Automatic Correction of Time-Varying Orbit Errors for Single-Baseline Single-Polarization InSAR Data Based on Block Adjustment Model

Abstract

Orbit error is one of the primary error sources of interferometric synthetic aperture radar (InSAR) and differential InSAR (D-InSAR) measurements, arising from inaccurate orbit determination of SAR platforms. Typically, orbit error in the interferogram can be estimated using polynomial models. However, correcting for orbit errors with significant time-varying characteristics presents two main challenges: (1) the complexity and variability of the azimuth time-varying orbit errors make it difficult to accurately model them using a set of polynomial coefficients; (2) existing patch-based polynomial models rely on empirical segmentation and overlook the time-varying characteristics, resulting in residual orbital error phase. To overcome these problems, this study proposes an automated block adjustment framework for estimating time-varying orbit errors, incorporating the following innovations: (1) the differential interferogram is divided into several blocks along the azimuth direction to model orbit error separately; (2) automated segmentation is achieved by extracting morphological features (i.e., peaks and troughs) from the azimuthal profile; (3) a block adjustment method combining control points and connection points is proposed to determine the model coefficients of each block for the orbital error phase estimation. The feasibility of the proposed method was verified by repeat-pass L-band spaceborne and P-band airborne InSAR data, and finally, the InSAR digital elevation model (DEM) was generated for performance evaluation. Compared with the high-precision light detection and ranging (LiDAR) elevation, the root mean square error (RMSE) of InSAR DEM was reduced from 18.27 m to 7.04 m in the spaceborne dataset and from 7.83~14.97 m to 3.36~6.02 m in the airborne dataset. Then, further analysis demonstrated that the proposed method outperforms existing algorithms under single-baseline and single-polarization conditions. Moreover, the proposed method is applicable to both spaceborne and airborne InSAR data, demonstrating strong versatility and potential for broader applications.

1. Introduction

Synthetic aperture radar (SAR) is a promising space observation technology [1]. Over the last decades, interferometric SAR (InSAR) and differential InSAR (D-InSAR) have advanced rapidly and are widely used in various fields, including high-precision digital elevation model (DEM) mapping [2,3], surface deformation monitoring [4,5], natural resource management, and topography change detection, all of which rely on InSAR phase information [6,7,8]. However, in practice, orbital errors often affect the accurate use of InSAR phase information in the aforementioned applications [9].

In airborne SAR platforms, atmospheric turbulence and navigation positioning errors can lead to changes in the relative positions of coherent antennas over time [10,11,12], making it difficult to estimate the accurate baseline at each moment or causing the estimated baseline error to vary unevenly, commonly referred to as time-varying orbit error. Although the current airborne SAR system can mitigate this error through motion compensation supported by the Global Navigation Satellite System (GNSS) and Inertial Measurement Unit (IMU), residual motion errors persist due to the limitations in real-time GNSS/IMU accuracy and compensation capabilities [12]. For spaceborne SAR platforms, satellites generally follow smoother orbital trajectories and possess precise orbital data, as seen with DLR’s TanDEM-X and ESA’s Sentinel-1, which exhibit almost negligible time-varying orbital errors. However, satellites like Japan’s Japanese Earth Resources Satellite-1 (JERS-1) and Advanced Land Observing Satellite (ALOS-1), Canada’s RADARSAT, and China’s GaoFen3 and LuTan-1 often cannot acquire accurate orbital positions in real time or do not provide precise orbit data [9,13,14,15,16]. Furthermore, in repeat-pass mode, the orbit errors of the two SAR images are independent and challenging to mitigate during the interferometric process, resulting in significant orbital error phases in the interferogram [17]. Therefore, developing robust models to estimate and correct time-varying orbit errors is crucial for the broader application of InSAR data.

Several methods have been proposed to achieve this goal, which can be categorized into four main types. The first category relies on an autofocus algorithm and does not require interferometric processing or external data [18], but it necessitates SAR images with a high signal-to-noise ratio (SNR). The second category is to process InSAR data using time-frequency analysis. Some researchers have employed multi-squint processing and utilized the differential phase between different sub-look interferograms for modeling to correct the orbital error phase [10,19,20]. Additionally, methods based on Fast Fourier Transform (FFT) or wavelet decomposition allow the interferometric phase to be converted into the frequency domain to estimate the orbital error phase [17,21,22]. The third category is based on ground control points (GCPs), such as using GNSS elevation and interferometric phase to establish observation equations. Nonlinear least squares are then employed to estimate the InSAR geometric baseline, or orbit errors are directly estimated by minimizing the residuals between InSAR and ground control point elevations [23,24]. However, many areas lack enough GNSS GCPs, leading most existing researchers to extract GCPs from external DEM or spaceborne light detection and ranging (LiDAR) data [25,26], such as the Ice, Cloud, and Land Elevation Satellite-2 (ICESat-2). The fourth category is the widely used rational function model, which typically represents orbit errors as a polynomial function of range and azimuth coordinates [27,28,29,30,31], including linear, high-order polynomial, and polynomial related to elevation.

