Relative Positioning in Europe: Influence of the GPS+Galileo Satellite Geometry

Transcription

1 Relative Positioning in Euroe: Influence of the GPS+Galileo Satellite Geometry Michaël Moins, Carine Bruyninx Royal Observatory of Belgium, Av. Circulaire 3, B-8 Brussels ABSTRACT While the uality of the signals successfully transmitted by GIOVE A, the first Galileo satellite launched at the end of last year, is currently being checked, a lot of work still has to be done before Galileo will be fully oerational. Our revious work, esecially within the context of the ure geometric asect, showed that the use of the additional Galileo constellation with resect to GPS only, imroves absolute ositioning based on code observables with about 4% in terms of formal errors when simulating urban conditions. For relative ositioning based on double difference carrier hase observables, the concet of RDOP (Relative Dilution of Precision) allowed to demonstrate that using GPS+Galileo, only half the observation time is sufficient to get similar recisions as with GPS only. These results were however obtained under error-free ideal conditions. The obective of this aer is to extend revious research by taking now different error sources into account, focusing only on results for relative ositioning. The comonents of the error budget will first be treated searately, and will be ut together afterwards when investigating the global error. INTRODUCTION Although we still have to wait a coule of years until Galileo, the Euroean version of GPS, will be fully oerational, several authors have investigated the imrovement of this additional GNSS system on different alications. In our revious work [], the effect of the geometry of the GPS+Galileo constellation on the recision of the ositioning was analyzed. Also a comarison with the current ositioning based on GPS was made. We showed that using Galileo in addition to GPS, a worldwide dulication of the number of daily visible satellites can be observed, yielding an imrovement of about 3 to 4 % for the Dilution of Precision ( DOP ). For relative ositioning, the combined system reached similar formal errors as stand-alone GPS, but needing only half of the observation time, and conseuently half the number of observations, in comarison with GPS only. These results were obtained in an ideal environment in which the effect of other error sources on the total error budget was neglected. In this aer we will take a closer look at the influence of the errors degrading ositioning on the reviously obtained results. Avila-Rodriguez et al [4] already showed some first results for absolute ositioning. In this aer we will first focus on each error source in articular, looking at ossible changes yielded by the use of a combined GPS+Galileo system. Afterwards, a full error budget will be brought together to look at the so-called total User Euivalent Range Error ( UERE ), which is a measure for the recision of single oint ositioning. Making the necessary calculations, an error budget for relative ositioning will be derived from this initial error budget. Since no error would remain when using double differences for relative ositioning, the model using single differences will be considered. Given this error budget for relative ositioning, a coule of results will be calculated within the region of the EPN (EUREF Permanent Network). Finally, to look at how the introduced error sources change

2 the uality of the relative ositioning, Relative DOP ( RDOP ) values will be used to calculate the horizontal and vertical relative ositioning error. The GPS satellite orbits have been created based on the broadcast navigation message, rovided by IGS. For Galileo, we considered a constellation of 7 satellites distributed over three orbits with right ascension angles of resectively, and, eually saced on these orbits by a mean anomaly of 6,, 8, 4,, 4, 8, or 6. Other initial values for orbital arameters were: a semi-maor axis of 9994 km, an inclination angle of 56, the eccentricity eual to, a rate of right ascension of a day, the argument of erigee eual to and finally a eriod of 4 h4m4s. For the calculation of the atmosheric errors, estimated values, also rovided by the IGS were used. IMPACT OF THE GPS+GALILEO SATELLITE GEOMETRY ON THE ERROR SOURCES The Error Budget for Absolute Positioning As well for code as for carrier hase observations, a certain number of systematic errors have to be taken into account when doing absolute ositioning. Deending on their roerties, those different error sources can be divided in following grous: ) signal roagation errors a. ionosheric refraction b. troosheric refraction c. multiath ) satellite errors a. clock bias b. orbital errors 3) receiver ranging error a. clock bias b. ranging errors The suare root of the sum of suares of these individual errors, the so-called User Euivalent Range Error ( UERE ), can be seen as a global error and as a measure of the recision for oint ositioning. Multilying this value with the Position Dilution of Precision ( PDOP ) conseuently rovides an aroximation of the osition error ([3] and []). The values of the reviously mentioned error sources deend on whether we are dealing with code or with carrier hase observations. From now on, we will therefore only consider carrier hase observations. Signal Proagation Errors When we talk about atmosheric errors, two comonents have to be taken into account: the trooshere and the ionoshere. For both error tyes, the International GNSS Service (IGS) makes available estimations. For the trooshere the IGS rovides us with Zenith Path Delay ( ZPD ) files for stations included in the IGS network. These weekly files contain values for the total ZPD. For the ionoshere, IONoshere ma EXchange (IONEX) files give us values of the Vertical Total Electron Content ( VTEC ) for a grid of oints reresenting the earth. Both roducts are giving values at zenith and can be converted to the tyical atmosheric errors using the aroriate maing function which deends on

