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Evaluating the accuracy of ionospheric range delay corrections for navigation at low latitude
Advances in Space Research 36 (2005) 546–551 www.elsevier.com/locate/asr
Evaluating the accuracy of ionospheric range delay corrections for navigation at low latitude A. Meza a
a,b
, M. Gende
a,b,*
, C. Brunini
a,b
, S.M. Radicella
c
Facultad de Ciencias Astrono´micas y Geofı´sicas, Universidad Nacional de La Plata, Argentina b Consejo Nacional de Investigaciones Cientı´ficas y Te´cnicas, Argentina c Aeronomy and Radiopropagation Laboratory, Abdus Salam ICTP, Italy
Received 30 September 2004; received in revised form 21 March 2005; accepted 19 April 2005
Abstract La Plata Ionospheric Model (LPIM) corrections in ionospheric range delay can be studied by analysing their effect on geodetic positioning. To evaluate these corrections, a scheme similar to that used in Satellite-Based Augmentation System (SBAS) is employed in this work. So, in this context an appropriate model for computing the ionospheric range delay and its error is required. By means of numerical simulation, the accuracy of LPIM for computing the ionospheric range delay correction at low latitude is evaluated. The slant range delays for the observing network are simulated using the NeQuick ionospheric model. The simulated ionospheric range delay correction applied to geodetic positioning is analysed at different levels of ionospheric activity as well as to solstices and equinoxes. The large errors of the ionospheric correction on geodetic positioning are obtained in high solar activity during the equinox. This work shows that the accuracy achieved using simulated data at low latitude is similar to the accuracy reported using real data at mid latitude. 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Ionosphere; Global Positioning System; Satellite-Based Augmentation Systems
1. Introduction Satellite-Based Augmentation Systems (SBAS) aim at the validation of the integrity of Global Positioning System (GPS) signals, making their use in critical services such as civil aviation possible. At present there are three different SBASs: the European Geostationary Navigation Overlay Service (EGNOS) (Benedicto et al., 1999), which covers the European Civil Aviation Conference region; the Wide Area Augmentation System (WAAS) (Loh et al., 1995), covering essentially continental USA and Canada; and the Multifunctional Transport Satellite Space-Based Aug*
Corresponding author. Tel.: +54 221 4213 677. E-mail address: [email protected] (M. Gende).
mentation System (MSAS) (Shimamura, 1999), which covers the Flight Instrumental Region associated to Japan. EGNOS and WAAS cover areas located at mid and high geomagnetic latitudes and MSAS covers mid-low latitude regions. The aim of SBAS is to obtain at least 7-m horizontal and vertical accuracy (Benedicto et al., 1999, http://gps.faa.gov/Programs/ WAAS/benefits-text.htm). Nowadays, there is a great interest in extending the SBAS facilities to low geomagnetic latitude regions. A fundamental SBAS requirement is not only to estimate corrections for the ionospheric range delay but also to make them available to users. In order to achieve this aim, three major problems need to be solved: firstly, to estimate ionospheric range delay corrections; secondly, to predict its value for future time periods;
0273-1177/$30 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2005.04.053
A. Meza et al. / Advances in Space Research 36 (2005) 546–551
and finally, to transmit the corrections to the users. The objective of this work is to solve the first problem by means of analysing the LPIM in geodetic positioning at low latitude based on the scheme used by SBAS. The Vertical Total Electron Content (VTEC) grid was computed by LPIM from simulated data. These values were interpolated and transformed to Slant Total Electron Content (STEC) to correct the ionospheric delay of the users GPS observations. The discussion is organised into three main parts: firstly, a short description of our GPS-based regional ionospheric model (LPIM) is provided; then, the numerical simulation scheme used to assess the accuracy of the LPIM-derived corrections is described; lastly, the effects of LPIM errors in positioning are evaluated and discussed.