The advantage of the first and second categories above is that they can remove time-varying orbital error phases without relying on external data. However, their performance is limited by azimuthal displacement and low coherence [10,17,18,26], particularly in forested areas where generalization is challenging. The third and fourth categories are currently the most widely used methods for removing the orbital error phase, assuming that the orbital error phase can be modeled using a physical model or a set of polynomial coefficients. However, modeling the significant time-varying component of orbit error along the track (azimuth) with a single set of polynomial coefficients proves difficult [16,31,32,33]. To overcome these problems, some studies proposed patch-based polynomial models [16]. Unfortunately, the criteria for segmenting blocks are often based on empirical methods, making them difficult to adapt for large-scale automated data processing, and unreasonable segmentation usually leads to a significant residual orbital error phase.

To overcome the challenges of inadequate block division and the difficulty of automating existing patch-based polynomial models, this paper proposes a block adjustment framework based on morphological feature segmentation to estimate the time-varying orbital error phase:

(1)

The differential interferogram is segmented into several overlapping blocks along the azimuth direction, reducing the spatial coverage of each block and enabling more accurate estimation of the orbital error phases using the polynomial model.

(2)

Automatic segmentation is achieved by analyzing the trends in the differential phase. In this paper, we study the changing trends of the time-varying orbital error phase in both the azimuth and range directions, using the peaks and troughs of the azimuthal differential phase profile as block boundaries.

(3)

The orbital error phase for each block is estimated simultaneously using a block adjustment model. Specifically, control points and connection points are first selected to establish the block adjustment model. Then, the model coefficients for each block are determined using an iterative weighted least squares algorithm. Finally, the orbital error phase for the entire InSAR interferogram is obtained through mosaicking.

The rest of this paper is organized as follows. Section 2 details the data preprocessing and the proposed block adjustment model for automatically estimating the orbital error phase. Section 3 describes the study area and datasets used in this study. In Section 4, we validate and assess the feasibility of the proposed method using spaceborne and airborne InSAR data characterized by time-varying orbit errors. Section 5 offers further discussion, while Section 6 presents the conclusions.

2. Methods

As previously mentioned, the core objective of this study is to effectively utilize the polynomial model for estimating the time-varying orbital error phase. As shown in Figure 1, the framework of the proposed method consists of three parts.

The first part involves automatic registration, multi-look, and interferometric processing of SAR data acquired at different times, along with estimating baseline parameters to remove the flat-earth phase from the interferogram. Subsequently, the external DEM is geocoded into the radar coordinate system, and the baseline is used to simulate the topography phase for subsequent differential interferometry. The second part focuses on dividing the differential interferogram into blocks by extracting the peaks and troughs of the azimuth profile, and the control points and connection points are selected based on specific criteria to establish a block adjustment model. The third part employs the iterative weighted least squares algorithm to solve the model coefficients, and then mosaics each block’s orbital error phase to obtain the total orbit error phase of the entire interferogram. These parts are elaborated in detail in the following sections.

2.1. Polynomial-Based Orbit Error Model

For InSAR data acquired in repeat-pass mode, the interferometric phase ${\varphi }_{int}$ can be expressed as:

$${\varphi }_{int}={\varphi }_{\mathit{flat}}+{\varphi }_{\mathit{top}}+{\varphi }_{def}+{\varphi }_{atm}+{\varphi }_{noise}$$

(1)

where ${\varphi }_{\mathit{flat}}$ is the flat-earth phase, ${\varphi }_{\mathit{top}}$ is the topographic phase, ${\varphi }_{def}$ is the deformation phase, ${\varphi }_{atm}$ is the atmospheric delay phase, and ${\varphi }_{noise}$ is the thermal noise phase. For InSAR and D-InSAR applications, it is first necessary to remove ${\varphi }_{\mathit{flat}}$ using a formula based on baseline parameters (length and tilt angle) and InSAR geometric parameters. However, inaccuracies in orbital vector measurements and limited sampling of known SAR geometric parameters often result in interferograms with residual flat-earth phases [2,26]. For D-InSAR, an external DEM is typically used for differential processing to remove ${\varphi }_{\mathit{top}}$. However, due to the low accuracy of SAR orbit determination and baseline estimation errors, significant orbit errors may still occur during topographic phase removal [26,34]. After removing the flat-earth and topographic phases, the derived differential phase can be expressed as:

$${\varphi }_{diff}={\varphi }_{orb}+{\varphi }_{atm}+{\varphi }_{def}+{\varphi }_{noise}$$

(2)

where ${\varphi }_{orb}$ is the time-varying orbital error phase. This study focuses solely on modeling the time-varying orbital error phase, assuming that ${\varphi }_{orb}$ primarily affects the differential phase, while ignoring the effects of atmospheric delay and the deformation phase.