3 the satellite elevation. Figure shows the mean satellite elevation for the GPS constellation as well as for the Galileo constellation. The mean elevation reresents a daily mean accounting for all visible satellites above an elevation cut off angle of 5. For both systems, the daily mean atterns are very similar from day-to-day and can be considered as reresentative, so later on, these values will also be considered when needing a yearly mean Figure : Worldwide distribution of the daily mean of the elevation [ ] of visible satellites using a 5 cut off, GPS only Galileo only 9 Figure shows that for aroximately 86 % of the earth surface, Galileo has larger mean elevations than GPS; all the differences amount from. 6 to.6. There are no significant differences in mean elevation values between single GPS and the combined GPS+Galileo system; the elevations differences amount from. 3 to.6 worldwide and from. 66 to.85 at Euroean level (see Figure ). As mentioned earlier, the IGS rovides a daily ionosheric grid containing the VTEC exressed in electron er suare meters [ el / m ] ([6] and [4]). Using the maing function m (elev) of the Klobuchar model, this VTEC is maed to the corresonding Slant Total Electron Content ( STEC ) and the ionosheric grou delay e iono on L (in meters) can be calculated as follows: [ + 6(.53 elev) ]* VTEC 4.8 e iono = m( elev)* * VTEC = * f f () with f the common L GPS & GALILEO freuency of 575.4MHz and elev the mean elevation of the satellites (for this model exressed in number of semicircles of 8 ). Our results show that the new and old daily means of the ionosheric ath delays, calculated resectively for the combined GPS+Galileo and for the single GPS system, are very similar. Within the Euroean region the differences have a mean value of 6.5mm and range from 49.8mm to 47.4mm. Worldwide, those values are ranging within an interval between 4.3cm and.83cm, with errors for the GPS-system being on average.35cm higher than those for the combined system. Highest differences are visible in vertical orientated area s around the euator (between and degrees of latitude). Figure 3 shows the daily mean ionosheric ath delay at resectively Euroean and

4 worldwide level for the combined system, with resective results of about. 4 to 4.6m and of about.6 to 7.36m Figure : Worldwide distribution of the difference in mean elevation [ ] of visible satellites between the combined GPS+Galileo and the single GPS system, using a 5 cutoff worldwide for Euroean region only Figure 3 : Worldwide distribution of daily mean of ionosheric ath delay [m] for the combined GPS+Galileo system, worldwide at Euroean level In the case of the trooshere, our rocedure will have to be slightly adated since the values for the total zenith ath delay are only rovided for some stations belonging to the IGS network. This network of stations does not contain enough data to interolate a comlete world-grid with the standard rocedures for interolation form MATLAB, but will be sufficient to make a ma for the troosheric ath delay for the Euroean region. The used maing function is the one from Black & Eisner, suitable for both hydrostatic as well as for wet delay. This function,. e troo = m( elev)* ezenith = * e.+ sin ( elev) zenith ()

