Abstract
The concept of the signal-in-space (SIS) root-mean-square (RMS) user range error (URE) is used to evaluate the performance of multiple global navigation satellite systems (GNSSs); however, a complete analytical derivation has not been published. This article describes the instantaneous SIS URE and the instantaneous SIS RMS URE, explains the role of the instantaneous SIS RMS URE in evaluating the statistical accuracy of GNSS signals, and provides an analytical derivation of the instantaneous SIS RMS URE. This derivation is then compared to the equations found in various papers and performance standards to illustrate how the equations, although appearing different, actually measure the same quantity with differing constraints.
1 INTRODUCTION
The concept of the signal-in-space (SIS) root-mean-square (RMS) user range error (URE) has been used to evaluate the performance of the Global Positioning System (GPS) for many years (Renfro et al., 2021). The SIS RMS URE is specified as a performance metric in several global navigation satellite system (GNSS) performance standards, and equations for the SIS RMS URE can be found as Equation (A-1) in the GPS Standard Positioning Service Performance Standard (SPS PS) (U.S. Department of Defense, 2020) and Equation (1) in the Galileo Service Definition Document (SDD) (European Union, 2023). Although the concept is explained either conceptually (Zumberge & Bertiger, 1996; Malys et al., 1997) or alongside intermediate expressions (Dieter et al., 2003; Montenbruck et al., 2018) in various locations, a complete analytical derivation does not appear to have been published.
2 INSTANTANEOUS SIS URE AND SIS RMS URE
2.1 Instantaneous SIS URE
To motivate the need for an analytical derivation, one must first consider the instantaneous SIS URE. This is the URE along a particular line of sight (LOS) from a space vehicle (SV) to a point on the surface of the Earth. The instantaneous SIS URE is formed by computing the SV position and clock offset at the time of interest from both the broadcast (predicted) orbit and a precise (truth) orbit. The differences in the SV position are then projected along the LOS, as illustrated in Figure 1.
Prediction error of SV position compared with the true position
This figure also illustrates the sign convention used throughout this article:
Component error offsets are defined as the broadcast position/clock minus the precise position/clock.
A positive clock offset represents an ahead clock.
The positive radial direction is directed away from the Earth.
The positive along-track direction is in the same semi-plane as the SV velocity vector.
The positive cross-track direction is formed by the cross-product of the radial and along-track directions (shown coming out of the page in Figure 1).
Figure 2 presents the geometry for a given user location. In Section 3.1, this figure is used as the basis for deriving a formula for the SIS RMS URE. The figure illustrates the concept of the field of view of an SV. From Figure 2, it can be seen that the field of view can be defined by two parameters: (1) the orbit radius rs and (2) the minimum elevation angle α0 (not explicitly shown in the figure). Note that the selection of these two parameters defines a maximum nadir angle θmax. The relationship between the elevation angle α and the nadir angle θ is given in Equation (7).
Geometry of the satellite range ρ from the SV antenna phase center S to the user location U, with elevation angle α, nadir angle θ, Earth radius re, and orbital radius rs The angle ∠SEU, where point E is the center of the Earth, serves as an alternate polar angle θ′. Point X is the intersection between an extension of the line 
and the outward radial vector from E normal to this extended line. R denotes the radial direction. Not shown is the azimuthal angle ϕ, which is referenced to the along-track direction.
For the instantaneous SIS URE, the clock error component may be either positive or negative, but will be constant across the field of view. Orbit error components, however, will vary (and may change sign) across the field of view. The reason for this trend can be inferred from Figure 2. Radial errors will be nearly uniform across the field of view, with the projection of the radial error along the vector from the SV to the user starting from 1.00 at a nadir angle of 0°, decreasing as the nadir angle increases. Using a GPS satellite orbit as an example, the radial error projection decreases to 0.971 at a nadir angle of 13.87° (corresponding to a user elevation angle of 2°). This projection depends on the cosine of the nadir angle. The effects of the along-track and cross-track errors depend on the sine of the nadir angle, ranging from 0 at a nadir angle of 0° to a maximum of 0.240 at a nadir angle of 13.87° (again using a GPS satellite orbit).
