Introduction

The IGS (International GNSS Service) has been providing precise GPS satellite orbit and clock products for more than 20 years (Johnston et al. 2017). This service enables the PPP (precise point positioning) technique that allows a single receiver to achieve centimeter to millimeter level positioning accuracy for daily static solutions (Malys and Jensen 1990; Zumberge et al. 1997; Kouba and Héroux 2001). After a good understanding of biases from satellites and receivers, the PPP-AR (ambiguity resolution) technique is then possible. To serve PPP-AR applications, a number of IGS analysis centers have developed their own satellite bias products, for instance, CNES/CLS (Center National d'Etudes Spatiales/ Collecte Localisation Satellites), CODE (Center for Orbit Determination in Europe), EMR (Natural Resources Canada), ESA (European Space Agency), GFZ (GeoForschungsZentrum), JPL (Jet Propulsion Laboratory), TUG (Graz University of Technology), TUM (Technical University of Munich) and WUHN (Wuhan University) (Duan et al. 2021; Ge et al. 2005; Geng et al. 2012; Li et al. 2018; Loyer et al. 2012; Schaer et al. 2018; Uhlemann et al. 2015). To combine bias products from individual analysis centers, a new IGS working group (PPP-AR) was created at the IGS workshop 2018 in Wuhan, China. As shown by Banville et al. (2020), a preliminary combination of satellite bias and clock products from six analysis centers over one week is confirmed to be successfully achieved.

The PPP-AR technique is carried out by fixing wide-lane and narrow-lane ambiguities sequentially. CNES/CLS provides GPS satellite HMW bias products based on C1W, C2W, L1W, and L2W signals in the Hatch-Melbourne-Wübbena (HMW) linear combination and provides dedicated satellite clock products, including satellite narrow-lane biases. Since 2018 CNES/CLS satellite products have been extended to Galileo satellites as well (Katsigianni et al. 2019). From December 2019, the CODE analysis center started to provide GPS/Galileo satellite orbit, integer clock and OSB products (Bock et al. 2009; Schaer et al. 2018; Villiger et al. 2019). Different than providing satellite HMW and narrow-lane biases, CODE provides bias products for individual signals. This contribution compares these two types of GPS satellite bias products and applies both of them for Sentinel-3 satellite orbit determination.

The Sentinel satellite missions are next-generation earth observation missions operated by the European Commission (EC) and the European Space Agency (ESA) as part of the Copernicus program. The goal is to support the joint ESA/EC initiative GMES (Global Monitoring for Environment and Security) (Aschbacher 2017). For example, to ensure accurate GMES services, orbit determination of Sentinel-3 satellites has a targeted uncertainty of less than 2 cm in the radial direction. The Sentinel-3 spacecraft, which are the focus of this research, hosts a POD (precise orbit determination) package comprising GPS and DORIS (Doppler Orbitography and Radiopositioning Integrated by Satellite) receivers and antennas, as well as a laser retroreflector (Jalabert and Mercier 2018; Štěpánek et al. 2020). The official precise Sentinel-3 satellite orbits are made available by the Copernicus Precise Orbit Determination (CPOD) Quality Working Group (QWG) based on onboard GPS measurements (Fernandez et al. 2016, 2014; Montenbruck et al. 2018; Peter et al. 2017). Kinematic (Jäggi et al. 2016) and reduced-dynamic (Bertiger et al. 1994, 2010; Bock et al. 2014; Flohrer et al. 2011; Montenbruck et al. 2005; Van Den IJssel et al. 2015) methods are optionally adopted by individual analysis centers within the CPOD QWG.

As part of CPOD QWG, we routinely generate Sentinel satellite orbit products based on the reduced-dynamic method. At the beginning, we used CODE final GPS satellite orbit and 5-s clock products. Sentinel data sampling was 10 s and we provided float-ambiguity orbit solutions. Mid of 2018, the half-cycle issue of carrier phase observations was corrected in the Sentinel-3 RINEX (Receiver INdependent EXchange format) file, and we started to provide ambiguity-fixed Sentinel-3 satellite orbits using CNES/CLS satellite products. To avoid long-time interval satellite clock interpolation, we reduced measurement sampling to 30 s, the clock product sampling provided by CNES/CLS. From the end of 2019, CODE published GPS satellite phase bias and 5-s sampling clock products. We can compare the performances of both types of products for Sentinel-3 satellite orbit determination. Also, we have the possibility to increase the Sentinel-3 data sampling again to 10 s. Therefore, the main goal of this contribution is as follows. First, describe satellite biases in undifferenced ambiguity resolution. Second, compare CNES/CLS products to CODE products and apply both of them for Sentinel-3 satellite orbit determination. Finally, analyze Sentinel-3 satellite orbits computed by different GPS products and different data samplings.

