Parameterized Modeling and Calibration for Orbital Error in TanDEM-X Bistatic SAR Interferometry over Complex Terrain Areas

Abstract

The TerraSAR-X add-on for Digital Elevation Measurements (TanDEM-X) bistatic system provides high-resolution and high-quality interferometric data for global topographic measurement. Since the twin TanDEM-X satellites fly in a close helix formation, they can acquire approximately simultaneous synthetic aperture radar (SAR) images, so that temporal decorrelation and atmospheric delay can be ignored. Consequently, the orbital error becomes the most significant error limiting high-resolution SAR interferometry (InSAR) applications, such as the high-precision digital elevation model (DEM) reconstruction, subway and highway deformation monitoring, landslide monitoring and sub-canopy topography inversion. For rugged mountainous areas, in particular, it is difficult to estimate and correct the orbital phase error in TanDEM-X bistatic InSAR. Based on the rigorous InSAR geometric relationship, the orbital phase error can be attributed to the baseline errors (BEs) after fixing the positions of the master SAR sensor and the targets on the ground surface. For the constraint of the targets at a study scene, the freely released TanDEM-X DEM can be used, due to its consistency with the TanDEM-X bistatic InSAR-measured height. As a result, a parameterized model for the orbital phase error estimation is proposed in this paper. In high-resolution and high-precision TanDEM-X bistatic InSAR processing, due to the limited precision of the navigation systems and the uneven baseline changes caused by the helix formation, the BEs are time-varying in most cases. The parameterized model is thus built and estimated along each range line. To validate the proposed method, two mountainous test sites located in China (i.e., Fuping in Shanxi province and Hetang in Hunan province) were selected. The obtained results show that the orbital phase errors of the bistatic interferograms over the two test sites are well estimated. Compared with the widely applied polynomial model, the residual phase corrected by the proposed method contains little undesirable topography-dependent phase error, and avoids unexpected height errors ranging about from −6 m to 3 m for the Fuping test site and from −10 m to 8 m for the Hetang test site. Furthermore, some fine details, such as ridges and valleys, can be clearly identified after the correction. In addition, the two components of the orbital phase error, i.e., the residual flat-earth phase error and the topographic phase error caused by orbital error, are separated and quantified based on the parameterized expression. These demonstrate that the proposed method can be used to accurately estimate and mitigate the orbital phase error in TanDEM-X bistatic InSAR data, which increases the feasibility of reconstructing high-resolution and high-precision DEM. The rigorous geometric constraint, the refinement of the initial baseline parameters, and the assessment for height errors based on the estimated BEs are investigated in the discussion section of this paper.

1. Introduction

During the TerraSAR-X add-on for Digital Elevation Measurements (TanDEM-X) mission, the bistatic interferometric mode was, for the first time, applied to a spaceborne SAR platform, which provides us with unprecedented high-quality interferometric data for topographic measurement of the Earth’s surface [1,2]. The bistatic configuration involves one of the two synthetic aperture radar (SAR) satellites transmitting the pulse signals, and both satellites simultaneously receiving the backscattered signals from the observed target. Because of the approximately simultaneous acquisitions and the close helix formation, TanDEM-X bistatic interferometry is only slightly affected by temporal decorrelation and atmospheric delay [3]. In addition, TanDEM-X bistatic system can acquire high-resolution interferometric images, which can be used to improve InSAR applications, such as the high-precision and high-resolution digital elevation model (DEM) reconstruction [2,4,5,6], subway and highway deformation monitoring [7,8,9,10], landslide and mudslide monitoring [11,12,13], glacier motion monitoring [14,15], and sub-canopy topography measurement [16,17,18,19]. However, during the interferometry, orbital error is unavoidable, because of the limited and inaccurate interferometric SAR (InSAR) geometric parameters. For example, the unexpected phase artifacts caused by orbital error distort the actual relationship between the phase and topographic height, restricting high-precision interferometric measurement. In particular, the orbital phase error over rugged mountainous areas is more complex, because it also includes the topographic phase error caused by the orbital error, which cannot be well corrected [4,11,20,21]. Therefore, to improve interferometric measurement, it is essential to eliminate the orbital phase error in TanDEM-X bistatic interferograms as much as possible.

