Assessment of Ionospheric TEC Estimation Uncertainty Using Single-Frequency Wideband Low-Elevation GNSS Signals

2 DATA DESCRIPTION

In this study, we used data collected by Septentrio PolaRxS receivers with zenith-facing antennas located in Haleakala, HI, and Toolik Lake, AK. The surveyed receiver coordinates are 20˚42’23”N, 156˚15’25”W, 3,060 m and 68˚37’39”N, 149˚35’53”W, 737 m, respectively. Figure 3 shows the receiver coordinates on a regional map of the northeastern Pacific region. L1CA, L2C, and L5 pseudorange data were collected at a 1-Hz rate. Data collected over one week from each receiver location are used in this analysis, specifically 11 June 2021 to 17 June 2021 for the Haleakala station and 30 September 2021 to 7 October 2021 (omitting 4 October 2021) for the Toolik Lake station. Both receiver stations are located in latitude ranges where close to half of the incident GPS signals within view are below 20˚ elevation. The Haleakala station has surrounding structures that act as multipath sources during low-elevation satellite passes. In approximately half of the daily receiver data files, pseudorange data from 0–2 satellites have been omitted as part of the data quality control, as they contain errors that cannot be accounted for. Other single pseudorange noise (PRN) data (PRN 25 on 15 June 2021, PRN 26 on 11-13 June 2021, and PRN 31 on 11 June 2021) have been removed owing to large fluctuations in oscillator bias from GPS ephemeris data that posed challenges in our analysis.

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Figure 3

Map of the northeastern Pacific region, with the locations of the Haleakala and Toolik Lake receiver stations denoted by red stars

3 METHODOLOGY

The approach used in this work utilizes Equation (2), the GNSS pseudorange measurement equation (Misra & Enge, 2006), to estimate the single-frequency TEC:

$\rho =r+I+T+c(\delta t_u-\delta t^s)+{\sigma }_M+{\sigma }_{\rho }$ 2

The pseudorange measurement, denoted by ρ, represents the apparent distance between a GNSS satellite and a receiver. The measurement includes several sources of delay and error. On the right-hand side of the equation, r denotes the true geometric range between the satellite and the receiver. The ionospheric delay, represented by I, results from the signal’s interaction with the ionized layers of the atmosphere, whereas the tropospheric delay, T, is due to signal slowing in the neutral atmosphere, including effects from humidity and pressure. Clock biases are also present: δt_u is the receiver clock bias, and δt_s is the satellite clock bias, both expressed in units of distance. Additional errors include σ_M, which accounts for multipath effects, where the GNSS signal reflects off surfaces before reaching the receiver, and ρ_p, which represents pseudorange measurement noise from sources such as thermal noise and receiver signal processing imperfections.

The block diagram in Figure 4 outlines the process for obtaining single-frequency TEC estimates. Factors such as the geometric range, satellite clock bias, and tropospheric delay are computed from satellite ephemeris and models. The receiver clock bias is calculated by utilizing high-elevation narrowband signals as an intermediate step and then denoising the resulting clock bias to remove narrowband signal uncertainty. We can then retrieve a single-frequency TEC estimation using the range corrections and wideband pseudorange data. This approach results in the relative slant TEC (sTEC), a measure of integrated electron density along a raypath that is offset because of receiver and satellite hardware biases (Zhong et al., 2016). The processes applied for range corrections, user clock bias, and single-frequency TEC are described in the following subsections.

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Figure 4

Block diagram of methodology for wideband single-frequency TEC estimation; Rx: receiver

3.1 Range Error Calculations

In this subsection, we describe the process of estimating and removing the satellite–receiver range, satellite clock, and tropospheric effects on the measured delay. As an example, we use data from a single pass of the GPS PRN 1 satellite on 11 June 2021 from the Haleakala station; plots of GPS range errors from this example are included in Figure 5, while their magnitudes and how they are obtained are described below.

