted signal is 600 M, which promises the high slant range resolution. The azimuth resolution of the system is about 0.5 m, range resolution is about 0.5 m, and the elevation resolution is about 2 m via the super-resolution algorithm. It is worth mentioning that although the position of each antenna is accurately measured on the ground, due to the disturbance of various factors, such as the difference between the geometric center of the antenna and the electrical phase center, there are still different errors during the position of each antenna phase center.

4.2. Experimental Results

This section will take the airborne array TomoSAR flight experimental data from a certain place in Sichuan Province of China as an example, to verify the effectiveness of the proposed method. The selected experimental scene is shown in Figure 7, and the scale is about 3 km × 2 km. There are many buildings and artificial facilities distributed in the selected scene. Therefore, there are enough prominent points that meet the conditions to choose.

4.2.1. Points Selection and Amplitude-Phase Inconsistency Information Extraction

Firstly, we use the above-mentioned Algorithm 1 to select prominent points, and obtain the position and complex value information of all the prominent points at all channels. The distribution of selected points in the 2D SAR images of ch1 and ch10 is shown in Figure 8a,b.

The number of selected points at each channel are shown in

Table 2. Number of selected prominent points.

Channel	1	2	3	4	5	6	7	8	9	10
Initial Points	5258	5475	5285	5581	5528	5552	5586	5312	5406	5328
Final Points	1714	1714	1714	1714	1714	1714	1714	1714	1714	1714

. At the beginning, the prominent points at each channel are selected separately. Additionally, some points in different channels do not meet the selection conditions in all channels because the scattering properties of some targets may vary with the angle of view. So, the number of initial points in each channel is not same. Finally, through screening, the prominent points with good scattering consistency at all channels are selected. Thus, the results of selected points at all channels become the same.

The next step is to extract the amplitude-phase inconsistency information at each prominent point using the above amplitude-phase inconsistency extraction Algorithm 2. The following Figure 9 shows the result of the inconsistency of the amplitude of ch1∼ch10. Blue points represent the result of extracting the amplitude information directly from the prominent point, and red points represent the result of subspace decomposition mentioned above. It can be obviously seen that after subspace decomposition, the distribution of amplitude points is more convergent, indicating that it is less easily disturbed by noise, and the extracted amplitude inconsistency results are more accurate.

The phase inconsistency information is extracted by the maximum interference spectrum method mentioned above. The results of ch1∼ch10 are shown in the Figure 10. It can be seen that due to the large errors, the phase inconsistency results have an obvious phase ambiguity phenomenon.

The results of the multi-channel amplitude-phase inconsistency at the prominent points obtained using the previous process are discretely distributed with the range. It is necessary to fit the changing curve for compensating the amplitude-phase inconsistency errors of the whole scene in the range. The fitting method mainly uses polynomial curve fitting and combines random sample consensus to avoid the interference of error points. Accurate fitting can also be performed by manually selecting feature points. The following Figure 11 and Figure 12, are the results of the polynomial fitting of the amplitude, and the phase inconsistency of each channel with the range. For phase inconsistency, the change is discontinuous due to the phase ambiguity. Therefore, it is necessary to increase the original phase inconsistency by one or two ambiguity periods $\left(2\pi \right)$, and then to fit the complete change curve.

It can be seen from the results that the amplitude inconsistency curve of each channel is quite different, and that the maximum difference at each different range bin is about 5 db. The phase inconsistency curve is generally similar to a power function, which meets with the theoretical situation. After fitting the amplitude-phase inconsistency curve of each channel, the multi-channel amplitude-phase compensation of the SLC data is performed according to the curve fitting results. Then, the spatially varying multi-channel amplitude-phase inconsistency errors calibration can be completed.