For the estimation and correction of orbital phase errors, two categories of methods have been developed. The first category is dependent on an empirical model, e.g., using a linear or quadratic polynomial model [22,23,24,25] to fit the phase ramps in the differential interferograms. For complex mountainous areas, since the orbital phase error usually presents topography-related features, an elevation-dependent term is added to the initial empirical model to improve the effectiveness of correcting the orbital phase error [11,26]. Since these kinds of empirical methods are simple and easy to implement, they have been widely applied in InSAR processing. In fact, these methods do not attempt to directly estimate the orbital phase error, but instead attempt to remove the phase ramps in the interferograms, according to the error characteristics. However, because of the helix formation of the twin satellites, the orbital phase error along the azimuth and range directions are often irregular in the TanDEM-X bistatic interferograms. In such a case, it is difficult to determine the order and coefficients of the empirical models. As a result, empirical model based fitting strategies have difficulty in accurately separating the orbital phase error from the interferometric phase, especially over rugged mountainous areas. The second category of methods focus on the modeling of the perpendicular baseline errors and the parallel baseline azimuth rate errors, to correct the orbital phase error in the residual phases [5,27]. This category of approach is dependent on the estimation strategy used for the initial perpendicular and parallel baselines. Specifically, the perpendicular baseline is often estimated by the state vectors of the two satellites or a frequency-based strategy, and the parallel baseline is obtained by the difference of the slant ranges from the master and slave tracks, or the fringe rate. Clearly, the parallel baseline and perpendicular baseline are independently estimated, and the frequency-based methods cannot easily take into account the rigorous InSAR geometric relationship. In addition, these estimation methods can be influenced by the approximation of the InSAR geometry, the size of the fast Fourier transform (FFT) window, and the flat terrain assumption, so that additional topography-correlated phase errors are introduced. This not only distorts the orbital phase error, but also increases the difficulty of the compensation of the phase error. In some cases, it can be expected that multi-baseline combinations are required to ensure a consistent estimation of the model parameters of interest [20,27,28]. Nevertheless, for most areas, multi-baseline interferometric pairs acquired by the TanDEM-X bistatic system are very rare.

Inaccurate extraction of the orbital phase error hinders TanDEM-X bistatic InSAR applications over rugged mountainous areas. Furthermore, the existing studies have provided very few effective solutions for the above problem. In this paper, based on the rigorous InSAR geometric relationship of the bistatic SAR satellites, we propose a parameterized model with respect to single-baseline InSAR. Through fixing the interferometric geometric parameters (i.e., the satellite flight height and slant range) of the master track, the orbital phase error is linked to baseline errors (BEs), including the baseline length error and the baseline tilt angle error. In addition, to avoid the undesirable phase bias caused by the height bias between the external DEM and the InSAR-measured height, the freely available global TanDEM-X DEM with a resolution of 90 m × 90 m (released by the German Aerospace Center, DLR) is used as the external DEM to perform differential interferometric processing. In such a case, the model parameters can be robustly estimated without the influence of unexpected height bias. The complexity of the orbital phase error indicates variable BEs, so that extracting the BEs along each range line is required [4,26,29]. During the parameter estimation, to solve this ill-posed problem, a weighted least-squares algorithm is adopted. The advantages of the proposed parameterized method embody two aspects: (i) the proposed parameterized method is able to extract accurate orbital phase error from the TanDEM-X bistatic interferograms over mountainous areas, and can ensure the reconstruction of a high-precision and high-resolution DEM; and (ii) the proposed method is able to separate and quantify the two components of the orbital phase error, i.e., the residual flat-earth phase error and the topographic phase error caused by the orbital error. To validate the parameterized model, TanDEM-X bistatic interferometric data from two test sites were used. One test site is located in Fuping County of Shanxi province, China, and the other is located in Hetang County of Hunan province, China. These two test sites have different degrees of topographic relief, which is beneficial for validating the performance of the proposed method.

The rest of this paper is arranged as follows. In Section 2, the interferometric processing of TanDEM-X bistatic InSAR data is reviewed, and the utilization of the freely published TanDEM-X DEM is highlighted. We then describe how the single-baseline parameterized model with respect to TanDEM-X bistatic InSAR is derived and how the estimation of the parameters is implemented along each range line. In Section 3, we introduce the TanDEM-X coregistered single look slant range complex (CoSSC) data over the two mountainous areas. We then explain how the orbital phase error of the TanDEM-X bistatic InSAR system is estimated by the proposed model, and we compare the proposed method with the widely used polynomial model. The corresponding parameter estimation results are also provided, to further demonstrate the performance and applicability of the proposed method. A discussion of the relevant issues is provided in Section 4. Our conclusions are drawn in Section 5.