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Figure 5

Composite of range correction terms for the PRN 1 satellite as it passes over the Haleakala station on 11 June 2021: (A) geometric range, (B) satellite clock bias, (C) satellite clock relativistic effects, (D), tropospheric delay, (E) dual-frequency derived ionospheric delay, and (F) receiver clock bias

3.1.1 Geometric Range Calculation

The geometric range is defined as the distance between the satellite antenna phase center at the time of signal transmission and the receiver antenna phase center at the time of reception. This quantity can be calculated using data from daily broadcast and precise satellite orbit solution GPS ephemeris files sourced from the Crustal Dynamics Data Information System (CDDIS) (International GNSS Service (IGS), 1992a, 1992b) and the surveyed receiver antenna location (Canadian Geodetic Survey, 2022). Broadcast ephemeris files have a 1-h temporal resolution, whereas precise ephemeris files have a 15-min time resolution. Precise ephemeris files have orbit information that is accurate to better than 5 cm and are released approximately 2 weeks after the completion of the GPS week. For each time epoch of a satellite pass, the broadcast ephemeris is first used as a rough estimate of satellite position coordinates. Precise ephemeris data are then used in an iterative process to account for satellite motion and Earth’s rotation, enabling an accurate calculation of the satellite’s position at the time of signal transmission. Note that the orbit data provided in precise ephemeris files are interpolated to the desired epochs using a 10th-order Lagrange polynomial. The geometric range is then computed using satellite coordinates at the time of transmission and receiver surveyed coordinates. Figure 5(a) shows an example range calculation for one satellite pass of PRN 1 on 11 June 2021 for the Haleakala station, with the elevation angle plotted as a dashed line. The magnitude of this pseudorange correction is on the order of 2 x 10⁷ m.

3.1.2 Satellite Clock Correction

Data for satellite clock correction (including correction for relativistic effects) is contained within broadcast and precise ephemeris data from CDDIS (International GNSS Service (IGS), 1992a, 1992b). Satellite clock bias information contained in precise ephemeris files is used to compute the satellite clock bias at the time of signal transmission. Satellite clock correction information is taken from the precise ephemeris files, with a temporal resolution of 15 min, and interpolated to the exact time of signal transmission via a Lagrange interpolating polynomial to the 10th degree. The satellite clock bias is on the scale of tens to hundreds of kilometers. This bias estimate is accurate to better than 0.1 ns, or 3 cm. Figure 5(b) shows an example of satellite clock correction for PRN 1 on 11 June 2021.

Satellite relativistic effects on orbiting clocks are minor clock errors and are computed from the broadcast ephemeris file data. Range error corrections from relativistic effects are on the scale of meters. Figure 5(c) shows an example of relativistic correction for PRN 1 on 11 June 2021 from the Haleakala station.

3.1.3 Tropospheric Delay Model

The tropospheric delay is computed using grid-wise Vienna Mapping Function 3 (VMF3) data with 1˚x1˚ and 6-h resolution (TU Wien, n.d.). The VMF3 tropospheric delay has a mean global root mean square error estimated as better than 5 cm at zenith (Ding & Chen, 2020; Landskron & Bohm, 2018). Surveyed receiver coordinates are inputted into the model, which then outputs wet and hydrostatic tropospheric parameters and mapping functions.

Tropospheric delays vary from meter-scale at high elevation angles up to tens of meters at low elevation angles. Because of the inverse relationship between elevation angle and tropospheric correction, this factor is very important for low-satellite-elevation estimates of TEC. Figure 5(d) shows an example tropospheric delay calculated for the Haleakala station on 11 June 2021 for PRN 1. Notably, because of the high altitude of the Haleakala station, modeled tropospheric delays will be smaller than those of other low-latitude stations; however, they will still be slightly higher than those of the Toolik Lake station.

