Air Moving-Target Detection Based on Sub-Aperture Segmentation and GoDec+ Decomposition with Spaceborne SAR Time-Series Imagery

Abstract

Air moving-target detection is crucial for national defense, civil aviation, and airspace supervision. Spaceborne synthetic aperture radar (SAR) provides high-resolution, continuous observations for this task, but faces challenges including target attitude variation-induced weak signals and Doppler defocusing from targets’ high-speed motion, which hinder target-background separation. To address this, we propose a novel method combining sub-aperture segmentation with GoDec+ low-rank decomposition to enhance signal-to-noise ratio and suppress defocusing. Critically, ADS-B flight data is integrated as ground truth for spatio-temporal validation. Experiments using Sentinel-1 SM mode SLC imagery across farmland, forest, and mountainous regions confirm the method’s effectiveness and robustness in real airspace scenarios.

1. Introduction

With the increasing demand for efficient and accurate moving-target detection in the fields of national defense security, civil aviation management, and airspace supervision, synthetic aperture radar has become an important technical means for moving-target detection due to its all-weather, high-resolution, and continuous observation capabilities. In particular, the spaceborne SAR system—characterized by wide-area coverage and regular revisit intervals—can continuously monitor large-scale areas and generate multi-temporal imagery, providing valuable data resources for capturing air moving targets. As a critical moving target in airspace, the real-time detection of aircraft is not only of strategic significance for national defense early-warning but also plays an essential role in ensuring civil aviation safety and effective airspace management.

Previously, the researches on using SAR for moving-target detection focuses on ground and maritime moving-target detection. In the field of moving-target detection, Ground Moving-Target Indication (GMTI) technology has formed several typical technical approaches. (1) Clutter suppression method based on Displaced Phase Center Antenna (DPCA). Wang et al. [1] achieved mainlobe clutter suppression by dual-channel phase center space offset, which laid a theoretical foundation for space-based radar moving-target detection. (2) Space-Time Adaptive Processing (STAP) technology. Melvin et al. [2] developed a unified theoretical model that provides mathematical support for multi-dimensional joint-domain filtering, but its high computational complexity limits its application on spaceborne platforms. (3) Along-Track Interferometry (ATI) detection framework. Livingstone et al. [3] and Chiu et al. [4] verified the feasibility of ATI in GMTI by experimental and statistical analysis.

However, these approaches are primarily designed for ground moving targets, and there are significant limitations when they are directly applied to air moving-target detection. On the one hand, the high speed of aerial targets results in larger Doppler frequency shifts, thereby substantially degrading the performance of conventional clutter suppression methods. On the other hand, the inherently weak scattering characteristics of aerial targets make their SNR much lower than those of ground targets, which further increases the difficulty of detection. To solve these problems, Yin et al. [5] proposed a sub-band dual-frequency conjugate processing method, combined with time-frequency analysis to achieve the detection and imaging of air moving targets under low signal-to-noise ratio. Based on dual-channel DPCA and adjacent correlation method, Li et al. [6] achieved low SNR air moving-target detection by improving Keystone transform and CFAR detector. Furthermore, an aerial high maneuvering target detection method suitable for spaceborne platforms is proposed [7], which uses the variable-scale Wigner–Ville distribution transform to coherently accumulate non-uniform moving targets. However, these methods still remain in the simulation experiment stage and lack of validation in real airspace environment.

Based on the above challenges and requirements, a new detection framework is proposed. The framework is based on spaceborne SAR time-series imagery, combined with sub-aperture segmentation and GoDec+ low-rank matrix decomposition for air moving-target detection, and complemented by ADS-B flight data for validation. The time-series imagery can effectively distinguish the dynamic difference between the moving target and the stationary background by multiple observations. The framework takes the pre-processed full-aperture SAR image sequence as input. Firstly, each image is segmented into two non-overlapping sub-apertures, effectively mitigating the defocusing effect and enhancing the SNR of moving targets. Subsequently, the GoDec+ algorithm is applied to decompose the sub-aperture image into a low-rank background component exhibiting spatio-temporal correlation and a sparse component representing motion uncertainty, thereby achieving a clear separation between target and background. Based on this decomposition, a CA-CFAR detector is constructed to perform target detection on the sparse component. Finally, fusing multi-temporal and single-temporal sub-aperture detection results improves the robustness of the output. For validation, by matching the detection output with ADS-B real-time flight data, the authenticity of the air moving targets is validated, ensuring the reliability and accuracy of the proposed method in practical applications.