VTEC ¼ cos z0 ; STEC
547
ð3Þ
where z 0 being the zenith distance of the signal at the piercing point. The ambiguities term, C sr , is estimated and reduced from Eq. (1) using P-code phase observations which are unambiguous but still less precise than /4. For regional purposes, the spatial and temporal variability of the VTEC on the shell is represented by a polynomial expansion dependent of the geographic longitude, kpp, and latitude, upp, of the signal piercing point, and the universal time, t, of the observation VTECðx; y; tÞ ¼ C 0;0 ðtÞ þ C 1;0 ðtÞ x þ C 0;1 ðtÞ y þ C 2;0 ðtÞx2 þ C 1;1 ðtÞ x y þ C 0;2 ðtÞ y 2 þ C 3;0 ðtÞ x3 þ C 2;1 ðtÞ x2 y þ C 1;2 ðtÞ y x2 þ C 0;3 ðtÞ y 3 þ þ C K;0 ðtÞ xK þ C K1;1 ðtÞ xK1 y
2. Description of the GPS-based regional La Plata Ionospheric Model (LPIM)
þ þ C 1;K1 ðtÞ y xK1 þ C 0;K ðtÞ y K ;
The GPS consists of a constellation of 27 radio navigation satellites and a ground control subsystem. The satellites broadcast two carrier signals at 1.5754 and 1.2276 GHz, named L1 and L2, respectively, modulated by the so-called P and the C/A codes (Seeber, 2003). Forming ionospheric observables from the GPS signals has been exhaustively discussed by Manucci et al. (1998). Basically, when L2 is subtracted from L1, the satellite-receiver geometrical range and all frequency independent biases are removed, and the so-called geometry free linear combination in meters,/4, is obtained /4 ¼ /1 /2 ¼ a STEC þ c ðsr þ ss Þ þ C sr þ t;
ð4Þ
where x = kpp k0 and y = upp u0 are the reduced longitude and latitude of the piercing point with respect to points k0 and u0 close to the geometric centre of the region; and K is the polynomial degree. The temporal coefficients Ci, j(t) are further expanded into Fourier series up to a maximum degree L XL 2p C i;j ðtÞ ¼ ai;j;0 þ ai;j;m cosðm tÞ þ bi;j;m m¼1 24 2p ð5Þ sinðm tÞ; 24 where ai, j, m and bi, j, m are the unknown coefficients of Fourier series.
ð1Þ
where /1 and /2 are the carrier phase observations at the frequencies L1 and L2; a is a constant; STEC is the Slant Total Electron Content; sr and ss are inter-frequencies electronic delays produced in the hardware of the receiver and the satellite, respectively; c is the speed of light in vacuum; C sr is the combination of both carrier phase ambiguities; and t is the L1–L2 combined measurement error. STEC is the integral of the electron density distribution, Ne, along the signal path from the satellite to the receiver Z STEC ¼ N e dl. ð2Þ pathsr
The LPIM (Brunini et al., 2003) assumes that the ionosphere is concentrated in a spherical shell of infinitesimal thickness, located at 450 km above the Earths surface. Within this approximation, the STEC along the signal path is converted into Vertical Total Electron Content (VTEC) at the point where the signal pierces the shell, using the approximate mapping function
3. Numerical simulation NeQuick model was used to simulate the geometryfree observable from the hypothetical network of GPS receivers were shown in Fig. 1. NeQuick (Hochegger et al., 2000) computes the ionospheric electron density as a function of solar activity, month, UT, height and geographic coordinates. It is a quick-run model for trans-ionospheric applications that allows to calculate vertical and slant TEC for any specified path. From 100 km up to the F2 peak, this model uses a modified DGR profile formulation (Di Giovanni and Radicella, 1990) which includes five semi-Epstein layers with modelled thickness parameters. It is also based on anchor points defined by foE, foF1, foF2 and M(3000)F2 values. NeQuick applies the diffusive equilibrium concept in a topside formulation based on a semi-Epstein layer governed by an empirical scale factor which is height dependent. The model has been adopted in the ionospheric specifications for the European Space
548
A. Meza et al. / Advances in Space Research 36 (2005) 546–551
(2) although the inter-frequency electronic delays and the random noise were assumed equal to zero for the data simulation (Eq. (6)), the estimation of non-zero values is allowed by Eq. (7), to account for any inconsistency between the LPIM and the NeQuick model.
4. Applying correction to the position estimates In order to understand how the ionospheric range delay error applies to the coordinates, we will start with the fundamental observation equation for GNSS single-point, single-epoch positioning (e.g., Seeber, 2003) ^ q0 ¼ ðcos E cos AÞ dn þ ðcos E sin AÞ de q þ ðsin EÞ dv þ c dt þ e;
Fig. 1. Distribution of the receivers in the area under study.