In statistical studies, the orbital error phase is modeled as a function of the relationship between azimuth and range coordinates, typically including linear, nonlinear, or high-order polynomials related to elevation. Among them, a quadratic polynomial with a set of coefficients is often used to model the orbital error phase as follows:

$${\varphi }_{orb}=a_0+a_{1}x+a_{2}y+a_{3}xy+a_4{x}^2+a_5{y}^2$$

(3)

where $a_i$ (i = 0, 1, ···, 5) is polynomial model coefficient, $x$ is the range coordinate, and $y$ is the azimuth coordinate. In most cases, the orbital error phase can be adequately estimated from the differential interferogram using a small number of control points (more than six). However, in case of significant time-varying orbit errors, the performance of existing polynomial models is relatively unsatisfactory, and relying solely on a set of coefficients may not accurately characterize the changes in the orbital error phase.

2.2. Block Adjustment Model

2.2.1. Automatically Divide Interferogram into Blocks

Aiming at the problems that a set of model coefficients cannot realize time-varying orbital error phase modeling and that the existing patch-based methods lack automation, this work analyzed the change patterns of the time-varying orbital error phase in azimuth and range directions, and proposed an automated segmentation method based on the azimuth profile of the differential interferogram. Figure 2a shows a differential interferogram derived from an L-band spaceborne SAR system, which has an obvious time-varying orbital error phase and an obvious non-uniform trend in both azimuth and range directions. To further analyze the changing pattern of the orbital error phase, differential phase profiles were plotted along the azimuth direction at the near and far ends of the differential interferogram, as shown in Figure 2b. The two profiles are nearly parallel, and the variation of the orbital error phase in the range direction at the same imaging time can be characterized by a linear or quadratic function, consistent with existing studies. However, in the azimuth direction, the orbital error phase exhibits a wave-like pattern, which is difficult to fit with a conventional polynomial function. Therefore, this study proposes a profile-based automatic segmentation method that divides complex orbital error phases into multiple simpler components. As illustrated by the dotted line in Figure 2b, by extracting the peaks and troughs of the profile for segmentation, the changes in the orbital error phase between the peaks and troughs follow a regular pattern and can be characterized by a simple polynomial function. Specifically, the differential interferogram is divided into n blocks using the coordinates indicated by the dotted lines, and each block is modeled using a set of polynomial coefficients, as depicted in Figure 2c.

To achieve accurate block segmentation, a second-order difference algorithm was employed to extract the peaks and troughs [35,36]. Specifically, first-order $dz\left(y\right)$ and second-order $d^{2}z\left(y\right)$ differential are performed on the azimuth profile as follows:

$$dz\left(y\right)=z\left(y\right)-z\left(y-1\right)$$

(4)

$$d^{2}z\left(y\right)=z\left(y\right)-2\mathrm{z}\left(y-1\right)+\mathrm{z}\left(y-2\right)$$

(5)

where $y$ represents the azimuth profile coordinate and $z$ represents the differential phase corresponding to the coordinate $y$. On this basis, the peaks and troughs are identified based on the following criteria:

$$\{\begin{array}{l}if \mathrm{d}\mathrm{z}\left(\mathrm{y}\right)>0 \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{d}\mathrm{z}\left(\mathrm{y}+1\right)<0 \mathrm{a}\mathrm{n}\mathrm{d} {\mathrm{d}}^2\mathrm{z}\left(\mathrm{y}\right)<0 \to \mathrm{y}:\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k} \\ if \mathrm{d}\mathrm{z}\left(\mathrm{y}\right)<0 \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{d}\mathrm{z}\left(\mathrm{y}+1\right)>0 \mathrm{a}\mathrm{n}\mathrm{d} {\mathrm{d}}^2\mathrm{z}\left(\mathrm{y}\right)>0 \to \mathrm{y}:\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h}\end{array}$$

(6)

Furthermore, it is important to note that although the time-varying orbital error phase predominates in the differential interferogram, other phases such as atmospheric delay, deformation, and noise are also present. To accurately extract the changing trend of the orbital error phase and segment it into blocks, one-dimensional filtering of the profile is necessary. On this basis, it is also necessary to obtain multiple profiles (at least two are recommended) along the azimuth direction for analysis, and to select the same peak and trough coordinates observed in different profiles as the boundaries of the blocks.

