Annals 48, 3, 2005+app1 439 ANNALS OF GEOPHYSICS, VOL. 48, N. 3, June 2005 Key words GNSS – Global Positioning System – sin- gle point positioning – ionospheric biases correction 1. Introduction The Global Positioning System (GPS) is the most developed and extended Global Navigation Satellite System (GNSS). Its field of application ranges from sub-centimetre level accurate geody- namic studies to decametre level stand-alone po- sitioning. Stand-alone positioning, perhaps the most extended use of GPS, is normally per- formed using inexpensive receivers that do not record the observed GPS signal measurements, but the coordinates. It is well known that the accuracy of GPS positioning is degraded by several biases (See- ber, 2003). Some of these biases are: delays caused by the atmosphere (ionosphere and tro- posphere), satellite clock and orbit inaccuracy and spurious signal reflection (multi-path). The main bias is the ionospheric delay, which is typ- ically one order of magnitude greater than any of the other afore mentioned biases (Mannucci et al., 1999). In order to mitigate the ionospheric bias ef- fect on positioning, GPS receivers use an em- bedded ionospheric model called Ionospheric Correction Algorithm (ICA) (Klobuchar, 1987). The GPS user community has developed sever- al alternative methods to correct this effect. Generally, these methods work in post-process- ing mode. Therefore, the recording of the ob- served satellite-to-receiver pseudoranges is re- quired. Other methods designed for real time positioning require at least a real time GPS cor- rection provider and a communication link be- tween the user and the provider. Ionospheric biases correction for coordinates derived from GPS single point positioning Mauricio Gende (1)(4), Elsa Mohíno Harris (2), Claudio Brunini (1), Sandro M. Radicella (3) and Miguel Herraiz (2) (1) Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La Plata, Argentina (2) Departamento de Geofísica y Meteorología, Universidad Complutense de Madrid, Spain (3) Aeronomy and Radiopropagation Laboratory, The Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy (4) Consejo Nacional de Investigaciones Científicas y Técnicas, Argentina Abstract Most GPS users employ low cost receivers. These receivers do not allow users to record the pseudorange that they observe, but the computed coordinates. This work presents an original and simple method to correct ionos- pheric biases introduced in GPS signals. The originality of this method is based on the fact that no pseudorange is needed to correct the biases, only the calculated coordinates are used. This distinguishes this method from oth- er classic alternatives. This paper evaluates the efficiency of the method with the use of real data. Mailing address: Dr. Elsa Mohíno Harris, Departamen- to de Geofísica y Meteorología, Universidad Complutense de Madrid, Avenida Complutense s/n, 28040 Madrid, Spain; e-mail: emohino@fis.ucm.es 440 Mauricio Gende, Elsa Mohíno Harris, Claudio Brunini, Sandro M. Radicella and Miguel Herraiz This paper will present a post-processing method to mitigate the effect of the ionospheric bias on positioning that does not require the ob- served pseudorange but only the user coordi- nates and the Pseudo Random Number (PRN) of the observed satellites. The method does not depend on the ionospheric model used and it accepts corrections from any algorithm that models the ionosphere. 2. Low precision positioning techniques A detailed review of the GPS characteristics, positioning algorithms and different biases af- fecting the observations can be found in several classical texts (e.g., Hofmann-Wellenhof et al., 2001). Three of the most common single-epoch positioning techniques will be briefly summa- rized in the following sections. 2.1. Stand-alone positioning This is the simplest positioning method in which coordinates are estimated based on range measurements derived from satellite to receiver travel time. Satellite positions, i.e. the epheme- ris, are known since they are encoded in the broadcasted information. Theoretically, having three simultaneous range measurements the problem can be solved by trilateration. However synchronization error due to the receiver clock adds one unknown and consequently at least one extra measurement is needed. In addition to the ephemeris information, the system broadcasts predicted ionospheric parameters in order to re- duce the ionospheric error using the ICA model. 2.2. Precise post-processing point positioning This technique differs form the previous one only by the fact that it includes post processed information. Precise ephemeris and satellite clock corrections replace the broadcast ones. An ionospheric delay estimation based on continuously updated global or regional ionos- pheric models replaces the ICA model. As a re- sult of including this improved information co- ordinates are more accurately determined (Héroux and Kouba, 1995). The main drawback of this technique is the need to record the ob- served pseudorange, which is not possible with many inexpensive receivers. 2.3. Differential GPS positioning (DGPS) The basic idea behind differential positioning (Blackwell, 1986) is to correct the observed pseudoranges in one GPS receiver (rover station) provided that these corrections could be deduced from another receiver placed in a fixed point with well known coordinates (reference station). DG- PS can work either in post-processing or real time mode. In the first case, two receivers observing and recording pseudorange measurements are needed simultaneously. In the second case, a cor- rection provider and a communication link be- tween the user and the provider are required. 