Introduction

For any GNSS, the capability of precise orbit determination (POD) and precise clock estimation (PCE) is an essential prerequisite for high-precision applications, such as real-time precise positioning service and its contribution to reference frame determination. In POD processing, one of the critical issues is the modeling of the solar radiation pressure (SRP) acting on satellites which is strongly related with the satellite surface geometric structure and optical and thermal properties. Theoretically, analytical SRP models can be established based on such metadata (Fliegel et al. 1992; Ziebart et al. 2002). However, such accurate metadata are not always available; thus, the model accuracy is always restricted due to the uncertainty of the metadata. Therefore, Beutler et al. (1994) developed the empirical ECOM model in the early 1990s for GPS satellites by considering a bias and periodic terms of one per revolution for the Sun direction (D), the solar panel axis (Y) and the orthogonal axis (B) directions with respect to a priori model. Afterward, Springer et al. (1999) found that the periodic terms in D and Y directions are not significant and the orbits could even be improved if they are ignored. This resulted in the so-called ECOM model which has been used by most of the IGS analysis centers. However, the classic ECOM model was confirmed insufficient for POD of GLONASS satellites and thus was extended by adding the second- and fourth-order periodic terms to the solar direction with nine parameters, referred as to ECOM2 model (Arnold et al. 2015). Not limited to the periodic terms of ECOM2, the extended ECOM2 model is provided by Arnold et al. (2015) for the SRP modeling of GNSS satellites. For Galileo satellite orbits, the ECOM2 SRP model provides a significant improvement compared to the ECOM model, while the ECOM2 model exhibits a slight degradation for BDS-2 IGSO and MEO orbits (Prange et al. 2017; Liu et al. 2019). Currently, the ECOM model is used for generating final BDS-3 orbit products of Wuhan University (WUM) (Guo et al. 2018). In principle, the performance of ECOM and ECOM2 SRP models for BDS-3 POD processing should be evaluated and investigated for possible improvement.

Based on the IGS published BDS data, a number of related investigations have been done for the BDS-2 satellites and BDS-3 experimental satellites (Steigenberger et al. 2013; Lou et al. 2014; Feng et al. 2014; Montenbruck et al. 2017; Guo et al. 2017; Zhang et al. 2017; Zhao et al. 2013, 2018; Chen et al. 2020). Duan et al. (2019) found that ECOM2 model exhibits better performance than the ECOM model for BDS-2 GEO and QZS-1 satellites. Furthermore, Wang et al. (2019) established a priori SRP model to enhance the ECOM for BDS-2 GEO satellites by using an empirical fitting approach. The importance of the SRP modeling for BDS-3 satellites has attracted increasing attention from researchers, with the utilization of the ECOM and ECOM2 models serving as the basis to ensure high orbit accuracy (Yan et al. 2019; Li et al. 2020; Duan et al. 2022). As of November 2020, there are about 200 IGS ground stations that can track B1I (1561.098 MHz) or B3I (1268.520 MHz) navigation signals, which enables the BDS-3 POD products of high quality. The current constellation and the global network provide a very promising opportunity for further study and investigation on the BDS-3 POD and PCE.

Motivation and the study objective

The empirical ECOM or ECOM2 force models have been widely applied by most of IGS analysis centers for GPS satellites. However, it should be adapted and optimized for each GNSS system or even individual satellites. We started with BDS-3 POD using the IGS Multi-GNSS Experiment (MGEX) stations in order to evaluate the performance of these empirical force models (Montenbruck et al. 2017; Johnston et al. 2017). We noticed a significant deficiency in the existing two empirical models for BDS-3 satellites in eclipse (ref. Fig. 3). The reason could be that the ECOM2 model is over-parameterized, as reported by Prange et al. (2017) for the BDS-2 satellites, or extra periodic terms have not been included (Springer et al. 1999). Building upon this research motivation, the primary objective of our study is to adapt and optimize the empirical force model for improving the orbit accuracy of BDS-3 satellites, especially over eclipse periods.