3.1.4 Dual-Frequency Ionospheric Delay Calculation

The dual-frequency ionospheric delay is calculated from high-elevation-angle signals as an intermediate step to obtain the receiver clock bias. The ionospheric delay for the L1CA signal, I_L1, can be found from Equation (3) (Misra & Enge, 2006):

$I_{L1}=\frac{f_{L2}^2}{f_{L1}^2-f_{L2}^2}({\rho }_{L2}-{\rho }_{L1})$ 3

In this equation, f_L1 and f_L2 are the carrier frequencies for the GPS L1 and L2 signals, whereas ρ_L1 and ρ_L2 are the receiver pseudoranges. For comparison, the dual-frequency TEC is also calculated using L1 and L2 pseudoranges based on Equation (4):

$TEC=\frac{f_{L2}^2{f}_{L2}^2}{40.3(f_{L1}^2-f_{L2}^2)}({\rho }_{L2}-{\rho }_{L1})$ 4

The magnitude of this correction typically ranges from close to zero to tens of meters. Errors in this computation can be primarily attributed to pseudorange noise and multipath errors; if high-quality antenna and receivers are utilized, these uncertainties are estimated to be better than approximately 0.1 and 0.15 m, respectively, when above a 30˚ elevation mask (Prochniewicz & Grzymala, 2021). If error independence is assumed, the root sum square of the combined errors is approximately 0.18 m for narrowband single-frequency signals and 0.4 m if dual-frequency uncertainty addition and scaling factors are considered (Misra & Enge, 2006). The equation for the dual-frequency-derived ionospheric delay error is shown in Equation (5):

${\sigma }_{I,L1}=\frac{f_{L2}^2}{(f_{L1}^2-f_{L2}^2)}\sqrt{{\sigma }_{\rho ,L1}^2+{\sigma }_{\rho ,L2}^2+{\sigma }_{M,L1}^2+{\sigma }_{M,L2}^2}$ 5

An example of the dual-frequency ionospheric delay is shown in Figure 5(e), where the increase in uncertainty at low elevation angles is evident.

3_2 Receiver Clock Bias Computation

The receiver clock bias is estimated by solving for cδt_u in the GPS pseudorange equation for all high-elevation satellites, as shown in Equation (6):

$c\delta t_u={\rho }_{L1}-(r+I_{L1}+T-c\delta t^s)$ 6

Range error estimates, including geometric range, satellite clock correction (including relativistic effects), tropospheric delay, and dual-frequency ionospheric delay, converted to L1 range delay, are computed for all satellites with L1CA and L2C signals available. An example of the resulting receiver clock bias from a single satellite overpass is shown in Figure 5(f). This receiver clock bias was obtained from PRN 1 on 11 June 2021 for the Haleakala station.

Receiver clock bias estimates from all satellites within each time epoch are averaged, and missing data points are interpolated to retrieve a single time series of clock bias estimates. Clock jumps and a linear trend are first removed from the receiver clock bias time series; then, a running average with a length of 6 s (360 samples) is applied for denoising. The linear trend and clock jumps are then added back to the smoothed clock bias for use in the single-frequency TEC estimation.

Note that only measurements from satellites with L1CA and L2C signals at elevation angles above 30˚ are used in this intermediate step to estimate the receiver clock bias. The elevation mask of 30˚ is used here to reduce narrowband measurement uncertainty at low elevation angles while retaining a larger amount of satellite signals for an accurate user clock bias estimate.

The error budget for the receiver clock bias is the square root of the sum of squares of the error contributions from the pseudorange, geometric range, dual-frequency-derived ionospheric delay, troposphere, and satellite clock errors. Under the assumption that these errors are independent, the combined error budget is shown in Equation (7). Note that contributions from pseudorange and multipath errors are included in the L1 ionospheric delay component:

${\sigma }_{rx,b}=\sqrt{{\sigma }_r^2+{\sigma }_T^2+{\sigma }_{I,L1}^2+{\sigma }_{sv,b}^2}$ 7

The receiver clock bias uncertainty is denoted by σ_rx,b, whereas contributing errors of geometric range, tropospheric delay, dual-frequency ionospheric delay, and satellite clock bias are denoted by σ_r, σ_T, σ_IL1, and σ_sv,b, respectively. The largest contribution to uncertainty is the dual-frequency ionospheric delay; however, because this method only uses high-elevation-angle measurements, the largest error sources (multipath and pseudorange noise) are minimized. The magnitude of the receiver clock correction varies for different receivers. Within our experiment, this correction is on the scale of tens to hundreds of kilometers.