2.2.2. Build Block Adjustment Model

Based on the above analysis, the changing trend of the orbital error phase in the range direction is relatively straightforward to model, and in the azimuth direction it can be modeled using a combination of multiple polynomials. In this work, the orbital error phase for each block is modeled using a cubic polynomial in azimuth and a quadratic polynomial in range. As shown in Figure 2a, although the trend of the orbital error phase in the range direction appears linear, statistical analysis reveals that quadratic polynomials exhibit better robustness and adaptability. In practice, the choice of polynomial degree can be adjusted based on the complexity of the orbital error phase. As shown in Figure 2d, for arbitrary block j, we extract m stable control points to establish a functional model:

$$\begin{array}{c}{L}_{GCP}=B_{GCP}{X}_{GCP}+\epsilon \\ \left[\begin{array}{c}{\varphi }_{\mathit{orb},1}\\ {\varphi }_{\mathit{orb},2}\\ ⋮\\ {\varphi }_{\mathit{orb},m}\end{array}\right]=\left[\begin{array}{ccccccc}1& x_1& y_1& x_1{y}_1& x_1^2& y_1^2& y_1^3\\ 1& x_2& y_2& x_2{y}_2& x_2^2& y_2^2& y_2^3\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ 1& x_m& y_m& x_m{y}_m& x_m^2& y_m^2& y_m^3\end{array}\right]\left[\begin{array}{c}{a}_0\\ a_2\\ ⋮\\ a_6\end{array}\right]+\left[\begin{array}{c}{\epsilon }_1\\ {\epsilon }_2\\ ⋮\\ {\epsilon }_m\end{array}\right]\end{array}$$

(7)

where $L_{GCP}$(${\varphi }_{orb,1}···{\varphi }_{orb,m}$) represents the orbital error phase of each control point, $B_{GCP}$ is the design matrix composed of the azimuth and range coordinates, $X_{GCP}$($a_0···a_6$) is the coefficient matrix, and $\epsilon$(${\epsilon }_1···{\epsilon }_m$) is the fitting residual. The functional model characterizes the relationship between polynomial coefficients and the orbital error phase. Furthermore, for any two blocks j and k with overlapping areas, a critical geometric constraint of this adjustment problem is that the differential phases after removing the orbital error phase should be identical, which can be expressed as follows:

$$\left[{\varphi }_{diff,j }- {\varphi }_{orb,j }\right]-\left[{\varphi }_{diff,k } - {\varphi }_{orb,k}\right]=0$$

(8)

For overlapping areas within the same interferogram, since the differential phases are identical, Equation (8) can be simplified as follows:

$${\varphi }_{orb,j }-{\varphi }_{orb,k}=0$$

(9)

As shown in Figure 2d, for any two blocks j and k, a specific overlapping area is defined to constrain the estimation of unknown coefficients between the different blocks. Therefore, by combining the model established with m control points and the constraints from n connection points, a block adjustment function model is established as follows:

$$V=BX-L$$

(10)

where $V$ represents the residual error vector of control points and connection points, $B$ is the design matrix, and $X$ is the unknown coefficients of all blocks. Specifically, for any block j and k, B, X, and L are expressed as follows:

$$B=\left[\begin{array}{cc}{B}_{GCP,j}& 0\\ B_{TP,j}& -B_{TP,k}\\ 0& B_{GCP,k}\end{array}\right], L=\left[\begin{array}{c}{\varphi }_{diff,j }\\ 0\\ {\varphi }_{diff,k }\end{array}\right]$$

(11)

$$X={\left[\begin{array}{cccccc}{a}_{0,j}& \cdots & a_{6,j}& a_{0,k}& \cdots & a_{6,k}\end{array}\right]}^T$$

(12)

where ${( )}^T$ represents the matrix transpose, and $B_{GCP}$ and $B_{TP}$ represent the design matrix of control points and connection points, respectively.

2.2.3. Time-Varying Orbital Error Phase Estimation

For coefficient estimation, the polynomial model for each block contains seven unknown coefficients, thus at least seven control points are required. To estimate the orbital error phase, this study extracts control points from an external DEM based on the following criteria: (1) Selection of pixels in non-forest areas. In forest areas, the interferometric phase tends to be noisier due to volume scattering and spatiotemporal decorrelation. Additionally, due to the penetrating ability of SAR signals, InSAR signals with different wavelengths have different phase center heights in forest scenes. (2) Limiting the slope to less than 10 degrees. In areas with steep slopes, InSAR technology is prone to geometric distortion. Furthermore, external DEMs generally have higher accuracy in gently sloped areas, making control points extracted from these areas more reliable. (3) Adaptive selection of pixels with higher interferometric coherence. The coherence reflects the quality of the interferometric phase, with pixels exhibiting higher coherence being less influenced by noise and other potential factors.