After the introduction of the empirical ECOM SRP models, the BDS-3 satellite information and POD data processing strategies are presented in detail. Then, the significant orbit quality degradations during eclipse periods using ECOM and ECOM2 models are investigated for BDS-3 CAST and SECM satellites, respectively. Afterward, an adapted SRP model is presented based on the parameterization analyses of the ECOM2 model to improve orbit quality and its performance is further evaluated for the estimated and predicted parts of BDS-3 orbits, respectively. Finally, the conclusions are summarized.

Empirical solar radiation pressure model for BDS-3 satellites

Fliegel et al. (1992) pioneered the development of an a priori model for the SRP modeling of GPS satellites and recommended the ROCK-T model with thermal re-radiation modeling improvement where a scaling factor of the model accelerations is estimated. In principle, a priori or analytical SRP model could be established based on sufficient satellite structure and surface data (Ziebart et al. 2002). However, such data are not always available and may have large uncertainty. Thus, solving for more than just a scaling factor of the model acceleration is necessary. Beutler et al. (1994) developed the nine-parameter empirical ECOM model which decomposes the perturbing acceleration into the right-handed Sun direction (D), solar panel axis (Y) and the orthogonal axis (B) directions. For each component, a first-order truncated Fourier series expansion is introduced as follows:

$$\begin{aligned} & a_{D} (u) = D_{0} + D_{c1} \cdot \cos (u) + D_{s1} \cdot \sin (u) \\ & a_{Y} (u) = Y_{0} + Y_{c1} \cdot \cos (u) + Y_{s1} \cdot \sin (u) \\ & a_{B} (u) = B_{0} + B_{c1} \cdot \cos (u) + B_{s1} \cdot \sin (u) \\ \end{aligned}$$
(1)

where \(a_{D}\), \(a_{Y}\) and \(a_{B}\) represent the accelerations in the three DYB directions, respectively; and \(u\) is the argument of latitude of the satellite. Constant (\(D_{0}\), \(Y_{0}\), \(B_{0}\)) and 1/rev periodic accelerations (\(D_{c1}\), \(D_{s1}\), \(B_{c1}\), \(B_{s1}\), \(Y_{c1}\), \(Y_{s1}\)) are estimated for the SRP model.

Springer et al. (1999) found that the first-order periodic terms in the D and Y directions are not significant for GPS satellite orbits and the following five-parameter ECOM model is developed and has been used at most of the IGS analysis centers:

$$\begin{aligned} & a_{D} (u) = D_{0} \\ & a_{Y} (u) = Y_{0} \\ & a_{B} (u) = B_{0} + B_{c1} \cdot \cos (u) + B_{s1} \cdot \sin (u) \\ \end{aligned}$$
(2)

where the SRP parameters consist of \(D_{0}\), \(Y_{0}\), \(B_{0}\), \(B_{c1}\), \(B_{s1}\). The five-parameter SRP model is referred to as ‘ECOM’ model. For Galileo In-Orbit Validation (IOV) satellites, obvious SRP modeling deficits have been identified for the ECOM model. Montenbruck et al. (2015) accordingly developed a priori SRP model for reducing this modeling deficiency.

It has been demonstrated that even-order and odd-order harmonic perturbations act only along the D and B direction, respectively, for GPS and GLONASS satellites (Arnold et al. 2015). The extended form of the ECOM2 model is thus proposed by Arnold et al. (2015).

$$\begin{aligned} & a_{D} (u) = D_{0} + \sum\limits_{i = 1}^{{n_{D} }} {[D_{c,2i} \cos (2i \cdot \Delta u) + D_{s,2i} \sin (2i \cdot \Delta u)]} \\ & a_{Y} (u) = Y_{0} \\ & a_{B} (u) = B_{0} + \sum\limits_{i = 1}^{{n_{B} }} {[B_{c,2i - 1} \cos ((2i - 1) \cdot \Delta u) + B_{s,2i - 1} \sin ((2i - 1) \cdot \Delta u)]} \\ \end{aligned}$$
(3)

where \(n_{D}\) and \(n_{B}\) are the user-defined upper limits. Different from the ECOM model, the new angular argument \(\Delta u\) is the difference between the argument of latitude of the satellite \(u\) and that of the Sun \(u_{s}\).

