Addressing Inaccurate Phase Center Offsets in Precise Orbit Determination for Agile Satellite Missions

2 PCOS IN POD

All results presented in this paper were obtained with the POD software PODCAST (Gutsche et al., 2022). The estimation within PODCAST relies on an extended Kalman filter (EKF) running forward and backward in time, whose results are then combined with a fixed-interval smoother. For the present study, the existing software was extended for the capabilities needed to estimate the lever arm. A brief overview of the employed observation model, specifically with respect to the lever arm, is given in this section. Gutsche et al. (2022) have provided a more detailed discussion of the software and the general processing steps of the POD approach.

Two common types of POD methods are kinematic and RD approaches. Kinematic POD relies solely on GNSS measurements, whereas the latter additionally employs rigorous modeling of the perturbing forces acting on the satellite. In RD POD, the state vector is augmented by additional parameters that allow for a deviation from the imperfect dynamic model (Wu et al., 1991). Depending on the constraints imposed on these parameters, the POD can be either more kinematic or more dynamic. It is important to note that as a consequence of modeling the orbit dynamics, the result of the RD POD is tied to the CoM, as opposed to kinematic POD, where the CoM position is inferred through the vector pointing from the CoM to the ARP.

In the following, the ionosphere-free combination is used for both code-phase and carrier-phase measurements to remove first-order ionospheric effects. The ionosphere-free combination for the carrier-phase measurement can be written as follows:

1

where Δt_r and Δt^s are the receiver and GNSS satellite clock offsets, λ_nl is the narrow-lane wavelength, and N_IF is the associated float ambiguity of the ionosphere-free combination (Hauschild, 2017a, 2017b). denotes the angle-dependent PVs for both the receiver and GNSS satellite, δt^rel describes the relativistic clock correction, is the phase wind-up, and ε summarizes all remaining errors, including the receiver noise. describes the distance between the mean phase center of the receiving antenna and the mean phase center of the transmitting antenna. This range can be described as follows:

2

Here, r_r is the CoM location of the satellite hosting the GNSS receiver that is determined in the RD POD, r^s is the location of the mean phase center of the GNSS antenna, and describes the line-of-sight vector in the inertial frame. The vectors of the ARP r_ARP,S, PCO r_PCO,S, and CoM location r_CoM,S given in the satellite-fixed reference frame S make up the lever arm l (see Figure 1). R_IS represents the rotation from the satellite-fixed reference frame to the inertial frame.

• Download figure
• Open in new tab
• Download powerpoint

FIGURE 1

Schematic of the vectors that make up the lever arm l

When investigating Equation (2), it becomes apparent that an error in the lever arm l, resulting from either the CoM location, ARP location, or PCOs, affects the range, and therefore, the GNSS observation is also influenced in the same way. As a result, one cannot discern a difference between these terms within the parameter adjustment. We introduce a combined correction to this lever arm Δl in the POD procedure, with the partial derivatives for both the code and carrier-phase measurement written as follows:

3

Incorporating the lever arm correction into the estimated state results in the combined state vector:

4

with the position r and velocity v of the LEO satellite, as well as the receiver clock offset Δt_r, and the float ambiguities of the ionosphere-free combination N_IF. The empirical accelerations Δa, which are estimated within RD POD (Wu et al., 1991), augment the state vector and are estimated here in the radial, along-track, cross-track frame (RTN). It is important to note that the estimated PCO reflects the optimal value in combination with the applied a priori PV map. As discussed by Rothacher et al. (1995), a shift in the PCO results in a new set of PVs. Thus, this property might necessitate further iterations to refine the PCO and PVs.

