Comparison of 4D tomographic mapping versus thin-shell approximation for ionospheric delay corrections for single-frequency GPS receivers over North America

Abstract

The majority of navigation satellite receivers operate on a single frequency. They compensate for the ionospheric delay using either an ionospheric model which typically only corrects for 50% of the delay or a thin-shell map of the ionosphere. A 4D tomographic imaging technique is used to map the free electron density over the full-height of the ionosphere above North America during autumn 2003. The navigation solutions computed using correction based upon the thin-shell and the full-height maps are compared in this paper. The maps are used to calculate the excess propagation delay on the L1 frequency experienced by GPS receivers at selected locations across North America. The excess delay is applied to correct the single-frequency pseudorange observations at each location, and the improvements to the resulting positioning are calculated. It is shown that the thin-shell and full-height maps perform almost as well as a dual-frequency carrier-smoothed benchmark and for most receivers better than the unfiltered dual-frequency benchmark. The full-height corrections perform well and are considerably better than thin-shell corrections under extreme storm conditions.

Introduction

The ionospheric delay is the largest source of positioning error to single-frequency Global Positioning System (GPS). GPS receivers can compensate for it with values calculated from thin-shell representations of the ionosphere or from a model. All receivers can use the Klobuchar (1987) model, and an increasing number can use Satellite-Based Augmentation Systems (SBASs) wherever these are available. For example, the following systems are available: Wide Area Augmentation System (WAAS) in North America, European Geostationary Navigation Overlay System (EGNOS) in Europe, and Multi-functional Satellite Augmentation System (MSAS) in Asia. Each of these systems approximates the ionosphere to a thin-shell. An alternative concept, tomographic imaging, is a medical imaging technique to create images of a parameter from integrated measurements. It is useful for ionospheric mapping because it allows the depth field of the object to be correctly represented; the ionosphere has a depth of several hundred kilometers. A comprehensive review of ionospheric imaging and data assimilation is given by Bust and Mitchell (2008). In this paper, a real-time mapping tomographic algorithm known as Multi-Instrument Data Analysis System (MIDAS) is used to produce ionospheric images. Theoretical studies of the advantages of using full-height rather than thin-shell images have already been carried out with MIDAS by Meggs and Mitchell (2006) for vertical delay and by Smith et al. (2008) for slant delay. Full-height ionospheric mapping has already successfully been used for dual-frequency GPS carrier-phase positioning over long baselines (Hernández-Pajares et al. 2000). It has also been shown to give good corrections for single-frequency GPS positioning even at times of strong geomagnetic activity when the ionospheric delay is most difficult to compensate for (Allain and Mitchell 2009).

The aim of this paper is to compare, using experimental data, the navigation accuracy that could be achieved with full-height tomographic images of electron density to correct for the unknown excess delay on single-frequency signals with what can be achieved with thin-shell representations of the ionosphere and unfiltered dual-frequency. This is carried out with a series of navigation calculations that are each performed using a different approach to the ionospheric corrections: (1) no correction, (2) Klobuchar model, (3) near-real-time thin-shell images, (4) near-real-time full-height images, (5) unfiltered dual-frequency, and (6) phase-filtered dual-frequency. The phase-filtered dual-frequency result is a benchmark that shows the best position that can be achieved at that instant without time averaging, whereas the unfiltered dual-frequency is of interest, for example using a moving receiver and not assuming phase continuity.

Method

This study covers all the days from Oct 1 to Dec 31, 2003, showing diurnal changes during a solar maximum year and a major storm period.

Global Positioning System receivers from five different sites in mainland North America were used to test the six different methods. Each receiver recorded dual-frequency data in Receiver Independent Exchange Format (RINEX) format at a sampling period of at most 30 s. For all the receivers that sampled at a period shorter than 30 s, the data were subsampled to a period of 30 s for consistency with the other receivers. This period of 30 s is much longer than the 1-s sampling period normally used for real-time positioning but is sufficient to compare the ionospheric corrections. The locations of these test sites are shown as red × marks in Fig. 1. The green + marks in Fig. 1 show the locations of dual-frequency GPS receivers that were used in the tomographic imaging to create the ionospheric corrections for methods 4 and 5.