For GLONASS satellites, a simplified form of the extended ECOM2 model is developed as follows:

$$\begin{aligned} & a_{D} (u) = D_{0} + D_{c2} \cdot \cos 2\Delta u + D_{s2} \cdot \sin 2\Delta u + D_{c4} \cdot \cos 4\Delta u + D_{s4} \cdot \sin 4\Delta u \\ & a_{Y} (u) = Y_{0} \\ & a_{B} (u) = B_{0} + B_{c1} \cdot \cos \Delta u + B_{s1} \cdot \sin \Delta u\quad \quad (\Delta u = u - u_{s} ) \\ \end{aligned}$$
(4)

where the second- and fourth-order harmonic terms (\(D_{c2}\), \(D_{s2}\), \(D_{c4}\), \(D_{s4}\)) are added for the D direction. Prange et al. (2017) proved that orbit misclosures of GLONASS satellites outside the eclipse season using the ECOM2 model were minimally increased compared with those using the ECOM model. Also, CODE suggests excluding the fourth-order terms in the D direction of the ECOM2 model for GLONASS satellites. Bury et al. (2020) found the sensitivity of Galileo satellites to the accelerations in the direction B and Y when the ECOM2 model is used. Galileo-IOV satellites suffer from the Sun-elevation-angle-dependent acceleration in the B direction, and Galileo-FOC satellites suffer from the constant Y bias. To avoid the confusion of naming, the simplified version given by (4) is called the ECOM2 model, and the extended version given by (3) is called the extended ECOM2 model.

For Galileo satellite orbits, ECOM2 also exhibits a significant improvement compared to the ECOM model, while the deficiency of the ECOM2 model is confirmed for the BDS-2 IGSO and MEO satellites (Prange et al. 2017; Liu et al. 2019). As to the ECOM2 model, we further introduce the sixth-order sine term in D direction, which can compensate for the SRP model deficiency. For operational BDS-3 satellites, we adapted the SRP model as follows.

$$\begin{aligned} & a_{D} (u) = D_{0} + D_{s4} \cdot \sin 4\Delta u + D_{s6} \cdot \sin 6\Delta u \\ & a_{Y} (u) = Y_{0} \\ & a_{B} (u) = B_{0} + B_{c1} \cdot \cos \Delta u + B_{s1} \cdot \sin \Delta u \\ \end{aligned}$$
(5)

where the additional fourth- and sixth-order terms (\(D_{s4}\), \(D_{s6}\)) are introduced in the D direction apart from five ECOM SRP parameters.

Data processing

We present the data and software package for the following test cases. The satellite information and DYB system scheme are given for a good understanding of the SRP modeling. Then, the processing strategy is provided for BDS-3 POD. We provide definitions of the deep eclipse period, eclipse period and non-eclipse period for comparison of SRP models. Apart from the orbit determination, the strategy for orbit prediction is introduced to further assess the SRP models.

Data and software package

MGEX observations data are fully integrated in the IGS network. GNSS observations are downloaded from three archives, i.e., the Institut National de l’Information Géographique et Forestière (IGN), the Bundesamt für Kartographie und Geodäsie (BKG) and the Crustal Dynamics Data Information System (CDDIS). In the IGS global network, there are about 200 ground stations used for BDS-3 POD. As the major differences are in the eclipsing periods, the data from day of year (DOY) 20 to 360, 2019 are processed. During this time span all the BDS-3 satellites experienced eclipse(s), the data available and the Sun elevation angles of the satellites are depicted in Fig. 1 where the low Sun elevation angles with its absolute value of less than 4° are also illustrated by shaded areas. For a better understanding of the DYB system scheme, the Sun–satellite–Earth reference frame is illustrated in Fig. 1.