3 MISSION SCENARIOS

In situ GNSS measurements for agile LEO satellites are not readily available. Thus, we must resort to simulated GNSS observations for this paper. To assess the influence of biases in the lever arm and differences in their estimation, both a non-agile and an agile mission scenario of 15-h duration were designed. The orbits for the non-agile and agile scenarios were both generated using the in-house orbit propagator ProP (Gutsche et al., 2022) with identical satellite properties and initial values for position and velocity. However, two different attitude profiles were utilized in the propagation and simulation of GNSS measurements. These measurements were obtained using a Spirent GSS9000 signal generator in combination with the geodetic-grade LION GNSS receiver (Gottzein et al., 2011). The orbit and satellite properties are based on HRWS, which was to be flown in a sun-synchronous dusk–dawn orbit at an altitude of approximately 514 km (Bartusch et al., 2021). The non-agile scenario corresponds to a mission with a constant attitude relative to the orbital frame. In this case, the satellite-fixed frame (S) is always aligned with the nadir frame (N), which is defined as follows:

5

The x-axis corresponds to the along-track direction (T), whereas the y-axis and z-axis are anti-parallel to the cross-track direction (N) and radial direction (R), respectively.

The simulated agile scenario uses a realistic attitude profile for the hybrid agility concept of HRWS (Bartusch et al., 2021). Over the course of one orbit revolution, which lasts approximately 95 min, the satellite performs a single maneuver during which it simultaneously rotates around all three axes: roll, pitch, and yaw. The roll angle approximately follows a ramp function at the start and end of the maneuver and remains constant in between. In contrast, the pitch and yaw angles continuously change during the 3- to 5-min-long maneuvers and can be approximated by a single sine or cosine wave. Each of these maneuvers has slight variations in magnitude, rotation rate, and rotation axes. However, for all of these attitude changes, all degrees of freedom are utilized, the duration of a single maneuver is always within 5 min, the magnitude of the rotation angle around each axis does not exceed 90°, and the absolute rotation rate is always smaller than 3°/s. In its nominal attitude, i.e., when no maneuver is performed, the satellite has a constant roll angle of −36° with respect to the nadir frame defined in Equation (5).

In summary, both scenarios are identical except for the attitude profiles, meaning that two different mission types are compared. Specifically, we highlight differences in the POD caused by inaccurate offsets and investigate whether the techniques for deriving the correct offsets for non-agile missions are also applicable to agile missions. Finally, the agile scenario is used to estimate the lever arm within the POD. Table 1 presents an overview of the models and parameters used for the simulation and the following POD procedure. For brevity, only the differences between the simulation and the POD processing are displayed. To generate the reference trajectory, a flat-plate macro-model was used to account for the attitude-dependent effects of aerodynamic drag and radiation pressure. The modeled characteristics of the HRWS satellite closely resemble those of TerraSAR-X (Buckreuss et al., 2003), which features an elongated body with a surface-mounted solar array. However, the utilized macro-model of HRWS accounts for the satellite’s increased size and mass (Spiridonova & Kahle, 2019). Within the highly reduced-dynamic (HRD) POD, the empirical accelerations fully compensate the non-gravitational forces, whereas the RD POD additionally relies on a simple cannonball model. For the latter, the average values of the generated lookup tables of the macro-model were used. Finally, the POD used the ionosphere-free combination and assumed a measurement noise of 1 m and 1 cm for the uncombined pseudorange and carrier-phase measurement, respectively.

For both scenarios described above, the boresight of the GNSS antenna is oriented opposite to the z-axis of the satellite-fixed frame. Hence, the antenna of the agile satellite will not be oriented towards the zenith during its nominal attitude mode. This results in some distinctive differences in the observed GNSS satellites between the two scenarios. The potential pass durations per GNSS satellite for both agile and non-agile scenarios based on software-in-the-loop simulations are displayed in Figure 2. Additionally, a non-agile scenario with an off-zenith pointing antenna was included to distinguish between changes induced by the constant roll angle offset and the agile maneuvers.