Fig. 1

Location of the test sites (×) and sites used for imaging (+). The grid is centered at 40°N 100°W

The satellite positions were calculated from the precise ephemerides obtained from the International GNSS Service (IGS). These were interpolated using a four-harmonic interpolation applied to two hours of the orbit to achieve an accuracy of 0.25 m in the satellite position. Satellite-specific clock biases were obtained from the IGS, and linear interpolation was used to estimate their values at the appropriate times. In order to implement calculations for the single-frequency positioning with no ionospheric correction, the P1 code was extracted from the RINEX file. Corrections were made for the Sagnac effect and the satellite relativistic clock bias in accordance with Ashby and Spilker (1995) The tropospheric delays were estimated from receiver altitude and satellite elevation using the approach of Spilker (1994). These corrections resulted in a set of partially corrected pseudorange observations for each site at the maximum sampling rate of the receivers, 30 s.

The only remaining correction was for the ionospheric delay. Several methods were used. The first method was to not correct for the ionospheric delay. The results from this method are labeled ‘uncorrected’. The five remaining approaches account for the ionosphere. The second method was to use the Klobuchar model, currently broadcast in the form of coefficients over the GPS L1 signal. The model represents the global ionosphere into eight coefficients that can be used by a receiver-based algorithm to make real-time corrections to the signal pseudoranges. The results from this model are labeled ‘Klobuchar’. The third method was to use MIDAS forced into thin-shell mode (labeled ‘thin-shell’). The fourth method was to use MIDAS in its normal full ionospheric height mode (labeled ‘full-height’). The algorithm first described in Mitchell and Spencer (2003) was implemented here with upgrades described in Spencer and Mitchell (2007). The input data came from dual-frequency receivers. The variation of the total electron content (TEC) along the line between a satellite and a receiver is computed from the related set of dual-frequency observations. Four-dimensional tomography is then carried out from the TEC variations and the intersections between the ray-paths and the grid to derive the free electron concentration within the grid. The distribution of the dual-frequency receivers across and around North America is shown in Fig. 1. The fifth and sixth methods were to use, respectively, unfiltered ionosphere-free pseudoranges (labeled ‘unfiltered dual’) and phase-filtered ionosphere-free pseudoranges (labeled ‘filtered dual’).

The receiver position was then calculated using a least-squares estimate applied to the corrected pseudorange observations.

Ionospheric maps

The area of coverage was extended far beyond the boundaries of North America to allow the calculation of the TEC even for low-elevation rays. The grids were centered at 40°N 100°W, allowing for a similar pixel size and shape to cover the area of study (Spencer and Mitchell 2007). The ranges were in latitude from −44° to +44° (west to east) in steps of 4°, in longitude from −44° to +44° (south to north) in steps of 4°. The 3D grid of the full ionospheric height tomography altitude range was from 100 to 1,500 km in steps of 50 km. The altitude range, for the 3D grid of the full-height images, was from 100 km to 1,500 km in steps of 50 km. The altitude of the thin-shell was 350 km, for coherence with the WAAS (Komjathy et al. 2005).

The vertical basis functions used for the full ionospheric height MIDAS inversion were three empirical orthonormal functions computed from a range of Epstein functions. The inversions were stabilized with a set of regularization factors preventing departure from a linear gradient.

Pseudoranges

The notations used here follow from Mannucci et al. (1999).

Klobuchar (1996) derives I f ⁻² the ionospheric group delay or phase advance in meters, with f the frequency of the signal in hertz, I the TEC in electrons per square meters, and I the ionospheric delay term in meter per square second:

$$ {I\;f^{ - 2} = 40.3\;f^{ - 2} \int N \;{\text{d}}l} $$

(1a)

or

$$ {I = 40.3\;\int {N\;} {\text{d}}l} $$

(1b)

Equation 1a shows the ionospheric delay can be calculated if the TEC is known, or measured if pseudorange observations are available for at least two frequencies.

The phase-filtered dual-frequency position is computed from phase-filtered ionosphere-free pseudoranges, here called P ₀. It is itself computed from four observables P ₁, P ₂, L ₁, and L ₂ as explained in the following equations. P ₁ and P ₂ are the pseudoranges from the precise P-code. L ₁ and L ₂ are the recorded carrier phases of the signal converted to distance units.