The rest of this paper is organized as follows. Section 2 introduces the datasets and experimental arrangement. Section 3 describes the implementation process and the theoretical foundations in detail. Section 4 implements a simulation experiment of spaceborne SAR air moving target, which provides experimental support for the subsequent result validation using real spaceborne SAR data. Section 5 shows the experimental results, and validates the results with the ADS-B truth data. Section 6 provides a discussion of the experimental results. Finally, Section 7 concludes the paper and outlines potential directions for future research.

2. Datasets and Experimental Arrangement

2.1. Sentinel-1 SM Mode SLC Data

In this study, SLC imagery acquired by the ESA Sentinel-1A satellite in SM imaging mode is selected as the experimental data. Equipped with a C-band SAR, Sentinel-1A offers all-weather, all-day earth observation capabilities, enabling the stable acquisition of high-resolution images under various meteorological conditions [8]. In addition, the Sentinel-1A satellite has a 12-day revisit period, which can provide temporally continuous observation data, and provide reliable data support for continuous tracking and spatio-temporal change analysis of moving targets [9,10].

The SM mode, one of the primary imaging modes of Sentinel-1A, employs a fixed incidence angle and a relatively narrow swath to ensure high spatial resolution and minimal geometric distortion. The SLC imagery preserves both amplitude and phase information of the radar echo signal, and its single-look processing effectively avoids the loss of target scattering characteristics may caused by multi-looking, thereby achieving high spatial resolution in both the range and azimuth directions [11,12]. Furthermore, the selection of SLC data is critical for the sub-aperture segmentation strategy employed in this study. The complex phase history within SLC data provides the necessary, uncompressed Doppler spectrum information, which is the fundamental requirement for precisely segmenting the full aperture into sub-apertures in the Doppler domain. This sub-aperture segmentation is essential to mitigate the severe defocusing effect induced by the high-speed motion of aerial targets. By reducing the effective coherent integration time per sub-aperture, the target’s energy becomes more concentrated, leading to a significant improvement in its SNR within the sub-aperture images. This enhanced SNR is a prerequisite for the effective detection of often weak air moving targets against complex backgrounds in the subsequent processing steps.

2.2. ADS-B Real-Time Flight Data

In this study, the truth validation data were acquired using the ADS-B system, which provides real-time broadcasts of aircraft flight status information. Specifically, during the SAR imaging period, flight data were recorded in real time and stored using timestamp-based nomenclature, resulting in a comprehensive flight dataset encompassing multiple temporal snapshots. Each data entry detailed key parameters for all flights at the corresponding timestamp, including aircraft identification codes, temporal information, latitude, longitude, ground status (i.e., whether landed), velocity, true track, and geometric altitude. These parameters enable the precise determination of each aircraft’s motion state over time.

Table 1. Example of ADS-B data.

ICAO24	Time (UTC)	Lon (°)	Lat (°)	OnGrnd (T/F)	Vel (kt)	Track (°)	Alt (ft)
a665aa	1,728,995,500	−89.8579	32.5007	F	237.49	188.97	12,519.66
a438f1	1,728,995,501	−74.5281	39.5485	F	271.35	52.24	8793.48
a665b9	1,728,995,501	−83.1155	28.9953	F	230.44	356.29	13,639.8
842194	1,728,995,500	140.6411	37.7729	F	209.76	191.89	12,466.32
a34e6a	1,728,995,500	−83.4240	29.3814	F	249.3	150.31	9928.86
a4ab49	1,728,995,501	−81.4450	31.1225	F	235.54	3.51	13,091.16
…	…	…	…	…	…	…

displays a segment of one recorded ADS-B data. Notably, to ensure strict temporal alignment between the ADS-B data and the SAR imaging period, data were continuously collected throughout the entire SAR imaging period rather than at an isolated moment.

This study further spatially superimposes the ADS-B flight trajectories onto the SAR image to generate a projection of the ADS-B flight trajectories on the SAR image. It should be pointed out that although SLC imagery is used in the process of moving-target detection, GRD imagery corresponding to the imaging period and coverage of SLC imagery is selected for trajectory superposition and visualization. This is because the GRD image, after geographic correction and projection transformation [13], provides a more intuitive and easier-to-match geographic coordinate system, which facilitates spatial alignment and visualization by superimposing geographic elements such as aircraft trajectories onto the image. First, the timestamp of the ADS-B data is filtered, and only the aircraft data recorded during the SAR image imaging period is retained. At the same time, according to the spatial coverage of SAR image and the height of satellite, the flight data is filtered in the spatial range, and the aircraft outside the image is eliminated. Figure 1 shows the projection effect of ADS-B flight trajectories on SAR image, providing an intuitive and effective means of validating the moving-target detection results.