• Download figure
• Open in new tab
• Download powerpoint

FIGURE 2

Possible durations to continuously observe GNSS satellites for an elevation cut-off angle of 5°

For the zenith-pointing antenna, most GNSS satellites are observed for 30–40 min, which is typical for LEO satellites. A change in orientation away from the zenith, indicated here by the roll angle ϕ, leads to a shift towards both longer and shorter passes. Some low-elevation GNSS satellites can be observed for a longer time; however, GNSS satellites located further towards the zenith and positive roll angles are observed for shorter durations. Although the current work treats the ambiguities as float values, resolving the integer ambiguities of the long-lasting passes could stabilize the POD solution. However, the GNSS satellites associated with these passes have low elevations above the Earth’s horizon. Thus, the longer distances traveled by their signals through the dispersive ionosphere could amplify unmodeled errors in the observations. Furthermore, a degradation in the cross-track direction due to the generally worse observation geometry is likely.

The agility of the satellite leads to an increase in the number of very short pass durations. Naturally, the short maneuvers interrupt otherwise longer-lasting satellite passes, leading to a generally weaker observation geometry. Further analysis showed that for some maneuvers with a large change in the off-zenith angle, the receiver tracked only four satellites. The position accuracy during these tracking gaps largely depends on the uncertainties of the dynamic model, which are usually driven by non-gravitational forces. Because of the very small area-to-mass ratio of 0.0013 m²/kg for HRWS, as compared with 0.0024 m²/kg for TerraSAR-X (Spiridonova & Kahle, 2019), and the short maneuver durations of less than 5 min, these drops in the number of observed satellites did not pose any significant problems to the POD.

4 INFLUENCE OF OFFSETS

The influence of erroneous PCO values for satellites with constant or slowly varying attitudes, referred to here as non-agile missions, has been covered by several studies and analyses of in-orbit data, particularly for the cross-track and radial directions. In this section, the effects of errors in the lever arm on the resulting CoM trajectory from the POD procedure are studied for both non-agile and agile mission scenarios. The GNSS measurements acquired from the hardware-in-the-loop simulations described in Section 3 are used for this purpose. Aside from the POD runs with error-free PCO values and parameterizations as displayed in Table 1, additional POD solutions were computed with 5-cm errors induced in the lever arm. This value is slightly more pessimistic than those identified in orbit for recent LEO missions (Montenbruck et al., 2018, 2021; Peter et al., 2020). To facilitate a comparison of the two scenarios, the errors for both agile and non-agile scenarios were induced in such a way that they conform to the x-, y-, and z-axis of the nadir frame during the satellite’s nominal attitude. In the case of the non-agile scenario, these axes coincide with those of the satellite-fixed reference frame.

The results for the POD runs of the non-agile scenario are depicted in Figure 3. For error-free values, the estimation error is within the centimeter level and does not exhibit any systematic bias. This result is due to the lack of any GNSS-related error sources, such as GNSS ephemerides, in the simulation. As a consequence, the POD performance itself is too optimistic for all of the investigated scenarios. However, the influence of different PCO errors on the POD solution becomes apparent. The POD errors in the radial and cross-track directions are smaller than the errors initially induced in the lever arm, because the force modeling in RD orbit determination can partially compensate for deficiencies in the PCO. The remaining mismatch between the orbit determined from the POD and the true CoM trajectory in the cross-track and radial directions can be compensated by additional accelerations in these directions (Montenbruck et al., 2018). These added accelerations are shown as constant biases in Figure 3(b) and sum to the total acceleration that the satellite would experience on the correct CoM orbit. More accurate force models permit tighter constraints on the empirical accelerations that further reduce this error and compensate for deficiencies in the modeling of the lever arm, as shown in Figure 3 by the lines corresponding to Δl_z,N of the RD scenario. Thus, the induced errors in the offset in the radial and cross-track directions lead to only a small increase in the estimation errors (see also Table 2). However, highly precise non-gravitational force models for a given satellite are a prerequisite. According to the relations described by Montenbruck et al. (2018), for the altitude of HRWS, an offset of 1 cm in the radial and cross-track directions induces mean accelerations of approximately −37 nm/s² and 12 nm/s², respectively. Hence, the achievable orbit leveling is essentially defined by the uncertainty of the modeled radial accelerations (Montenbruck et al., 2021). Nevertheless, the HRD case already provides notable dynamic constraints that partially mitigate the PCO error in the radial and cross-track directions. In contrast to the cross-track and radial components, the error in the lever arm in the along-track direction cannot be compensated by any of the estimated accelerations and directly maps to the estimation error in this direction.