PPP-AR (ambiguity resolution)

We describe pseudorange (\(P\)) and phase (\(L\)) measurements between one receiver (subscript \(r\)) and one satellite (superscript \(s\)) on frequency \(l\) as

$$\begin{aligned} P_{r,l}^{s} & = \rho_{r,l}^{s} + c\left( {dt_{{\text{r}}} - dt^{{\text{s}}} } \right) + c\left( {d_{r,l} - d_{l}^{s} } \right) + I_{r,l}^{s} + e_{r,l}^{s} \\ L_{r,l}^{s} & = \rho_{r,l}^{s} + c\left( {dt_{{\text{r}}} - dt^{{\text{s}}} } \right) + c\left( {b_{r,l} - b_{l}^{s} } \right) - I_{r,l}^{s} + \lambda_{l} N_{r,l}^{s} + \varepsilon_{r,l}^{s} \\ \end{aligned}$$
(1)

where \(\rho\) represents the geometric distance, \(c\) the speed of light, \(dt_{{\text{r}}}\) and \(dt^{{\text{s}}}\) the receiver and satellite clock offsets, \(I\) the ionospheric delay, \(\lambda\) the wavelength, \(N\) the phase ambiguity, \(e_{r}^{s}\) and \(\varepsilon_{r}^{s}\) the other corrections and error sources of pseudorange and phase observations, for instance, tropospheric delays or multipath errors. Furthermore, \(d\) and \(b\) represent code and phase biases. Since \(I\) is dispersive, ionosphere-free (\({\text{IF}}\)) linear combination of dual-frequency measurements is widely used to eliminate the first-order ionospheric delays,

$$\begin{aligned} P_{{r,{\text{IF}}}}^{s} & = \rho_{{r,{\text{IF}}}}^{s} + c\left( {{\text{d}}t_{r} - dt^{s} } \right) + e_{{r,{\text{IF}}}}^{s} \\ L_{{r,{\text{IF}}}}^{s} & = \rho_{{r,{\text{IF}}}}^{s} + c\left( {{\text{d}}t_{r} - dt^{s} } \right) + c\left( {b_{{r,{\text{IF}}}} - b_{{{\text{IF}}}}^{s} } \right) + \frac{1}{{f_{1}^{2} - f_{2}^{2} }}\left( {f_{1}^{2} \lambda_{1} N_{r,1}^{s} - f_{2}^{2} \lambda_{2} N_{r,2}^{s} } \right) + \varepsilon_{{r,{\text{IF}}}}^{s} \\ \end{aligned}$$
(2)

where subscript \({\text{IF}}\) denotes the ionosphere-free combination. The \({\text{IF}}\) code biases are assumed to be zero according to the IGS clock datum definition. Receiver phase bias (\(b_{{r,{\text{IF}}}}\)) is usually estimated together with receiver clock offset (\({\text{d}}t_{{\text{r}}}\)) as a common parameter (\({\text{d}}\hat{t}_{{\text{r}}}\)). In this case, the code receiver clock offset differs from that of the phase. However, the difference is within one narrow-lane cycle and can be neglected except for time transfer applications. Satellite phase bias (\(b_{{{\text{IF}}}}^{s}\)) is considered to be part of the ambiguity parameter in the float solution. So, in principle, there are two reasons preventing an \({\text{IF}}\) ambiguity from an integer value. First, that the satellite phase bias is unknown, second, that the \({\text{IF}}\) ambiguity itself cannot be expressed in the form \(\lambda_{{{\text{IF}}}} N_{{{\text{IF}}}}\) where \(N_{{{\text{IF}}}}\) is an integer ambiguity. To resolve the \({\text{IF}}\) ambiguity, the widely used method is to introduce the integer wide-lane ambiguity (\(N_{wl} = N_{1} - N_{2}\)) into \({\text{IF}}\) ambiguity and then try to fix the deduced narrow-lane ambiguity,

$$\frac{1}{{f_{1}^{2} - f_{2}^{2} }}\left( {f_{1}^{2} \lambda_{1} N_{r,1}^{s} - f_{2}^{2} \lambda_{2} N_{r,2}^{s} } \right) = c\frac{{f_{2} }}{{f_{1}^{2} - f_{2}^{2} }}N_{r,wl}^{s} + \frac{c}{{f_{1} + f_{2} }}N_{r,1}^{s}$$
(3)

where the second term \(N_{r,1}^{s}\) on the right-hand side is the narrow-lane ambiguity, with a wavelength of about 10.7 cm for GPS frequency \(f_{1}\) and \(f_{2}\). Therefore, an \({\text{IF}}\) ambiguity can be resolved by fixing wide- and narrow-lane ambiguities to integer values sequentially.