For the overlapping areas between adjacent blocks, the azimuth length of the overlap is set to one-tenth of block j, and connection points are evenly selected at fixed intervals, as shown in Figure 2d. Note that, based on the analysis of Equations (8) and (9), the modeling of connection points is independent of the differential phase, so no additional conditions are needed for their selection. After establishing the block adjustment model using control points and connection points, the model coefficients of the orbital error phases are estimated through an iterative weighted least-squares adjustment, as follows:

$$\widehat{X}={\left(B^{T}PB\right)}^{-1}{B}^{T}PL$$

(13)

where $\widehat{X}$ represents the estimated model coefficients, P represents a diagonal weight matrix, ${( )}^{-1}$ represents the matrix inverse, and ${( )}^T$ represents the matrix transpose. In the iterative process, the robust estimate IGG is used to design and update $P$ [26]:

$$P\left(v_n^b\right)=\{\begin{array}{cc}1& \left|v_n^b\right|<1.5{\sigma }^0\\ {\displaystyle \frac{1}{\left|v_n^b\right|+\eta }}& 1.5{\sigma }^0<\left|v_n^b\right|<2.5{\sigma }^0\\ 0& \left|v_n^b\right|\ge 2.5{\sigma }^0\end{array},{\sigma }^0={\displaystyle \frac{v^b{P}^{b-1}}{n-1}}$$

(14)

where $v_n^b$ represents the correction value of the b-th iteration, $\eta$ is a minimum value set to avoid the denominator being zero ($\eta ={10}^{-12}$), and ${\sigma }^0$ is the standard deviation. The iteration is terminated when the difference between the two calculated model coefficients $\left|X_i-X_{i+1}\right|<\delta$, and $\delta ={10}^{-6}$ is the threshold set in this study. Subsequently, the orbital error phase of each block is estimated separately, followed by mosaicking to obtain the orbital error phase of the entire interferogram.

3. Study Area and Data

This section introduces the two test sites and presents the datasets utilized in this study, including InSAR data, land use land cover (LULC) maps, external DEM, and LiDAR validation data.

3.1. Study Area

Two test sites (Figure 3) characterized by different climate types and terrains were selected to evaluate the effectiveness of the proposed method in estimating the time-varying orbital error phase. The first test site (Figure 3a) is located in Hunan Province, China, which is located inland and has a humid subtropical monsoon climate. The terrain of this test site is undulating, and the overall terrain is mainly hilly, with altitudes ranging from 0 m to 800 m. In addition, the land cover includes towns, water bodies, farmland, and forests. The second test site (Figure 3b) is located in Krycklan, northern Sweden, with a temperate coniferous forest climate, and coniferous and deciduous broadleaf forests primarily cover it. The terrain of the test site is relatively flat, with altitudes ranging between 50 m and 350 m.

3.2. Datasets

3.2.1. Spaceborne and Airborne InSAR Data

This study used spaceborne and airborne InSAR data acquired in repeat-pass mode to verify the effectiveness of the proposed method. The red rectangle in Figure 3a is the coverage of China’s LuTan-1 repeat-pass SAR data, which was acquired on 16 April and 20 April 2023. The LuTan-1 SAR system operates in the L-band and uses two identical satellites in a follow-up mode to achieve 4 days of return mapping [37]. The single-look complex (SLC) SAR image acquired by this satellite exhibits azimuth and the range resolution of the data is 4.78 m and 1.66 m, respectively. Figure 3b shows the coverage of airborne data, which is a P-band multi-baseline interferometric dataset obtained via the airborne E-SAR system on 14 October 2008. This dataset consists of six SAR acquisitions, combined into five interferometric pairs, with a temporal baseline between 34 min and 71 min.

Table 1. Details of the spaceborne and airborne InSAR data acquired under the repeat-pass model used in this study.

Test Sets	SAR Data	Date	Temporal Baseline	Kz	Azimuth × Range (m)	Band
				(rad/m)
Hunan	1 (master)	16 April 2023	4 day	0.016~0.032	4.78 × 1.66	L
	2	20 April 2023
krycklan	0103 (master)	14 October 2008	34~71 min	0.007~0.080	0.70 × 1.50	P
	0101	14 October 2008
	0105	14 October 2008		0.005~0.073	0.70 × 1.50	P
	0107	14 October 2008		0.020~0.135	0.70 × 1.50	P
	0109	14 October 2008		0.040~0.180	0.70 × 1.50	P
	0111	14 October 2008		0.050~0.250	0.70 × 1.50	P

lists the detailed parameters of both the spaceborne and airborne data.

3.2.2. LULC Maps

The land use land cover (LULC) maps utilized in this study are global classification products produced annually by Esri, Impact Observatory, and Microsoft teams. These maps are created using Sentinel-2 images through deep learning models [38], which include nine categories such as water, trees, and crops. At present, land cover categories with a resolution of 10 m from 2017 to 2023 have been continuously produced, achieving a classification accuracy of 85%. This study used classification products obtained in 2022, which are mainly used to mask areas such as forests and waters to avoid affecting orbit error modeling.