These relationships are illustrated in Figure 3 and Figure 4, which present the instantaneous SIS URE for an SV at GPS altitude versus azimuth and user elevation angle, for different broadcast errors. Put another way, these plots illustrate the instantaneous SIS URE across the field of view of the SV when looking down from above. Figure 3 shows the URE contributions for positive 1-m errors in the radial, along-track, cross-track, and clock components, whereas Figure 4 shows URE contributions for pairs of 1-m errors (e.g., 1-m radial and 1-m cross-track errors at the same time). Note that each plot is scaled to its individual URE range, to better show the variations in URE across the field of view. The plots are oriented such that the center of the plot is along the radial direction from the SV to the center of the Earth, with the vertical axis of the plot aligned with the SV’s along-track vector. Note that errors in the along-track and cross-track directions lead to a distribution of positive and negative instantaneous SIS URE values across the field of view. Table 1 provides the range of URE values for each component or component pair. Note that the values in the first four rows of Table 1 correspond to the error projection limits derived from Figure 2 earlier in this section.
URE contributions for different 1-m errors (a) Radial (b) Along-track (c) Cross-track (d) Clock
URE contributions for different pairs of 1-m errors (a) Radial + Along-track (b) Radial + Cross-track (c) Radial + Clock (d) Along-track + Cross-track (e) Along-track + Clock (f) Cross-track + Clock
URE Ranges for 1-m Error Components for an SV at GPS Altitude
2.2 Real-World Observations
Under nominal conditions, the variations across the field of view are small, but under extreme conditions, the variations can become more dramatic. One such case is the error observed with GPS SV number (SVN) 59/ pseudorandom noise (PRN) 19 for a brief period early on June 17, 2012 (Figure 5), when invalid Earth orientation parameters were erroneously uploaded (Gruber, 2012). In this case, the variation due to the error in the predicted orbit exceeded ±400 m. Note that there exists a line through the field of view representing a set of user locations for which the instantaneous SIS URE has a value of zero. The errors in the ephemeris components and the clock for this case are provided in Table 2. Note that because the errors are predominantly in the along-track and cross-track components, the distribution in Figure 5 is similar to that in Figure 4(d).
UREs for SVN 59/PRN 19 on June 17, 2012
Component Errors For GPS SVN 59/PRN 19 at 0030Z on 17 June 2012
2.3 SIS RMS URE
The preceding description of the instantaneous SIS URE demonstrated that obtaining a statistical average of the URE over the field of view of the SV is a complex undertaking. The different orbit and clock error components have different effects and variations across the field of view:
Clock component errors are constant across the field of view.
Radial component errors are nearly, but not completely, constant across the field of view and keep the same sign.
Along-track and cross-track component errors vary and change sign across the field of view, and their effects on the instantaneous SIS URE represent a small fraction of the component error.
The instantaneous SIS URE is important to the individual user. However, system operators and those who analyze the performance of a system over large areas need a single quantitative value that can represent the SV signal accuracy at a given instant in time. This can be addressed by computing the instantaneous SIS RMS URE (hereafter, SIS RMS URE) across the surface of interest at a given instant in time. Intuitively, this value should be dependent on the minimum user elevation angle because the effects of radial, along-track, and cross-track errors differ as the nadir angle varies. Additionally, Table 1 illustrates that the effect of the along-track and cross-track errors are de-weighted in comparison to radial and clock errors. Considering the geometry between the edge of the field of view and the SV, this de-weighting should in some way be dependent on the orbit radius.
3 ANALYTICAL APPROACH
The SIS RMS URE is a statistical combination of all possible instantaneous SIS URE values for a given SV at a given moment in time, assuming an ensemble of users uniformly distributed over the surface of the Earth. The SIS RMS URE can be calculated in a brute-force manner or in an analytical fashion. In the brute-force approach, a set of values is produced on a sufficiently dense grid (similar to those plotted above), and then the RMS of the set is computed. For purposes of computation efficiency, the analytical approach is far more efficient and is the focus of the remainder of this article. A full derivation of the SIS RMS URE is presented in Section 3.1. The result of the derivation is the following equation:
1
where:
2
3
4
5
Here, c is the speed of light, re = 6371 km is the mean Earth radius, rs is the satellite orbital radius, and α0 is the minimum user elevation angle. ΔR, ΔA, ΔC, and ΔT are the broadcast errors in the radial, along-track, cross-track, and clock components, respectively. The coefficients w and w′ weight the error components’ contribution to the SIS RMS URE. The w coefficient is associated with orbit errors in a single component (i.e., one coefficient each, all dependent on w, for the radial, along-track, and cross-track components), whereas w′ is associated with cross terms (of which only the radial-clock term is nonzero). These coefficients vary with the GNSS being considered as functions of the orbital radius and minimum elevation mask.