Agency ESA satellite navigation and positioning programmes and, more recently, by ITU-R Recommendation. The NeQuick source code is available at http://www.itu.int/ITU-R/software/studygroups/rsg3/databanks/ionosph. From Eqs. (1) and (2) and using the NeQuick electron density function, the simulated /4 values are defined as Z /4NQ ¼ a N eNQ dl. ð6Þ pathsr
From Eq. (6), a dataset of geometry free observations with 30-s sampling rate was simulated for the observing GPS network were shown in Fig. 1. The satellite positions were computed from the GPS ephemeris. It should be noted that all the biases and random noise in Eq. (1) were assumed equal to zero in the simulated observations. Using this simulated dataset, the expansion coefficients together with the inter-frequency electronic delays of the VTEC expansion equations (4) and (5) can be adjusted by least squares. The corresponding observation equation follows from Eqs. (1)–(6): a /4NQ ¼ VTEC þ c ð~sr þ ~ss Þ þ ~t. ð7Þ cos z0 In this way, we obtain the VTEC as a function of time, latitude and longitude. It should be noted that the above simulation scheme assumes that: (1) the ambiguity term C sr of Eq. (1) has been correctly estimated and removed from the observations;
ð8Þ
^ is the observed pseudo-range corrected by the where q ionospheric and tropospheric delays, by relativistic effects and by satellite clock errors; q0 is the satellite-receiver geometrical range, computed from the a priori receiver coordinates satellites ephemerid precise, E and A are the satellite elevation and azimuth, and cos E Æ cos A, cos E Æ sin A, sin E, the corresponding direction cosines of the receiver-to-satellite direction with respect to the north-east-vertical local coordinate system; c is the speed of light in vacuum; dn, de, dv are the corrections to the a priori receiver coordinates in the local system, dt is the receiver clock error; and e represents the observation random error. If n satellites are simultaneously observed, the observation equation (8) can be arranged in a linear equation system Dq ¼ A Dx þ e;
ð9Þ
where Dq is a n · 1 matrix containing the observed minus computed ranges; Dx is a 4 · 1 matrix containing the positioning and time unknown dn, de, dv, dt; e is n · 1 matrix containing the observation random errors; and A is n · 4 matrix containing the direction cosines and the time unknown coefficient cos E1 cos A1 cos E1 sin A1 sin E1 c .. .. .. .. A¼ ð10Þ . . . . . cos En cos An cos En sin An sin En c Provided that four or more satellites are simultaneously observed (i.e., n P 4), the linear equation system (Eq. (9)) can be solved by least squares Dx ¼ ðAt AÞ1 At Dq
ð11Þ
Eq. (11) expresses the key relation of this work, because it establishes a link between the unmodelled errors and their effects on positioning. If the a priori receiver coordinates were perfectly known, the receiver
A. Meza et al. / Advances in Space Research 36 (2005) 546–551
clock were in perfect synchrony with the GPS time, and all the systematic errors affecting the observations were perfectly corrected, the last term of Eq. (11) should be Dq = 0, and this Eq. (11) will lead to a solution Dx = 0, except for the presence of observation random errors. In this work, Dq will be considered to be affected only by the ionospheric range delay error, i.e., Dq = a STEC. In order to assess the effects of this error on coordinates and clock estimates, Eq. (11) will be used.
549
(4) the NeQuick slant ionospheric range delays effect on positioning was computed by solving the positioning problem discussed in Section 4. Fig. 2 shows the ionospheric range delay and its LPIM corrections on the user receiver coordinates and clock, for April 20, 2000. For navigation purposes, clock error is not relevant but it is presented here because it takes a large part of the ionospheric range delay. 5.3. Origin of the discrepancies
5. Results 5.1. Chosen scenery The area chosen for the study is the African sector located between 5 and 45 longitude and 7 and 13 latitude (Fig. 1). From the ionospheric point of view, this is a critical area due to its location close to the Equatorial Anomaly Region. As a consequence, a great ionospheric variability is expected, depending on the azimuth and elevation of the observed satellites. Eight GPS receivers marked with asterisks in Fig. 1 compose the hypothetical-tracking network. A user marked with a triangle in Fig. 1, located in the central part of the region covered by the tracking stations, is also considered. The study was undertaken in the years 1996 and 2000, which correspond to periods near minimum and maximum solar activity, respectively. The solstice and equinox, two epochs in which the ionosphere presents different behaviour, were chosen to carry out the analysis. Due to the impossibility for the NeQuick to reproduce small day by day variability but smooth ionospheric behaviour, selecting one day per season is enough to fulfil the task. The selected dates were the 20th of April and the 20th of July. 5.2. Discrepancies between the ionospheric range delay and its LPIM corrections In order to evaluate the accuracy of the LPIM ionospheric range delay correction, we proceeded as follows: (1) using the methodology explained in Section 3, a time-dependent Grid Ionospheric Vertical Delay (GIVD) was computed at each Ionospheric Grid Point (IGP) of a 5 · 5 grid covering the hypothetical SBAS region (Fig. 1); (2) the User Ionospheric Vertical Delays (UIVD) were obtained by interpolation from the nearby GIVD; (3) the UIVD were converted into slant ionospheric range delay, and their effect on positioning was computed by solving the positioning problem discussed in Section 4;
This section determines the origin of the discrepancy between ionospheric range delay and its LPIM corrections. The approximations made in order to distinguish and quantify the individual contributions are reviewed. The causes that produce deficient ionospheric corrections can be subdivided into three categories: the interpolation error when obtaining a VTEC value from GIVDs, LPIMs mismodelling of the VTEC estimate and finally, the deficient conversion mapping function between VTEC and STEC values. 5.3.1. Interpolation error All SBAS systems provide GIVD values with 5 spacing. As users need VTEC values in the pierce points, they must interpolate between IGP. This carries an associated error that can be estimated in a simple way by comparing an arbitrary interpolated value from a 5 · 5 grid generated by the NeQuick model, with the exact value for that point directly provided by the model. As the LPIM model is not used in this test, this error is free from the errors listed in Section 5.3.2. Tests were done simulating the ionospheric conditions of years 1996 and 2000 (low and high solar activity) for solstice and equinox. In order to get the worst cases, the values located in the centre of each grid cell, the point which is separated 2.5 horizontally and vertical from each near neighbour value of the 5 · 5, were compared with its near neighbours in the 5 · 5 grid. One grid was calculated for each hour of the day. The test covered from latitude 7 to 13 and from longitude 5 to 45. Our results show that, in this region, the error has a 99% of probability to be below 0.6 m in the worst epoch (equinox with high solar activity) at the worst hour of the day (14 UT), see Table 1. 5.3.2. LPIM mismodelling LPIM cannot provide an exact value of VTEC value due to several reasons: (a) it uses an approximate function to relate STEC and VTEC values, Eq. (3); (b) it estimates the interfrequency electronic delays for the satellites and the receivers, Eq. (1);
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Fig. 2. Discrepancies between ionospheric range delay and its LPIM corrections on the user receiver coordinates and clock.