3.2.3. External DEM Data

FABDEM (Forest and Buildings removed Copernicus DEM) eliminates the building and tree height bias in the Copernicus GLO 30 DEM, thereby providing a more accurate representation of the true Earth surface elevation in forested areas compared to other global DEMs [39], with an absolute vertical accuracy of 1.12 m in built-up areas and 2.88 m in forested areas. In this study, FABDEM was used to assist in obtaining differential interferograms of the Hunan test site. Additionally, it is utilized to extract control points based on the criteria outlined in Section 2.

3.2.4. LiDAR Verification Data

Since its launch in September 2018, the ICESat-2 satellite has been gathering global LiDAR elevation footprint points and generating land and vegetation height products. As shown by the yellow dots in Figure 3a, we utilized the ICESat-2 surface elevation data acquired in 2023, and subjected it to quality filtering to assess the accuracy of the InSAR DEM after removing orbital error phases from the spaceborne InSAR data. For the Krycklan test site, the Swedish Defense Research Agency acquired a high-precision LiDAR digital terrain model (DTM) during the BIOSAR2008 mission with a resolution of 1 m × 1 m. In this study, the LiDAR DTM was resampled using cubic spline interpolation to make its resolution the same as that of airborne InSAR data, and then was used to verify the effectiveness of the proposed method on airborne P-band SAR data.

4. Results

4.1. Hunan Test Site

After performing the co-registering of SAR images, differential interferometric processing is carried out, including adaptive spectral filtering of the SLC data, multi-look, differential interferogram filtering using Goldstein filtering, and differential interferogram unwrapping using the Minimum Cost Flow (MCF) algorithm [40,41]. Although the SAR data acquisition time interval is 4 days and more than half of the test site is located in forest areas, the resulting interferogram still maintains good quality, with an average coherence of approximately 0.8, as shown in Figure 4a. However, the precise satellite orbit data of this system has a severe lag, and the real-time orbit vector parameters of some data have non-negligible randomness and noise. The time-varying orbit error limits the emergency mapping and monitoring application, highlighting the need for an effective method to eliminate it.

Figure 4b shows the differential interferogram derived by InSAR and the external DEM. It is evident that the orbital error phase dominates the differential phase, exhibiting a noticeable non-uniform trend in both the azimuth and range directions. To further analyze the changing pattern of the time-varying orbit error phase, profiles of the differential interferogram were plotted along the azimuth and range directions, with the positions of the selected profiles shown in Figure 4a. Figure 4c shows the two azimuth profiles after one-dimensional mean filtering, revealing a relatively complex trend and a significant wavy shape. Conversely, the profiles in the range direction show a more uniform change, linearly transitioning from near range to far range, as shown in Figure 4d. Figure 4c shows the extracted peaks and troughs used for segmentation. Subsequently, a block adjustment model was constructed according to the process shown in Figure 1, with the estimated orbital error phase shown in Figure 5b. Overall, the corrected result closely resembles the original differential interferometric phase (Figure 5a), with no noticeable discontinuities between blocks. This demonstrates the effectiveness of the proposed method using connection point constraints. Figure 5c shows the differential interferogram after correcting for the resultant time-varying orbital error phase, and it is evident that the differential interferometric phase has been significantly reduced, with the main phase components consisting of the residual terrain phase and possible atmospheric delay phase.

To further validate the effectiveness of the proposed method, we focus on the accuracy of the DEM retrieved from the interferometric phase after removing the orbital error phase. Due to the significant atmospheric delay phase, the elevation difference between the InSAR DEM and the external terrain product is substantial, as shown in the northeast part of Figure 6b. Afterward, the InSAR DEM was quantitatively validated using the spaceborne ICESat-2 elevation, with the corresponding elevation error histogram shown in Figure 6c. Before correction, the root mean square error (RMSE) of the InSAR DEM was 18.27 m, and the mean absolute error (MAE) was 0.65 m, indicating that time-varying orbital errors severely impacted the InSAR DEM. After correction, the DEM elevation errors fall within a more reasonable interval, with an RMSE of 7.04 m. Furthermore, the standard deviation (STD) after correction showed significant improvement compared to before correction. Note that, since this study did not focus on calibrating the atmospheric delay errors, the MAE of the corrected InSAR DEM increased to 5.41 m. Nonetheless, quantitative and qualitative evaluations still demonstrate that the proposed method can effectively and automatically remove the time-varying orbital error phase in spaceborne InSAR data.