3.1 Derivation of a Globally Averaged URE with an Elevation Cutoff
The URE of a GNSS satellite is calculated by considering the satellite’s position and clock errors in the radial, along-track, cross-track, and time frame. For a given user location, the range error δrange can be expressed as follows:
6
where θ and ϕ are the polar and azimuthal angles, respectively, of the user in a coordinate system centered on the satellite antenna phase center. Here, the satellite nadir direction serves as the z-axis (to which the polar angle is referenced), the along-track direction serves as the x-axis (to which the azimuth is referenced), and the cross-track direction serves as the negative y-axis. The x, y, and z vectors are not the same as the radial, along-track, and cross-track vectors, but are defined in terms of these vectors and serve both as a means of connecting Figure 2 to Equation (6) and as a reference for θ and ϕ. ΔR, ΔA, and ΔC are the satellite position errors in the radial, along-track, and cross-track directions, respectively. ΔT is the satellite clock error, and c is the speed of light. Equation (6) is a result of projecting the radial, along-track, and cross-track errors into the LOS, assuming that they are small compared with the distance between the satellite and user.
The geometry of a range measurement is shown in Figure 2. Also shown are the satellite elevation angle α and an alternate polar angle θ′, which is sometimes more convenient than θ for expressing the user location. The Earth and satellite orbit radii are re and rs, respectively. Because the interior angles of the triangle EUS must add up to π, one can deduce that α + θ + θ′ = π / 2. From triangle EXS, one can deduce that ∠XEU + θ + θ′ = π / 2, from which it follows that ∠XEU = α. From the geometry, one can also conclude the following:
From this, we obtain the following relation:
7
where γ = re / rs. Because the globally averaged URE is computed by integrating over θ′, while the range error depends on θ, a relationship between θ and θ′ is needed. Using the relationship α = π / 2 – θ – θ′, we obtain the following:
From this, one can deduce the following relation:
Squaring both sides, we obtain the following:
Rearranging terms leads to the following:
Because it will be useful later on to express cos θ′ in terms of sin θ, the preceding equation is rewritten as follows:
8
This expression can subsequently be written as follows:
which can be rewritten as the following expression:
by using the quadratic formula for γ cos θ′. Note that the positive sign has been chosen in front of the radical to ensure that cos θ′ is positive when θ is 0. This requirement is based on the assumption that the user is on the same side of the Earth as the satellite.
If the elevation is constrained to be at least α0, Equation (7) implies that sin θ will be constrained between 0 and γ cos α0 and cos θ′ will be constrained between 1 and ul, where we have the following:
9
If there is no constraint on the elevation (other than the obvious constraint that it must be non-negative), then sin θ will be constrained between 0 and γ, which will yield the following constraint:
When computing the RMS statistics of the range error, it is useful to square the range error:
10
Using this expression, one can compute the RMS of the URE over all possible user locations within the view of the satellite. Because both orbit and clock contributions are included, this approach gives the total URE, which can be written as follows:
11
where the integration is weighted by the solid angle differential dΩ about the center of the Earth, which is equivalent to assuming an ensemble of users evenly distributed over the surface of the Earth. The integration over ϕ can be accomplished by first noting the following:
This gives us the following:
which can be written as follows:
12
where:
Letting u = cos θ′ and writing sin2 θ in terms of θ′, one obtains the following for w:
which can be rewritten as follows:
This expression can be easily integrated to obtain the following:
which can be simplified as follows:
13
If no elevation constraint is used, then we have ul = γ and the following:
For w′, one proceeds by writing the following relation:
This can be integrated by parts to obtain the following:
14
which, in the absence of an elevation constraint (ul = γ), can be simplified as follows:
3.2 Application to Earlier Examples
Using the orbital radius rs = 26560 km and elevation mask α0 = 2° specified by the GPS SPS PS, the SIS RMS URE for the example cases in Figures 3 and 4 can be calculated. The results are shown in Table 3. Among the single 1-m offset examples, the relative de-weighting of the along-track and cross-track components is evident in the fact that the corresponding SIS RMS URE values are much smaller than the value of the SIS RMS URE for the radial component. The constant offset imparted by a clock error is reflected in the 1-m SIS RMS URE for the clock offset example. The fact that the radial and clock offset result in URE values with opposing signs is reflected in a lower SIS RMS URE value when those two offsets are combined. For the real-world example from June 17, 2012, note that the SIS RMS URE value might obscure the fact that the individual SIS URE values were both positive and negative and that the maximum values were ±400 m. Whether that is the case for a specific anomaly would depend on the nature of the orbit error.