2. Methods

2.1. Differential Interferometric Processing for TanDEM-X Bistatic InSAR Data

According to the InSAR geometric relationship, the TanDEM-X interferometric phase ${\phi }_{int}$ can be written as:

$${\phi }_{int}=-\frac{p·2\pi }{\lambda } \left(R_{1 }-R_2\right)$$

(1)

Where R₁ is the slant range from the master antenna center to the ground target; R₂ is the slant range from the slave antenna center to the ground target; p is determined by the interferometric mode, so when repeat-pass interferometry is used, p = 2, and when dual-antenna interferometry is used, p = 1; and $\mathsf{\lambda }$ is the wavelength of the onboard SAR signal. Because of the simultaneous acquisition and the close helix formation of satellites, the temporal decorrelation, the deformation signals, and the atmospheric delay can all be ignored. Therefore, the composition of the bistatic interferometric phase ${\phi }_{int}$ can be expressed as:

$${\phi }_{int}={\phi }_{flat}+{\phi }_{topo}+{\phi }_{noi}$$

(2)

where ${\phi }_{flat}$ is the flat-earth phase; ${\phi }_{topo}$ is the actual topographic phase; and ${\phi }_{noi}$ is the phase errors caused by the thermal noise.

To obtain the actual topographic phase, the flat-earth phase needs to be removed by the use of the flat-earth formula [4,11,29], which is linked to the baseline parameters (i.e., the baseline length and the baseline tilt angle) and the InSAR geometric parameters (i.e., the slant range $R_1$ from the master antenna center to the illuminated ground target, and the look angle ${\theta }_0$ of the master antenna in the sight-to-light direction). However, due to the limited precision of the current sensors’ navigation system and the helix formation of the TanDEM-X bistatic system, InSAR geometries are time-varying. In addition, the known InSAR geometric parameters are limited and inaccurate [1,4,20,30]. As a result, residual flat-earth phase error remains, which dominates phase ramps in the interferogram. It is worth noting that, when topographic phase exists, the residual flat-earth phase error is difficult to estimate because it is easily concealed by the topographic phase. At this time, differential InSAR processing based on an external DEM needs to be performed to achieve the purpose of exposing the orbital phase error [4,11,26,29]. However, because of the imperfect InSAR geometric parameters, significant topographic phase error caused by the orbital error is also produced during the removal of the topographic phase.

Although the single-baseline parameterized model proposed in literature [4] can be used to solve the phase ramps, it is aimed at the airborne InSAR case, and is not suitable for the spaceborne InSAR case, because the two kinds of interferometric geometries have significant differences, such as the curvature of the Earth and the flight altitude. Thus, it is necessary to build a parameterized model for the correction of the orbital phase error of TanDEM-X bistatic InSAR. In order to avoid the undesirable errors contributed by the simplification of the topographic phase formula [4,29,31,32], the removal of the flat-earth phase and topographic phase can be simultaneously carried out, based on the published global TanDEM-X DEM and the initial InSAR geometric parameters. Therefore, the simulated interferometric phase ${\phi }_{sim-int}$ can be expressed as:

$${\phi }_{sim-int}=-\frac{p·2\pi }{\lambda } \left(R_1-R_{2\_sim}\right)$$

(3)

where $R_{2\_sim}$ is the simulated slave slant range:

$$R_{2\_sim}=\sqrt{R_1^2+B^2-2B{R}_1\cos \left(\frac{\pi }{2} \pm \alpha \mp \theta \right)}$$

(4)

where B is the initial baseline length between the master and slave antenna centers; $\alpha$ is the initial baseline tilt angle; and $\theta$ is the look angle with respect to the external DEM, which can be calculated by the use of the spaceborne InSAR geometry:

$$\theta ={\cos }^{-1}\left[\frac{R_1^2+H^2-R_h^2}{2R_{1}H}\right]$$

(5)

where H is the flight height of the master satellite, i.e., the distance between the master antenna center and the Earth’s center; and $R_h$ is the distance between the Earth’s center and the observed target on the external DEM. After differential processing using Equation (4), the differential phase ${\phi }_{diff}$ can be written as:

$${\phi }_{diff}={\phi }_{res-DEM }+{\phi }_{OE}+{\phi }_{noi}$$

(6)

where ${\phi }_{res-DEM}$ is the phase caused by the difference between the actual bistatic InSAR-measured topographic height and the external DEM. For the TanDEM-X bistatic interferogram, differing from other DEM data, the published TanDEM-X DEM (with a resolution of 90 m × 90 m) as an external DEM is suitable, because their measured heights are consistent, which can avoid unexpected height bias. Note that a height difference still remains, which is contributed by some topographic details (such as local gullies and ridges) that cannot be described by the freely published TanDEM-X DEM. ${\phi }_{OE}$ is the orbital phase error.

