SUCCESS RATES FOR INTEGRATED GPS AND GALILEO AMBIGUITY RESOLUTION. Kees de Jong ABSTRACT RESUMO

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1 SUCCESS RATES FOR INTEGRATED GPS AND GALILEO AMBIGUITY RESOLUTION Kees de Jong Ezelsveldlaan, 6 RV Delft, The Netherlands kdj@hetnet.nl ABSTRACT Modernized GPS and the future Euroean Galileo system will offer unrecedented accuracy and availability for recise ositioning alications. For these alications, the ambiguities of the carrier observations have to be resolved to their integer values. The ambiguity success rate is a valuable design arameter to determine if the ambiguities can indeed be resolved correctly to their integer values. In this contribution, an overview is given of the modernized GPS and Galileo signals, followed by the models used for recise ositioning over short, long and intermediate distances and the comutation of ambiguity success rates. Finally, success rates are comuted for a number of observation scenarios, ranging from current dual-frequency GPS to integrated trile-frequency GPS and Galileo. Keywords: GPS, Galileo, ambiguity resolution, ambiguity success rate. RESUMO A modernização do GPS e o futuro sistema Euroeu denominado Galileo irão oferecer acurácia e disonibilidade nunca até então obtidas em alicações de osicionamento de recisão. Para essas alicações, as ambigüidades das observações de fase têm que ser resolvida como inteiras. A taxa de sucesso da solução da ambigüidade é um arâmetro que determina se as ambigüidades odem de fato ser solucionadas corretamente. Uma visão geral dos sinais GPS modernizado e do Galileo será aresentada neste artigo, seguida elos modelos usados ara osicionamento reciso de linhas bases com distâncias curtas, longas e intermediárias e do cálculo da razão do sucesso da ambigüidade. Finalmente, razões de sucesso são calculados ara vários cenários de observações, desde o caso do GPS com dula freqüência até a freqüência trila do GPS e Galielo integrados. Palavras Chaves: GPS, Galileo, solução da ambigüidade, razão de sucesso da ambigüidade.. INTRODUCTION Precise ositioning with Global Navigation Satellite Systems (GNSS), such as the US GPS and the future Euroean Galileo, requires resolution of the ambiguities of the carrier hase observations to their integer values. The ambiguity success rate is defined as the robability of correctly fixing the ambiguities to their integer values. To comute the success rate, no actual data is required; it is therefore a valuable design arameter. In this contribution, the success rate will be used to evaluate the exected erformance of modernized GPS and Galileo for a number of observation scenarios, which are a function of the number of satellite systems used, the number of frequencies and the length of the baseline vectors considered. First a brief overview will be given of the modernized GPS and Galileo signals, followed by the measurement and stochastic models used to comute the ambiguity success rates. Next, the success rates will be given for a number of observation scenarios, followed by conclusions.. MODERNIZED GPS Between 978 and 985, exerimental so called GPS Block I satellites were launched. In 989 the first oerational Block II satellite was brought into orbit and the system reached full oerational caability in 994. Figure gives an overview of the status of the current 8-satellite constellation. Although the constellation is erforming well, a number of satellites are on single string, which means there is no redundancy to erform one or more critical tasks. Fortunately, for the eriod new launches have already been scheduled to maintain constellation health, see Table. Revista Brasileira de Cartografia Nº 54

2 Age [years] Current age, oerational Current age, not oerational Mean mission duration Design life GPS satellite number [PRN code] Figure GPS satellite configuration status (March 3). Table : GPS satellite launch schedule. Year Launches 3 3 IIR 4 IIR, IIR-M 5 3 IIR-M 6 IIR-M, IIF Over the years, the mean mission duration (MMD) of the satellites has increased considerably. From a contractual MMD of six years and a design life of 7.5 years, the MMD has gone u to 9.6 years for the block II satellites to over years for the block IIA satellites. For the latest block IIR and the new block IIR-M satellites, the design life is years and the current MMD is nearly eight years. The future block IIF satellites, to be launched from 6 onwards, will have a design life of years and an MMD of years. Originally these values were higher, but with the increased oerational life san of the current satellites this would result in delaying further modernization into the comletely redesigned GPSIII, which should rovide users with the best ossible satellite navigation system for the next 3 years. Mass-market alications usually rely on tracking the C/A code on the L carrier (575.4 MHz). For recision alications (cm accuracy or better), highend dual-frequency receivers are in use. Over the years about 5, of these receivers have been sold. Before 994 the P-code on the L carrier (7.6 MHz) was freely available. After the encrytion of the P-code into the secret Y-code, referred to as Anti-Soofing (AS), manufacturers of civil receivers develoed (semi-) codeless techniques to acquire and track the L signals anyway. However, a major drawback of the current aroach to L signal recovery is that the secial techniques that are required for it, due to the encrytion of the P-code, results in a significant degradation of the signal to noise ratio. Unlike Selective Availability (SA), the intentional degradation of the GPS signals, which was discontinued in May, AS will remain switched on. The announcement of a second and third signal frequency therefore received a warm welcome from the civil GPS community. These signals will be modulated on the current L frequency and the new L5 frequency of MHz. The original lans aimed at a modulation of the L-carrier with the same C/A-code as the L-carrier. However, current GPS signals already date back to the 97's and better alternatives are available. The Euroean initiative to develo Galileo may have been another reason to look into a modernized GPS signal structure. Current discussions aim at develoing and imlementing a new civil GPS code, called L CS (for Civil Signal), see [USCG, 3]. The new code should allow for signal tracking in adverse conditions when even the current L C/A-code signals can no longer be tracked, e.g., at low elevations or even inside buildings. Details of the new signal can be found in [Fontana et al., ]. Comared with resent semicodeless techniques to get around Anti-Soofing, measurement erformance will imrove considerably. Less cycle slis are to be exerienced on L and satellites can be tracked further down to the horizon. Concerning measurement recision, a level similar to resent L C/A-code erformance is anticiated. Eventually, the L CS-code might relace the L C/A-code in single-frequency mass-market alications. Dual-frequency range measurements are essential in high-recision alications to account for ionosheric delays and to enhance carrier hase ambiguity resolution. High-end civil dual-frequency systems will therefore be based on L C/A-code and L CS-code. It is not yet clear if the dual-frequency market will exand once the new L civil signal is available and if this, in addition, will give rise to rice dros in receiver equiment. GPS satellites that will transmit the new L signal will be launched from 4 onwards. Full oerational caability (FOC) is anticiated for. By that time there will be about 8 satellites transmitting the new L signal. The L5 signals will be transmitted by the Block IIF satellites, the first of which will be launched in 6. The L5 signals will be different from the L CS, see again [USCG, 3], and robably more similar to the current P-code, i.e., they will have a chiing rate that is times that of the current C/A and the L CS codes. This high chiing rate code will offer high erformance ranging caabilities as well as more robust carrier tracking for high-recision alications. It is anticiated that the L5 code measurement recision will be a few times better than the resent L C/A-code recision, see [van Dierendonck and Hegarty, ]. Within years, GPS will turn into a civil trilefrequency system. It is exected that high-end receivers, used for differential and relative ositioning in recise geodetic and navigational alications, eventually will track code (seudo range) and carrier hase on all three frequencies L, L and L5, see Figure. Revista Brasileira de Cartografia, N o 54