Fig. 1
figure 1

DYB system scheme within the Sun–satellite–Earth reference frame (top) and the variation of the Sun elevation angle \(\beta\) (bottom) for BDS-3 satellites during the time period from DOY 20 to 360 in 2019. The Sun–satellite–Earth reference frame provides the visual representation of \(\beta\) angle above the orbit plane, the elongation angle (\(\varepsilon\)) and the argument of latitude of the satellite relative to the argument of latitude of the Sun (\(\Delta u\)). The DYB frame presented here is referenced from the work of Bury et al. (2020)

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BDS-3 satellite information observed are shown in Table 1. BDS-3 satellites are provided by two manufacturers, i.e., the China Academy of Space Technology (CAST) and the Shanghai Engineering Center for Microsatellites (SECM). The satellites are located in three orbit planes, namely A, B and C. The variations of the Sun elevation angle relative to the plane A, B and C are represented by green, purple and red lines, respectively, in Fig. 1. Each satellite is assigned a specific orbital slot number based on its location within the orbit plane. The orbital slot numbers, such as A02 for satellite C207 in orbit plane A, are summarized in Table 1. Additionally, we have added the information, including the COSPAR-ID, orbit latitudes and satellite mass.

Table 1 Information of operational BDS-3 MEO satellites available for the POD processing
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For the SLR validation, the satellite identification (ID) number provided by the International Laser Ranging Service (ILRS) and the SLR retroreflectors are listed in Table 1. SLR data are used as a quality calibration due to their high accuracy and unambiguous nature (Marini and Murray 1973). Concerning the SLR validation, SLR residuals are calculated as the difference between the associated SLR measurements and the range from the microwave-based satellite position to a specific SLR station. Unlike orbit overlaps, the SLR validation is an external indicator for assessing orbit accuracy. BDS-3 MEO C202, C206, C207, C208 satellites equipped with laser retroreflector arrays can be tracked by the SLR stations from the ILRS (Pearlman et al. 2002). The SLR station coordinates are fixed to a priori reference frame SLRF2014, and the station displacements are corrected consistently with the microwave-based solution. The tropospheric delays, relativistic effects and the laser retroreflector array offsets with respect to the center of mass of satellites are corrected in the SLR measurements (Petit and Luzum 2010).

The PANDA (Positioning And Navigation Data Analyst) software package has been steadily improved over decades since it was developed at Wuhan University (Liu and Ge 2003; Shi et al. 2008; Guo et al. 2018; Zhao et al. 2018). The PANDA software package can be flexibly and accurately extended for supporting new features. Currently, the software can support data processing for various purposes, for example GNSS and low Earth orbit (LEO) satellites POD, PCE, precise point positioning (PPP), with a comparable accuracy as most of the IGS Analysis Centers’ software packages (Zhao et al. 2005; Guo et al. 2017). All the GNSS data processing is performed based on the PANDA software package in this study.

Processing strategy

For the POD processing, 48-h data with 600 s sampling rate and 7° cutoff elevation angle are processed. The coordinates of ground stations are constrained to IGS published station coordinates of the weekly solutions in order to achieve an alignment with respect to a well-defined reference frame IGS-14 (ftp://igs-rf.ensg.eu/pub/IGS14/). The constraints are set to 0.2 mm for the station coordinates. After the preprocessing, about 250 IGS ground stations, of which about 100 stations track dual-frequency BDS-3 signals, are used to estimate initial orbit state parameters. The important aspects regarding parameters, force models and observation modeling are listed in Table 2. The zenith hydrostatic tropospheric delay is precisely determined by using the Saastamoinen model (Saastamoinen 1973) as refined by Davis et al. (1985). The residual zenith wet tropospheric delays (ZWDs) are estimated based on the Global Pressure and Temperature 3 (GPT3) model (Landskron and Böhm 2018). Using the empirical mapping coefficients derived from Vienna Mapping Function 3 (VMF3), the hydrostatic and wet mapping functions could be provided by the GPT3 model (Landskron and Böhm 2018). The associated coefficients are derived from monthly mean pressure-level data of European Centre for Medium-Range Weather Forecasts (ECMWF) Reanalysis-Interim (ERA-Interim) fields (https://apps.ecmwf.int/datasets/data/interim-full-mnth/levtype=pl/). As the correction of higher-order ionospheric delays remaining in the dual-frequency ionosphere free (IF) combined observations is needed for multi-GNSS POD, the second- and third-order ionospheric items are considered as well (Chen et al. 2019).