3.3 Error Budget Summary

Table 1 provides a summary of range correction and uncertainty magnitudes for the methods described in this section.

3.4 TEC Estimation

After the geometric range, tropospheric delay, satellite clock bias, and receiver clock bias have been estimated, the single-frequency ionospheric delay and TEC can be solved directly from the GPS pseudorange equation. Note that the resulting product is the relative sTEC, which means that unknown hardware biases, also known as code biases, are still present in the TEC estimates (Zhong et al., 2016). This remaining bias stands to be resolved in future work, as the goal of this study is to evaluate the reduction of TEC uncertainty from pseudorange noise and multipath at low elevation angles:

$I_{L5}={\rho }_{L5}-(r+T+c(\delta t_u-\delta t^s\left)\right)$ 8

$TEC=\frac{f^2}{40.3}{I}_{L5}$ 9

Figure 6 shows the resulting single-frequency wideband TEC estimates compared with results obtained via the dual-frequency method for multiple GPS satellite passes over the Haleakala ground-based receiver. At low elevation angles (below ~20˚), narrowband pseudorange uncertainty overwhelms the dual-frequency TEC signal, whereas the wideband estimate remains relatively stable. Additionally, in Figure 6(b), the presence of unknown code biases is evident from the TEC offset observed between the two estimation methods.

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Figure 6

Single-frequency wideband-derived TEC compared with dual-frequency narrowband TEC for multiple satellite passes over the Haleakala station on 11 June 2021 for GPS (A) PRN 1, (B) PRN 18, (C) PRN 25, (D) PRN 26, (E) PRN 27, and (F) PRN 30

These plots show the relative sTEC, meaning that hardware (code) biases remain in the estimates.

4 RESULTS

The single-frequency TEC estimation methodology described in Section 3 was applied to the L1CA, L2C, and L5 received signals and compared with the code-based dual-frequency TEC for the purpose of this analysis. Figure 7(a) and 7(b) show an analysis of two one-week receiver data sets from the Haleakala and Toolik Lake stations, respectively. The standard deviations of sTEC estimates were taken from continuous portions of satellite passes within 1˚ elevation angle bins. Subsequently, the standard deviations within all elevation angle bins were averaged by using weights corresponding to the number of samples within each bin. Scattered points in Figure 7 depict the average uncertainty level for different TEC estimation methods. The yellow points correspond to the dual-frequency pseudorange computation of sTEC whereas the red, blue, and purple points correspond to single-frequency sTEC estimations based on the L1CA, L2C, and L5 signals, respectively.

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Figure 7

Standard deviation of sTEC estimates and sample size from one week of satellite passes over the (A) Haleakala and (B) Toolik Lake stations, binned by 1˚ satellite elevation angle

The Haleakala data show that the L5 single-frequency estimates improve the signal uncertainty by a factor of ~3 at high satellite elevation and by a factor of ~7 at elevations below 30˚ when compared with dual-frequency methods. From the Toolik Lake data, we see that the L5 single-frequency estimates improve the signal uncertainty by a factor of -5 at high satellite elevations and by a factor of -3 at elevations below 30˚ when compared with dual-frequency methods. Interestingly, for the Toolik Lake data at very low elevation, the L2C single-frequency TEC estimates have slightly lower uncertainty estimates than the L5 estimates, which requires further investigation. The latitude difference between the Haleakala and Toolik Lake stations is evident from the plotted sample size shown in Figure 7(a) and 7(b). Fewer satellites are available a elevation angles higher than ~73˚ for the Toolik Lake station, whereas satellite passes are available for elevation angles up to ~88˚ at the Haleakala station. Sample sizes for each station are dependent on the satellite pass geometry, with the peak of satellite passes corresponding to a longer time spent at specific elevations. Few data are available from the Toolik Lake station below 5˚ elevation owing to a pre-existing elevation mask.