To resolve wide-lane ambiguities, the Hatch-Melbourne-Wübbena (HMW) linear combination is widely used (Hatch 1982, Melbourne 1985; Wübbena 1985).

$$\begin{aligned} {\text{MW}}\left( {P_{r,1}^{s} ,P_{r,2}^{s} ,L_{r,1}^{s} ,L_{r,2}^{s} } \right) & = \frac{{f_{1} L_{1} - f_{2} L_{2} }}{{f_{1} - f_{2} }} - \frac{{f_{1} P_{1} + f_{2} P_{2} }}{{f_{1} + f_{2} }} \\ & = \lambda_{{{\text{wl}}}} N_{{r,{\text{wl}}}}^{s} + \lambda_{{{\text{wl}}}} ({\text{mw}}_{{r,{\text{wl}}}} - {\text{mw}}_{{{\text{wl}}}}^{s} ) + \varepsilon_{{r,{\text{wl}}}}^{s} \\ \end{aligned}$$
(4)

where \(mw_{{r,{\text{wl}}}}\) and \({\text{mw}}_{{{\text{wl}}}}^{s}\) denote the HMW bias for receiver and satellite. The HMW linear combination is both geometry- and ionosphere-free. Only the ambiguity parameter and the HMW bias terms remain in the equation. To resolve the wide-lane ambiguity \(N_{{r,{\text{wl}}}}^{s}\), satellite HMW bias \({\text{mw}}_{{{\text{wl}}}}^{s}\) needs to be considered in advance. CNES/CLS computes daily satellite HMW biases based on GPS C1W, C2W, L1W, L2W signals. Stations with only C1C signal are corrected to C1W in advance using CODE DCB products (Dach et al. 2009). Different than that from CNES/CLS, CODE first computes OSB correction of each code signal, for instance, C1C, C1W and C2W (Villiger et al. 2019). Then, OSB corrections of C1W and C2W are applied in the HMW linear combination. The determined HMW bias products are then assumed to contain only phase biases. We need to mention that this assumption is not physically true but is consistent with satellite clock products.

While wide-lane ambiguities are fixed, narrow-lane ambiguities can be deduced by using (3). To fix narrow-lane ambiguities, we need to consider satellite phase biases (\(b_{{{\text{IF}}}}^{s}\)). CNES/CLS computes dedicated satellite clock products, including satellite phase biases, whereas CODE computes daily satellite narrow-lane biases \(b_{IF}^{s}\) independently from satellite clock products. Then, by combining with the determined CODE satellite HMW bias products, satellite phase biases are solved on each phase signal. Details regarding CODE phase bias and integer clock estimates will be soon published by CODE (from the header of the CODE daily bias file).

Comparison of CNES/CLS and CODE satellite bias products

There are small differences when applying CODE and CNES/CLS products for PPP-AR applications. We use GPS satellite HMW biases from CNES/CLS in the HMW linear combination as

$${\text{MW}}\left( {P_{{r,{\text{C1W}}}}^{s} ,P_{{r,{\text{C2W}}}}^{s} ,L_{{r,{\text{L1W}}}}^{s} ,L2_{{r,{\text{L2W}}}}^{s} } \right) + \lambda_{{{\text{wl}}}} \left( {{\text{mw}}_{{{\text{wl}}}}^{s} } \right) = \lambda_{{{\text{wl}}}} N_{{r,{\text{wl}}}}^{s} + \lambda_{{{\text{wl}}}} ({\text{mw}}_{{r,{\text{wl}}}} ) + \varepsilon_{{r,{\text{wl}}}}^{s}$$
(5)

where \(P_{{r,{\text{C1W}}}}^{s}\), \(P_{{r,{\text{C2W}}}}^{s}\),\(P_{{r,{\text{L1W}}}}^{s}\),\(P_{{r,{\text{L2W}}}}^{s}\) are code and phase observations on individual signals. CNES/CLS GPS satellite HMW bias products (\({\text{mw}}_{{{\text{wl}}}}^{s}\)) are used to correct HMW measurements. The receiver HMW biases \({\text{mw}}_{{r,{\text{wl}}}}\) are estimated as epoch-wise parameters to capture the potential variation of receiver HMW biases. In this case, we need to select a reference ambiguity to cope with the singularity between receiver HMW biases and wide-lane ambiguities. The reference ambiguity is directly fixed to the closest integer value and all the other wide-lane ambiguities (\(N_{{r,{\text{wl}}}}^{s}\)) can be fixed accordingly. The \({\text{IF}}\) float solution using CNES/CLS products can be run in parallel to the wide-lane ambiguity resolution,