Table 1 Values of the 99% probability positioning error for the different cases at UT = 14 h
Interpolation error (m) LPIM mismodelling (m)
Equinox 1996
Solstice 1996
Equinox 2000
Solstice 2000
0.2 0.6
0.1 0.5
0.6 2.5
0.3 0.8
(c) it uses a smooth mathematical representation of the real VTEC, Eqs. (4) and (5). The total LPIM mismodelling error can be estimated by comparing VTEC values obtained by LPIM with the VTEC values directly provided by the NeQuick model. Tests under the same condition in Section 5.3.1 were carried out, with the exception that in this case the comparison was done in identical 5 · 5 grid to avoid any interpolation error. Results show that this error has a 99% of probability to be below the values shown in Table 1. For this work, the worst cases were chosen, that means that the comparison is made at about UT = 14 h for Equinoxes and Solstices in 1996 and 2000. 5.3.3. Mapping function associated errors In order to obtain an ionospheric correction for each GPS satellite, the interpolated VTEC values must be transformed into STEC for each satellite-receiver line. The relation used for this transformation is the geometric mapping function shown in Eq. (3). The following
Fig. 3. Total and mapping function error.
A. Meza et al. / Advances in Space Research 36 (2005) 546–551
strategy was developed to isolate and estimate the error associated with the mapping function. As a first step, the STEC value using NeQuick model was calculated. Then the VTEC value for the same point was also calculated using NeQuick again and transformed to a slant value using Eq. (3). Finally, to determine the impact of the mapping function in the user position, the difference STECNQ cos1 z0 VTECNQ was introduced as Dq in Eq. (11). Fig. 3 shows the total ionospheric error as well as the influence of the error due to the mapping function on vertical position component, in the four analysed cases.
551
In all the cases that were analysed, a user can expect an error below 5 m. This accomplishes the 25 feet (7.6 m) that WAAS requires and the 6 m of accuracy that WAAS consistently demonstrates in mid latitudes using real GPS data (http://users.erols.com/dlwilson/ gpswaas.htm). All the results presented in this paper come from a numerical simulation; therefore larger error values would be expected when using real data. An extensive work with real data for this region is presently being carried out. Finally, taking into account the error in Table 1 and Fig. 3, the LPIM model and the mapping function would have to be improved in order to achieve the accuracy obtained in this study.
6. Conclusions This work focuses on the estimation of the accuracy of ionospheric range delay corrections at low latitude. The accuracy was evaluated comparing the ionospheric range delay and its correction. The comparison was not performed at the STEC level, but on the effects of ionospheric error on user coordinates. The benefit of this approach is that it shows the accuracy of the corrections in the position, where the user needs it, and not in the pseudoranges, where they do not have a clear meaning for navigation purposes. The height and the clock of the GPS receiver are the most affected component by the error on LPIM corrections. The LPIM ionospheric model generally works well in a low latitude area but has got some problems representing the ionospheric condition in high solar activity during the equinox, see Table 1. The interpolation errors do not play an important role, since they are about three times smaller than the LPIM mismodelling.