SIS RMS URE Values for Sample Cases
4 PERFORMANCE STANDARD EQUATIONS AND COEFFICIENTS
The derivation presented in Section 3.1 allows us to confirm the SIS RMS URE equations presented in the standards documents. At the time of this writing, three performance standards present a form of the SIS RMS URE equation: as Equation (A-1) in the GPS SPS PS, as Equation (1) in the Galileo SDD, and as an unnumbered equation in Sections A.2.1.2 and B.1 of the GLONASS (Global’naya Navigatsionnaya Sputnikovaya Sistema, Russian for GNSS) Open Service Performance Standard (OS PS) (Central Research Institute of Machine Building, 2020). The BeiDou Open Service Performance Standard (Chinese Satellite Navigation Office, 2021) does not directly provide these equations for either its medium (MEO) or geostationary (GEO), inclined geosynchronous (IGSO) Earth orbit SVs. Instead, the BeiDou equations below were derived using Equations (1)–(5), the respective orbital radius, and the 5° elevation mask specified in the OS PS assertions. The SIS RMS URE equations for GPS, Galileo1, and GLONASS2, as per the relevant performance standards, are listed below as well as the derived equations for BeiDou:
15
16
17
18
19
Equations (15)–(19) are algebraically equivalent, but their coefficients vary between systems because of the different orbital radii and elevation masks. To more easily compare between the different GNSSs, Equations (20)–(24) show the published equations above in the same form as Equation (1). From this set of equations, the published values for w and w′ can be determined as shown in Table 4. The values in Table 4 may appear to imply that 
; however, while this approximation is very good for MEO and GEO, it becomes progressively worse at lower altitudes.
20
21
22
23
24
Coefficients from Standards Documents in Terms of Coefficients in Derivation
Given Equations (1)–(5), the coefficients w and w′ can be derived for various GNSSs. Figure 6 and 7 show how the coefficients in Equation (1) vary with SV orbital radius, for elevation masks of 0°, 2°, and 5°. Vertical dashed lines mark the orbital radii for GPS, Galileo, GLONASS, and BeiDou (both MEO and GEO). Horizontal dashed lines indicate the coefficient values derived from the equations published for GPS, Galileo, and GLONASS. For GPS and Galileo, the small discrepancies between the published and predicted values are likely due to differences in the number of significant figures kept. The discrepancies in the values for GLONASS, however, are larger, because the equation in the OS PS is a modified form of an empirical GPS equation.
w coefficient versus orbital radius for 0°, 2°, and 5° elevation masks, along with values from GNSS performance standards
w′ coefficient versus orbital radius for 0°, 2°, and 5° elevation masks, along with values from GNSS performance standards
These coefficients are a function of both the orbital radius and the minimum elevation angle assumed between the user and the SV. (Alternatively, the angle limiting the field of view could be the maximum off-nadir angle, but the existing standards speak to the minimum elevation angle.) Table 5 provides the values of the coefficients for known GNSSs and regional navigation satellite systems (RNSSs) for different minimum elevation angle assumptions. The GPS values for a 2° elevation mask correspond to the values in the GPS SPS PS, which is consistent with the conditions specified in the performance standard. Equation (1) in the Galileo SDD is specified to be valid for a 5° mask, but the coefficients for Galileo in Table 4 do not quite match those for the 5° mask in Table 5. The values in the SDD correspond to a slightly higher elevation mask of 5.3°; it is worth noting, though, that Equation (1) keeps a higher number of significant figures than the other systems’ performance standards and is also noted to be an approximation formula. As mentioned previously, the published coefficients for GLONASS are different than would be expected for its 5° elevation mask because of the difference in orbital radius versus that implicitly assumed in its empirical equation.