4.2.2. Three-Dimensional Imaging Results

In this section, the effectiveness of the proposed multi-channel spatially varying amplitude-phase inconsistency errors calibration method is verified by comparing the 3D imaging results of the actual data. The elevation resolution is obtained using the compressed sensing OMP algorithm [36]. Firstly, a narrow region is selected in the experimental scene, as shown in the red frame area in the Figure 7, covering the complete range direction and a small amount of azimuth length. Then, the fixed amplitude-phase calibration method (method 1) and the proposed method (method 2) in this paper are used to perform the calibration. The fixed amplitude-phase calibration method selects a small flat area, as shown in the yellow frame in the Figure 7, and calculates the amplitude inconsistency according to the average amplitude of each channel in the area. Then, the maximum interference spectrum method is used to calculate the phase inconsistency. Additionally, the calculated amplitude-phase inconsistency is used to compensate for the entire area. Figure 13a,b shows the azimuth cumulative results of the point cloud after the elevation unmasking process of the selected area after calibration using method 1 and method 2, respectively. The horizontal axis is the range bin, and the vertical axis is the elevation unmasking after OMP. The color brightness indicates the number of points. The red line in the figure represents the elevation fluctuation of the ground, and the actual ground should be relatively flat. It can be seen that after calibration by method 1, the reconstruction results have a large degree of fluctuation before and after the ground plane. After using the method 2 in this paper to calibrate the spatially varying amplitude-phase errors of the selected area, the reconstruction results are relatively flat, indicating that the reconstruction position is more accurate, which shows the effectiveness of the method.

In the following, Figure 14 is a comparison of the three-dimensional reconstruction results of two single buildings in the imaging scene. Figure 14a,b are the 2D SAR images of the two buildings that are located in the green frames ① and ② in Figure 7, respectively. Figure 14c,d are the 3D reconstruction point cloud of buildings after the calibration via method 1. Figure 14e,f are the 3D reconstruction point clouds of buildings after the calibration via method 2. It can be seen that there are multiple spurious targets (in red circles) in the reconstruction results after calibration via method 1, while the reconstruction results after the calibration via method 2 exist only one building. There is no ghosting problem after the calibration via method 2. It is worth mentioning that there are still some stray points in the point cloud after calibration, which may be caused by multiple scattering between adjacent building facades. Here are the original point cloud results without any further post-processing.

Figure 15a,b shows the interferograms of adjacent channels and their histograms at the area of building 1 after calibration by method 2, proposed in this paper. It can be seen that the interferograms of adjacent channels after spatially varying amplitude-phase inconsistency errors calibration are roughly consistent. It is worth mentioning that the baselines between ch1 and ch2 and between ch6 and ch7 are longer, so there are some differences in their interferograms from others, but the centers of the histograms are consistent with others. Additionally, the interferogram between ch5 and ch6 is not shown because the baseline between them is zero.

Figure 16a is the optical image of observed large-scale scene, and Figure 16b is the point cloud of the 3D reconstruction after the proposed multi-channel amplitude-phase inconsistency errors calibration method. The color depth represents the height of the point cloud. It can be seen that the reconstructed point cloud information is basically consistent with the actual surface.

In addition, some ground control points (GCPs) are arranged in another experimental scene during the flight experiment. Additionally, the true heights of these points are precisely measured using the Differential Global Position System (DGPS). Repeating the same procedure, the effectiveness of the proposed method can also be confirmed by comparing the reconstructed heights of these GCPs. Figure 17a is the distribution of GCPs. Figure 17b is the comparison results of the GCPs’ heights. It can be seen that the positioning errors are large after compensation via the fixed calibration method, and that the average error is about 20 m. After calibration via the proposed method, the average positioning error of these points is only about 1 m.