• Download figure
• Open in new tab
• Download powerpoint

FIGURE 3

POD results for the 15-h non-agile mission scenario for an error-free lever arm, as well as a 5-cm error introduced into various axes of the lever arm (a) Estimation error of the position in the radial, along-track, and cross-track directions (b) Empirical accelerations in the radial, along-track, and cross-track directions. Each injected error corresponds to a different axis in the nadir frame in the nominal attitude mode. GPS: Global Positioning System

Figure 4 displays the POD results obtained for the agile scenario with both the correct and erroneous PCO values. Similarly, the dynamic model can partially compensate for the errors in the offsets from the CoM, and remaining inconsistencies cause biases in the empirical accelerations. However, the distinction between the cross-track and radial components is not as clear, as the maneuvers blur the clear separation between the accelerations in the cross-track and radial directions. Even the dynamically more constrained RD solution cannot fully compensate for the induced PCO error and exhibits large variations in all three directions. Equivalent to the non-agile mission, the along-track error directly maps to the achieved position accuracy. The statistics of all POD solutions are summarized in Table 2. It becomes evident that both agile and non-agile mission types clearly benefit from more precise force modeling in the presence of systematic biases in the PCO values. Accurate force models are likely even more crucial for agile missions, as frequent attitude changes can cause quick perturbations of non-gravitational forces. Furthermore, these maneuvers can induce GNSS tracking problems, such as phase breaks, thereby increasing the reliance of POD on the dynamic model. On top of this, the satellite position is specifically important while the SAR image is taken, i.e., during the satellite maneuvers.

• Download figure
• Open in new tab
• Download powerpoint

FIGURE 4

POD results for the 15-h agile mission scenario for an error-free lever arm, as well as a 5-cm error introduced into various axes of the lever arm (a) Estimation error of the position in the radial, along-track, and cross-track directions (b) Empirical accelerations in the radial, along-track, and cross-track directions. Each injected error corresponds to a different axis in the nadir frame in the nominal attitude mode.

Figure 5 depicts the post-fit residuals in the forward filter for the agile HRD scenario with Δl_y,N = 5 cm. One can clearly see distinctive increases in the residuals during the maneuvers, which are not present for the non-agile scenario. This result likely indicates that the mismodeled PCO cannot be absorbed by other state parameters when the satellite rotates rapidly. Instead, these slews could render the lever arm observable and make its simultaneous estimation feasible.

• Download figure
• Open in new tab
• Download powerpoint

FIGURE 5

Post-fit residuals for a forward EKF of agile HRD POD with Δl_y,N = 5 cm, as shown in Figure 4

5 OBSERVABILITY ANALYSIS

In the previous section, the influence of inaccurate lever arms utilized in the POD was demonstrated for agile satellites. To alleviate the strict precision requirements for PCO values and non-gravitational force models, we investigate the possibility of directly estimating the lever arm without the need to employ any non-gravitational force models when the satellite performs an attitude maneuver. The observability of these values when jointly estimated with empirical accelerations is explored based on software-simulated GNSS measurements. Because the rotation around the yaw axis typically has the fewest limitations and, more importantly, rotating around this axis does not significantly affect the visibility and tracking of the GNSS satellites, the observability is examined on the grounds of a simple 10-min yaw maneuver that can be described as follows:

6

The yaw angle follows a simple sine function with amplitude A and period T, as well as a ramp-up and ramp-down period to avoid discontinuities in the first-order derivative. Two sets of maneuvers were simulated using a Spirent GSS9000 simulator. Either the amplitude A or period T in Equation (6) was held constant at a reference value of A_ref = 90° or T = 600 s, respectively. The other parameter was adjusted to vary the maximum rotation rate given by , where ω = 2π/T.