4.2. Krycklan Test Site

In this study, the processing flow for airborne P-band InSAR data is the same as that for spaceborne InSAR data. During the BIOSAR2008 mission, five interferometric pairs were combined, with the master image named 0103. The airborne P-band interferometric data were all acquired under clear weather conditions, with a temporal baseline of only tens of minutes, allowing the atmospheric delay errors in the interferograms to be ignored. To analyze the changing pattern of time-varying orbit error, the differential interferometric phase profiles of the five interferometric pairs were plotted along the blue dotted lines in Figure 7a, as shown in Figure 7b. Compared to spaceborne InSAR data, the orbital error phase of the airborne SAR exhibits greater time variability and complexity. This can be attributed to: (1) the lower stability of the airborne platform compared to the spaceborne platform, which is more susceptible to airflow; and (2) the influence of uncompensated residual motion errors, flight altitude, and slant range measurement errors.

Figure 8a1–a5 shows the differential interferograms of the five interferometric pairs. In addition to the orbital errors phase, there is a significant residual topographic phase caused by the height difference between the external DEM and the InSAR height, which should be considered for the selection of peaks and troughs. As shown in Figure 7b,c, accurately identifying the peaks and troughs of each differential interferogram using just one profile is challenging. Therefore, for each interferometric pair, multiple profiles were plotted along the azimuth direction, and their common peaks and troughs were extracted as references for segmentation. Figure 8b1–c5 shows the estimated airborne time-varying orbital error phases and corrected differential interferometric phases using the proposed method. After correction, the phase trend in the original differential phase is eliminated, and the residual phase is mainly distributed between ±1 rad, indicating the successful removal of the orbital error phase.

To evaluate the effectiveness of the proposed block adjustment model, the corrected interferometric phase is converted into elevation, as shown in Figure 9a. Figure 9b–f shows the differences between the InSAR DEMs estimated from five interferometric pairs and the LiDAR DTM. It can be observed that except for the interferometric pair 0301–0101, the other interferograms demonstrate excellent performance, with different values in most areas close to zero. Note that the bulges in Figure 9b–f are mainly caused by the scattering phase centers in forest areas and unmasked water regions. Additionally, compared to Figure 8c1–c5, the values in Figure 9b–f exhibit an opposite trend, which can be attributed to the long baseline having a smaller phase height conversion factor relative to the shorter baseline [12,17], as shown in Table 1. Figure 10a–e shows the elevation error histograms of the estimated DEM and airborne LiDAR DTM for the five interferometric pairs, both before and after removing the time-varying orbital error phase. Before correction, the maximum DEM error in the P-band exceeds 50 m, with RMSEs ranging between 7.83 m and 14.79 m and MAEs between 0.58 m and 9.56 m. Obviously, the existence of time-varying orbit errors seriously affects the accuracy of InSAR DEM estimation. After correction, the DEM elevation errors are distributed within a reasonable range, mainly concentrated within ±10 m. Compared with the errors before correction, the RMSEs have been reduced to between 3.36 m and 6.02 m, reflecting an average improvement of 59.32%. Additionally, the mean of the error histogram of the corrected InSAR DEM is greater than zero, which is consistent with existing studies [12,42], as the phase center of the SAR signal typically lies between the canopy top and the ground surface. In summary, the proposed method effectively removes the centennial orbital error phase in the airborne InSAR interferogram and improves the accuracy of InSAR DEM.

5. Discussion

5.1. Advantages of the Proposed Method

Many studies have attempted to achieve orbit error estimation using polynomial models or exploiting multiple baseline data constraints. To verify the robustness of the proposed time-varying orbit error estimation method under single-baseline and single-polarization conditions, it is compared with the following three cases:

Case 1: A nonlinear polynomial model is applied to each interferometric pair to fit the orbital error phase, which is a common approach in most studies [27,28,29,30,31]. Given the complexity of time-varying orbit error, a cubic polynomial is used for estimation.

Case 2: A polynomial model based on empirical blocking. Du et al. proposed a patch-based polynomial model that divides the SAR image into multiple blocks along the azimuth and range directions, with the number of blocks in each direction being less than three [16]. In this study, given that the airborne data exhibit small and simple changes in range direction, the differential interferometric phase is divided into five equal blocks along the azimuth direction for modeling.

Case 3: For the time-varying orbit error model, Wang et al. proposed a multi-baseline InSAR orbit error correction model [12]. Typically, through the mutual constraints between multi-baseline InSAR data, multi-baseline parameterized models in all directions are successively established for absolute phase calibration.

Table 2. The accuracy of the InSAR DEM derived using different orbit error correction methods.