$$\begin{gathered} P_{{r,{\text{IF}}}}^{s} + c{\text{d}}\hat{t}^{s} = \rho_{{r,{\text{IF}}}}^{s} + c{\text{d}}t_{r} + e_{{r,{\text{IF}}}}^{s} \hfill \\ L_{{r,{\text{IF}}}}^{s} + c{\text{d}}\hat{t}^{s} - c\frac{{f_{2} }}{{f_{1}^{2} - f_{2}^{2} }}N_{{r,{\text{wl}}}}^{s} = \rho_{{r,{\text{IF}}}}^{s} + c{\text{d}}\hat{t}_{r} + \frac{c}{{f_{1} + f_{2} }}N_{r,1}^{s} + \varepsilon_{{r,{\text{IF}}}}^{s} \hfill \\ \end{gathered}$$
(6)

where \({\text{d}}\hat{t}^{s}\) is the CNES/CLS satellite clock product (including satellite phase bias), \({\text{d}}\hat{t}_{r}\) includes both receiver clock offset and receiver phase bias. As we explained above (3), differences between receiver clock offset in pseudorange and phase equations are within one narrow-lane cycle and are generally tolerable considering the noise and multipath error of pseudorange measurements. Thus, we estimate a common receiver clock offset for both phase and pseudorange measurements. All the fixed wide-lane ambiguities in (5) can be introduced into the \(IF\) float solution, and the deduced narrow-lane ambiguities can be resolved according to a certain ambiguity resolution algorithm (for instance, the LAMBDA method (Teunissen et al. 1997, 2002)).

For CODE products, we apply OSB corrections on each signal in the HMW linear combination as

$$\begin{aligned} {\text{MW}} & \left( {P_{{r,{\text{C1W}}}}^{s} + {\text{OSB}}_{{{\text{C1W}}}}^{s} ,P_{{r,{\text{C2W}}}}^{s} + {\text{OSB}}_{{{\text{C2W}}}}^{s} ,P_{{r,{\text{L1W}}}}^{s} + {\text{OSB}}_{{{\text{L1W}}}}^{s} ,P_{{r,{\text{L2W}}}}^{s} + {\text{OSB}}_{{{\text{L2W}}}}^{s} } \right) \\ & = \lambda_{{{\text{wl}}}} N_{{r,{\text{wl}}}}^{s} + \lambda_{{{\text{wl}}}} ({\text{mw}}_{{r,{\text{wl}}}} ) + \varepsilon_{{r,{\text{wl}}}}^{s} \\ \end{aligned}$$
(7)

where OSB is the correction of satellite biases on individual signals. Then, similar as for (5) wide-lane ambiguities can be fixed to integer values. The \({\text{IF}}\) float solution using CODE products is

$$\begin{gathered} P_{{r,{\text{IF}}}}^{s} \left( {P_{{r,{\text{C1W}}}}^{s} + {\text{OSB}}_{{{\text{C1W}}}}^{s} ,P_{{r,{\text{C2W}}}}^{s} + {\text{OSB}}_{{{\text{C2W}}}}^{s} } \right) + c{\text{d}}t^{s} = \rho_{r,IF}^{s} + c{\text{d}}t_{r} + e_{{r,{\text{IF}}}}^{s} \hfill \\ L_{{r,{\text{IF}}}}^{s} \left( {L_{{r,{\text{L1W}}}}^{s} + {\text{OSB}}_{{{\text{L1W}}}}^{s} ,L_{{r,{\text{L2W}}}}^{s} + {\text{OSB}}_{{{\text{L2W}}}}^{s} } \right) + c{\text{d}}t^{s} - c\frac{{f_{2} }}{{f_{1}^{2} - f_{2}^{2} }}N_{{r,{\text{wl}}}}^{s} = \rho_{{r,{\text{IF}}}}^{s} + c{\text{d}}t_{r} + \frac{c}{{f_{1} + f_{2} }}N_{r,1}^{s} + \varepsilon_{{r,{\text{IF}}}}^{s} \hfill \\ \end{gathered}$$
(8)

Satellite phase biases of each signal are corrected by OSB values. All the fixed wide-lane ambiguities in (7) are introduced as known and the deduced narrow-lane ambiguities can be fixed to integer values. We would like to mention that integer wide-lane ambiguities in (8) may differ by integer cycles compared to those in (6). Thus, this must have to be carefully considered if we intend to mix or combine the usage of CNES/CLS and CODE bias products.