Following Ruggaber and Brembeck (2021), we analyzed the observability of each system for the EKF using the covariance matrix of the corresponding weighted least-squares estimator, which can be calculated as follows:

7

where H is the design matrix and P and R are the covariance matrices of the estimation error and measurement error, respectively. Unfortunately, this approach is not applicable when the empirical forces are modeled as a first-order Gauss– Markov process, as this process type cannot be captured in a batch least-squares estimator. Instead, similar to recent studies that introduce the empirical accelerations as piecewise constant parameters (Mao et al., 2021; Montenbruck et al., 2021), we treated the accelerations as constant for the duration of the observability analysis. A duration of 30 min was selected to capture the cross-correlations between the various parameters and to ensure that temporal changes in the empirical accelerations are not ignored.

The state vector is split into parameters that are solved over the entire duration and epoch-wise receiver clock offsets Δt_r to correctly account for possible cross-correlations or overly optimistic estimates. This approach results in an adapted design matrix and state transition matrix for and a design matrix for the receiver clock offsets:

8

where z_k denotes the modeled observations at epoch k and c is the speed of light.

The full design matrix H required for Equation (7) is assembled over a duration of 30 min as follows:

9

where and must be linearized around the ground truth state used in the simulation, rather than the estimated state (Ruggaber & Brembeck, 2021). The ambiguities are assumed to be fixed, meaning that they are introduced as predetermined values. This approach will lead to overly optimistic formal errors but will effectively capture the behavior of the separability between the different estimated parameters. Finally, the inverse in Equation (7) is calculated using rank-revealing singular value decomposition (SVD) to detect singular or near-singular matrices.

The Pearson correlation coefficient r between the lever arm corrections and the position and empirical accelerations in the RTN frame is extracted for all maneuver runs from the covariance matrix defined in Equation (7) and displayed in Figure 6(a). The resulting formal errors of the lever arm, normalized by their respective maximum values, are depicted in Figure 6(b). In both plots, the lever arm correction in the z-direction (≈ radial) is not shown, as this state is unobservable when rotating only around the yaw axis. This state is expressed as a near-singular matrix in the SVD. Both figures show that as the maximum rotation rate increases, the cross-correlation between the parameters and formal errors exhibits a significant decrease. For very low rotation rates, the x-offset is highly correlated with the along-track direction and is essentially unobservable. The same result occurs for the y-offset, which cannot be separated from an error in the cross-track position or a bias in the cross-track acceleration. However, even small rotation rates above 1°/s lead to a decorrelation between the lever arm components, position, and empirical accelerations. In this case, the majority of the correlation coefficients fall within the range of |r| < 0.3, which can be considered to enable a clear separation from other values and is marked as a blue shaded area in Figure 6(a).

• Download figure
• Open in new tab
• Download powerpoint

FIGURE 6

Observability analysis for a single 10-min maneuver around the satellite’s yaw axis as defined in Equation (6) with varying maneuver amplitudes and periods (a) Correlation between the estimated position, empirical accelerations, and lever arm correction. Areas with |r| < 0.3 are shaded in blue. (b) Normalized formal error for the estimated lever arm corrections

The standard deviation of Δl_x and Δl_y, as displayed in Figure 6(b), strongly decreases with increasing rotation rate. Variations in the period T and amplitude A yield similar results, although the formal error for larger absolute yaw angles ( T = 600 s) shows slightly better results. This effect could be attributed to the slightly stronger correlation between the offsets and position in the along-track and cross-track directions.

Although only rotations around the yaw axis have been considered thus far, the presented findings confirm that simple maneuvers can make the estimation of the offsets directly observable. Additionally, even better separation of the lever arm and accelerations might be achieved when the empirical accelerations are constrained toward zero, which we neglected in this analysis in order to investigate the worst-case scenario.

Looking ahead, we propose exploring the effect of maneuvers around all degrees of freedom on the observability of the lever arm. However, this would require maneuvers around the pitch axis and yaw axis, where more limitations usually apply with respect to attitude changes, such as a maximum rotation angle. Additionally, these rotations would alter the observed GNSS satellites, potentially degrading the observation geometry and inducing changes in the tracking behavior of the GNSS receiver. Therefore, a more detailed and comprehensive study is required to thoroughly investigate these aspects and derive optimally designed calibration maneuvers.