The ADS-B-based validation method, leveraging continuously acquired real-time flight information, provides an independent and precise reference for the SAR air moving-target detection results. By employing spatio-temporal matching between the ADS-B data and the SAR detection results, the study can reliably determine whether the moving targets detected in SAR images correspond to the aircraft in flight at those moments. Aircraft status parameters (e.g., ground presence, flight speed, and altitude) further reinforce target identification. To mitigate deviations caused by imaging errors and data latency, a reasonable spatial tolerance range was established. If the target point was consistent with the ADS-B record within the pre-set spatial tolerance bound, the target was deemed highly likely to be a genuine aircraft.

2.3. Experimental Arrangement

To assess the algorithm’s adaptability across different environments, three groups of SAR time-series imagery (see

Table 2. Data parameters and grouping.

Group	Resolution (m × m, Range × Azimuth)	Satellite-Illuminated Area	Major Background	Number of Scenes	Time Range of Imagery
I	3.2 × 4.1	Chicago	Farmland	36	5 January 2024–28 February 2025
II	3.2 × 4.1	Sao Paulo	Forest	32	18 January 2024–24 January 2025
III	3.2 × 4.1	Sao Paulo	Mountains	35	6 January 2024–17 February 2025

) were constructed to cover diverse ground scattering characteristics corresponding to farmland, forest, and mountainous scenes. All datasets were acquired in SM mode with VV polarization and processed as SLC data. Figure 2a–c depict the characteristic ground features for each group: (a) shows a farmland area with regular structural patterns and high textural contrast, (b) shows a forested region with pronounced volume scattering, and (c) shows mountainous terrain characterized by terrain-induced shadowing.

3. Technical Framework and Theoretical Introduction

The technical framework proposed in this study comprises three main modules: image pre-processing, moving-target detection, and result validation, as illustrated in Figure 3. The proposed framework firstly focuses on the precise radiometric calibration and geometric alignment of the raw SAR image sequence, thereby ensuring a stable and consistent input data foundation for subsequent processing. Next, a combined method that integrates sub-aperture segmentation with GoDec+ low-rank matrix decomposition is employed to extract air moving targets. Finally, ADS-B truth data is utilized to objectively validate the detection results, establishing a complete “processing–detection–validation” technical framework.

In the data pre-processing stage, the primary objective is to address the signal distortion caused by radiometric heterogeneity in spaceborne SAR time-series imagery, thereby providing a reliable data foundation for subsequent processing [14]. Effective preprocessing of Sentinel-1 SLC data, crucial for applications like moving-target detection, typically involves standardized steps to handle the complex raw data and ensure consistency [15]. Specifically, the raw SLC complex data is first converted to amplitude. These amplitude images are then transformed into the log-domain, which compresses the dynamic range and enhances the discernibility of weak target signals. Subsequently, mean filtering is applied to suppress speckle noise. In addition, radiometric calibration is performed on time-series images to achieve consistent radiometric intensity, thereby minimizing the effects of varying imaging environments and sensor conditions [16]. Finally, a feature-based registration algorithm is utilized to strictly align the time-series images, ensuring that the same object or target corresponds to the same image coordinate across different temporal acquisitions. The efficiency and robustness of such preprocessing pipelines are active areas of research, especially for large-scale time-series analysis [17].

In the subsequent moving-target detection module, a sub-aperture segmentation strategy is employed to mitigate the defocusing effect of the moving target. Further combined with the GoDec+ low-rank matrix decomposition technique, the sparse target component is effectively separated from the background clutter. Finally, the CFAR detector is applied to the separated sparse component to achieve accurate air moving targets extraction.

In the final result validation module, the detection results of this method are validated and objectively evaluated by matching the extracted air moving targets with the ADS-B real-time flight data.

The following sections provide a detailed explanation of the main theories and key steps.

3.1. Sub-Aperture Segmentation

Sub-aperture segmentation is an SAR signal decomposition method based on Doppler frequency domain processing [18]. Its theoretical foundation stems from the synthetic aperture characteristic of SAR imaging. In the process of obtaining full-aperture images by the SAR system, the information of each pixel is not instantaneously collected, but formed by integrating the echo signals of multiple azimuth angles through a finite synthetic aperture time. These echo signals essentially correspond to different Doppler frequency components, and the sub-aperture segmentation technique uses the decomposition ability of the Doppler domain to divide the full-aperture data to obtain sub-aperture images at different observation angles. These sub-aperture images reflect the scattering characteristics of the target in different observation time windows.