The comparison of satellite HMW biases is easily possible since they are independent of satellite orbits and clock offsets. Figure 1 shows the daily GPS satellite HMW bias products from CNES/CLS (left panel), CODE (middle panel) and the difference between the two (right panel). The time interval covers doy (day of year) 266–365 2019. CNES/CLS GPS satellite HMW bias products are nearly constant over time, with a mean STD of 0.019 wide-lane cycles. In contrast, CODE GPS satellite HMW bias products are discontinuous from day to day. The reason is that CNES/CLS constrains satellite HMW biases tightly to the estimates of the previous day while CODE satellite HMW bias products seem to be independent from day to day. If we remove the daily datum difference (for instance using G01 as reference satellite) from CODE HMW biases and subtract integer wide-lane cycles, then the mean STD of CODE GPS satellite HMW bias products is 0.020 wide-lane cycles. Similarly, in the comparison of CNES/CLS and CODE HMW bias products, we remove the reference and integer-cycle differences since only the fractional part is critical for the wide-lane ambiguity resolution. As shown in the right panel, differences are very close to zero, with a mean RMS of 0.01 wide-lane cycles.

Fig. 1
figure 1

CNES/CLS, CODE (formed by OSB values) daily GPS satellite HMW biases and the fractional differences (CNES/CLS-CODE) of these two. The reference and integer-cycles difference are removed from CNES/CLS-CODE. For CODE products, daily integer-cycles and reference differences are corrected when calculating STD value of each satellite

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A direct comparison of satellite narrow-lane biases is not easily possible since satellite narrow-lane biases are correlated with satellite orbit and clock products, as well as with integer wide-lane ambiguities. Moreover, CNES/CLS provides no satellite narrow-lane biases but incorporates them into satellite clock offsets. We use the following method to compare CODE and CNES/CLS products in the narrow-lane ambiguity resolution. As seen from (6) and (8), different GPS satellite bias products should result in different narrow-lane ambiguities.

$$\Delta \rho_{{r,{\text{IF}}}}^{s} + c\Delta {\text{d}}t^{s} + c\frac{{f_{2} }}{{f_{1}^{2} - f_{2}^{2} }}\Delta N_{{r,{\text{wl}}}}^{s} + {\text{IF}}\left( {{\text{OSB}}_{{{\text{L1W}}}}^{s} ,{\text{OSB}}_{{{\text{L2W}}}}^{s} } \right) = \frac{c}{{f_{1} + f_{2} }}\Delta N_{r,1}^{s}$$
(9)

where \(\Delta \rho_{{r,{\text{IF}}}}^{s}\) and \(\Delta {\text{d}}t^{s}\) denote differences of satellite orbits (in the radial direction) and clock offsets, \(\Delta N_{r,wl}^{s}\) denotes the difference of integer wide-lane ambiguity in cycles computed from CNES/CLS \(({\text{mw}}_{{{\text{wl}}}}^{s} )\) and CODE (\({\text{MW}}\left( {{\text{OSB}}_{{{\text{C1W}}}}^{s} ,{\text{OSB}}_{{{\text{C2W}}}}^{s} ,{\text{OSB}}_{{{\text{L1W}}}}^{s} ,{\text{OSB}}_{{{\text{L2W}}}}^{s} } \right)\)) bias products. \({\text{IF}}\left( {{\text{OSB}}_{{{\text{L1W}}}}^{s} ,{\text{OSB}}_{{{\text{L2W}}}}^{s} } \right)\) represents the \({\text{IF}}\) satellite phase biases using CODE OSB products and \(\Delta N_{r,1}^{s}\) denotes the resulting narrow-lane ambiguity difference. Since satellite products from CODE and CNES/CLS have different datum, for instance satellite clock and bias products, we take G01 as a reference and remove the reference from all the other satellites. If the narrow-lane ambiguity of a satellite can be fixed both in (6) and (8) by using CNES/CLS and CODE products, respectively, then the sum of the left hand side of (9) should be equal to certain integer narrow-lane cycles. Therefore, the fractional part of the total difference is a representation of the difference between CNES/CLS and CODE narrow-lane phase bias. We call the fractional part of the total difference “narrow-lane” bias difference. In fact, this method can be used between two CNES/CLS-like or two CODE-like products for any other constellation as well. Comparison can be made at any sampling that is greater than or equal to that provided by the clock products. Furthermore, because the comparisons are solely based on satellite products, we do not need to process any measurement.