6 ESTIMATION OF THE PCO

The observability of the lever arm in the presence of rapid attitude changes has been shown based on simulated GNSS measurements. In the present section, we investigate the effect on the agile POD when the state vector is extended to incorporate the lever arm. Specifically, the accuracy of the estimated lever arm and the impact on POD performance are assessed based on the simulated agile scenario using the Spirent signal generator and LION receiver. To achieve this, several Monte Carlo simulations were run, which involved the definition of 60 randomly distributed vectors on a unit sphere. These vectors represent the knowledge error of the lever arm. The length of these vectors was varied to study the influence of the error magnitude, specifically, |Δl| = [1, 2, 3, 4, 5] cm. Finally, each offset error was introduced into the initial values of the POD parameterization, and simulations were run both with and without simultaneous estimation of the lever arm correction, resulting in a total of 600 POD runs. The configuration is equivalent to the HRD POD presented in Table 1 and is identical for all simulations presented here. Differences between each POD run were restricted to the varying offset error and the estimation of the lever arm, which is estimated as a random constant in the x-, y-, and z-directions of the satellite-fixed reference frame.

The convergence of the PCO estimates within the forward EKF is shown in Figure 7 to further demonstrate that slews render the PCO observable. It should be noted that in contrast to the displayed results, the smoothed values obtained from the combination of the forward and backward solutions are constant over time. Some observed small-scale variations over the entire orbital arc, caused by the limited numerical precision, are at least one order of magnitude smaller than the formal error and were therefore deemed uncritical for the final PCO estimate. The temporal evolution of the PCO exhibits a prominent decrease in all errors during the first attitude maneuver approximately 50 min into the simulation, further showcasing the increase in observability. The formal errors and the x-component of the PCO continue to converge toward smaller values, whereas the bounds and mean values associated with the y- and z-components increase again after several hours. The latter result is likely caused by a systematic effect, where the biases in the offsets are already compensated for by additional empirical accelerations, making it challenging for the EKF to decorrelate the values after longer durations. For the single POD run using the RD configuration, the error in all three directions of the lever arm is on the millimeter level, indicating that better non-gravitational models benefit the PCO estimation. To recapitulate the results of Section 5, another option to better distinguish between the empirical accelerations and the y- and z-components of the PCO is to perform faster and more frequent rotations around all spacecraft axes, potentially as part of dedicated calibration maneuvers.

• Download figure
• Open in new tab
• Download powerpoint

FIGURE 7

Lever arm estimation error of the forward EKF in all three directions based on Monte Carlo simulations with |Δl| = 5 cm (a) PCO error of all PODs and the mean instantaneous value over time (b) Mean estimation error of all Monte Carlo simulations, bounded by the minimum and maximum values. The mean formal error and the estimation error of a single RD POD are also displayed.

Figure 8 displays the root mean square (RMS) accuracy of the final POD solution obtained with the HRD parameterization for all executed Monte Carlo simulations. A linear trend in the RMS error (RMSE) values is evident in all directions when no correction is simultaneously estimated in the parameter adjustment. In this case, the along-track direction is the most sensitive to errors in the lever arm, closely followed by the cross-track direction. RMSEs are not as pronounced in the radial direction; however, estimations are usually the most difficult for this direction if there is not a proper knowledge of the radial accelerations. When the correction in the offset Δl is also estimated in the HRD scenario, a significant improvement can be seen in the POD accuracy. This is clearly the case for the radial and along-track directions. In the cross-track direction, a large percentage of the RMSE values obtained without estimation of the lever arm is superior to the estimated counterpart. A possible explanation for this result could be that the leading bias in this direction is driven by the neglected solar radiation pressure and aerodynamic drag. As shown in Section 4, the introduction of non-gravitational models could potentially weaken this effect. Nonetheless, the obtained POD accuracies in all directions are consistently close to or even below 1 cm. Moreover, no outliers are present, and the estimation delivers reliable results, independent of the initial magnitude of the lever arm error. The POD with the estimation of the offsets gives better results for almost all Monte Carlo simulations, especially in the radial and along-track directions.