References Benedicto, J., Michel, P., Ventura-Traveset, J. EGNOS: project status overview. Air & Space European 1 (1), 58–64, 1999. Brunini, C., Van Zele, M.A., Meza, A., et al. Quiet and perturbed ionospheric representation according to the electron content from GPS signals. J. Geophys. Res. 108 (k2), doi: 10.1029/ 2002JA009346, 2003. Di Giovanni, G., Radicella, S.M. An analytical model of the electron density profile in the ionosphere. Adv. Space Res. 10 (11), 27–30, 1990. Hochegger, G., Nava, B., Radicella, S.M., et al. A family of ionospheric models for different uses. Phys. Chem. Earth 25 (4), 307–310, 2000. Loh, R., Wullschleger, V., Elrod, B., et al. The US Wide-Area Augmentation System (WAAS) 42(3), 410–435, 1995. Manucci, A.J., Wilson, B.D., Yuan, D.N., et al. A global mapping technique for GPS-derived ionospheric total electron-content measurements. Radio. Sci. 33, 565–582, 1998. Seeber, G. Satellite Geodesy, second ed. Walter de Gruyter, 2003. Shimamura, A. MSAS (MTSAT satellite-based augmentation system) project status. Air & Space Europe 1 (2), 63–67, 1999.
Evaluating the accuracy of ionospheric range delay corrections for navigation at low latitude A. Meza a
a,b
, M. Gende
a,b,*
, C. Brunini
a,b
, S.M. Radicella
c
Facultad de Ciencias Astrono´micas y Geofı´sicas, Universidad Nacional de La Plata, Argentina b Consejo Nacional de Investigaciones Cientı´ficas y Te´cnicas, Argentina c Aeronomy and Radiopropagation Laboratory, Abdus Salam ICTP, Italy
Received 30 September 2004; received in revised form 21 March 2005; accepted 19 April 2005
Abstract La Plata Ionospheric Model (LPIM) corrections in ionospheric range delay can be studied by analysing their effect on geodetic positioning. To evaluate these corrections, a scheme similar to that used in Satellite-Based Augmentation System (SBAS) is employed in this work. So, in this context an appropriate model for computing the ionospheric range delay and its error is required. By means of numerical simulation, the accuracy of LPIM for computing the ionospheric range delay correction at low latitude is evaluated. The slant range delays for the observing network are simulated using the NeQuick ionospheric model. The simulated ionospheric range delay correction applied to geodetic positioning is analysed at different levels of ionospheric activity as well as to solstices and equinoxes. The large errors of the ionospheric correction on geodetic positioning are obtained in high solar activity during the equinox. This work shows that the accuracy achieved using simulated data at low latitude is similar to the accuracy reported using real data at mid latitude. 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Ionosphere; Global Positioning System; Satellite-Based Augmentation Systems
1. Introduction Satellite-Based Augmentation Systems (SBAS) aim at the validation of the integrity of Global Positioning System (GPS) signals, making their use in critical services such as civil aviation possible. At present there are three different SBASs: the European Geostationary Navigation Overlay Service (EGNOS) (Benedicto et al., 1999), which covers the European Civil Aviation Conference region; the Wide Area Augmentation System (WAAS) (Loh et al., 1995), covering essentially continental USA and Canada; and the Multifunctional Transport Satellite Space-Based Aug*
Corresponding author. Tel.: +54 221 4213 677. E-mail address: [email protected] (M. Gende).
mentation System (MSAS) (Shimamura, 1999), which covers the Flight Instrumental Region associated to Japan. EGNOS and WAAS cover areas located at mid and high geomagnetic latitudes and MSAS covers mid-low latitude regions. The aim of SBAS is to obtain at least 7-m horizontal and vertical accuracy (Benedicto et al., 1999, http://gps.faa.gov/Programs/ WAAS/benefits-text.htm). Nowadays, there is a great interest in extending the SBAS facilities to low geomagnetic latitude regions. A fundamental SBAS requirement is not only to estimate corrections for the ionospheric range delay but also to make them available to users. In order to achieve this aim, three major problems need to be solved: firstly, to estimate ionospheric range delay corrections; secondly, to predict its value for future time periods;
0273-1177/$30 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2005.04.053
A. Meza et al. / Advances in Space Research 36 (2005) 546–551
and finally, to transmit the corrections to the users. The objective of this work is to solve the first problem by means of analysing the LPIM in geodetic positioning at low latitude based on the scheme used by SBAS. The Vertical Total Electron Content (VTEC) grid was computed by LPIM from simulated data. These values were interpolated and transformed to Slant Total Electron Content (STEC) to correct the ionospheric delay of the users GPS observations. The discussion is organised into three main parts: firstly, a short description of our GPS-based regional ionospheric model (LPIM) is provided; then, the numerical simulation scheme used to assess the accuracy of the LPIM-derived corrections is described; lastly, the effects of LPIM errors in positioning are evaluated and discussed.