When discussing the Doppler frequency, it is necessary to distinguish the echo frequency modulation characteristics of two types of targets, namely, stationary targets and moving targets [19]. The ’frequency modulation’ here refers to the rate at which the frequency of the echo signal changes over time, which is mainly caused by the Doppler effect. For stationary targets, the Doppler frequency modulation of the echo is mainly determined by the motion parameters of the SAR platform. The expression is:

$$K_{ra}=-{\displaystyle \frac{2v^2}{\lambda R_c}}$$

(1)

where v is the sensor speed, $\lambda$ denotes the radar wavelength, and $R_c$ represents the slant range from the sensor to the target at t = 0.

Since the relative motion characteristics of the stationary target remain unchanged, the echo signal exhibits a consistent frequency modulation characteristic throughout the entire synthetic aperture time, thus exhibiting a highly coherent Doppler response in each sub-aperture image.

For the moving target, in addition to the platform motion, the target’s own motion, including velocity and acceleration, will introduce additional Doppler modulation in the echo signal, and the frequency modulation is expressed as:

$$K_{ta}=-{\displaystyle \frac{2}{\lambda R_c}}\left(v_x^2+v_y^2+x_0{a}_x+y_0{a}_y-2v{v}_x\right)$$

(2)

where $v_x$ and $v_y$ denote the target’s velocity in the azimuth and range directions, $a_x$ and $a_y$ are the acceleration, and $x_0$, $y_0$ represent the target’s initial coordinates. According to the frequency modulation formula, compared with the stationary target, the echo signal of the moving target will introduce additional frequency modulation effect due to the motion of the target itself, resulting in the dynamic change of the Doppler frequency with time. This change causes the echo of the target to produce additional phase modulation in the azimuth direction, and the SAR system usually assumes that the target is in a stationary state during the imaging process [20], so it fails to compensate for the phase change of the moving target, which in turn causes the azimuth defocusing in the full-aperture image and affects the imaging quality.

Therefore, the full-aperture image can be divided into different frequency bands in the Doppler domain, and the signals in each frequency band can be constructed into corresponding sub-aperture images. By shortening the synthesis time of each sub-aperture, the frequency modulation of the moving target in each sub-aperture is approximately constant, which can significantly reduce the phase error accumulation caused by the target motion, thereby reducing the defocusing effect [21].

The implementation of the sub-aperture segmentation process comprises several key steps, including the conversion of the full-aperture image to the Range-Doppler (R-D) domain, estimation and correction of the weighting function, the spectrum segmentation, and the final inverse Fourier transform for spatial domain reconstruction. The processing flow is illustrated in Figure 4.

3.2. GoDec+ Low-Rank Matrix Decomposition

Low-Rank Matrix Decomposition (LRMD) has been widely applied in radar signal processing and related fields. Its fundamental concept is to decompose the observed data into a low-rank background component and a sparse anomaly component, thereby enabling effective target extraction. Traditional LRMD methods, such as Robust Principal Component Analysis (RPCA) [22], typically rely on nuclear norm minimization to constrain the low-rank component and utilize the ${\mathit{\ell }}_1$-norm to model sparse noise. However, these approaches incur high computational costs for large-scale data and exhibit limited robustness when faced with complex noise interference.

To overcome these limitations, the Go Decomposition (GoDec) algorithm was proposed, which employs a greedy algorithm to approximate the solution of the LRMD problem, thus accelerating the computation [23]. In GoDec, the observed matrix X is decomposed as the sum of a low-rank matrix L, a sparse matrix S, and a Gaussian noise matrix G.

$$X=L+S+G,\mathrm{rank}\left(L\right)\le r,\mathrm{card}\left(S\right)\le k$$

(3)

where $\mathrm{rank}\left(L\right)\le r$ constrains the rank of the low-rank component, and $\mathrm{card}\left(S\right)\le k$ limits the number of nonzero entries in the sparse component. GoDec updates the low-rank component via randomized approximate projection and employs Hard Thresholding to update the sparse component, thereby significantly improving efficiency in processing large-scale data. Nonetheless, while the model performs well when the noise is Gaussian and the anomalies are sparse, its robustness is often insufficient under non-Gaussian noise conditions (e.g., Laplacian noise, salt-and-pepper noise) or when severe anomalies (e.g., occlusions) are present.

To address these issues, Guo et al. introduced the GoDec+ algorithm [24]. The core idea of GoDec+ is to incorporate the Maximum Correntropy Criterion (MCC) to replace the traditional ${\mathsf{\ell }}_1$-norm for modeling sparse noise. Specifically, the GoDec+ model modifies the matrix decomposition form as follows:

$$X=L+N,\mathrm{rank}\left(L\right)\le r$$

(4)

where the noise component N subsumes the roles of both S and G from the original GoDec model.