• Download figure
• Open in new tab
• Download powerpoint

FIGURE 8

RMS accuracy of the agile HRD POD under the presence of a varying lever arm error Δl with and without estimation of its value

These improvements were obtained with no constraints or prerequisites on the non-gravitational force modeling of the agile spacecraft. Our results further hint at additional improvements when a more dynamic POD approach is employed, such as a more detailed modeling of the forces acting on the satellite in combination with more tightly constrained empirical accelerations. This approach would also benefit the POD accuracy in view of the more frequent tracking gaps induced by the maneuvers.

7 ESTIMATION OF THE PCO OF SENTINEL-6A

As mentioned previously, to the best of the authors’ knowledge, no in-orbit GNSS observations are publicly available for any agile satellite. However, all products required for POD, as well as reference solutions, are available for many of the Sentinel satellites of the Copernicus program (Thépaut et al., 2018). The altimetry satellite Sentinel-6 Michael Freilich, or Sentinel-6A, performed several short yaw maneuvers in 2021 and 2022 (Cullen, 2023). In this section, these slews are used to further validate the possibility of more reliably estimating the PCO in the presence of rapid attitude changes. The results of this method are then compared with the values derived by several other works (Kobel et al., 2022; Montenbruck et al., 2021, 2022). These studies indicate inconsistencies of approximately 7–8 mm in the PCO of the y-direction in the satellite frame. The difference in the y-offset can be explained by a yaw bias of –0.43° (Montenbruck et al., 2022) and has been applied in the solution of the Copernicus POD (CPOD) service (CPOD Team, 2023).

Compared with HRWS, Sentinel-6A orbits Earth at a significantly higher altitude of approximately 1336 km (Cullen, 2023). At this altitude, the uncertainties from non-graviational forces are significantly smaller, as the atmospheric drag only plays a minor role; the main uncertainties originate from the solar radiation pressure and Earth radiation pressure. As a result, the empirical accelerations in both the RD and HRD POD of Sentinel-6A can be more tightly constrained. For the more dynamic POD, we created a simplified computer-aided design model based on the assembly drawing and surface properties of Sentinel-6A (Cullen, 2023). Based on this geometric model, we generated lookup tables for both the infrared and visible spectrum using the ray-tracing technique of the open-source software developed by Zardaín et al. (2020). We considered instantaneous thermal re-radiation for all multilayer-insulation-covered surfaces, as reported by Hackel (2019), as well as for the surface-mounted part of the solar array, as suggested by Montenbruck et al. (2021). In contrast, we assumed no re-radiation for the two deployable solar panels, as the heat exchange between the two sides (Cullen, 2023) would approximately cancel out the induced forces. The HRD POD makes use of a cannonball model, with the average cross-sectional area and optical coefficients extracted from the ray-tracing model. In both cases, constant accelerations over the entire orbit arcs were estimated in the along-track and cross-track directions. Additionally, once-per-revolution accelerations were estimated in these directions to better compensate for residual errors in the non-gravitational force modeling. The full parameterizations used in the HRD and RD POD of Sentinel-6A are summarized in Table 3.

In the normal attitude mode, Sentinel-6A performs yaw steering to align the satellite’s x-axis with the velocity vector relative to the Earth’s surface (Cullen, 2023; Donlon et al., 2021). Hence, the yaw angle with respect to the nadir frame varies by approximately ±4° for the duration of one orbit. On several days in 2021 and 2022, the satellite performed larger attitude maneuvers in which the satellite was either rotated by +90° or –90° to support altimeter boresight pointing or flipped by 180° for longer durations to support POD analysis (Cullen, 2023). Figure 9 depicts the satellite yaw angle with respect to the orbital plane on the days that were used to estimate the lever arm correction. The ±90° yaw biases are applied for approximately 40 min and are induced by 5-min slews, whereas the 180° bias lasts for four days and is initiated by 10-min slews. In both cases, the attitude changes occur at a maximum yaw rate of 0.5°/s and are considerably slower than the previously investigated agile scenarios. To ensure that the EKF is in a steady state when the slews occur, the processed orbit arcs span 4 h before and after the first and last attitude maneuver of each day.