Expressing P ₁ and P ₂ with P ₀, I f ⁻², and ε the noise which includes multipath:

$$ {P_{1} = P_{0} + I\;f_{1}^{ - 2} + \varepsilon_{1} } $$

(2a)

$$ {P_{2} = P_{0} + I\;f_{2}^{ - 2} + \varepsilon_{2} } $$

(2b)

and expressing L ₁ and L ₂ with P ₀, I f ⁻², n the integer ambiguity, and λ the carrier wavelength :

$$ {L_{1} = P_{0} - I\;f_{1}^{ - 2} + n_{1} \lambda_{1} } $$

(3a)

$$ {L_{2} = P_{0} - I\;f_{2}^{ - 2} + n_{2} \lambda_{2} } $$

(3b)

gives two expressions for I:

$$ {{\frac{{P_{ 1} - P_{ 2} }}{{f_{ 1}^{ - 2} - f_{ 2}^{ - 2} }}} = I + {\frac{{\varepsilon_{ 1} - \varepsilon_{ 2} }}{{f_{ 1}^{ - 2} - f_{ 2}^{ - 2} }}}} $$

(4a)

$$ {{\frac{{L_{1} - L_{2} }}{{f_{1}^{ - 2} - f_{2}^{ - 2} }}} = - I + {\frac{{n_{1} \lambda_{1} - n_{2} \lambda_{2} }}{{f_{1}^{ - 2} - f_{2}^{ - 2} }}}} $$

(4b)

Equation 4a gives I with a noise term. Equation 4b gives I with an offset term from the integer ambiguity. The integer ambiguity stays constant while the satellite is visible apart from large and sudden changes called cycle slips. The offset term of (4b) depends on the integer ambiguity and stays also constant apart from similar changes. As the changes of the offset term of (4b) are large and sudden, they are easily detectable. The offset between cycle slips is taken as a weighted mean of the differences between the first and second solutions mentioned earlier, i.e., the value of I is computed by fitting (4b) into (4a). The cosecant of the elevation angles is used for the weights. This way, the weights are correlated with the signal to noise ratio (Klobuchar 1996). We are then left with:

$$ {P_{1} - I\;f_{1}^{ - 2} = P_{0} + \varepsilon_{1} } $$

(5a)

$$ {L_{1} + I\;f_{1}^{ - 2} = P_{0} + n_{1} \lambda_{1} } $$

(5b)

Again, Eq. 5a gives P ₀ with a noise term and (5b) gives P ₀ with an offset term. P ₀ is, as mentioned earlier for I, computed by fitting (5b) into (5a) with a similar weighted mean.

The unfiltered dual-frequency position is computed from unfiltered ionosphere-free pseudoranges, here called P _ε . It is itself computed from two observables P ₁ and P ₂ as explained in the following equation. Neglecting the importance of the noise term in (2a, 2b) gives:

$$ {P_{\varepsilon } = {\frac{{P_{1} f_{2}^{ - 2} - P_{2} f_{1}^{ - 2} - \varepsilon_{1} f_{2}^{ - 2} + \varepsilon_{2} f_{1}^{ - 2} }}{{f_{2}^{ - 2} - f_{1}^{ - 2} }}}} $$

(6)

Comparing (6) with (2a, 2b) shows the unfiltered dual-frequency pseudorange P _ε is about three times noisier than either of the single-frequency pseudoranges P ₁ or P ₂. If we consider ε ₁ and ε ₂ to be uncorrelated and both with a Gaussian distribution of the same standard deviation of Δε, then the resulting standard deviation of Δε _ε will be such as:

$$ {\Updelta \varepsilon_{\varepsilon } = {\frac{{\sqrt {f_{2}^{ - 4} + f_{1}^{ - 4} } }}{{\left( {f_{2}^{ - 2} - f_{1}^{ - 2} } \right)}}}\,\Updelta \varepsilon \approx 3.0\Updelta \varepsilon } $$

(7)

The uncorrected position is computed using P ₁.The other positions are computed from ionosphere-corrected pseudoranges, here called $P_{0}^{\prime }$ similarly to (2a):

$$ {P_{0}^{\prime } = P_{1} - I\;f_{1}^{ - 2} } $$

(8)

The ionospheric delay term I is here calculated using (1b), with N the electron density from the model or the map, and dl from the ray-path to voxel intersection.