Compared with conventional methods, GoDec+ presents distinct advantages that align well with the demands of processing spaceborne SAR time-series imagery for air moving-target detection. Firstly, its computational efficiency is crucial. Given the large volume of multi-temporal data (e.g., dozens of scenes per group in Table 2), methods relying on repeated full SVD become prohibitively expensive. GoDec+ overcomes this bottleneck by leveraging a combination of half-quadratic (HQ) optimization and the Greedy Bilateral (GreB) update strategy [25] significantly accelerating the decomposition process. Secondly, GoDec+ offers enhanced robustness and parameter flexibility. The integration of the MCC provides a theoretically sound mechanism for suppressing severe anomalies and non-Gaussian noise, which are inherent challenges in complex SAR scenes. Furthermore, GoDec+’s explicit control over the rank of the low-rank background component and the sparsity level of the target component is particularly beneficial. This tunability allows us to effectively adapt the decomposition across diverse background environments encountered in our experiments—from the relatively homogeneous scattering of farmland (Group I) to the complex volume scattering of forests (Group II) and the strong terrain-induced variations in mountainous regions (Group III). By balancing background suppression fidelity with the retention of potentially weak moving-target signals through parameter adjustment, GoDec+ provides a more effective sparse target representation for subsequent CA-CFAR detection compared to less flexible alternatives.

3.3. Moving-Target Detection

To overcome the defocusing caused by the high-speed motion of aerial targets in SAR images and mitigate background interference, this paper proposes two complementary detection strategies based on sub-aperture segmentation: full temporal sub-aperture GoDec+ decomposition and single temporal sub-aperture GoDec+ decomposition. Both strategies are coupled with a CA-CFAR detector, which dynamically sets detection thresholds based on the local background noise average within a sliding window, thereby ensuring a constant false alarm rate under diverse noise conditions [26]. The detection output is fused by the detection results of the two strategies.

• Multi-Temporal Sub-Aperture GoDec+ Decomposition
   In the multi-temporal approach, each pre-processed SAR image is first divided into two non-overlapping sub-apertures along the azimuth direction. The rationale of this partitioning is that, although a finer division of the full-aperture image can shorten the sub-aperture integration time and thereby mitigate the defocusing of moving targets, it inevitably leads to a significant degradation in azimuth resolution. Consequently, to achieve a balance between reducing target defocusing and maintaining adequate azimuth resolution, this study opts to divide the full-aperture image into two non-overlapping sub-apertures. The sub-aperture images are then sequentially arranged to form a sub-aperture image sequence with dimensions M × N × T, where M is the number of azimuth pixels, N is the number of range pixels, and T is the number of temporal frames. This sequence is input to the GoDec+ low-rank matrix decomposition algorithm, which simultaneously extracts the low-rank background and separates the sparse component representing the moving targets. Finally, a sliding window-based CA-CFAR detector is applied to the sparse component, where adaptive thresholding and morphological filtering are utilized to suppress isolated false alarms, thus yielding multi-temporal moving-target detection results.
• Single-Temporal Sub-Aperture GoDec+ Decomposition
   The single-temporal strategy also commences with sub-aperture segmentation. Specifically, each scene is divided along the azimuth into two non-overlapping sub-aperture images, which are treated as independent observation samples. A data matrix of size M × N × 2 is then constructed, and the GoDec+ low-rank decomposition algorithm is employed to separate the background from the target. This process effectively extracts the sparse component containing the features of high-speed moving targets. Subsequently, a CA-CFAR detector is applied to the sparse image for target localization and detection, enabling rapid identification of moving targets within a single scene.
• Detection Results Fusion
   To exploit both the multi-temporal information and the local spatio-temporal characteristics of individual scenes, the detection results from the two strategies are compared and fused. Specifically, a composite decision function is formulated as follows:

  $$D\left\{\mathrm{final}\right\}=D\left\{\mathrm{multi}\right\}\vee D\left\{\mathrm{single}\right\}$$

  (5)

  where $D\left\{\mathrm{multi}\right\}$ and $D\left\{\mathrm{single}\right\}$ represent the detection outcomes from the multi-temporal and single-temporal approaches, respectively, and ∨ denotes the logical ’OR’ operator.

This fusion strategy maximizes the retention of potential moving targets. The multi-temporal approach is effective in capturing the overall characteristics of continuously or periodically moving targets but may exhibit reduced sensitivity to transient or rapidly varying targets. In contrast, the single-temporal approach focuses on local spatio-temporal features, enabling the prompt detection of sudden targets, but it is relatively less robust noise suppression and multi-temporal verification. By complementing each other, these two strategies jointly enhance the robustness and accuracy of the moving-target detection.