To quantify the improvement in uncertainty obtained by single-frequency methods compared with the dual-frequency method, we used observations from both receiver stations, as shown in Figure 7. The uncertainty reduction percentage is calculated as the percentage of standard deviation reduction for the single-frequency methods compared with that of the dual-frequency method. The percentage of uncertainty reduction is shown as a function of elevation angle for the Haleakala and Toolik Lake stations in Figures 8(a) and 8(b), respectively. Red, blue, and purple lines show uncertainty reduction percentages for the single-frequency L1CA, L2C, and L5 TEC estimates, respectively. Almost all trends show an uncertainty level reduction that increases with decreasing elevation angle, with the exception of the L5 single-frequency TEC for the Toolik Lake data set. At all elevation angles for the Haleakala data, the L5 TEC has the highest percentage of uncertainty reduction, peaking at the lowest elevation angles, below 20˚, at just below 90%. For the Toolik Lake data, the L5 TEC has the highest level of uncertainty reduction, at approximately 80% for high elevation angles and 70%—80% below an elevation of approximately 10˚.

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Figure 8

Single-frequency TEC uncertainty reduction percentage compared with dualfrequency TEC, computed from scattered (A) Haleakala and (B) Toolik Lake data shown in Figure 7

Figures 9 and 10 show a spatial analysis of one-week data sets from the Haleakala and Toolik Lake stations, respectively. Skyplots A and B show the binned weighted averaged standard deviation of TEC for the narrowband dual-frequency method and the wideband single-frequency method, respectively. Standard deviation color bars are set to unity for comparison between uncertainty levels at each site. Plots C and D contain sample size values for each spatial bin for the narrowband and wideband methods, respectively. Note that for this analysis, all satellite PRNs with L1CA and L2C signals are included in plots A and C.

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Figure 9

Binned spatial analysis of one week of pseudorange-only TEC data from the Haleakala station: (A) binned standard deviation of narrowband dual-frequency (L1CA, L2C) TEC estimations, (B) binned standard deviation of wideband single-frequency (L5) TEC estimations, (C) number of samples within bins for narrowband dual-frequency (L1CA, L2C) TEC estimations, (D) number of samples within bins for wideband single-frequency (L5) TEC estimations

Elevation angle bins have a 5˚ width, while azimuthal angle bins have a 15˚ width.

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Figure 10

Binned spatial analysis of one week of pseudorange-only TEC data from the Toolik Lake station: (A) binned standard deviation of narrowband dual-frequency (L1CA, L2C) TEC estimations, (B) binned standard deviation of wideband single-frequency (L5) TEC estimations, (C) number of samples within bins for narrowband dual-frequency (L1CA, L2C) TEC estimations, (D) number of samples within bins for wideband single-frequency (L5) TEC estimations

Elevation angle bins have a 5˚ width, while azimuthal angle bins have a 15˚ width.

For the Toolik Lake station, the maximum binned standard deviation from the dual-frequency TEC estimates is 9.94 TECu (A), whereas the maximum binned standard deviation for the L5 TEC is 0.672 TECu (B). There are a total of ~ 5.0 x 10⁶ 1-Hz L1/L2 TEC samples (C), compared with ~ 3.5 x 10⁶ 1-Hz L5 TEC samples (D). All skyplots show higher uncertainty levels at lower elevation angles. The Haleakala station has a maximum binned standard deviation of 4.63 TECu for the narrowband dual-frequency TEC (A) and a maximum binned standard deviation of 1.53 TECu for the wideband single-frequency TEC (B). The total sample size for narrowband signals is ~ 4.5 x 10⁶ (C), slightly larger than that of the wideband samples, ~ 3.1 x 10⁶ (D).