Figure 2 shows daily differences of GPS satellite narrow-lane biases between CNES/CLS and CODE products. We observe some large differences every day, while most of the differences are very close to zero, with a mean RMS of 0.01 narrow-lane cycles. As informed by CNES/CLS within the CPOD QWG discussion, one or several GPS satellites in some days have no narrow-lane ambiguities fixed in the generation of satellite clock products. As a consequence, CNES/CLS satellite clock products do not contain narrow-lane biases for these satellites and users should not use them for undifferenced ambiguity resolution. We display all these satellites provided by CNES/CLS with blue dots in Fig. 2, which is about 3% of the total number. Obviously, we can detect all these satellites correctly by using our comparison method, and in fact, we find more satellites with large narrow-lane bias difference. With a detailed analysis, we find that the additionally detected outliers are mainly caused by the one Block IIA satellite (PRN 18, SVN 34 with green dots in Fig. 2) as satellite clock products of this satellite cannot be well determined, especially during eclipse seasons (Duan and Hugentobler 2021). Moreover, satellite clock products of Block IIA satellites from both analysis centers are not available during post-shadow recovery periods. The amount of narrow-lane bias differences larger than 0.05 narrow-lane cycles is about 3% of the total number. 2% is caused by the Block IIA satellite, while 1% is caused by other GPS satellites. All the satellites that CNES/CLS indicates to have no narrow-lane biases are not considered in the statistics. Therefore, it is best not to use this Block IIA satellite in the ambiguity resolution.

Fig. 2
figure 2

GPS satellite narrow-lane bias differences between CODE and CNES/CLS products. Blue dots denote satellites for which CNES/CLS confirmed that satellite clock products do not contain narrow-lane biases. Green dots represent differences for satellite PRN 18. RMS value is the RMS of differences excluding all the outliers

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Applications in Sentinel-3 satellite POD

As announced by Fernández (IGSMAIL-7886), precise orbit products, RINEX observation files and platform metadata of Sentinel-3 satellites were officially released on January 14, 2020. Users can access all the data from the ESA open access data Hub. As a short summary, we collect here some information that is significant for satellite POD. Sentinel-3 satellite reference frame is defined with + X-axis perpendicular to the launch vehicle interface plane and oriented from launch vehicle toward the satellite, + Z-axis parallel to the launch vehicle interface plane and pointing toward the perpendicular of the panel supporting the altimeter reflector, and Y-axis completing the right-handed orthogonal frame. Sentinel-3 satellite attitude can be determined by the 1-s sampling quaternions measured by the star trackers. The onboard GPS raw measurements are reformatted and available as RINEX file with a sampling of 1 s. The total mass of satellite, the center of gravity, location of GPS antenna and phase center variation (PCV) of Sentinel-3 satellites are shown in Table 1. The information on the mass varies slowly with time. Updated values can be found in the mass history file. Each Sentinel-3 satellite carries two GPS receivers and only one (GPSA) provides GPS measurements. The latest PCV values of the GPSA receiver are determined by ambiguity-fixed phase residuals (Montenbruck et al. 2018).

Table 1 Mass, center of gravity, location of GPS antenna, and phase center offset on August 9, 2019, from the Sentinel-3 POD document. X, Y and Z are in the satellite reference frame
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Optical and infrared properties of each Sentinel-3 satellite surface are given by Montenbruck et al. (2018). With this detailed information, solar radiation pressure (SRP) can be described by a box-wing macro model, as shown for one plane (Duan et al. 2020),

$${\text{acc}} = - \frac{A}{M}\frac{{S_{0} }}{c}\cos \theta \left[ {\left( {\alpha + \delta } \right)e_{{\text{D}}} + \frac{2}{3}\left( {\delta + \kappa \alpha } \right)e_{{\text{N}}} + 2\rho \cos \theta e_{{\text{N}}} } \right]$$
(10)

where \({\text{acc}}\) represents the acceleration, \(A\) the surface area, \(M\) the total mass of the satellite, \(S_{0}\) the solar flux, \(c\) the vacuum velocity of light, \(\kappa\) the thermal re-radiation factor (0 for solar panels and 1 for satellite body surfaces in this contribution), \(\alpha\), \(\delta\), \(\rho\) the fractions of absorbed, diffusely scattered, and specularly reflected photons. Furthermore, \({\mathbf{e}}_{{\text{D}}}\) denotes the Sun direction, \({\mathbf{e}}_{{\text{N}}}\) the surface normal vector, and \(\theta\) the angle between both vectors. Earth radiation is modeled similarly by using satellite infrared properties. The amount of earth-reflected visible solar radiation and thermal radiation is calculated from the CERES (Clouds and Earth’s Radiant Energy System) (Priestley et al. 2011). Atmospheric drag for each satellite surface is computed based on the MSISe-90 (extended Mass Spectrometer and Incoherent Scatter data) density model (Hedin 1991). Solar flux and geomagnetic activity data are obtained from the SWPC (Space Weather Prediction Center) of the US NOAA (National Oceanic and Atmospheric Administration). Orbit modeling options are given in Table 2.