2.2. Parameterized Modeling for the Orbital Phase Error of the TanDEM-X Bistatic InSAR System

According to the above, the production of orbital phase error is mainly associated with the InSAR geometric parameters. So a derivation for orbital phase error with respect to the relevant geometric parameters is implemented, as follows:

$${\phi }_{OE}=-\frac{p·2\pi }{\lambda } \left(\frac{\partial {\phi }_{diff}}{\partial R_1}·\Delta R_1+\frac{\partial {\phi }_{diff}}{\partial {\theta }_0}·\Delta {\theta }_0+\frac{\partial {\phi }_{diff}}{\partial B}·\Delta B+\frac{\partial {\phi }_{diff}}{\partial \alpha }·\Delta \alpha \right)$$

(7)

where $\frac{\partial {\phi }_{diff}}{\partial R_1}$, $\frac{\partial {\phi }_{diff}}{\partial {\theta }_0}$, $\frac{\partial {\phi }_{diff}}{\partial B}$, and $\frac{\partial {\phi }_{diff}}{\partial \alpha }$ represent the partial derivatives of the differential phase with respect to the slant range of the master antenna, the look angle of the master antenna, the baseline length and the baseline tilt angle, respectively. $\Delta R_1$, $\Delta {\theta }_0$, $\Delta B$, and $\Delta \alpha$ are the errors of the slant range, look angle, baseline length and baseline tilt angle, respectively.

It can be found that all the unknown parameters in Equation (7) are to be estimated, suggesting that this is an under-determined problem. To solve this issue, the geometry ($R_1$, H) of the master track is fixed, i.e., $R_1$ and H are free of errors [4,29]. This strategy can map the relative phase observations into fixed interferometric geometry. Therefore, a parameterized model for the orbital phase error in a single bistatic interferogram is linked to the BEs ($\Delta B$, $\Delta \alpha$):

$${\phi }_{OE}=-\frac{p·2\pi }{\lambda } \left(\frac{\partial {\phi }_{diff}}{\partial B}·\Delta B+\frac{\partial {\phi }_{diff}}{\partial \alpha }·\Delta \alpha \right)$$

(8)

where

$$\{\begin{array}{c}\frac{\partial {\phi }_{diff}}{\partial B}=\frac{p·2\pi }{\lambda }·\frac{B-R_{1}cos \left(\pi /2 \pm \alpha \mp \theta \right)}{R_{2\_sim}}\\ \frac{\partial {\phi }_{diff}}{\partial \alpha }=-\frac{p·2\pi }{\lambda }·\frac{B{R}_{1}sin \left(\pi /2 \pm \alpha \mp \theta \right)}{R_{2\_sim}}\end{array}$$

(9)

In Equation (9), parameter $\theta$ consists of the designed look angle ${\theta }_0$ and the look angle difference $\Delta \theta$ caused by the topography. As mentioned above, joint removal of the flat-earth phase and topographic phase leads to two components for the orbital phase error, i.e., the residual flat-earth phase error ${\phi }_{res-flat}$ and the topographic phase error caused by the BEs ${\phi }_{BE-topo}$.

$${\phi }_{OE}={\phi }_{res-flat}+{\phi }_{BE-topo}$$

(10)

According to Equation (8), these two components can be separated and quantified. Clearly, the residual flat-earth phase error is contributed to by the designed look angle ${\theta }_0$, and the topographic phase error caused by BEs is characterized by look angle difference $\Delta \theta$.

It is worth noting that, if the actual topographic height is flat and close to the reference height, only removing the flat-earth phase error is sufficient, and differential processing can be neglected. As a result, the orbital phase error is dominated by the residual flat-earth phase error. Nevertheless, this situation is rare.

2.3. Parameter Estimation for the BEs

The main idea of the proposed method is to transfer the estimation for the orbital phase error to the estimation for the BEs ($\Delta B$, $\Delta \alpha$). Considering the limited precision of the twin satellite-mounted navigation systems and the uneven baseline changes caused by the helix formation of TanDEM-X bistatic system [1,27,30], the BEs tend to be time-varying. In such a case, it is predictable that building a single parameterized model for a whole interferogram cannot easily take into account the inhomogeneous influence of the time-varying BEs on the interferograms. Hence, the parameterized model is built along each range line.