4. Simulation Experiments

The purpose of this chapter is to verify the influence of target motion parameters, including velocity and altitude changes, on imaging position offset and defocusing effect through simulation experiments. The experiment is divided into two parts. Firstly, the geometric-motion model of spaceborne SAR moving target is established. Then, based on the model, the point target imaging simulation experiment is conducted to comprehensively analyze the influence of target motion parameters on imaging quality.

4.1. Spaceborne SAR Moving-Target Model

The relative motion relationship between the spaceborne SAR platform and the moving target has multi-dimensional coupling characteristics, and it is necessary to establish a composite model including orbital dynamics and target kinematics. In the imaging simulation experiment, the squint equivalent distance model is often used to reasonably simplify the complex geometric relationship, as illustrated in Figure 5.

In establishing the model, a three-dimensional Cartesian coordinate system $X-Y-Z$ is defined, where the $X-Y$ plane represents the ground plane and the Z-axis is perpendicular to the ground. The radar platform is set to move at the equivalent speed V, $\varphi$ is the equivalent angle of view, and the platform altitude is H. The spatial coordinates of the target point P at the initial time t = 0 is ($x_0,y_0,z_0$), and the corresponding initial slope distance is $R_c$. Considering the motion of the target in the azimuth, range and vertical directions, the velocity components ($v_x,v_y,v_z$) and the acceleration components ($a_x,a_y,a_z$) are introduced respectively.

At time t, the platform is located at ($Vt$, 0, H), and the target’s position is updated to ($x_t,y_t,z_t$). The instantaneous slope distance $R\left(t\right)$ between the platform and the target is given by the three-dimensional Euclidean distance:

$$R\left(t\right)=\sqrt{{(Vt-x_t)}^2+{(0-y_t)}^2+{(H-z_t)}^2}$$

(6)

Substituting the target’s updated coordinates

$$\begin{array}{cc}\hfill x_t& =x_0+v_{x}t+{\displaystyle \frac{1}{2}}{a}_x{t}^2\hfill \end{array}$$

(7a)

$$\begin{array}{cc}\hfill y_t& =y_0+v_{y}t+{\displaystyle \frac{1}{2}}{a}_y{t}^2\hfill \end{array}$$

(7b)

$$\begin{array}{cc}\hfill z_t& =z_0+v_{z}t+{\displaystyle \frac{1}{2}}{a}_z{t}^2\hfill \end{array}$$

(7c)

The slope distance becomes

$$R(t)=\sqrt{{\left[Vt-\left(x_0+v_{x}t+\frac{1}{2}{a}_x{t}^2\right)\right]}^2+{\left(y_0+v_{y}t+\frac{1}{2}{a}_y{t}^2\right)}^2+{\left(H-z_0-v_{z}t-\frac{1}{2}{a}_z{t}^2\right)}^2}$$

(8)

Figure 6 illustrates the geometric model of spaceborne SAR observing the aerial target, intuitively revealing the projection offset phenomenon that occurs when targets possess altitude information. Specifically, for ground target, the imaging position directly corresponds to the vertical projection of the target onto the ground. In contrast, for aerial target with altitude, a spatial separation arises between the actual three-dimensional position projected onto the ground (denoted as $P^{\prime }$) and the projection as it appears in the SAR image (denoted as $P^{″}$). In other words, the target’s altitude introduces an additional projection offset, resulting in a noticeable discrepancy between the target’s imaged and expected positions.

4.2. Simulation of Air Moving Targets

In the section, the influence of target motion parameters on imaging position offset and defocusing effect is further verified by simulation experiments. Based on the simulation parameters of Sentinel-1 satellite (see

Table 3. Simulation parameters of the spaceborne SAR system.

Parameter	Value
Satellite altitude (km)	693
Incident angle (°)	20
Look angle (°)	0
Radar frequency (MHz)	5405
Range sampling rate (MHz)	47
Range bandwidth (MHz)	42
Azimuth bandwidth (Hz)	1398
Wavelength (cm)	5.56
Pulse width (µs)	6
Antenna size (m × m, range × azimuth)	12.3 × 0.821

), the simulation scene of point target with multiple motion modes is constructed, and the Range–Doppler algorithm (RDA) is applied to process the echo data of each point target. In order to effectively separate the influence of each motion component, the experiment adopts the Zero Look Angle observation mode to simplify the geometric relationship and clearly reveal the independent contribution of each motion component to the imaging performance.