5 is recommended to be used for satellite elevation angles larger than 7, and is known for its comutation simlicity mainly due to the fact that no meteorological data is necessary ([4] and [5]). The mean satellite elevation values elev are hereby no longer exressed in number of semicircles of 8, but in degrees like usual. The results show that over the Euroean region the troosheric error on the combined GPS+Galileo system is in average about.9cm smaller than the error on GPS only. A minimum of.7cm and a maximum of 6.3cm, are observed differences over Euroe. Figure 4 shows the daily mean troosheric ath delay values for GPS+Galileo; they vary between 3. 4 and 4.63m with a mean value of 4.7m Figure 4 : Distribution of daily mean of troosheric ath delay [m] for the combined GPS+Galileo system, Euroe Finally, the last signal roagation error to discuss is the multiath error. Exerimental research using the multiath characteristics of ermanent GPS stations [9] showed that the hase multiath can reach u to a value of to 3 cm for the slant delay. Within the Euroean region, we will therefore consider a common multiath error of 3 cm for the remaining art of this aer. Satellite and Receiver Errors As for the case of the multiath error, for the remaining art of this aer, common values will also be assigned to all receiver and satellite errors. Based on the IGS web-ages [8] a value of 5 cm is used for the satellite orbit error. IGS final clock roducts have an accuracy smaller than of.ns, euivalent with a satellite as well as a receiver clock error of 3 cm. Finally, receiver ranging errors, aearing when measuring carrier hases, will not be considered since these errors seems to be negligible (less than one millimeter) for high uality receivers [7]. Numerical Overview of the Error Sources Table gives an overview of the all the errors and their values that will be considered further on within the Euroean region for the combined GPS+Galileo system.. The total UERE, i.e. the suare root of the sum of suares of the individual errors, will range between 4.m and 6.54m.

6 [m] satellite clock error. 5 satellite orbit error. 3 ionosheric ath delay troosheric ath delay multiath. 3 receiver clock error. 3 receiver ranging error / UERE Table : Overview of all the error values and total UERE at Euroean level The osition error is obtained by multilying the UERE with the PDOP. Figure 5 shows Euroean mas of the horizontal and vertical osition error for single GPS system as well as for the combined GPS+Galileo system (c) Figure 5 : Distribution of daily mean ositioning error [m] at Euroean level, horizontal osition error, single GPS system horizontal osition error, combined GPS+Galileo system (c) vertical osition error, single GPS system (d) vertical osition error, combined GPS+Galileo system (d) The results in Figure 5a for the GPS system show horizontal ositioning errors ranging between 4. 9 and 5.7m, while for the combined system, Figure 5b, euivalent values are all below 3.87m. Figure

7 5c and Figure 5d show similar results for the case of the vertical ositioning errors, with values ranging from to 9.3m for the GPS system, while euivalent values for the combined system are all below 6.m. For both systems as well in the case of vertical as horizontal ositioning errors, we observe in both cases a mean imrovement of about 34 % for the Euroean region, ranging between 3 and 4%. This imrovement for the Euroean region is showed in Figure Figure 6 : distribution of imrovement ratio [%] for horizontal and vertical ositioning error RELATIVE POSITIONING Single Difference Carrier Phase Model for Relative Positioning, between two receivers and instead of one single receiver osition is the main difference between absolute and relative ositioning. Solving this model reuires the use of observables as single ( SD ) or as double differences ( DD ), yielding to a decrease or elimination of the influence of some error sources. In this aer, we relace the DD model by SD because, using our assumtions, the DD model would lead to a full elimination of all errors. The model for relative ositioning using SD of the observables (3) will therefore be used for the remaining art of this aer. Calculating the vector ( X Y, Z ) = ( X X, Y Y, Z Z ) Φ = Φ Φ = ρ + cδ + λa I + T + MP + ε Φ (3) Φ is the SD of the carrier hase observable, ρ the SD of the aroximate geometric distances between receivers and and satellite, λ and c the resective wavelength and seed (= seed of light) of the signal, while A finally is the SD non-integer ambiguity term. Remaining terms of euation (3) δ, I, T, and MP are the SD of the error terms, reresenting resectively receiver clock error, ionoshere ath delay, trooshere ath delay and multiath. As for absolute ositioning, an a riori estimated osition ( X, Y, Z) for the unknown osition of receiver is given, allowing to rewrite this unknown osition as ( X, Y, Z) = ( X + X, Y + Y, Z + Z ). Rewriting the known osition of reference receiver the same way, Taylor series exansion will afterwards be executed around ( X, Y, Z ) and ( X, Y, Z ) for resective terms belonging to given receivers and. Receiver being the reference station with known coordinates, will conseuently yield to a simlification of the model euation since X = Y = Z =.