The BEs in each range line are estimated using a weighted least-squares algorithm, details of which can be found in [4,11]. After the estimation along each range line, the whole ${\phi }_{OE}$ is reconstructed by the estimated BEs along the azimuth direction according to Equation (8).

To better explain the proposed method, a detailed flowchart for the TanDEM-X bistatic InSAR processing and parameterized model is provided in Figure 1.

3. Results

3.1. TanDEM-X Bistatic InSAR Data

In this study, two test sites with rugged and steep topography were selected. The first test site is located in Fuping County of Shaanxi province, China, as shown in Figure 2a. Most of this study area is flat, with an elevation of about 300 m, and only a small part of the area features significant terrain (in the upper-left corner of the image), where the elevation ranges from 300 m to 1400 m. The flat topography can be used to test the magnitude of the phase error caused by topography. The second test site features mountainous terrain, with the elevation ranging from 50 m to 1400 m. This test site is located in Hetang County, Hunan province, China, as shown in Figure 2b. It can be seen that the second test site has more complex terrain than the first test site, and the mountainous areas of these two test sites are relatively fragmented, which is beneficial for verifying if the proposed method is able to correct the orbital phase error in TanDEM-X high-resolution interferograms over rugged mountainous areas.

TanDEM-X bistatic CoSSC data from the two test sites were acquired in 2013. The corresponding interferometric parameters are listed in

Table 1. Interferometric parameters of the TanDEM-X bistatic InSAR data.

Study Area	Fuping, Shaanxi	Hetang, Hunan
Wavelength (m)	0.0315	0.0315
Acquisition date of images	3 September 2013	20 November 2013
Range × azimuth (m)	1.36 × 2.18	1.36 × 1.59
Center slant range (m)	674,513.588	589,952.677
Incidence angle (°)	42.554	31.304
Flight altitude of the master SAR sensor (m)	6,883,695.906	6,885,019.494
Initial baseline length (m)	157.350	109.178
Initial baseline tilt angle (°)	18.987	17.945

. Firstly, the TanDEM-X bistatic CoSSC data were directly used to conduct bistatic interferometry. The interferometric pairs were multi-looked, and the final resolutions were processed as 12 m × 12 m (azimuth × ground range). At this time, the size of these two interferograms in the SAR coordinates are processed as 3333 × 2863 and 4448 × 2621 (azimuth × range), respectively. The flat-earth phase and topographic phase were then simultaneously calculated by the use of Equation (3), where the nominal geometric parameters listed in Table 1 and freely published TanDEM-X DEM (90 m × 90 m) were used. To obtain the differential phases, the flat-earth phase and topographic phase were simultaneously subtracted from the interferometric phases over two test sites. Finally, after performing adaptive phase filtering [33,34,35,36], the minimum cost flow (MCF) algorithm was applied to the above differential phases to perform phase unwrapping processing [7,17,37,38]. During the phase unwrapping, reference points located in flat areas were chosen that met the requirement of high coherence. In addition, based on the InSAR geometric relationship and the provided interferometric parameters, the initial baseline parameters were calculated, i.e., the corresponding baseline length and baseline tilt angle were 157.350 m and 18.987° for the Fuping test site and 109.178 m and 17.945° for the Hetang test site. This data preparations allowed the building of the parameterized model.

Note that the use of the published TanDEM-X DEM can eliminate the phase bias caused by the different radar signals in measuring the height of observed targets, in areas such as vegetation, snow or desert [4,14,18,29,39,40,41,42]. This can ensure that the estimation of the orbital phase error in the TanDEM-X bistatic interferograms is not affected by the phase bias. Therefore, it is reasonable to speculate that the differential phase in flat bare areas should be close to zero, whereas the differential phase in areas of topographic relief will be contributed to by the terrain details (such as gullies and ridges) due to the low resolution of the published TanDEM-X DEM.

3.2. Results of the Orbital Phase Error Estimation and Correction

As the quadratic polynomial model considering an elevation-related term (which is hereafter referred to as the PolyH_fit method) has been widely used to fit the orbital phase error in differential interferogram, the results obtained by the PolyH_fit method and the proposed method are presented and compared.