In the simulation, different target models are constructed by varying the motion parameters of the point targets. Specifically, the following three types of point targets are set: a stationary target $P1$ as a reference; a target $P2$ with range velocity only; and a target $P3$ with both range and azimuth velocities. Since acceleration typically induces negligible variations in the instantaneous imaging process, a constant velocity model is adopted to reduce simulation complexity and computational burden. In addition, for each target, an elevated variant $P{1}^{\prime }$, $P{2}^{\prime }$, $P{3}^{\prime }$ is generated by introducing altitude variations, thereby enabling validation of the effects of target altitude on projection position offsets. The coordinates of all targets on the $X-Y$ plane are consistent, so as to eliminate initial position factors and only observe the imaging offset caused by motion and altitude changes.

Table 4. Motion parameters of targets.

Target	Velocity (m/s × m/s, Range × Azimuth)	Altitude (m)
$P1$	0 × 0	0
$P2$	250 × 0	0
$P3$	50 × 250	0
$P{1}^{\prime }$	0 × 0	9000
$P{2}^{\prime }$	250 × 0	7500
$P{3}^{\prime }$	50 × 250	6000

lists the motion parameters of each target in detail, in which the vertical motion component is not considered to simplify the analysis dimension, and the influence of plane motion and altitude change is mainly studied.

Figure 7 shows the original echo signals of targets. After processing with the RDA, the imaging results for the targets are obtained, as shown in Figure 8.

In Figure 8, the stationary target $P1$ is focused as a bright spot, while $P{1}^{\prime }$ exhibits a displacement in the range direction relative to the bright spot due to altitude variation. The range moving target $P2$ shows a significant offset in the azimuth direction. In addition to the azimuth offset caused by the range velocity, the altitude variant counterpart $P{2}^{\prime }$ also superimposes the range displacement. The target $P3$, which possesses both range and azimuth velocities, displays azimuth offset accompanied by defocusing effects. When altitude variation is introduced $P{3}^{\prime }$, the displacement and defocusing become more pronounced, indicating that the superposition effect of motion and altitude variation has a greater impact on imaging quality.

In summary, the simulation results demonstrate that range velocity causes azimuth offset, azimuth velocity leads to defocusing, and the change of the target altitude will also cause the offset of the imaging position. These findings not only validate the influence of target motion parameters on the imaging quality of spaceborne SAR systems, but also provide both theoretical and experimental support for subsequent moving-target detection and result validation using real spaceborne SAR data.

5. Experimental Results

In this section, the performance of the proposed algorithm is systematically validated and analyzed using real spaceborne SAR time-series imagery under various typical scenarios. Through multiple experiments, the algorithm’s ability to accurately localize moving targets in complex backgrounds is comprehensively evaluated, and the detection results are verified using ADS-B flight data.

Three typical scenarios are analyzed (see Figure 9):

• Group I (Farmland Scenario): A 250 × 350-pixel ROI cropped from an 19 October 2024 scene;
• Group II (Forest Scenario): A 150 × 250-pixel ROI cropped from a 1 November 2024 scene;
• Group III (Mountainous Scenario): A 500 × 600-pixel ROI cropped from an 8 October 2024 scene.

First, the ROI was segmented along the azimuth direction into two non-overlapping sub-apertures for each experimental group. This sub-aperture segmentation process, shown in Figure 10a–c for Groups I–III respectively, significantly suppresses the defocusing effect in the resulting sub-images.

Subsequently, the GoDec+ decomposition algorithm was applied to each group’s sub-aperture images to separate background components from moving-target signals. The decomposition results are presented in Figure 10d–f, where each subfigure contains two components: the left image represents the low-rank background, while the right image shows the sparse component with enhanced moving targets.

Finally, a CA-CFAR detector was applied to process both the full-temporal and single-temporal sub-aperture decomposition results for each group. The fused detection outputs, shown in Figure 10g–i, comprise binary detection images in their left portions with white pixels indicating target points, and corresponding overlays on original imagery in the right portions that enhance visualization of moving-target distributions.

To validate the detection result, the detected targets were spatially overlaid on the ADS-B flight trajectory, as shown in Figure 11. In the overlay, the long line represents the vertical ground projection of the flight trajectory, the red dot indicates the detected target positions, and the green dot indicates the projection position of the flight trajectory at the central moment of image acquisition.

After removing ground-parked aircraft and flights outside the SAR imaging coverage through spatio-temporal filtering, the valid target counts and detection results are summarized in

Table 5. Detection performance statistics.

Group	SAR Imaging Date	ADS-B Targets	Valid Targets	Detection Targets	Detection Rate
I	19 October 2024	15	12	8	0.67
II	1 November 2024	14	10	7	0.7
III	8 October 2024	17	15	9	0.6
Mean	-	15.3	12.3	8	0.65

. For the Group I SAR image acquired on 19 October 2024, 8 of 12 valid targets were successfully detected. For the Group II acquisition dated 1 November 2024, 7 detections were achieved from 10 valid targets. For the Group III image from 8 October 2024, 9 targets were identified among 15 valid aircraft. This yields an average detection rate of 65% across all groups.