Positioning

All other calculations are carried out in the same way as in Allain and Mitchell (2009), taking P as the phase-filtered ionosphere-free pseudorange or unfiltered ionosphere-free pseudorange or ionosphere-corrected pseudorange or uncorrected pseudorange.

Results

Figure 2 shows an example of a tomographic image over part of North America. The example has been chosen arbitrarily and is for 18:00 UT on the October 27, 2003. The 3D image has been integrated through and contoured to show the spatial distribution of vertical TEC on Fig. 2a. The TEC shows a gradient to the southern center of the map as would be expected for this time and location. Figure 2b shows a cross section of the free electron density at grid longitude 0°. It shows the electron density profile reaches a maximum at around 350 km for all grid latitudes from −8° to 16°. Figure 3 shows the thin-shell image at the same time. The TEC values are similar to the TEC from the full-height image, slightly lower overall.

Fig. 2

Full-height image of October 27, 2003, 18:00 UT:TEC from a vertical integration (a), cross section of the free electron density on the 0° grid longitude plane (b)

Fig. 3

Thin-shell image of October 27, 2003, 18:00 UT

Figures 4 and 5 show the mean and 95th percentile positioning errors for the five test sites (see Fig. 1) for all the days of this study. Comparing the uncorrected results in each of the five graphs (red curve), the largest errors occur at local midday, with all sites experiencing values of about 8 m (95% 13 m) but DRAO, in the northwest, values of about 7 m (95% 12 m) and CVMS, in the southeast, values of about 9 m (95% 16 m). The Klobuchar model reduces these errors to mean values of about 5, 4, and 6 m, respectively (95% 13, 13 and 15 m), but on the local evening, the largest errors occur around 18:00 local time (around 00:00 UT). The thin-shell and full-height ionospheric maps reduce these errors to almost constant mean values of around 1 m for the northerly sites DRAO and NRC1, 1.5 m for TMGO, at the center, and 2 m for the two southerly sites UNR1 and CVMS with a 95th percentile of around 5 m for all stations. Whereas the phase-filtered dual-frequency results is similar for all locations: mean around 0.5 m (95% 1.5 m), the unfiltered dual-frequency is most site dependent, going from mean around 1.5 m (95% 3 m) for the northerly sites to a mean of almost 6 m (95% 16 m) for CVMS. The unfiltered dual-frequency positioning for CVMS has the worst mean of all but the uncorrected positioning, then only during the daytime, and has the worst 95th percentile of all positionings.

Fig. 4

Average absolute positioning error for DRAO (a), NRC1 (b), TMGO (c), UNR1 (d), and CVMS (e)

Fig. 5

95th Percentile absolute positioning error for DRAO (a), NRC1 (b), TMGO (c), UNR1 (d), and CVMS (e)

October storm

Figures 6, 7, 8, 9, and 10 show the positioning errors for all 5 test sites from October 27 to November 2, 2003. The scale is different whether the values are below or above 10 m. Each curve shows positioning error calculated on a point-by-point basis then averaged (mean) over all of the points for the hour. At all times and for all stations, the dual-frequency positions are similar. All stations have a filtered dual-frequency position and an unfiltered dual-frequency position similar to their seasonal averages, within 1 m for the filtered dual-frequency and around 1, 1.5, 2.5, 3, and 6 m, respectively, for DRAO, NRC1, TMGO, UNR1, and CVMS. This shows that the dual-frequency positions were unaffected by this storm. The accuracy is similar on all days shown but October 29–30 for all stations and for all single-frequency positions. The uncertainties on the thin-shell and full-height positions are both around 1.5 m for all stations but for CVMS at around 2 m, showing positioning performances close to filtered dual-frequency for both. The uncertainty on the uncorrected position shows the diurnal cycle of the Sun-driven ionosphere with a local midday peak around 12 m. The uncertainty on the Klobuchar position is similar to the uncorrected position, showing the correction has, on these days, little effect on the quality of the positioning.