Table 2 Orbit modeling options
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We have repeated our operational processing for Sentinel-3A/B satellites from doy 266 2019 to doy 365 2019 using CNES/CLS (grg) and CODE (cod) GPS satellite products, respectively. Satellites indicated by CNES/CLS to have no narrow-lane bias in the satellite clock products are removed from the ambiguity resolution and the associated ambiguities are estimated as float values. The Block IIA satellite PRN 18 is not used in the ambiguity resolution for both products. Settings of processing are shown in Table 3. Orbit arc length is 30 h, with 3 h of the previous day and 3 h of the next day. Days with maneuvers and large data gaps are excluded from the analysis. One solar radiation scaling factor, one drag coefficient, and one set of sine/cose empirical parameters in along- and cross-track directions are estimated for every arc. Velocity changes are introduced every 15 min in radial, along- and cross-track components with a constraint of 500 nm/s. Observation sampling is 30 s and we set observation sampling to 10 s as well when using CODE products. All the processing schemes are listed in Table 4. Ambiguity fixing rates, orbit overlaps, orbit differences compared to the external combined results (COMB), and SLR residuals are taken to assess the precision and accuracy of Sentinel-3 satellite orbits.

Table 3 Settings of Sentinel-3 POD
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Table 4 Processing schemes
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As shown in Fig. 1, the fractional parts of HMW biases for both products are very similar. Therefore, the wide-lane fixing rates are nearly identical for cod-30 s and grg-30 s solutions, which are both more than 98%. The narrow-lane fixing rates using individual products and observation samplings are shown in Fig. 3. The mean fixing rates are all above 95%. There is no clear difference between grg and cod products using 30 s sampling observations. The cod-10 s results show a slightly higher narrow-lane fixing rate of about 0.5% than for 30 s sampling results. The reason is that having more observations leads to higher reliability and robustness of parameter estimates, which is reflected by lower formal errors.

Fig. 3
figure 3

Narrow-lane ambiguity fixing rates for Senitnel-3A (S3A) and Sentinel-3B (S3B) satellites using grg and cod products for the measurement sampling of 30 s and 10 s

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Orbit overlap comparison is traditionally employed in orbit determination to assess the internal consistency of a specific processing scheme. We compute Sentinel-3 satellite orbits using 30 h of GPS measurements centered on the day of the daily solution. Each solution of one arc overlaps with the solution of the previous arc for 6 h. The RMS of the 6-h overlap differences is computed from the 3D position differences. As shown in Fig. 4, the 3D mean RMS values for 30-s solutions using grg and cod GPS products improve from 18–20 to 10–12 mm by fixing ambiguities to integer values. The cod-30 s-ambfix solution shows slightly better results than the grg-30 s-ambfix solution. This could be partly because of the usage of the same Bernese GNSS software as CODE. Both cod-10 s-float and cod-10 s-ambfix solutions show notable improvements in overlaps. However, this does not mean the corresponding orbits achieve the same improvement in precision. Because the constraints are kept fixed, the estimated pulses are relatively more determined by the observations in the case of 10 s sampling since 3 times more observations are then used. This results in higher variability of these pulses rendering the orbit slightly less dynamic than the cod 30 s orbit. Therefore, a smaller overlap RMS is expected. But still, the overlap comparison between float and ambiguity-fixed solutions of the same observation sampling can be used as an indicator to assess the orbit internal consistency. RMS of orbit overlaps improves from 13 to 7 mm by fixing ambiguities to integer values for 10 s observation sampling results.

Fig. 4
figure 4

6-h overlap consistency of 3D position difference for Sentinel-3A/B orbit solutions (mean RMS for doy 266–365/2019)

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Then, we take the combined Sentinel-3 orbits (COMB) of all the Sentinel analysis centers (AIUB, CLS, CNES, CPOD, DLR, ESOC, EUM, GFZ, JPL, TUD, TUM) as external true orbits to assess the accuracy of our individual solutions. 24-h daily orbits centered on the 30-h arc are compared to the COMB orbits of the same day. Mean RMS of orbit differences for doy 266–365/2019 is computed in radial, along- and cross-track directions, as displayed in Figs. 5 and  6. All the ambiguity-fixed solutions show notable improvements compared to the respective float solutions, especially in the along- and cross-track directions. For instance, the along-track mean RMS for the Sentinel-3A cod-30 s solution improves from 17 to 11 mm by fixing ambiguities to integer values. In addition, the cod-10 s-ambfix solution shows an overall improvement of about 10% over the cod-30 s-ambfix solution. So, it is helpful to increase measurement sampling to 10 s if 5-s integer GPS clock products are available.