Table 2 Processing strategy for the BDS-3 POD
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For BDS-3 satellites, the satellite-end IF code biases are considered to be zero in the IGS satellite clock datum definition. For each BDS-3 tracking ground station, the inter-system biases (ISB) are estimated in the GPS/BDS-3 integrated POD processing. The IF float ambiguity is expressed as the combination of wide and narrow lane for ambiguity fixing (Ge et al. 2008). The block-specific satellite phase center offset (PCO) values provided by the Test and Assessment Research Center of China Satellite Navigation Office (CSNO/TARC) are applied into the BDS-3 satellite antenna phase center correction (https://lists.igs.org/pipermail/igsmail/2019/007778.html).

The BDS-3 MEO satellites enter the shadow at the Sun elevation angle of about ± 12.9°, which represents the eclipse period. Over the whole eclipse period, results (Fig. 3) confirm that a significant degradation of orbit quality occurs in eclipse when the Sun elevation angle stays at [− 4°, 4°], which is referred to as a deep eclipse in the following. For CAST and SECM satellites, ECOM and ECOM2 have different performances over the deep eclipse (ref. Figure 2). This degradation in deep eclipse may be caused by an asymmetric satellite body, self-shadowing effects, potential radiator effect, thermal radiation and others (Arnold et al. 2015; Duan et al. 2020). Because of the uncertainty for these details, we focus on the enhancement of the modeling of perturbing forces to significantly reduce these effects in deep eclipse.

Fig. 2
figure 2

RMS values of overlapping orbit differences during the non-eclipsing (top) and deep eclipse (bottom) periods in 2019 for BDS-3 satellites

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Apart from the orbit validation, we further evaluate the orbit prediction performance using different SRP models. After each session orbit determination is completed, we can achieve the initial orbit state information, including the satellite positions, satellite velocities and the force model parameters. Using the initial orbit state derived from orbit determination, the predicted orbits can be generated through the orbit integrator. For the orbit integration, a combination of the Runge–Kutta and Admas–Bashforth–Moulton numerical methods is applied (Nordsieck 1962; Munthe-Kaas 1999). The starting point is provided by using the single-step Runge–Kutta method for the following multi-step integration. Then, the multi-step Admas–Bashforth–Moulton method is used for predicting and correcting the orbit integration in order to predict orbits forwards.

Results

We confirm that the ECOM and ECOM2 are not suitable for BDS-3 satellites and then analyze the contribution of model parameters for further model improvement. Afterward, the feasibility of the adapted SRP model is demonstrated by the overlapping orbit differences and SLR validation. Additionally, results show that the orbit prediction can be improved over the eclipsing periods due to the adapted SRP model.

Orbit quality of ECOM and ECOM2 models

Orbit differences of the overlapping period of two adjacent solutions are employed to assess the quality of the BDS-3 satellite orbits using the ECOM and ECOM2 models. Orbit overlaps could be regarded as an indicator of internal consistency and used for evaluating the orbit precision. An overlapping orbit arc of 24 h is compared for the two adjacent 48-h orbit solutions. RMSs of the overlapping orbit differences are calculated for estimated orbits and they are shown in Fig. 2 for the non-eclipsing and deep eclipse periods, respectively. The non-eclipsing period means the time period for a satellite when the Sun elevation angle is not within [− 12.9°, 12.9°].

From Fig. 2, ECOM and ECOM2 have similar performance especially in the cross-track and along-track components in terms of RMS of overlapping orbit differences over the non-eclipsing period, with ECOM having a slightly smaller RMS for both types of satellites in the radial-track component. However, for BDS-3 satellites within deep eclipse periods, the RMSs of both SRP models are increased obviously. On average, for SECM satellites, ECOM2 shows a rather large RMS in the radial direction, whereas ECOM is worse than ECOM2 in the other two directions. For CAST satellites, the precision of ECOM are degraded in comparison with ECOM2.