VTEC ¼ cos z0 ; STEC
547
ð3Þ
where z 0 being the zenith distance of the signal at the piercing point. The ambiguities term, C sr , is estimated and reduced from Eq. (1) using P-code phase observations which are unambiguous but still less precise than /4. For regional purposes, the spatial and temporal variability of the VTEC on the shell is represented by a polynomial expansion dependent of the geographic longitude, kpp, and latitude, upp, of the signal piercing point, and the universal time, t, of the observation VTECðx; y; tÞ ¼ C 0;0 ðtÞ þ C 1;0 ðtÞ x þ C 0;1 ðtÞ y þ C 2;0 ðtÞx2 þ C 1;1 ðtÞ x y þ C 0;2 ðtÞ y 2 þ C 3;0 ðtÞ x3 þ C 2;1 ðtÞ x2 y þ C 1;2 ðtÞ y x2 þ C 0;3 ðtÞ y 3 þ þ C K;0 ðtÞ xK þ C K1;1 ðtÞ xK1 y
2. Description of the GPS-based regional La Plata Ionospheric Model (LPIM)
þ þ C 1;K1 ðtÞ y xK1 þ C 0;K ðtÞ y K ;
The GPS consists of a constellation of 27 radio navigation satellites and a ground control subsystem. The satellites broadcast two carrier signals at 1.5754 and 1.2276 GHz, named L1 and L2, respectively, modulated by the so-called P and the C/A codes (Seeber, 2003). Forming ionospheric observables from the GPS signals has been exhaustively discussed by Manucci et al. (1998). Basically, when L2 is subtracted from L1, the satellite-receiver geometrical range and all frequency independent biases are removed, and the so-called geometry free linear combination in meters,/4, is obtained /4 ¼ /1 /2 ¼ a STEC þ c ðsr þ ss Þ þ C sr þ t;
ð4Þ
where x = kpp k0 and y = upp u0 are the reduced longitude and latitude of the piercing point with respect to points k0 and u0 close to the geometric centre of the region; and K is the polynomial degree. The temporal coefficients Ci, j(t) are further expanded into Fourier series up to a maximum degree L XL 2p C i;j ðtÞ ¼ ai;j;0 þ ai;j;m cosðm tÞ þ bi;j;m m¼1 24 2p ð5Þ sinðm tÞ; 24 where ai, j, m and bi, j, m are the unknown coefficients of Fourier series.
ð1Þ
where /1 and /2 are the carrier phase observations at the frequencies L1 and L2; a is a constant; STEC is the Slant Total Electron Content; sr and ss are inter-frequencies electronic delays produced in the hardware of the receiver and the satellite, respectively; c is the speed of light in vacuum; C sr is the combination of both carrier phase ambiguities; and t is the L1–L2 combined measurement error. STEC is the integral of the electron density distribution, Ne, along the signal path from the satellite to the receiver Z STEC ¼ N e dl. ð2Þ pathsr
The LPIM (Brunini et al., 2003) assumes that the ionosphere is concentrated in a spherical shell of infinitesimal thickness, located at 450 km above the Earths surface. Within this approximation, the STEC along the signal path is converted into Vertical Total Electron Content (VTEC) at the point where the signal pierces the shell, using the approximate mapping function
3. Numerical simulation NeQuick model was used to simulate the geometryfree observable from the hypothetical network of GPS receivers were shown in Fig. 1. NeQuick (Hochegger et al., 2000) computes the ionospheric electron density as a function of solar activity, month, UT, height and geographic coordinates. It is a quick-run model for trans-ionospheric applications that allows to calculate vertical and slant TEC for any specified path. From 100 km up to the F2 peak, this model uses a modified DGR profile formulation (Di Giovanni and Radicella, 1990) which includes five semi-Epstein layers with modelled thickness parameters. It is also based on anchor points defined by foE, foF1, foF2 and M(3000)F2 values. NeQuick applies the diffusive equilibrium concept in a topside formulation based on a semi-Epstein layer governed by an empirical scale factor which is height dependent. The model has been adopted in the ionospheric specifications for the European Space
548
A. Meza et al. / Advances in Space Research 36 (2005) 546–551
(2) although the inter-frequency electronic delays and the random noise were assumed equal to zero for the data simulation (Eq. (6)), the estimation of non-zero values is allowed by Eq. (7), to account for any inconsistency between the LPIM and the NeQuick model.
4. Applying correction to the position estimates In order to understand how the ionospheric range delay error applies to the coordinates, we will start with the fundamental observation equation for GNSS single-point, single-epoch positioning (e.g., Seeber, 2003) ^ q0 ¼ ðcos E cos AÞ dn þ ðcos E sin AÞ de q þ ðsin EÞ dv þ c dt þ e;
Fig. 1. Distribution of the receivers in the area under study.