Fig. 5
figure 5

Orbit differences in radial, along-track and cross-track components compared to the COMB orbits for Sentinel-3A satellite (mean RMS for doy 266–365/2019)

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Fig. 6
figure 6

Orbit differences in radial, along-track and cross-track components compared to the COMB orbits for Sentinel-3B satellite (mean RMS for doy 266–365, 2019)

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For a fully independent assessment of the orbit quality, SLR measurements (Pearlman and Degnan 2002; Pearlman et al. 2019) are used to compare with the expected satellite-station distance based on the given orbit solutions. The nine SLR stations (except Mt. Stromlo) and the LRR (Laser Retroreflector) information are the same as given by (Montenbruck et al. 2018). The screening threshold is 10 cm. Figures 7 and 8 show SLR residuals of Sentinel-3A and Sentinel-3B satellite orbits as a function of time. Ambiguity resolution improves SLR residuals for all cases. For Sentinel-3A, the STD value is improved by about 10%, 13% and 11% for grg-30 s-ambfix, cod-30 s-ambfix and cod-10 s-ambfix solutions compared to the respective float solutions. The improvement for Sentinel-3B is about 16%, 23% and 13% accordingly. The cod-10 s-ambfix solution does not show clear improvement compared to the cod-30 s-ambfix solution in SLR residuals. In general, both CNES/CLS and CODE GPS products work well for ambiguity resolution in Sentinel-3 satellite orbit determination.

Fig. 7
figure 7

SLR residuals of Sentinel-3A satellite orbits for doy 266–365, 2019

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Fig. 8
figure 8

SLR residuals of Sentinel-3B satellite orbits for doy 266–365, 2019

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Summary and conclusions

In this contribution, we compare CNES/CLS satellite bias products to those from CODE and apply both of them for Sentinel-3 satellite POD. CNES/CLS provides satellite HMW biases for the wide-lane ambiguity resolution and dedicated satellite clock products (including satellite phase bias) for the narrow-lane ambiguity resolution. CODE provides satellite bias products as OSB corrections on each code and phase signal. We find that the CNES/CLS satellite HMW biases are nearly constant over time because CNES/CLS daily HMW bias estimates are tightly constrained to those of the previous day. The CODE satellite wide-lane biases seem, however, to be independent from day to day. If we compare only the fractional part of satellite HMW biases between CNES/CLS and CODE we observe small differences, with a mean RMS of 0.01 wide-lane cycles. A direct comparison of satellite narrow-lane biases is, however, not easily possible. We compute the total difference caused by satellite orbits, clock offsets, integer wide-lane ambiguities and narrow-lane biases by using different GPS satellite products in the narrow-lane equation. Then, the resulting differences of narrow-lane ambiguities represent the difference of narrow-lane biases. We find that this method can detect all those satellites that are indicated by CNES/CLS to have no narrow-lane biases in the satellite clock products. Furthermore, we find that on average 3% more satellites have differences larger than 0.05 narrow-lane cycles, with 2% caused by the Block IIA satellite and 1% caused by other satellites. The reason is that Block IIA satellite clock offsets are not well determined during eclipse seasons.

Then, we apply both products for Sentinel-3 satellite POD. To avoid satellite clock interpolation, we first set observation sampling to 30 s, the clock products sampling provided by CNES/CLS. The narrow-lane ambiguity fixing rates using CNES/CLS and CODE products are both above 95%. 3D RMS of orbit overlap using CNES/CLS and CODE GPS products improves from 18–20 to 10–12 mm by fixing ambiguities to integer values. By comparing to the external COMB orbits, we observe similar improvements. When checking SLR residuals, the STD values of ambiguity-fixed Sentinel-3 satellite orbits using CNES/CLS and CODE products are reduced by about 10%, 13% for Sentinel-3A and 16%, 23% for Sentinel-3B, respectively. Therefore, both CNES/CLS and CODE GPS products lead to high ambiguity success-fixing rates in Sentinel-3 satellite POD.

In addition, we process 10 s sampling observations using CODE products since CODE clock products have a sampling of 5 s. The narrow-lane ambiguity fixing rate increases by 0.5% compared to that of the 30 s solutions. As a consequence, RMS, by comparing to the external COMB products, improves by another 10%. For the SLR residuals, 10 s solutions show almost the same STD values as that of 30 s solutions. High sampling observations can improve orbit accuracy mainly in the along-track direction due to the slightly higher narrow-lane ambiguity fixing rate.