Figure 3 shows the time series of overlapping orbit differences for the BDS-3 CAST and SECM satellites. A large disagreement of overlapping orbits occurs at the boundary of adjacent days when either ECOM or ECOM2 is used for deep eclipses. For the CAST satellite C202, the fluctuations of overlapping orbit differences can reach up to 20 cm in the radial-track component using the ECOM model, and on average, the fluctuations of ECOM2 are larger than that of ECOM outside the eclipse period. For SECM satellite C208 inside deep eclipses, ECOM has fluctuations of about 20 and 50 cm in the cross-track and along-track components, respectively, while the fluctuations are within 10 and 20 cm, respectively, for the ECOM2 model. These results mean that neither ECOM nor ECOM2 can provide precise orbits for BDS-3 satellites and thus should be significantly improved at least in the deep eclipse periods.

Fig. 3
figure 3

Time series of overlapping differences for estimated orbits during the time period from DOY 22 to 166 in 2019 for BDS-3 CAST (i.e., C202, top) and SECM (i.e., C208, bottom) satellites. The eclipse periods are depicted by the shaded areas

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Parameterization analyses over deep eclipse

In order to assess the significance of SRP model parameters, we carry out a numerical evaluation by performing POD with different sets of parameters. Apart from the different arguments for ECOM and ECOM2, their difference is that ECOM2 has the additional second- and fourth-order harmonic terms in the D direction. To analyze the impact of these terms, POD solutions are conducted by using four parameterizations in the D direction of the ECOM2 model, as indicated in Table 3. From their overlapping orbit RMS during the eclipsing periods, we are able to know whether and how a SRP parameter contributes to POD. RMSs for the four SRP models over the deep eclipsing periods are summarized in Fig. 4.

Table 3 Comparison of schemes for the SRP modeling
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Fig. 4
figure 4

RMS values of the overlapping differences for estimated orbits derived from the POD schemes using the ECOM2 model, M1, M2 and the adapted model, respectively, during the deep eclipse periods in 2019 for BDS-3 satellites

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Compared with the ECOM2 model, the RMS in the radial direction is significantly reduced from 3.7 to 2.4 cm when the cosine terms are excluded (M1). Using M2 (further ignoring the second-order sine term in M1, only the fourth-order sine term is kept), the RMS in the radial-track component can be improved by about 0.5 cm compared to M1. However, the cross-track and along-track components of M2 still have RMSs of about 4 and 8 cm, respectively. With the adapted model, the RMS values are reduced by 0.3, 1.2 and 2.5 cm in the radial-track, cross-track and along-track components, respectively, compared to that for model M2.

In order to further confirm the significance of the fourth- (Ds4) and sixth-order (Ds6) sine terms, the estimates are compared with their STDs. The STD is the square root of variance for the estimate. From the perspective of adjustment, the significance of the estimate could be confirmed when the estimate exceeds its STD value. Figure 5 shows the daily estimates and their STD during the deep eclipse periods from DOY 150 to 360 in 2019. For Ds4 and Ds6, most of the estimates exceed their STDs during the eclipse period. For all the BDS satellites in deep eclipse, the mean square root of variance and the mean estimate for the high-order sine terms are summarized in Table 4. The mean square root of variance for the CAST satellites is 0.5 and 0.7 nm/s2 for Ds4 and Ds6, respectively. Their mean estimate can reach about 3.2 and 5.1 nm/s2, respectively. For the SECM satellites, Ds4 has a mean estimate of 1.3 nm/s2, which exceeds its mean square root of the variance of 0.5 nm/s2. The mean square root of the variance and the mean estimate for Ds6 are 0.6 and 4.0 nm/s2, respectively.

Fig. 5
figure 5

Daily estimates and square root of the variance for the fourth- (left) and sixth-order (right) sine SRP terms in the D direction for the BDS-3 CAST satellite, i.e., C202 (top) and the BDS-3 SECM satellite, i.e., C208 (bottom)

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Table 4 Mean estimate and the mean square root of variance for Ds4 and Ds6, respectively, for BDS satellites over the deep eclipsing period (unit: nm/s2)
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Validation of the empirical SRP models

In order to demonstrate the advantage of the adapted SRP model, the POD data processing is also carried out using the adapted SRP model with all the other processing parameters kept the same as for POD using ECOM and ECOM2 models. The overlapping orbit RMS is calculated in the same way for non-eclipsing and deep eclipsing periods, and is plotted in Fig. 6 together with that of ECOM and ECOM2 for comparison.