Fig. 6

Positioning errors for DRAO from October 27 to November 2, 2003

Fig. 7

Positioning errors for NRC1 from October 27 to November 2, 2003

Fig. 8

Positioning errors for TMGO from October 27 to November 2, 2003

Fig. 9

Positioning errors for UNR1 from October 27 to November 2, 2003

Fig. 10

Positioning errors for CVMS from October 27 to November 2, 2003

October 29–30 was one of the strongest storms ever recorded. Figures 11 and 12 represent, respectively, the full-height and thin-shell images of the ionosphere at 22:20 UT on October 29, 2003, and Figs. 13 and 14 represent, respectively, the full-height and thin-shell images of the ionosphere at 21:10 UT on October 30, 2003. All four TEC maps show that the ionosphere was very disturbed on these days, showing strong variations over small distances, with an overall gradient to the southwest reaching TEC values of almost 180 TEC units at these times. For both full-height images, a cross section of the free electron density at grid longitude −8° is shown on Figs. 11b and 13b. Both cross sections show that the peak height was the same between −8° and 0° grid latitude and between 8° and 16° grid latitude, with strong changes around 4° grid latitude. On the 29th, the peak height was just below 300 km around 12° grid latitude and just below 400 km around −4° grid latitude. This explains a difference in TEC of up to about 70 TEC units at −4° grid latitude −16° grid longitude between the full-height and the thin-shell images. On the 30th the change in peak height are even more important, about 250 km around 12° grid latitude and above 400 km around −4° grid latitude. There is also a very high electron density, about half the value of the maximum, in the top side of the ionosphere at 12° longitude, giving the electron density profile two strongly separated maxima. This explains differences between the full-height and the thin-shell images of around 60 TEC units in several locations, up to 70 TEC units at 12° grid latitude −8° grid longitude.

Fig. 11

Full-height image of October 29, 2003, 22:20 UT:TEC from a vertical integration (a), Cross section of the free electron density on the −8° grid longitude plane (b)

Fig. 12

Thin-shell image of October 29, 2003, 22:20 UT

Fig. 13

Full-height image of October 30, 2003, 21:10 UT:TEC from a vertical integration (a), cross section of the free electron density on the −8° grid longitude plane (b)

Fig. 14

Thin-shell image of October 30, 2003, 21:10 UT

The very complex structure of the ionosphere on these days means it was difficult to correct for the ionosphere. Table 1 summarizes the maximum positioning errors for all stations on both days. NRC1, in the northeast, is the station least affected by the disturbances as they occurred mainly in the southwest: the uncertainties on the uncorrected and Klobuchar positions are only slightly higher than that of the surrounding days and the uncertainties on the thin-shell and full-height positions both reach up to around 10 and 6 m, respectively. All other remaining stations are significantly more affected. For these stations and for both days, all uncorrected positioning errors reach values between 30 and 53 m, but for DRAO on the 29th, the value is 23 m. For these stations and for both days, the Klobuchar positioning errors remain within values lower than the worse uncorrected positions, all between 30 and 45 m, but for DRAO on the 29th, the value is 23 m and for CVMS on the 30th, it is 19 m. For these stations and for both days, the thin-shell positions remain within 20–41 m but for CVMS on the 30th at 9 m. The thin-shell positions are an improvement over the Klobuchar position on most occasions but for TMGO on the 29th, 2 m worse, and for UNR1 on the 30th, 10 m worse. All full-height values remain within 15 m, showing an improvement over the thin-shell positions on most occasions but for DRAO on the 30th, 12 m worse, reaching 34 m. This can be explained by the fact that it was both below a region of the ionosphere particularly difficult to image and at the edge of the main group of receivers used for the inversion.

Table 1 Maximum positioning errors (m) for all stations on local October 29–30

November storm

November 20 was a moderate storm day. Figures 15 and 16 represent, respectively, the full-height and thin-shell images of the ionosphere at 20:20 UT on that day. Both TEC maps show that the ionosphere was relatively disturbed on that day, showing again strong variations over small distances, with a line of enhancement from grid latitude −4° grid longitude 24° to grid latitude 8° grid longitude 4° with TEC values reaching 180 TEC units. A cross section of the free electron density at grid latitude 12° is shown on Fig. 15a, showing that the peak height was more or less around 400 km for all grid latitudes shown. This explains a difference in TEC of up to about 40 TEC units at 4° grid latitude 12° grid longitude between the full-height and the thin-shell images.