3.2.1. Fuping Test Site

Figure 3a represents the unwrapped differential phase after the differential processing. We can see that there is a significant error trend (i.e., phase screen) from the lower-left corner to the upper-right corner in the original interferogram. Figure 3b,c display the estimated orbital phase error obtained using the PolyH_fit method and the proposed method, respectively. It can be clearly seen that the phase trends of these two estimates are highly consistent with the original interferogram. In addition, the magnitude of the orbital phase error is significant. The results obtained after correction by the above two methods are shown in Figure 3d,e. The residual phase corrected by the proposed model (see Figure 3e) tends to be zero, on the whole, whereas the residual phase corrected by the PolyH_fit method (see Figure 3d) retains some local topography-related errors, suggesting the difference of the performance of above two methods. The phase difference between Figure 3d,e is shown in Figure 3f, where the magnitude is mainly concentrated in the range of [−0.7 rad, 0.7 rad], especially in the mountainous areas. According to phase-to-height conversion (PtoHC) calculation, the phase difference can cause height errors of around −6 m to 3 m.

To further demonstrate the effectiveness of correction by the proposed method, two profiles (marked by the black dotted lines) from the azimuth and range directions were selected from Figure 3d,e. The use of freely global TanDEM-X DEM indicates that it is reasonable that the residual phase should be very close to zero except for topographic relief areas. In other words, the smaller the residual phase, the better the correction effectiveness. So the true residual phase is assumed to be zero. As a result, the root-mean-square error (RMSE) of the residual phase can be calculated. Once the correction is biased, the RMSE of the residual phase will become larger. As displayed in Figure 4, for the PolyH_fit method, the RMSE of the residual phase is 0.36 rad in the azimuth direction and 0.11 rad in the range direction, whereas, after correction by the proposed method, the RMSE is 0.24 rad in the azimuth direction and 0.05 rad in the range direction. Clearly, in both the azimuth and range directions, the superiority of the proposed method is clear.

3.2.2. Hetang Test Site

The unwrapped differential phase at the Hetang test site shows a significant error trend (i.e., phase screen) from the lower-left corner to the upper-right corner, as shown in Figure 5a. The orbital phase error estimated by the PolyH_fit method and the proposed method is shown in Figure 5b,c, respectively. Although both two estimates show good similarity with the original interferogram, the residual phases (in Figure 5d,e) after orbital phase error correction present significant differences (see Figure 5f). The proposed method shows a better correction performance for the orbital phase error, and some local topography-related residuals still exist in Figure 5d. In Figure 5f, the interval of difference is between −1.0 rad and 1.0 rad, and after PtoHC calculation, the height errors range from −10 m to 8 m. Clearly, in orbital phase error correction, the proposed method is more superior to the PolyH_fit method.

To validate the correction effectiveness in a complex terrain case, a local mountainous area (marked by the black rectangular box) and two profiles (marked by the black dotted lines) were selected from Figure 5d,e. The corrected differential phase maps over the mountainous area of interest are enlarged in Figure 6, where the topographic details, such as ridges and valleys, can be clearly identified. After correction by the PolyH_fit method, an obvious topography-related phase trend remains (see Figure 6a). In contrast, after correction by the proposed method, the phase map does not show an improper topography-dependent trend (see Figure 6b), which conforms to the expectation of using the TanDEM-X DEM. The main reason for these different results is that the PolyH_fit method is empirical, and the added elevation-related term cannot completely absorb the topographic phase error caused by orbital error, which influences the high-precision InSAR DEM inversion. Differing from the empirical model, the proposed model is a parameterized derivation according to rigorous InSAR geometry, and its parameter estimation considers both the composition and error characteristics of the orbital phase error.

According to the transverse and longitudinal profile-line maps in Figure 7, the RMSEs (0.23 rad for the azimuth direction and 0.18 rad for the range direction) of the residual phases of the proposed method are all smaller than those obtained with the PolyH_fit method. The two profiles of the proposed method are also closer to zero in the mountainous areas, which is more in line with the actual situation.

From the results obtained for the two test sites, the widely used PolyH_fit method fails to separate the accurate orbital phase error from the interferometric phase over mountainous areas. In contrast, the proposed method shows a better performance in the estimation and correction of the orbital phase error in the TanDEM-X bistatic interferograms, which is beneficial for high-precision and high-resolution InSAR applications.