Fig. 6
figure 6

RMS values of overlapping orbit differences during the non-eclipsing (top) and deep eclipse (bottom) periods in 2019 for BDS-3 satellites

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For the non-eclipsing period, there is no big difference between each other in the cross- and along-track components for the three orbit solutions, except that ECOM2 has a slightly worse degradation in the radial-track component. For the CAST satellites, the averaged RMS values of overlapping orbit differences over the deep eclipse periods are 3.9, 7.3 and 15.9 cm for the ECOM model and 3.0, 3.0 and 5.9 cm for the ECOM2 model in the radial-track, cross-track and along-track component, respectively. Using the adapted SRP model, the orbit precision of BDS-3 CAST satellite orbits can be improved to 1.6, 2.7 and 4.6 cm in the three track-related components. For BDS-3 SECM satellites, the orbit precision are 2.2, 5.0 and 9.8 cm for the ECOM model and 4.3, 3.2 and 5.8 cm for the ECOM2 model in the radial-track, cross-track and along-track component, respectively. The precision of orbit differences can be improved to 1.6, 2.9 and 5.3 cm, respectively, in the three directions due to the adapted SRP model.

Figure 7 shows the time series of overlapping orbit differences using ECOM, ECOM2 and the adapted SRP model. For the CAST satellites with deep eclipse, the large fluctuations using ECOM or ECOM2 models in the radial direction are maintained within 10 cm through the adapted SRP model. The fluctuations in the cross-track and along-track components using the ECOM model which is of about 20 and 50 cm, respectively, are reduced by the adapted model to the magnitude of about 10 and 20 cm, respectively. For the BDS-3 SECM satellites, more significant fluctuations of about 40 cm in the radial direction using the ECOM2 model is reduced by the adapted SRP model to less than 10 cm over the deep eclipse period. Similarly, the ECOM errors of about 20 and 50 cm, respectively, in the cross-track and along-track components are mostly reduced to the level of that of the ECOM2 model.

Fig. 7
figure 7

Time series of overlapping differences during the time period from DOY 22 to 166 in 2019 for BDS-3 CAST (i.e., C202, top) and SECM (i.e., C208, bottom) satellites. The eclipse periods are depicted by shaded areas

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Figure 8 shows the time series of SLR residuals for orbits derived from the ECOM, ECOM2 and the adapted SRP model, respectively. For the BDS-3 CAST satellite, the ECOM has significant systematic effects within the deep eclipsing area compared to the non-eclipsing part. For the BDS-3 SECM satellite, larger systematic effects are visible in the SLR residuals of ECOM2 compared to that for other two models. To compare the ECOM, ECOM2 and the adapted SRP orbit solutions, the average statistical results of all satellites are listed in Table 5. For BDS-3 CAST satellites, the STD and RMS values of SLR residuals are about 11 cm in the case of ECOM. The accuracy can be improved by a factor of up to 2 due to the adapted SRP model. For BDS-3 SECM satellite, the RMS value of SLR residuals is 8.6 cm for the ECOM2 model. Using the adapted SRP model, the RMS value can be improved by about 2 cm. For the non-eclipsing period, a loss of accuracy of about 0.6 cm is visible for the ECOM2 orbit solution compared to that for the adapted SRP model, which confirms a slight degradation in the precision of estimated orbits, as shown in Fig. 6.

Fig. 8
figure 8

SLR residuals of estimated orbits during the time period from DOY 150 to 360 in 2019, for the BDS-3 CAST (i.e., C202) and BDS-3 SECM (i.e., C208) satellite, respectively. The eclipse periods are depicted by the shaded areas

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Table 5 SLR validations of estimated orbits over the deep eclipsing period from DOY 233 to 310 and the non-eclipsing period from DOY 20 to 360 in 2019 (unit: cm)
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Figure 9 illustrates the SLR residuals of the BDS-3 satellites with respect to the elongation angle for the ECOM, ECOM2 and the adapted ECOM orbit solutions, respectively. SLR residuals are fitted by a polynomial of 6 degrees, which is depicted as a red line. This fitting line highlights the fluctuations in the slope of SLR residuals with respect to the elongation angle. If the slope becomes steeper, it indicates a greater occurrence of orbital errors in the radial direction. For the CAST satellite, the slope of SLR residuals becomes steeper for the ECOM model when the elongation angle is close to 0°. For the SECM satellite, the ECOM2 model exhibits a steeper slope over the elongation angle span of [120°,180°]. The adapted SRP model can significantly reduce the steepness of these slopes, which confirms its positive contribution.