2.4. Comparison between the three techniques Table I shows typical values for horizontal and vertical errors that result from point posi- tioning, precise point positioning and differen- tial positioning. Although these particular val- ues were acquired during only a 24 h period with a sample rate of 30 s, they represent typi- cal errors for each technique. 3. Algorithm to mitigate the effect of the ionos- pheric bias on positioning The idea of the proposed correction algo- rithm can be summarized as follows: coordi- Table I. Standard deviation error for the different positioning techniques. Horizontal Vertical error (m) error (m) Point positioning 3.5 12.5 Precise point positioning 2 3 Differential positioning 0.2 0.75 441 Ionospheric biases correction for coordinates derived from GPS single point positioning nates could be corrected from ionospheric error by a geometric propagation of an ionospheric biases estimation. Note that in this process no observed range is needed. 3.1. Error propagation into coordinates The basic equation of observation for each range measurement is c t t S R= - - -t t d d ot ^ h (3.1) tt being the satellite-to-receiver observed pseudorange, ρ the true geometric range, c the speed of light in vacuum, δ t S and δ tR the syn- chronization errors of the satellite and receiver clocks respectively, ν is the total budget of re- maining error that biases the measurements. If d0= +t t t and d Rt , cos n- ad + cos coseR R- -bd cdo , where cosα, cosβ, cosγ are the director a cosine, (3.1) transforms into . cos cos cos c t n e c t S R R R R 0- + + = - - + - + t t d o ad bd cdo d t (3.2) If there are simultaneous observations to n satel- lites, using matrix notation (3.2) transforms into A x=t∆ ∆ (3.3) with cos cos cos cos cos cos c c A n n n 1 1 1 f f f f f f f f f f f f = - - - - - - a a b b c c R T S S S S S SS V X W W W W W WW and , , , .n e tx R R R R T = d d do d∆ ^ h This linear system can be solved by least squares x A A A T T 1 = t∆ ∆ - ^ h . (3.4) Finally, the position vector xt , can be found by iteration using x x x0= + ∆t . Where x0 has a first guess value or is to the value obtained in the previous iteration step. It is important to note that eq. (3.4) repre- sents the way that errors are propagated in the coordinate. If t 0S0 = == dt tt , ν = νionosphere and ionosphere=t o∆ ∆ then the error on the coordi- nates due to the ionosphere can be easily com- puted using eq. (3.4) x A A A T T 1 ionosphere ionosphere= o∆ ∆ - ^ h . (3.5) 3.2. Method to correct ionospheric biases at coordinate level Ionospheric bias is proportional to Slant Total Electron Content (STEC). This relation can be ex- pressed as: K STEC1ionosphere #=o∆ , where K1 is a constant and STEC is defined as the integral of free electrons density along the satellite-receiver ray path. There are several ways to estimate STEC values, the present work used Global Ionospheric Maps (GIMs) produced by the Center for Orbit Determination in Europe (CODE) (Schaer et al., 1996). GIMs provide worldwide grids of Vertical Total Electron Content (VTEC); they are com- puted using a global network of dual-frequency GPS receivers. CODE GIMs are available at . In order to convert VTEC values into STEC val- ues an appropriate mapping function was used: M K ESTEC 1 2 2cos1 -= VTEC , ^ h where K2 is a constant and E is the satellite elevation. Having access to STEC estimations from GIMs an ionospheric correction for each pseudo- range can be computed as M VTECionosphere CODE GIMs #=o∆ -u . (3.6) Using eq. (3.5) the impact of those STEC esti- mations on the coordinates is obtained x A A A T T ionosphere ionosphere 1 = o∆ ∆ - u u^ h . (3.7) Users with a low-cost GPS receiver will not have access to each pseudorange measurement, but only to the final biased coordinates xt which can be written as x ionosphere= x + x∆ +t x _other biases+ ∆ where x is the exact or true value, evidently not accessible, and usually >xionosphere∆ > > x _other biases∆> . Finally, if xionosphere∆u is provided it can be subtracted from xt and a more accurate position can be found 442 Mauricio Gende, Elsa Mohíno Harris, Claudio Brunini, Sandro M. Radicella and Miguel Herraiz . x x x x x x x _ improved ionosphere ionosphere ionosphere other biases = + + - = + - ∆ ∆ ∆ ∆ t t u u (3.8) Compared to the methods presented in Sections 2.2 and 2.3, the one proposed in this paper does not require access to the pseudorange. This is a clear advantage for situations where pseudor- anges are not available, for instance when inex- pensive Original Equipment Manufacturer (OEM) GPS boards are used for vehicle tracking. The algorithm presented in this paper takes it for granted that the satellite geometry contained in matrix A is the same for the user biased coor- dinates xt , and for the estimated ionospheric cor- rection, xionosphere∆t . If this assumption is wrong the correction will lose precision; thus, caution must be taken to ensure that the satellites involved in the positioning algorithm are the same as those involved in the correction procedure. 