8 This way, euation (3) will be linearized in order to solve for the unknown arameters: Φ cδ λa + I T (4) Every observation can be written as euation (4), yielding to a model reresented by the matrix euation L = AX + v, with X as the vector ( X, Y, Z ) of the unknowns, A the design matrix containing all coefficients of the unknowns, L the vector containing all observations and finally v as the vector of residuals. Using this observation model we can comute the associated covariance matrix T of the unknowns Σ X = ( A Σ L A) and convert it to its toocentric euivalent Σ T. The covariance matrix of the observables Σ L is not a unit matrix, but the mathematical correlations between the SD measurements are taken into account. This covariance matrix Σ X of the unknowns rovides information on the recision of the solution. The RDOP (Relative DOP ) is similar to the PDOP value for the case of absolute ositioning, but will be calculated in a different way with the formula, (4): RDOP = trace σ ( Σ ) X SD (5) Contrary to the comutation of the PDOP, the observations are accumulated over sessions varying between / and 4 hours, using a 6 -second measurement interval. Going back to euation (5), σ SD is the uncertainty of a SD measurement. This definition imlies that the RDOP will not deend on the a riori variance σ of the carrier hase measurements, since for the matrix Σ X as well as for the value σ SD, a factor σ can be set aart, whereas those factors aearing in denominator and nominator of euation (5) can be removed. Conseuently, we will not have to make assumtions about this value. The units of RDOP are meters/cycle. Theoretically, the uncertainty of a SD measurement multilied by RDOP will therefore yield a relative osition error, and a closer look at this error will be taken. Error Budget of the Single Difference Carrier Phase Model MP All SD between receivers and are calculated as =. Conseuently, a roerty of using SD, is the elimination of the satellite clock error within the model. Because of the use of fixed reresentative values for receiver clock error and multiath, these errors are also eliminated using SD and will not have to be taken into account for the comutation of the uncertainty of a SD measurement. In comarison with the errors aearing in Table, only the satellite orbit error and ionosheric and troosheric errors will therefore be used. For the atmosheric errors, their SD will now be considered, while the orbital error of a satellite for the case of relative ositioning will be eual to its euivalent error for the case of absolute ositioning, multilied by d / with d the baseline length between receiver and exressed in kilometers, [6]. After elimination of some errors conform with SD roerties, and other errors yielded by the use reresentative values for them used in this aer, the error budget for relative ositioning, as we assume it here, will finally consist of: ) SD of signal roagation errors a. ionosheric refraction b. troosheric refraction ) satellite orbital errors = ρ X X ρ X Y Y ρ Z Z Y ρ i Z

