SISRE is defined as the difference between the true satellite position and time and those broadcast by a navigation message. In this concept, for each satellite, an individual SISRE comprises the vector of orbit errors \(d_{orb} = \left( {dA, dC, dR} \right)\) in the along-track \(\left( {dA} \right)\), cross-track \(\left( {dC} \right)\) and radial \(\left( {dR} \right)\) directions coupled with the clock errors \(dt\) at an arbitrary epoch. For statistics, the global average SISRE is computed as the RMS of the instantaneous SISRE for all satellites and epochs. Accordingly, the global average SISRE is an RMS statistic of the orbit and clock errors that are projected in the signal propagation direction (line-of-sight to the users), which can be simply expressed as

$$SISRE = \sqrt {(W_{R} \cdot dR - dT)^{2} + W_{A,C}^{2} \cdot \left( {dA^{2} + dC^{2} } \right)}$$

(1)

where \(W_{R}\) and \(W_{A,C}\) are the constellation-specific weighted factors (or projection factors) for the radial components and along/cross-track components, respectively.

To evaluate the orbit errors in the total positioning error budget, the orbit-only contribution to the SISRE is expressed as follows:

$$SISRE_{orbit} = \sqrt {W_{R}^{2} \cdot dR^{2} + W_{A,C}^{2} \cdot \left( {dA^{2} + dC^{2} } \right)}$$

(2)

Note that \(W_{R}\) and \(W_{A,C}\) are the satellite-specific value and epoch-specific value due to the time-varying geometry. For simplicity, we use the constant \(W_{R}\) of 0.98 for both constellations, while \(W_{A,C}^{2}\) is set to 0.016 and 0.020 for the Galileo constellation and GPS constellation, respectively [2].

The precise orbit and clock products with an accuracy of 2–3 cm provided by the International GNSS Service (IGS) [17] can be used as a reference for the broadcast ephemeris error statistics. However, the compatibility and consistency of the orbit and clock information should be taken into account when the comparing broadcast ephemeris and precise products.

Note that all precise orbit products provided by the IGS are referred to the center of mass (CoM) of the spacecraft, while the broadcast orbits of the Galileo and GPS are referred to the antenna phase center (APC) [18]. Consequently, the known phase center offset (PCO) for every satellite should be utilized for the correction of CoM to APC. In this study, the phase center locations for the E1 and E5a frequencies as provided in GSA [19] has been employed for this purpose. For the newly available launch-9 satellites, which do not have access to their PCO, the average PCO of nominal FOC satellites is applied for these four satellites.

Conventionally, the MGEX precise products provide the clock offset of Galileo based on the signal combination E5a/E1, while the clock references differ in two types of Galileo navigation messages. The first type of navigation message is the freely accessible navigation (FNAV) message, which is referred to the signal combination E5a/E1, and the second type of navigation message is the integrity navigation (INAV) message, which is referred to the combination E5b/E1. To be consistent with that of precise clock products, the MGEX differential code bias (DCB) estimates can be used to convert the clock offset of the INAV message \(\Delta t_{INAV}\) to the datum E5a/E1 as

$$\Delta t_{IF} \left( {E1,E5a} \right) = \Delta t_{INAV} \left( {E1,E5b} \right) + \frac{{f_{E5a}^{2} }}{{f_{E1}^{2} - f_{E5a}^{2} }}DCB_{C1XC5X}^{sat} - \frac{{f_{E5b}^{2} }}{{f_{E1}^{2} - f_{E5b}^{2} }}DCB_{C1XC7X}^{sat}$$

(3)

where \(\Delta t_{IF} \left( {E1,E5a} \right)\) is the clock offset of the precise product generated from the observation of the ionosphere-free (IF) combination E5a/E1; \(f_{{{\text{E}}1}}\), \(f_{{{\text{E}}5{\text{a}}}}\), and \(f_{{{\text{E}}5{\text{b}}}}\) are the carrier frequencies of signals E1, E5a, and E5b, respectively, and the code observations correspond to C1X, C5X, and C7X, respectively.

Similar with the time group delay (TGD) parameter in the navigation message of GPS and BDS, the broadcast group delay (BGD) is one of the Galileo broadcast parameters that is commonly employed to compensate the inter-frequency biases of satellites for single-frequency users. Every Galileo satellite broadcasts unique BGD offsets, i.e., \(BGD\left( {E1,E5a} \right)\) and \(BGD\left( {E1,E5b} \right)\), and they are contained on FNAV and INAV messages according to the Galileo interface control document (ICD) [20]. The direct relationships between BGD parameters and DCB estimates are expressed as follows:

$$\left\{ {\begin{array}{*{20}l} {BGD\left( {E1,E5a} \right) = - \frac{{f_{E5a}^{2} }}{{f_{E1}^{2} - f_{E5a}^{2} }}DCB_{C1XC5X}^{sat} } \hfill \\ {BGD\left( {E1,E5b} \right) = - \frac{{f_{E5b}^{2} }}{{f_{E1}^{2} - f_{E5b}^{2} }}DCB_{C1XC7X}^{sat} } \hfill \\ \end{array} } \right.$$

(4)

To investigate the quality of Galileo BGD parameters, DCB estimates with an accuracy of 0.1 ns provided by the German Aerospace Center (DLR) were applied as reference values. Figure 1 shows the statistics of the differences between BGDs and DCBs for each Galileo satellite from 2015 to 2018. The BGD parameters for the entire constellation show agreement with the DCB estimates. The mean biases of their differences are near zero and the standard deviation (STD) is ± 0.5 ns, which support the relationship between BGDs and DCBs, as shown in Eq. 4. The older IOV E11 shows the exception that the mean biases are 0.5 ns below zero, which can be attributed to the actual BGD that has changed from the ground calibration value over time.

Consequently, for single frequency positioning, the Galileo satellites clock \(\Delta t_{SV}\) can be corrected by BGD parameters or the more accurate DCB estimates with Eq. 5.

$$\left\{ {\begin{array}{*{20}l} {\Delta t_{SV} \left( {E1} \right) = \Delta t_{INAV} \left( {E1,E5b} \right) - BGD\left( {E1,E5b} \right)} \hfill \\ {\Delta t_{SV} \left( {E5b} \right) = \Delta t_{INAV} \left( {E1,E5b} \right) - (\frac{{f_{E1} }}{{f_{E5b} }})^{2} BGD\left( {E1,E5b} \right)} \hfill \\ {\Delta t_{SV} \left( {E5a} \right) = \Delta t_{FNAV} \left( {E1,E5a} \right) - (\frac{{f_{E1} }}{{f_{E5a} }})^{2} BGD\left( {E1,E5a} \right)} \hfill \\ \end{array} } \right.$$

(5)

Due to the lack of definition of BGD correction for E5 observation in Galileo ICD, we utilize the DCB estimates instead for \(\Delta t_{SV} \left( {E5} \right)\) compensation, that is,

$$\Delta t_{SV} \left( {E5} \right) = \Delta t_{FNAV} \left( {E1,E5a} \right) + \frac{{f_{E5a}^{2} }}{{f_{E1}^{2} - f_{E5a}^{2} }}DCB_{C1XC5X}^{sat} + DCB_{C1XC8X}^{sat} .$$

(6)

where C8X refers to the code observation of the E5 signal.