Agency ESA satellite navigation and positioning programmes and, more recently, by ITU-R Recommendation. The NeQuick source code is available at http://www.itu.int/ITU-R/software/studygroups/rsg3/databanks/ionosph. From Eqs. (1) and (2) and using the NeQuick electron density function, the simulated /4 values are defined as Z /4NQ ¼ a N eNQ dl. ð6Þ pathsr
From Eq. (6), a dataset of geometry free observations with 30-s sampling rate was simulated for the observing GPS network were shown in Fig. 1. The satellite positions were computed from the GPS ephemeris. It should be noted that all the biases and random noise in Eq. (1) were assumed equal to zero in the simulated observations. Using this simulated dataset, the expansion coefficients together with the inter-frequency electronic delays of the VTEC expansion equations (4) and (5) can be adjusted by least squares. The corresponding observation equation follows from Eqs. (1)–(6): a /4NQ ¼ VTEC þ c ð~sr þ ~ss Þ þ ~t. ð7Þ cos z0 In this way, we obtain the VTEC as a function of time, latitude and longitude. It should be noted that the above simulation scheme assumes that: (1) the ambiguity term C sr of Eq. (1) has been correctly estimated and removed from the observations;
ð8Þ
^ is the observed pseudo-range corrected by the where q ionospheric and tropospheric delays, by relativistic effects and by satellite clock errors; q0 is the satellite-receiver geometrical range, computed from the a priori receiver coordinates satellites ephemerid precise, E and A are the satellite elevation and azimuth, and cos E Æ cos A, cos E Æ sin A, sin E, the corresponding direction cosines of the receiver-to-satellite direction with respect to the north-east-vertical local coordinate system; c is the speed of light in vacuum; dn, de, dv are the corrections to the a priori receiver coordinates in the local system, dt is the receiver clock error; and e represents the observation random error. If n satellites are simultaneously observed, the observation equation (8) can be arranged in a linear equation system Dq ¼ A Dx þ e;
ð9Þ
where Dq is a n · 1 matrix containing the observed minus computed ranges; Dx is a 4 · 1 matrix containing the positioning and time unknown dn, de, dv, dt; e is n · 1 matrix containing the observation random errors; and A is n · 4 matrix containing the direction cosines and the time unknown coefficient cos E1 cos A1 cos E1 sin A1 sin E1 c .. .. .. .. A¼ ð10Þ . . . . . cos En cos An cos En sin An sin En c Provided that four or more satellites are simultaneously observed (i.e., n P 4), the linear equation system (Eq. (9)) can be solved by least squares Dx ¼ ðAt AÞ1 At Dq
ð11Þ
Eq. (11) expresses the key relation of this work, because it establishes a link between the unmodelled errors and their effects on positioning. If the a priori receiver coordinates were perfectly known, the receiver
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clock were in perfect synchrony with the GPS time, and all the systematic errors affecting the observations were perfectly corrected, the last term of Eq. (11) should be Dq = 0, and this Eq. (11) will lead to a solution Dx = 0, except for the presence of observation random errors. In this work, Dq will be considered to be affected only by the ionospheric range delay error, i.e., Dq = a STEC. In order to assess the effects of this error on coordinates and clock estimates, Eq. (11) will be used.
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(4) the NeQuick slant ionospheric range delays effect on positioning was computed by solving the positioning problem discussed in Section 4. Fig. 2 shows the ionospheric range delay and its LPIM corrections on the user receiver coordinates and clock, for April 20, 2000. For navigation purposes, clock error is not relevant but it is presented here because it takes a large part of the ionospheric range delay. 5.3. Origin of the discrepancies
5. Results 5.1. Chosen scenery The area chosen for the study is the African sector located between 5 and 45 longitude and 7 and 13 latitude (Fig. 1). From the ionospheric point of view, this is a critical area due to its location close to the Equatorial Anomaly Region. As a consequence, a great ionospheric variability is expected, depending on the azimuth and elevation of the observed satellites. Eight GPS receivers marked with asterisks in Fig. 1 compose the hypothetical-tracking network. A user marked with a triangle in Fig. 1, located in the central part of the region covered by the tracking stations, is also considered. The study was undertaken in the years 1996 and 2000, which correspond to periods near minimum and maximum solar activity, respectively. The solstice and equinox, two epochs in which the ionosphere presents different behaviour, were chosen to carry out the analysis. Due to the impossibility for the NeQuick to reproduce small day by day variability but smooth ionospheric behaviour, selecting one day per season is enough to fulfil the task. The selected dates were the 20th of April and the 20th of July. 5.2. Discrepancies between the ionospheric range delay and its LPIM corrections In order to evaluate the accuracy of the LPIM ionospheric range delay correction, we proceeded as follows: (1) using the methodology explained in Section 3, a time-dependent Grid Ionospheric Vertical Delay (GIVD) was computed at each Ionospheric Grid Point (IGP) of a 5 · 5 grid covering the hypothetical SBAS region (Fig. 1); (2) the User Ionospheric Vertical Delays (UIVD) were obtained by interpolation from the nearby GIVD; (3) the UIVD were converted into slant ionospheric range delay, and their effect on positioning was computed by solving the positioning problem discussed in Section 4;