9 Set-u of Calculations and Results Following the same rincile as for HDOP and VDOP, we considered horizontal and vertical comonents for the RDOP values. Those values are calculated for several baselines between EPN stations, as shown in Figure 7, subdivided in three grous deending on their orientation. Figure 7 : Ma of Euroe showing considered baselines The results show that all observations accumulated over sessions of hours, using a SD model, show the same imrovement of 3% for horizontal as well as for vertical comonents in comarison with GPS only. Note that the errors for horizontal baselines, i.e. baselines between stations with eual latitudes, are systematically less than those for the vertical as well as for the diagonal baselines. This is mainly due by the fact that for these horizontal baselines, atmosheric errors of both baseline stations don t often differ much. Nevertheless, the imrovement of 3% was similar for all kinds of baselines. This imrovement is shown in Figure 8b where the RDOP imrovement ratio is shown for horizontal and vertical comonents all together. The magenta and red colored lines in Figure 8a reresent the resective RDOP values for the combined GPS+Galileo and for the single GPS system..5 5 RDOP RDOP imrovement ratio Figure 8 : RDOP values for single GPS (red) and combined GPS+Galileo (magenta) system RDOP imrovement ratio, horizontal and vertical comonents ut together

10 The osition errors were considered searately for the horizontal and vertical comonents; Figure 9, also show an imrovement of about 3% for all comonents of the osition error. horizontal comonent of ositioning error horizontal comonent of ositioning error vertical comonent of ositioning error vertical comonent of ositioning error (c) (d) Figure 9 : Comonential Relative Positioning error [m] for a given system horizontal comonent of osition error, single GPS system horizontal comonent of osition error, combined GPS+Galileo system (c) vertical comonent of osition error, single GPS system (d) vertical comonent of osition error, combined GPS+Galileo system CONCLUSION This aer comared the theoretical ositoning error of GPS only with the error of the future GPS+Galileo combined system. Using IGS roducts to comute atmosheric errors with adeuate maing functions, no big changes were observed for the values of these errors when considering them individually. Nevertheless, the imrovement in DOP values, showed in revious work [], imly a similar imrovement for the case of the aroximate ositioning error. This is true as well for the case of absolute ositioning considering PDOP values, with an imrovement ranging from 3 to 4 %, as for the case of relative ositioning with SD considering RDOP values, with an imrovement of about 3 %.

11 REFERENCES [] S. Daghay, M. Moins, C. Bruyninx, Y. Rolain, F. Roosbeek Imact of the Combined GPS+GALILEO Satellite Geometry on Positioning Precision, Proceedings of EUREF 5, -3 June 5, Vienna, Austria (in ress). [] C.C. Goad Investigation Of An Alternate Method Of Processing Global Positioning Survey Data Collected In Kinematic Mode, GPS-Techniues Alied to Geodesy and Surveying, Aril 988. [3] G.J. Husti Global Positioning System een inleiding, Series on Mathematical Geodesy and Positioning, Delft University Press,. [4] J.A. Avila-Rodriguez, G.W. Hein, S. Wallner, T. Schueler, E. Schueler, M. Irsigler Revised Combined Galileo/GPS Freuency and Signal Performance Analysis, Proceedings of ION GNSS 5, 3-6 Setember 5, Long Beach, California, USA. [5] J. Guo, R.B. Langley A New Troosheric Proagation Delay Maing Function for Elevation Angles Down to, Proceedings of ION GNSS 3, 9- Setember 3, Portland, Oregon, USA. [6] C. Rizos Princiles and Practice of GPS Surveying,!, The University of New South Wales, Sidney, Australia, 999. [7] J. Ashaee, N. Ashaee Basics of High-Precision Global Positioning System, " # $!%"#!, Javad Navigation Systems, San Jose, California, USA, 998. [8] IGS Product Table, &"!# # ##!, International GNSS Service, 5. [9] H. van der Marel Comutation and Analysis of Antenna and Multiath Characteristics of Permanent GPS Stations Proceedings of IGS Worksho, 8 May 6, Darmstadt, Germany (in ress) [] G. Strang, K. Borre Linear Algebra, Geodesy and GPS, Wellesley Cambridge Press, 997.