• Download figure
• Open in new tab
• Download powerpoint

FIGURE 9

Yaw angle of Sentinel-6A with respect to the nadir frame for several days in which a yaw flip or yaw bias occurs

DOY: day of year

Figure 10 illustrates the estimated PCO in the y (≈ – cross-track) and x (≈ along-track) directions in the satellite-fixed reference frame. All days with the attitude maneuvers depicted in Figure 9 are displayed, as well as one day during which no slew was performed. For reference, the three-dimensional (3D) RMSE of the POD solution compared with the combined solution of the CPOD Quality Working Group (QWG) is also shown.

• Download figure
• Open in new tab
• Download powerpoint

FIGURE 10

Estimated PCO correction in the x-direction (top) and y-direction (middle), as well as the 3D RMSE compared with the combined solution of the CPOD QWG for a variety of days and parameterizations

The weighted mean values are indicated by purple lines. G: GPS; E: Galileo

It is evident that the day with the nominal attitude, void of any large attitude maneuvers, exhibits the largest uncertainties in the estimated lever arm corrections. The value in the y-direction is only driven by the constraints applied to the empirical accelerations, and the x-direction result is unobservable in this attitude mode. For the days in which one or several slews occur, the formal errors of both Δl_x and Δl_y are significantly reduced, even though the processed orbit arcs are considerably shorter. The estimated values for different parameterizations of one day agree well with each other and do not indicate a major discrepancy between the phase centers of the ionosphere combination of L1/L2 and E1/E5a. The variations between the various days are larger and could be explained by the fact that only one or two attitude maneuvers occur during each orbit arc. Additionally, as discussed in Section 5, the rotation rates of the maneuvers are slower (ψ_max ≈ 0.5°/s) than the rates that would be required to clearly separate the estimated PCO from the empirical accelerations. Therefore, the estimated PCOs are still linked to the applied constraints in the cross-track direction and might vary between days. Further investigation showed that, while single values are slightly sensitive to the POD parameterization, the weighted mean values in the x- and y-directions are not strongly affected. Overall, the mean value of −7.3 mm in the y-direction agrees well with the values reported by other studies. Montenbruck et al. (2021) suggested a correction between −7 and −8 mm, whereas Kobel et al. (2022) reported a value of −8.5 mm, and the yaw bias of −0.43° (Montenbruck et al., 2022) that is applied in the CPOD solution (CPOD Team, 2023) amounts to Δl_y = −7.1 mm. This provides further evidence that attitude maneuvers can support the calibration of PCOs. The change in the 3D RMSE with respect to the combined CPOD QWG solution when correcting for the PCO is not conclusive, as the combined solution of this timespan did not yet apply the proposed yaw bias (CPOD Team, 2022). However, Kobel et al. (2022) showed that the correction in the y-component reduces the satellite laser ranging residuals of Sentinel-6A.

Further improvements in the estimation of the lever arm can most likely be achieved by performing integer ambiguity resolution, as modeling inconsistencies can be partially absorbed by float ambiguities (Jäggi et al., 2009). Integer resolution would aid in identifying these inconsistencies and provide a geometrically more constrained solution in the cross-track and along-track directions (Montenbruck et al., 2018), which could benefit the PCO estimation during attitude maneuvers. Additionally, the generation of PV maps could lead to improvements, as these are inherently tied to the employed dynamic model (Montenbruck et al., 2021) and are affected by an incorrect assumption of the PCO (Jäggi et al., 2009). Finally, the presented results are based on maneuvers that were not specifically designed to estimate the PCO. Dedicated calibration maneuvers involving several sequential attitude changes to attenuate the correlation between the offsets and the empirical accelerations would most likely lead to a superior performance of the direct PCO estimation.

CONFLICT OF INTEREST

The authors declare that there are no conflicts of interest regarding the publication of this paper.