This section determines the origin of the discrepancy between ionospheric range delay and its LPIM corrections. The approximations made in order to distinguish and quantify the individual contributions are reviewed. The causes that produce deficient ionospheric corrections can be subdivided into three categories: the interpolation error when obtaining a VTEC value from GIVDs, LPIMs mismodelling of the VTEC estimate and finally, the deficient conversion mapping function between VTEC and STEC values. 5.3.1. Interpolation error All SBAS systems provide GIVD values with 5 spacing. As users need VTEC values in the pierce points, they must interpolate between IGP. This carries an associated error that can be estimated in a simple way by comparing an arbitrary interpolated value from a 5 · 5 grid generated by the NeQuick model, with the exact value for that point directly provided by the model. As the LPIM model is not used in this test, this error is free from the errors listed in Section 5.3.2. Tests were done simulating the ionospheric conditions of years 1996 and 2000 (low and high solar activity) for solstice and equinox. In order to get the worst cases, the values located in the centre of each grid cell, the point which is separated 2.5 horizontally and vertical from each near neighbour value of the 5 · 5, were compared with its near neighbours in the 5 · 5 grid. One grid was calculated for each hour of the day. The test covered from latitude 7 to 13 and from longitude 5 to 45. Our results show that, in this region, the error has a 99% of probability to be below 0.6 m in the worst epoch (equinox with high solar activity) at the worst hour of the day (14 UT), see Table 1. 5.3.2. LPIM mismodelling LPIM cannot provide an exact value of VTEC value due to several reasons: (a) it uses an approximate function to relate STEC and VTEC values, Eq. (3); (b) it estimates the interfrequency electronic delays for the satellites and the receivers, Eq. (1);
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Fig. 2. Discrepancies between ionospheric range delay and its LPIM corrections on the user receiver coordinates and clock.
Table 1 Values of the 99% probability positioning error for the different cases at UT = 14 h
Interpolation error (m) LPIM mismodelling (m)
Equinox 1996
Solstice 1996
Equinox 2000
Solstice 2000
0.2 0.6
0.1 0.5
0.6 2.5
0.3 0.8
(c) it uses a smooth mathematical representation of the real VTEC, Eqs. (4) and (5). The total LPIM mismodelling error can be estimated by comparing VTEC values obtained by LPIM with the VTEC values directly provided by the NeQuick model. Tests under the same condition in Section 5.3.1 were carried out, with the exception that in this case the comparison was done in identical 5 · 5 grid to avoid any interpolation error. Results show that this error has a 99% of probability to be below the values shown in Table 1. For this work, the worst cases were chosen, that means that the comparison is made at about UT = 14 h for Equinoxes and Solstices in 1996 and 2000. 5.3.3. Mapping function associated errors In order to obtain an ionospheric correction for each GPS satellite, the interpolated VTEC values must be transformed into STEC for each satellite-receiver line. The relation used for this transformation is the geometric mapping function shown in Eq. (3). The following
Fig. 3. Total and mapping function error.
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strategy was developed to isolate and estimate the error associated with the mapping function. As a first step, the STEC value using NeQuick model was calculated. Then the VTEC value for the same point was also calculated using NeQuick again and transformed to a slant value using Eq. (3). Finally, to determine the impact of the mapping function in the user position, the difference STECNQ cos1 z0 VTECNQ was introduced as Dq in Eq. (11). Fig. 3 shows the total ionospheric error as well as the influence of the error due to the mapping function on vertical position component, in the four analysed cases.
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In all the cases that were analysed, a user can expect an error below 5 m. This accomplishes the 25 feet (7.6 m) that WAAS requires and the 6 m of accuracy that WAAS consistently demonstrates in mid latitudes using real GPS data (http://users.erols.com/dlwilson/ gpswaas.htm). All the results presented in this paper come from a numerical simulation; therefore larger error values would be expected when using real data. An extensive work with real data for this region is presently being carried out. Finally, taking into account the error in Table 1 and Fig. 3, the LPIM model and the mapping function would have to be improved in order to achieve the accuracy obtained in this study.
6. Conclusions This work focuses on the estimation of the accuracy of ionospheric range delay corrections at low latitude. The accuracy was evaluated comparing the ionospheric range delay and its correction. The comparison was not performed at the STEC level, but on the effects of ionospheric error on user coordinates. The benefit of this approach is that it shows the accuracy of the corrections in the position, where the user needs it, and not in the pseudoranges, where they do not have a clear meaning for navigation purposes. The height and the clock of the GPS receiver are the most affected component by the error on LPIM corrections. The LPIM ionospheric model generally works well in a low latitude area but has got some problems representing the ionospheric condition in high solar activity during the equinox, see Table 1. The interpolation errors do not play an important role, since they are about three times smaller than the LPIM mismodelling.
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