Overall uncertainty levels were higher for the Haleakala station at higher elevation angles. This result could be due to ambient multipath sources near the receiver. Spatial locations with high uncertainty levels were consistent for both TEC estimation methods; however, the wideband single-frequency uncertainty estimates are consistently lower at all locations. The Toolik Lake station has fewer ambient multipath sources; consequently, larger uncertainty levels are concentrated at lower elevation angles, where the ground is a main source of multipath. These uncertainty levels are higher compared with that of the Haleakala skyplots. At both locations, the availability of L1CA and L2C signals benefits the spatial coverage owing to a higher proportion of satellites with signal availability, as GPS PRNs follow similar paths in the sky from day to day within a one-week period.

5 CONCLUSION

This study presented analyses of two one-week GPS receiver data sets from receiver stations located in Haleakala, HI, and Toolik Lake, AK, to test a methodology that utilizes single-frequency wideband pseudorange data to estimate TEC at low elevation angles. Within this methodology, we used high-elevation-angle L1CA and L2C pseudorange data and range corrections (geometric range, satellite clock bias, and tropospheric correction) to compute the receiver clock bias. From this, we can directly calculate the relative sTEC using L5 pseudorange and range corrections. This method enables the estimation of TEC at low elevation angles with lower uncertainty levels than code-based dual-frequency methods. This improved uncertainty is achievable by omitting the use of narrowband signals for less susceptibility to pseudorange noise and large multipath errors.

Key findings from this study come from analyses of relative sTEC estimates using the single-frequency method applied to GPS L1CA, L2C, and L5 pseudorange measurements. These results were compared to the relative sTEC from dual-frequency (L1CA and L2C) pseudorange-derived methods. Binned standard deviations of single-frequency wideband TEC estimates are consistently lower than those produced by the dual-frequency pseudorange-differencing method. TEC estimation uncertainty improvements obtained via the wideband single-frequency method range from a factor of ~3–5 at high elevations to ~3–7 at low elevations (Figure 7). Additionally, the percentage of uncertainty reduction was quantified for all single-frequency methods (L1CA, L2C, and L5) compared with dual-frequency pseudorange methods (Figure 8). In most conditions, the L5 single-frequency method performed the best among all methods, and in most single-frequency cases, the uncertainty reduction increased with decreasing elevation angle. Notably, for the Toolik Lake station, the single-frequency TEC estimation method using L2C signals has a slightly higher standard deviation reduction than the single-frequency L5 method at very low elevation angles (below ~10˚), which requires further investigation. From skyplots of binned standard deviation (Figures 9 and 10), we analyzed spatial differences between narrowband dual-frequency and wideband single-frequency TEC estimation methods. Because of the smaller number of GPS satellites transmitting L5 wideband signals, the wideband single-frequency method had worse spatial coverage over the selected one-week timeframe. However, analysis showed a consistent decrease in TEC estimate uncertainty for all elevations and azimuthal directions for single-frequency wideband TEC estimation methods as opposed to dual-frequency narrowband methods.

Future work will investigate the removal of satellite and receiver hardware code biases from relative TEC estimates. Moreover, we would like future work to include analyses of uncertainty reduction using single-frequency CMC methods and a comparison of the bias corrections required for absolute TEC using such techniques. Additionally, investigation into mapping functions that can adequately convert between vertical TEC and sTEC at very low elevation angles will be explored. The thin-shell approximation model is commonly used, which attributes all ionospheric delay on GNSS signals to a single ionospheric pierce point at which the incident signal passes through a constant altitude (Mannucci et al., 1998; Odijk, 2002). Because very low-elevation signals have longer path lengths through the ionosphere with distributed ionospheric delay, the thin-shell approximation is less accurate. At elevation angles below ~10˚, this mapping function becomes less accurate (Mannucci et al., 1998). Alternatively, sTEC measurements at very low elevation angles could be applied to three-dimensional tomography models of the ionosphere, offering signal geometry diversity that would be better for discerning the vertical distribution of TEC from ground station data.

HOW TO CITE THIS ARTICLE

Evans, M. C., Breitsch, B., & Morton, Y. J. (2026). Assessment of ionospheric TEC estimation uncertainty using single-frequency wideband low-elevation GNSS signals. NAVIGATION, 73. https://doi.org/10.33012/navi.737

CONFLICT OF INTEREST

The authors state no conflicts of interest.