3.3. Composition of the Orbital Phase Error

In this section, the orbital phase error in each test site are quantified as two components, i.e., the residual flat-earth phase error (i.e., Figure 8a,c) and the topographic phase error caused by orbital error (i.e., Figure 8b,d), as mentioned above. It can be seen that, for both the Fuping test site and the Hetang test site, the magnitude of the residual flat-earth phase error is larger than that of the topographic phase error caused by orbital error. Clearly, the residual flat-earth phase dominates the error trend and the magnitude of the phase ramps, whereas the topographic phase error caused by orbital error contributes to the local variation. Once the topographic phase error caused by orbital error in mountainous areas is concealed by the residual flat-earth phase error, the elevation-related term is easily ignored when using an empirical model. From Figure 8, it can be seen that the topographic phase error caused by orbital error ranges from 15 rad to 45 rad for the Fuping test site and from 5 rad to 60 rad for the Hetang test site. It is worth noting that, even though most areas of the Fuping test site are flat, the topographic phase error caused by orbital error (about 15 rad) is also significant because the corresponding elevation is around 300 m from the reference height. For mountainous areas with larger elevation, the influence of the topographic phase error caused by orbital error is more significant and complex. Visually, the topographic phase error caused by orbital error is similar to a diminished version of the topographic phase. This confirms that the topographic phase error caused by orbital error is dependent on the elevation of the topography. Rugged mountainous areas actually increase the complexity of the orbital phase error, which is one of the main reasons why the orbital phase error is difficult to estimate with the widely used PolyH_fit method. Furthermore, according to the PtoHC calculation (noting that PtoHC factors of the two test sites are lower than 0.2) [31], the maximum height error due to the topographic phase error caused by orbital error is 225 m for the Fuping test site and 300 m for the Hetang test site. It is therefore clear that the impact of the topographic phase error caused by orbital error on InSAR height measurement is considerable. The proposed parameterized model itself implies a topography-dependent contribution to absorb the false topographic phase, which is derived from rigorous InSAR geometric derivation. This demonstrates that the proposed method is suitable for orbital phase error correction.

3.4. Results of the BEs Estimation

Figure 9 shows that the estimated BEs present a linear trend along the azimuth direction, indicating the time-varying characteristic of the BEs. The magnitude of the baseline length errors is from −36.69 m to −36.44 m for the Fuping test site, and from −33.04 m to −33.02 m for the Hetang test site, as shown in Figure 9a,c. It can also be seen that, in Figure 9b,d, the estimated baseline tilt angle errors range from 0.081° to 0.088°, and from 0.057° to 0.068°, respectively. These results reveal that the initial interferometric baselines contain significant uncertainties. The reasons for the significant bias can be divided into two aspects. Firstly, only limited interferometric parameters are provided, such as limited state vectors or slant ranges, and they cannot involve each range line in the azimuth direction. In addition, during the refined co-registration, although the offsets between the master image and slave image are eliminated, some of the known parameters corresponding to the slave image (such as the slant range and the distance between the center of the slave SAR antenna and the center of the Earth) are not adjusted. In other words, the interferometric parameters are not mapped into the reference geometry. Consequently, when these parameters are used during the removal of the flat-earth phase and topographic phase, significant phase screens are caused.

4. Discussion

4.1. Refinement for the Initial Baseline Parameters

From the above, it is clear that the solution for the orbital phase error in TanDEM-X bistatic interferograms is associated with the BEs. In the proposed method, the use of the TanDEM-X DEM constrains the topographic height, and then the geometry ($R_1$, H) of the master track is fixed, which suggests that accurate positions for the slave track in InSAR geometry need to be determined. However, it can be proved that using estimated BEs can compensate the initial positions of the slave track [42]. As a result, after the BE compensation, the means of the refined baseline parameters are 120.795 m and 19.072° for the Fuping test site and 76.148 m and 18.030° for the Hetang test site, as shown in

Table 2. Refinement of the baseline parameters of the TanDEM-X bistatic interferograms over the Fuping and Hetang test sites.

Baseline Parameter	Baseline Length (m)		Baseline Tilt Angle (°)
	Fuping	Hetang	Fuping	Hetang
Refined baseline parameters	120.795	76.148	19.072	18.030

. Accurate estimation of the baseline parameters is important for high-precision height measurement via InSAR. Actually, this is not easy, especially in a complex terrain case. One feasible solution is based on the constraint of the external height data. However, using ground control points (GCPs) is impractical because how to select representative GCPs in rugged mountainous areas is difficult to determine. In addition, it can be difficult to access some rugged and steep areas to set up GCPs. When using Ice, Cloud, and land Elevation Satellite (ICESAT) points, their reliability is uncertain, mainly due to the influence of topographic slopes. Furthermore, sparse GCPs are difficult to achieve the time-varying BEs [29]. In contrast, using the freely published TanDEM-X DEM can provide efficient and available points to estimate BEs in each range line. However, due to the lack of real values for the baseline parameters, it is difficult to determine the accuracies of the refined baseline parameters. Thus, relevant comparisons were not performed in this study.