Fig. 15

Full-height image of November 20, 2003, 20:20 UT : TEC from a vertical integration (a), cross section of the free electron density on the 12° grid longitude plane (b)

Fig. 16

Thin-shell image of November 20, 2003, 20:20 UT

Figures 17, 18, 19, 20, and 21 show the positioning errors for all five stations from November 18–24. No observations were available for CVMS on the 18th until 20:03:30 UT, which explains the blank before then on Fig. 21. All stations have a filtered dual-frequency position and an unfiltered dual-frequency position similar to their seasonal averages. This shows that the dual-frequency positions were again unaffected by this storm. The uncorrected positioning errors give a good estimate of the ionization: on the 18th and 19th, they reach up to around 6 m for DRAO and 7 m for the other stations; on the 20th, ionization enhancements are shown, negative for DRAO with almost no daytime peak, limited for UNR1, again reaching up to 7 m, and strong for NRC1, TMGO, and CVMS, reaching up to 20, 13, and 25 m, respectively; and slowly settling on the 21st and 22nd with maxima above average on the 23rd and 24th around 11 m for all stations but DRAO around 9 m. Of particular interest on the 18th, 19th, and 21st were the good performances of the Klobuchar positions, all well within 5 m, even 2 m for NRC1. It, however, went back to normal on the following 3 days with maximum errors above average around 8 m. All stations have the full-height and thin-shell positions similar to their seasonal averages on all days but the November 20. They perform similarly on that day, with maximum errors around 14 and 8 m for NRC1 and CVMS and around 3 m for all other stations.

Fig. 17

Positioning errors for DRAO from October 27 to November 2, 2003

Fig. 18

Positioning errors for NRC1 from October 27 to November 2, 2003

Fig. 19

Positioning errors for TMGO from October 27 to November 2, 2003

Fig. 20

Positioning errors for UNR1 from October 27 to November 2, 2003

Fig. 21

Positioning errors for CVMS from October 27 to November 2, 2003

Conclusion

Different methods of correcting for the ionospheric delay have been compared for five stations around North America from October to December 2003. The first method uses no correction, the second uses the Klobuchar (1987) model, the third and fourth use, respectively, thin-shell and full-height real-time images from a four-dimensional inversion algorithm called MIDAS, and the fifth method uses unfiltered dual-frequency corrections. The benchmarks use carrier-smoothed dual-frequency corrections. The results were presented as hourly mean and hourly 95th percentile of the absolute error. It was shown that, while the Klobuchar model compensates for most of the ionospheric delay, the thin-shell and full-height MIDAS images perform almost as well as the dual-frequency carrier-smoothed benchmark and for most receivers better than the unfiltered dual-frequency benchmark.

Using MIDAS, thin-shell and full-height images give a position on average within 1.5 m and 95% of the time within 5 m, when the values are 0.5 and 1.5 m for the filtered dual-frequency, 4 and 13 m for the Klobuchar model, and up to 6 and 16 m for the unfiltered dual-frequency benchmark. The full-height corrections perform well and are considerably better than thin-shell corrections under extreme storm conditions: the positions of all but one receivers remained within 15 m during the October 29–30, 2003 storm. These results confirm those from Allain and Mitchell (2009), where Klobuchar model, full-height imaging, and carrier-smoothed dual-frequency corrections were also compared for Europe. The main differences are that the storm enhancements were mainly positive and strong over North America rather than negative over Europe and that no significant losses of lock on the carrier-smoothed dual-frequency positioning were detected for this North America set of receivers. The unfiltered dual-frequency positioning error is particularly site dependent, with an average from 1.5 to 6 m and a 95th percentile from 3 to 16 m. This is unlike the filtered dual-frequency benchmark, because the noise, which includes multipath, for the same satellite and the same receiver is very different on separate frequencies. This difference is greatly amplified by the dual-frequency calculation and makes the unfiltered dual-frequency pseudorange and positioning very noisy. A more thorough study on this point, i.e., the effect of multipath on dual-frequency pseudorange, would be valuable in the future.

The greatest error appears to be for a location at the edge of the network used for the MIDAS inversion while there was an important ionospheric structure above. This was probably due to the generally lower accuracy of the MIDAS images in these areas and to the intrinsic, numerical, and computational difficulties of imaging certain features.

This study has shown the following: the full-height and thin-shell images both allow very good corrections of a quiet ionosphere; they even give a better positioning than unfiltered dual-frequency as they do not amplify the noise on the observations; the full-height images allow significantly better corrections of the ionospheric delay than the thin-shell images during extreme storms.