5. Discussion

The processing results of the actual flight data of the airborne array TomoSAR verify the effectiveness of the proposed method. Firstly, by comparing the multi-channel amplitude inconsistency results extracted before and after subspace decomposition in Figure 9, it can be seen that the amplitude inconsistency results after subspace decomposition are more convergent. The method can effectively reduce noise interference and improve the accuracy of the amplitude inconsistency estimation results. Secondly, the comparison experiment of the 3D reconstruction results in Figure 13 is carried out by using the narrow region distributed along the range direction. Because the range is far away from forward and backward, there is a large spatial variability. The traditional fixed amplitude-phase inconsistency errors calibration method and the spatially varying amplitude-phase inconsistency errors calibration method proposed in this paper are used before the 3D reconstruction. It can be seen that the reconstructed terrain in Figure 13a after method 1 exhibits great fluctuation, which is quite different from the actual flat terrain. The reconstructed terrain in Figure 13b after compensation using the proposed method 2 is similar to the actual terrain, indicating that this method can effectively calibrate the spatially varying amplitude-phase inconsistency errors and improve the positioning accuracy of 3D reconstruction.The comparisons of the GCPs’ heights in Figure 17 also confirm that the proposed method can improve the positioning accuracy of 3D reconstruction. In addition, the 3D reconstruction of two single buildings in Figure 14 is carried out after the calibration of fixed and spatially varying amplitude-phases, respectively. It can be seen from the results in Figure 14c–f that there are multiple spurious targets in the 3D reconstruction point cloud after fixed amplitude-phase calibration, while there are no spurious targets in the point cloud results after the spatially varying amplitude-phase calibration. In addition, it can be seen from the interferograms of adjacent channels in Figure 15a,b that phase inconsistency errors can been calibrated better when using the proposed method. These experimental results show that the proposed method can more accurately calibrate the amplitude-phase inconsistency errors, effectively suppress the generation of spurious targets in the 3D reconstruction results, and improve the accuracy of the 3D reconstruction results. As mentioned above, the proposed method can effectively calibrate the spatially varying amplitude-phase inconsistency errors in multi-channel SLC data, thereby improving the relative positioning accuracy of TomoSAR 3D reconstruction, and reducing the generation of spurious targets, which is significant for the 3D reconstruction of large-scale scenes. Although the proposed method has the ability to calibrate the spatially varying amplitude-phase errors, there are still some tasks to be completed. Firstly, the proposed method only considers the spatial variation in the range direction, and the spatial variation in the azimuth direction cannot be ignored when the motion errors are large. Subsequent research needs to consider the spatial variation of the azimuth. In addition, this method needs to use prominent points, so it is more suitable for urban areas. The calibration method in the natural area with less prominent points needs further study.

6. Conclusions

The airborne array TomoSAR system has a broad application prospect in the field of 3D reconstruction due to its unique advantages. However, because of the disturbance of phase center errors, antenna pattern inconsistency, and other factors, there are amplitude-phase inconsistency errors in the SLC data after TomoSAR multi-channel 2D imaging. These errors degrade the 3D imaging results, especially the elevation reconstruction results, resulting in a series of problems such as spurious targets and a decline in reconstruction accuracy. In addition, due to the diversity and instability of error sources, the amplitude-phase inconsistency errors will vary with the range. Thus, the traditional internal calibration method can solve the problem. Additionally, the commonly used external calibration methods need to lay a large number of calibrators, which is difficult to achieve in some districts. Therefore, in view of the above problems, this paper proposes a multi-channel, spatially varying amplitude-phase inconsistency errors calibration method based on prominent points by using the subspace decomposition method and the maximum interference spectrum method. Firstly, the prominent points are automatically selected in the multi-channel SLC data, and then the multi-channel amplitude-phase inconsistency information at each point are extracted using the subspace decomposition method and the maximum interference spectrum method, respectively. Finally, the errors are compensated by the curve of the amplitude-phase inconsistency of each channel with the range after the curve fitting. Real airborne flight data experiments show that compared with the method of only fixed amplitude-phase inconsistency calibration, the proposed method can effectively improve the position accuracy of 3D reconstruction and suppress the generation of spurious targets. The results show that this method has broad prospects in the application of TomoSAR large-scale scene 3D imaging. In future research, we will continue to focus on the multi-channel amplitude-phase inconsistency errors calibration method, with wider application scenarios and higher calibration accuracies.