Accuracy (m)	0101			0105			0107			0109			0111
	RMSE	STD	MAE	RMSE	STD	MAE	RMSE	STD	MAE	RMSE	STD	MAE	RMSE	STD	MAE
Before correction	14.79	14.45	3.12	12.10	11.28	9.56	9.22	9.15	1.13	7.83	7.81	0.58	11.94	11.68	2.46
Case 1	11.75	11.72	0.91	9.78	9.74	0.83	7.08	7.07	0.21	5.83	5.64	1.15	6.03	5.81	1.26
Case 2	9.75	9.70	0.94	6.91	6.82	1.17	5.07	4.96	1.15	4.55	4.41	1.02	4.75	4.58	1.29
Case 3	6.62	6.32	1.95	5.39	5.06	1.86	3.84	3.20	2.12	3.37	2.68	2.05	3.26	2.59	1.98
Proposed method	6.02	5.82	1.52	4.82	4.33	2.12	3.95	3.43	1.96	3.57	2.95	2.02	3.36	2.62	2.11

shows the performance indicators of InSAR DEM estimated by different methods in the Krycklan test site after removing the orbital error phase, alongside the accuracy of the InSAR DEM before correction for reference. Among the methods used, the DEM estimated using only a polynomial model (Case 1) for removing orbital error phases shows the worst accuracy, which is consistent with expectations because it is difficult for a single set of polynomial coefficients to accurately characterize the time-varying orbit error of the entire InSAR interferogram. In addition, the accuracy of the polynomial model based on fixed blocks in Case 2 is better than that of Case 1, but it is still terrible compared to Case 3 and Case 4. This is attributed to unreasonable blocking, which can lead to the polynomial model either overfitting or underfitting, preventing an accurate estimation of the orbital error phase. Compared with the multi-baseline model in Case 3, the method proposed in this study achieves nearly equivalent success in estimating the orbital error phase. Notably, the proposed method can achieve optimal modeling of the time-varying orbit error phase using single-baseline single-polarization InSAR data, demonstrating greater versatility and effectiveness.

5.2. Limitations and Future Improvements

In this study, we proposed an automated block adjustment model for the time-varying orbital error phase estimation. To validate the effectiveness of the proposed method, experiments were conducted using spaceborne and airborne InSAR data from two distinct test sites, each characterized by different climate types and terrains, demonstrating satisfactory results for most applications. However, the following limitations and future improvements need further research to generalize the proposed method.

Firstly, this method lacks the capability to distinguish the orbital error and the atmospheric delay phase, which is also a challenge in current related research. In the case of a significant atmospheric delay phase, the proposed method encounters challenges in accurately extracting wave peaks and troughs for automated separation. Although Figure 5 demonstrates the proposed method’s successful extraction of the blocks and estimates of the orbital error phase, this success is partly due to the fact that the InSAR data only has a 4-day temporal baseline and the atmospheric delay phase is small. A promising alternative is to employ existing methods to initially estimate the atmospheric delay phase, followed by iterative estimation of the orbital error phase, so that both error phase components can be removed from the interferogram at the same time.

Secondly, the proposed method relies on extracting control points from an external DEM. While this study implemented strict criteria to obtain more accurate elevation control points, challenges remain in areas with significant terrain variations or where flat, bare-earth surfaces are absent within the InSAR data coverage. Utilizing spaceborne LiDAR data, such as that from the global ecosystem dynamics investigation (GEDI) and ICESat-2, could provide more precise elevation information which can serve as control points for the proposed method. In future work, we will further investigate how to use spaceborne LiDAR as control points to estimate and correct the time-varying orbital error phase.

Finally, in regions where land is discontinuous or separated by large bodies of water, such as coastal areas, the interferogram contains extensive water bodies and isolated islands and the proposed method could face the following challenges: (1) how to determine segmentation boundaries on discontinuous profiles, and (2) how to select overlapping areas and model them. In this case, it may be necessary to create multiple block adjustment models based on land cover boundaries. Additionally, due to the significant penetration capability of SAR signals in forested scenes, the elevations obtained through InSAR are neither at the canopy top nor at the true bare ground [12,42]. Therefore, in areas completely covered by forests and when it is challenging to obtain sufficient bare ground points as GCPs, it is necessary to consider using control points located within the forest areas and to establish a block adjustment model that takes penetration depth into account.

Based on these limitations, future work will focus on reducing reliance on high-precision control points and enhancing the robustness and applicability of the model across different regions and conditions, thereby achieving generalization and large-scale application.

6. Conclusions

This study proposes a new method for estimating the time-varying orbital error phase through a block adjustment model, achieved by dividing the entire SAR interferogram into several blocks along the azimuth direction. Each block is modeled using a separate polynomial model and solved jointly using an iterative weighted least squares algorithm. To validate the effectiveness of the proposed method, experiments are conducted using spaceborne and airborne InSAR data with different orbital error phase trends, and the results show that the time-varying orbit error phase estimated based on the empirical blocking method is unacceptable. Fortunately, the proposed method can achieve automatic segmentation, which enables the block adjustment model to estimate and correct the time-varying orbital error phase more effectively. The results indicate that, regardless of whether spaceborne or airborne InSAR data are used, the accuracy of the estimated DEM improves significantly after removing the orbital error phase using the proposed method. These promising results establish a solid foundation for future investigations into applying the proposed method for large-scale block adjustments to remove time-varying orbit errors.