4. Results 4.1. Data set The results shown in this section are based on nearly 360000 samples. Two GPS stations were used, located one at geomagnetic mid lat- itude, California, and the other one at geomag- netic low latitude, Galapagos Islands. Two time periods were analysed, the solstice and the equinox. For each period a set of thirty days with non perturbed ionospheric conditions was selected. All data correspond to a high solar ac- tivity period, the year 2001. To avoid problems due to poor signal-to-noise ratio, an elevation mask of 10° was imposed. Epochs with poor satellite geometry distribution (dilution of pre- cision greater than 6) were not processed. In order to evaluate the performance of the algorithm four positioning strategies were used: 1) stand-alone positioning without any correc- tion (raw); 2) stand-alone positioning applying ICA correction (ICA); 3) stand-alone position- ing using the ionospheric free linear combina- tion (ion free); 4) coordinates correction using a ionospheric model (CC). The first approach represents a situation where the user has no access to any ionospheric correction. The second one corresponds to the user that applies the ionospheric correction pro- vided by the GPS system. This should be the standard way of point positioning, but unfortu- nately not all GPS receivers apply the ICA cor- rections. In the third case the user has a double frequency receiver and can eliminate almost all the ionospheric bias (Hartmann and Leitinger, 1984). Note that this latter case is extremely un- usual since double frequency receivers are very expensive and usually are not used in point posi- tioning. This situation was included only to show the magnitude of the error that still remains even when more that 99% of the ionospheric effect was eliminated. The last approach uses the method introduced in this paper. In this case GIMs maps are used in order to estimate VTEC. In order to mitigate other sources of biases both precise ephemerides and a tropospheric model were used in the four presented tech- niques. 4.2. Algorithm performance Table IIa,b presents the accuracy of each po- sitioning method for Galapagos and California respectively. Results agree with some well known facts about the ionosphere. Raw errors clearly show that the ionosphere is more active during the equinox and at low latitudes. ICA and CODE errors demonstrate that, because of its complexity, it is more difficult to estimate the ionosphere during the equinox and at low lati- tudes. Results also agree with the positioning the- ory. Ion free errors are independent of the epoch but depend on the noise of the P code, which is clearly higher in the Galapagos receiver. As ex- pected, due to the geometric dilution factor the vertical component is the one that tends to absorb the main part of the ionosphere. It is worth noting that the method works well, in fact, though the aim of the present work does not aim at comparing or assessing the quality of any particular ionospheric model, it provided a better result that the ICA correc- tions. CC columns show similar results com- pared to ion free ones for the vertical, with the exception of the equinox period in Galapagos, probably due to the complexity of the VTEC for that region in that period. But larger dis- 443 Ionospheric biases correction for coordinates derived from GPS single point positioning crepancy values arise for the horizontal compo- nents. This is probably due to the fact that in or- der to model the horizontal components cor- rectly, good horizontal gradients of VTEC must be provided (Gende et al., 2003). CC method was also compared against post process precise point positioning. Differences are in the order of 10 cm for the horizontal component and in the order of 40 cm for the vertical component. Figure 1 shows four his- Fig. 1. Histograms of vertical errors for Galapagos during the solstice. Table IIa,b. Coordinate errors in meters for each positioning technique at Galapagos Islands (a) and at Cali- fornia (b). Solstice Equinox Raw ICA CC Ion free Raw ICA CC Ion free North 1.72 1.58 1.30 0.68 3.41 3.56 2.62 0.70 East 1.45 1.24 1.29 0.88 2.12 1.78 1.84 0.98 Vertical 10.26 3.89 2.56 2.16 15.52 4.24 4.58 2.21 North 2.32 2.04 0.66 0.70 2.50 1.71 1.21 0.67 East 1.01 0.80 0.54 0.52 1.18 0.82 0.67 0.52 Vertical 7.78 2.12 1.30 1.42 11.90 2.89 1.92 1.26 a b 444 Mauricio Gende, Elsa Mohíno Harris, Claudio Brunini, Sandro M. Radicella and Miguel Herraiz tograms of vertical errors for Galapagos station. Data correspond to the first week of July 2001 and around 18000 samples are plotted. Each graph illustrates epoch by epoch residuals for raw, ICA, CC and ion free technique. The first plot clearly illustrates the fact that the iono- sphere increases the height of the GPS station. The second one shows that the ICA model par- tially corrects this effect. The third plot corre- sponds to the positioning approach presented in this paper. Noticeably better than the first two approaches, it slightly tends to overcorrect the ionospheric effect. The last plot presents a bet- ter shape but has some outlier values; these are probably due to the P code signal combination necessary to make ion free solution. 5. Conclusions Results show that the proposed methodology is capable of improving GPS positioning accura- cy even when no pseudoranges are recorded. The method presented in this work improves the ac- curacy of positioning with low cost GPS equip- ment without any extra expense. Improvements have the same magnitude as those ones obtained in precise point positioning; approximately 7 m for the horizontal components and 11 m for the vertical. This represents an improvement of 80% if no correction is applied or 40% if ICA correc- tion is applied. Although the proposed method- ology cannot present results as good as DGPS, it can be an alternative for users who, needing higher accuracy do not have access to a DGPS correction provider and cannot record pseudor- anges for precise point positioning. This can be the case for many low-priced Original Equip- ment Manufacturer (OEM) GPS boards that are typically used in vehicle tracking. Acknowledgements Data from the IGS (International GPS Ser- vice) permanent network of GPS receivers were used in this work. 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