6. Discussion

The role of SNR is to measure the intensity of the moving-target signal relative to the background noise. High SNR means that the moving target is easier to be accurately identified and measured. In this experiment, the target region is defined as a fixed 5 × 5 pixel window centered on the detected peak energy point of the target. To prevent target energy interference with background estimation, a 20-pixel-wide buffer zone is established immediately surrounding this target region. Beyond this buffer zone, the background region is selected as a uniform 20 × 20 pixel area where strong scatterers are eliminated through morphological operations. This spatial configuration effectively isolates background contamination while enabling accurate characterization of the target’s net signal power relative to the purified background noise level. The final SNR is computed based on the ratio of the net signal power in the target region to the noise level characterized from the background region. The specific formula is

$$\mathrm{SNR}=10{log}_{10}\left({\displaystyle \frac{P_{\mathrm{s}}-P_{\mathrm{b}}}{P_{\mathrm{n}}}}\right)$$

(9)

where $P_{\mathrm{s}}$ denotes the total power in the target region, $P_{\mathrm{b}}$ represents the background power component within the target region, and $P_{\mathrm{n}}$ signifies the noise power in the pure background region. Figure 12 shows the SNR of the moving target in full-aperture image and the corresponding sub-aperture image. It can be seen that by dividing the full-aperture image into sub-aperture images, the defocusing effect is effectively reduced and the SNR of the moving target is improved.

By combining sub-aperture segmentation with GoDec+ decomposition, the moving-target signal is effectively separated from the complex background. Moreover, the long smearing trace in SAR image extends along the azimuth direction, indicating that the target has a significant azimuth velocity, which is consistent with the ADS-B flight trajectory. In addition, the spatial distribution of the detected target shows congruence with the actual aircraft, although there is a distance between them. That is because the flight trajectory reflects the vertical ground projection position, and the additional offset caused by the altitude of the aircraft is not corrected, there is still a spatial distance between the detection result and the flight projection, so the deviation is theoretically reasonable.

Given the limited dataset size, comprehensive large-scale statistical testing was precluded. The analysis focused on a representative SAR image acquired from Group I on 19 October 2024, exhibiting characteristic mixed terrain of farmland, woodland, and anthropogenic structures. Spatio-temporal filtering confirmed 12 valid aerial targets (trajectory projections in Figure 13), with 8 targets successfully detected (annotated green) and 4 missed detections (annotated red). The detection rate was calculated as

$$\mathrm{DR}={\displaystyle \frac{N_{detected}}{N_{total}}}\times 100\%$$

(10)

where $N_{detected}$ denotes successfully detected targets and $N_{total}$ the total valid targets. This yielded a DR of 67%. The observed missed detections are attributed to radar cross-section (RCS) attenuation associated with nose or tail orientations relative to the radar, compounded by imaging position displacement exceeding the detection window due to high radial velocities.

The false positive case analysis reveals that anomalous signals in the original scene, when processed through GoDec+ decomposition, produce background and target components (see Figure 14) attributed to a ground-moving train. Subsequent CFAR detection results (see Figure 15) further validate this conclusion through spatially parallel alignment between the target trajectory and railway track, with azimuthal displacement of the train caused by radial velocity, ultimately demonstrating detection failure scenarios triggered by ground moving targets in high-clutter artificial environments.

These experimental findings, combined with the statistical data in Table 5, objectively demonstrate the method’s practical utility for routine civil aviation monitoring while clarifying applicability boundaries regarding RCS sensitivity and localization displacement of high-radial-velocity targets.

7. Conclusions

This paper proposes an air moving-target detection method that integrates sub-aperture segmentation with GoDec+ low-rank matrix decomposition, and validates the results using real-time ADS-B flight data. The experimental results demonstrate that the proposed algorithm effectively detects and localizes air moving targets in three typical scenarios, including farmland, forest, and mountainous region. While the proposed method demonstrates robust detection across diverse terrains, its effectiveness is significantly influenced by aircraft attitude, showing reliable detection when the fuselage faces the radar but reduced sensitivity during nose or tail orientations. Additionally, the framework does not yet leverage multi-channel clutter suppression capabilities available in advanced SAR systems. Future work will integrate multi-channel techniques (e.g., DPCA) with our sub-aperture GoDec+ framework to further improve clutter suppression and detection sensitivity, particularly for low SNR targets in complex clutter environments.