4.2. Assessment for Height Errors Based on the Refined Baseline Parameters

Baseline parameter refinement is important for improving PtoHC during high-precision topographic measurement. In this section, taking the Hetang test site with rugged mountainous terrain as an example, we describe how the refined baseline parameters from Section 4.1 can be used to compensate the PtoHC factor, which can conduct mutual conversion between the phase and height [31]. Compared to PtoHC factor (mean value: 0.152) before compensation, the refined PtoHC factor decreases by about 0.044, indicating that the initial value is over-estimated. Using PtoHC factors before and after compensation, the height errors converted from the residual phase (Figure 5e) were assessed, as shown in Figure 10a. The results show that their mean values are nearly equal to zero; the standard deviation (std) of the height errors is 2.22 m when using the initial PtoHC factor, while the std becomes 3.03 m for the height errors converted by the refined PtoHC factor. This reveals that, before PtoHC factor compensation, the statistical accuracy of height errors with respect to TanDEM-X DEM is under-estimated. Based on the above, it can be concluded that the proposed method can not only mitigate the orbital phase error for high-precision DEM reconstruction, but it can also be used to accurately assess the accuracy of the height measurement.

4.3. Impact of the Selection of External DEM on the Proposed Method

The proposed method uses the global TanDEM-X DEM as an external DEM to constrain the topographic height. One main reason for this choice is that the TanDEM-X DEM is freely available. This not only represents good feasibility, but also eliminates the height bias between the external DEM and the TanDEM-X InSAR-measured height. If other global DEMs were used, there could be height bias between the external DEM and the TanDEM-X InSAR-measured height, which is caused by the different measurement positions and the changes of the ground surface. Thus, in the following, two cases are separately considered. For change or deformation of the ground surface, we selected the Shuttle Radar Topography Mission (SRTM) DEM with a resolution of 30 m as the external DEM, to test its impact. Since the difference of the accuracies of these two DEMs over the Hetang test site is only 1.5 m, and their mean values are highly consistent, the estimated orbital phase error shows little difference when compared with that in Figure 5c. In addition, the RMSEs of the differences between the two estimated BEs are 0.008 m and 1.7 × 10⁻⁵. Therefore, so small differences can be neglected. As shown in Figure 10b, the magnitude of the height errors is larger than when using the TanDEM-X DEM, both before and after the PtoHC factor compensation. This suggests that local differences, such as local changes or deformation of the ground surface, are mainly reflected in the corrected residual phase. Obviously, in this case, the proposed method is little affected. However, it is worth noting that the effectiveness is different when significant height bias caused by the different measurement positions exists. This situation mainly arises in dense forest areas or snow areas. Due to the similar error trends, the resulting phase bias is mainly absorbed by the parameterized model [29]. Furthermore, the corrected residual phase may be affected. As a result, the proposed method cannot easily detect the actual height bias. This is also dependent on the magnitude of the height bias. A similar situation has been analyzed in the literature [29]. Here, because of the lack of suitable external DEM data, relevant validation will be implemented in our future work.

5. Conclusions

In this paper, we have proposed a novel parameterized method to estimate and correct the orbital phase error in TanDEM-X bistatic InSAR. Under the condition of little temporal decorrelation and atmospheric delay, the orbital phase error is rigorously linked to the BEs through fixing the geometry of the master track and constraining the topographic height with the freely published TanDEM-X DEM. To ensure the estimation of the time-varying BEs, the parameterized model is built along each range line and then estimated. Two study areas with rugged mountainous topography were selected to validate the performance of the proposed method. The results showed that the orbital phase error estimated by the proposed method is more accurate than that estimated by the widely used PolyH_fit method. Benefiting from the use of the TanDEM-X DEM, the residual phases are reliable, and actual topographic details, such as ridges and valleys, can be clearly identified after the correction. In addition, the two components of the orbital phase error, i.e., the residual flat-earth phase error and the topographic phase error caused by orbital error, are separated and quantified. Based on the above, the proposed method is able to accurately estimate and correct the orbital phase error in TanDEM-X bistatic SAR interferometry. In our future work, the proposed method will be used to achieve high-precision and high-resolution DEM reconstruction.