Fig. 9
figure 9

SLR residuals as a function of the elongation and the Sun elevation angles (color) for the BDS-3 CAST (i.e., C202, top) and BDS-3 SECM (i.e., C208, bottom) satellites, respectively. The vertical color bar indicates the absolute value of the Sun elevation angle

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Orbit prediction

Results show that the BDS-3 satellite orbit during eclipse periods estimated by actual observation can be improved by the new SRP model. Thus, the orbits predicted based on the estimated ones can also benefit from the adapted SRP model. In order to assess the quality of the predicted orbits, they are compared with the estimated orbits from the next POD with one-day latency. In the two-day orbit solution, the middle part of the estimated orbits is taken as a reference for comparing the predicted orbits. The schematic for the assessment of predicted orbits is illustrated in Fig. 10. The 6-h orbit arc predicted from the starting point of orbit prediction is usually available for real-time services. Averaged RMS values of overlapping orbit differences are summarized in Table 6 for the 6-h predicted orbit arc. Using the adapted SRP model, the averaged RMS values of 6-h predicted orbits are about 3, 5 and 10 cm in the radial-track, cross-track and along-track components during the non-eclipsing periods. For eclipse periods, the averaged RMS values for CAST satellites are about 7, 14 and 26 cm in the three directions, respectively, using the ECOM model. The ECOM2 could obtain the predicted precision of about 6, 6 and 13 cm in the three directions for SECM satellites. The three-dimensional averaged precision for all satellites could reach about 3, 5 and 12 cm, respectively, due to the adapted SRP model over the deep eclipse period.

Fig. 10
figure 10

Schematic for the assessment of predicted orbits in the 2-day orbit solution

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Table 6 Averaged RMS values of overlapping differences for 6-h predicted orbit arc over the non-eclipsing and deep eclipsing time periods from DOY 20 to 360 in 2019 (unit: cm)
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Conclusions

In this contribution, we evaluated the ECOM and ECOM2 SRP models for BDS-3 POD and found significant fluctuations in the overlapping orbit differences within the eclipsing periods. On average, ECOM2 has worse performance in the radial direction for the BDS-3 SECM satellites, whereas ECOM degrades the agreement of the overlapping orbits in the cross-track and along-track directions. After we confirmed that ECOM and ECOM2 must be optimized for BDS-3 satellites, the periodic terms of the ECOM2 model parameters in the solar direction, i.e., the D direction, are carefully analyzed. The significances of the fourth-order and sixth-order sine terms along the D direction have been confirmed by the parameterization analyses over deep eclipse periods. Therefore, an adapted SRP model with the fourth- and sixth-order sine terms in the D direction is developed for BDS-3 satellites. The adapted SRP model is then evaluated through a comparison of the overlapping orbit differences and SLR observations. The result shows that the fluctuations of overlapping orbit differences in the radial direction using ECOM or ECOM2 can be reduced from 20 cm to less than 10 cm and the corresponding SLR residuals are also improved by a factor of 2 due to the adapted SRP model. For the cross-track and along-track components, large overlapping orbit differences can be reduced to within 10 and 20 cm for the adapted SRP model in comparison with that for ECOM. Furthermore, we investigate the impact of the adapted SRP model on orbit prediction. For the 6-h predicted arc within deep eclipses, the averaged RMS values for CAST satellites are about 7, 14 and 26 cm in the radial-track, cross-track and along-track components, respectively, using the ECOM model. For SECM satellites, the ECOM2 has the predicted precision of about 6, 6 and 13 cm in the three directions, respectively. Because of the adapted SRP model, for all satellites, the three-dimensional precision of the 6-h predicted orbit arc can be improved to about 3, 5 and 12 cm, respectively.