Coefficients for GNSSs/RNSSs at a Variety of Minimum Elevation Angles
In a similar manner, the values in Table 5 may be compared with values in other references to verify the assumptions used in developing the equations. The equation in Zumberge & Bertiger (1996) corresponds to GPS at a 0° minimum elevation angle. The equation in Malys et al. (1997) is not quite algebraically identical to the others, but the w coefficient corresponds to GPS at a 0° minimum elevation angle. A set of equivalent (although not identically defined) coefficients can be found in Table 4 of Montenbruck et al. (2018), whose values agree with those presented in Table 5. From this, it can be seen that the apparent disagreement between various forms of the SIS RMS URE equation is simply a result of using different orbital radii and elevation masks.
Finally, because Equation (1) is generalized with respect to orbital radius and minimum elevation angle, it can be applicable for multiple GNSSs. This is useful for systems that do not publish a form of the SIS RMS URE equation in their own performance standards, but do specify a minimum elevation angle with their performance assertions. For instance, BeiDou (Chinese Satellite Navigation Office 2021), GLONASS (Central Research Institute of Machine Building 2020), and NAVIC (Indian Space Research Organization Satellite Center, 2017) all use a 5° elevation mask, whereas signals from the quasi-zenith satellite system (QZSS) (Cabinet Office, 2022) are only receivable at a minimum elevation angle of 10°. Note that the non-geostationary QZSS orbits provide a particularly interesting challenge for the calculation of SIS RMS UREs. Given that the distance of the SV from the center of the Earth is constantly changing, it is necessary to re-calculate the coefficients at each epoch. By using the generalized form of Equation (1), one can gain an improved understanding of GNSS performance specification documents.
5 CONCLUSIONS
The SIS RMS URE provides a useful metric for measuring the statistical accuracy of the broadcast orbit and clock data provided to the user. This metric is used extensively in various GNSS performance standards.
The differences between two kinds of UREs, i.e., the instantaneous SIS URE and SIS RMS URE, were explored. When examining an anomaly, it is important to recall that the SIS RMS URE is a statistical quantity and that the statistics can sometimes obscure important diagnostic information. Examining the instantaneous SIS URE across the SV field of view, particularly in relation to the SV direction of motion, can provide useful information regarding where to search for a root cause.
The derivation presented for the SIS RMS URE equation illustrates a dependence on the orbital radius of the constellation of interest and the selection of an assumed minimum elevation angle from the user to the SV. The resulting equation is consistent with the equations provided in the GPS SPS PS and Galileo SDD. Finally, the coefficients for several GNSS/RNSS constellations were derived for multiple minimum elevation angles. This work both illustrates the analytical derivation concepts and provides a means for verifying the implementation of the equations found in other reports.
AUTHOR CONTRIBUTIONS
The derivation in Section 3.1 was developed by Jason Drotar.
HOW TO CITE THIS ARTICLE
Renfro, B., Drotar, J., Finn, A., Stein, M., Reed, E., & Villalba, E. (2024). An analytical derivation of the signal-in-space root-mean-square user range error. NAVIGATION, 71(1). https://doi.org/10.33012/navi.630
Footnotes
↵1 The Galileo SDD uses the opposite sign convention for radial offsets compared with the convention used here; thus, the radial-clock cross-term in the above equation has the opposite sign of the equation in the SDD.
↵2 The GLONASS OS PS includes more than one equation for the SIS RMS URE. These equations could not be reconciled. The equation presented here is found in Appendix B, Section B.1, with the coefficient sin2 α = 0.0225, as is specified “for assessment.” The other variation can be found in Appendix A, Section A.2.1.2.
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
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