3 L5 (E5a) E5b f [MHz] L 7.6 E f [MHz] E L E f [MHz] GPS civil signal Galileo civil signal Galileo commercial signal GPS recision signal Galileo recision signal Non--civil signal Figure GPS and Galileo signals. The new GPS signals will affect both the sace and user segment, i.e., both satellites and receivers. In addition, the GPS control segment will be modernized as well. New software will be installed to imrove the quality of the GPS orbit and clock arameters, transmitted by the satellites in their navigation message. This will also be accomlished by extending the number of tracking stations. Finally, ehemeris and clock data will be uloaded to the satellites more frequently. 3. GALILEO In March, the Euroean Commission gave the green light for the develoment of Galileo, the Euroean counterart of GPS. Galileo should be oerational in 8, when modernized GPS will have reached initial oerational caability (IOC) with 8 satellites. From a technical oint of view, Galileo will be very similar to (modernized) GPS. According to [Salgado et al., ] the constellation will consist of 7 satellites, distributed evenly and regularly over three orbital lanes. The constellation is robably enhanced by three active sare satellites. The rojected altitude is slightly larger than for GPS. Table gives an overview of the Galileo orbits. Table : Galileo orbital arameters. Orbital lanes 3 Satellites/lane 9 Semi-major axis 99 km Inclination 56 Eccentricity ~ The frequency and signal lan of Galileo are not yet final. Since the frequency sectrum is extremely crowded, see e.g. [NTIA, 3], only limited bandwidth is available. However, the roosal described in [Hein et al., ; Hein et al., ] is believed to serve as the current baseline. Like modernized GPS, Galileo will emloy three bands in the frequency sectrum (see also Figure ). Table 3 gives an overview of the frequency bands that may be used for Galileo. Table 3: Galileo frequencies. Identifier Frequency E-L-E MHz E5b 7.4 MHz E MHz At the uer end of the L-band sectrum, a small band is available on either side of the GPS L- band. It is roosed to overlay the GPS L signal by using the very same center frequency of MHz. An overlay signal, with most ower concentrated in the E and E side bands, is lanned to be reserved for the so-called Public Regulated Service. An Oen Service signal, also overlaid, would occuy a smaller band, in the GPS L signal band. See again Figure. Using secial signal rocessing techniques, most of the ower of the transmitted signal will be in the sidebands, to avoid interference with GPS L. The resulting signal will rovide seudo-range measurements with a recision somewhere between that of GPS L C/A-code and future P-code. Revista Brasileira de Cartografia, N o 54 3

4 At the lower end of the L-band sectrum, directly adjacent to the GPS L5-band, the E5-band has been reserved for Galileo. In the E5-band it is also ossible to create an overlay with the GPS L5 signal. However, this band is wide enough to avoid this, if required. It is exected that the large bandwidth will result in a signal that has a erformance comarable to the L5 GPS P-code. Due to the recent decision to shift the center frequency in the E5b-band from.5 to 7.4 MHz, some signal will be outside the E5-band. Finally a reservation exists for Galileo in the mid-range of the L-band sectrum, as indicated in Figure. The E6-band is 4 MHz wide and is to be shared with radar alications. The signal in this band will be available for commercial services, as oosed to the other two signals, that are freely available (but remember that the current GPS L signal was also not meant for use by the civilian community). Measurement erformance of the E6 signal will be comarable to that of E-L-E. More details on the roosed Galileo signals and frequencies are available from the Galileo website at htt:// 4. PRECISE POSITIONING ionosheric (seudo-) observations, I, with covariance matrix Q. The measurement model becomes I B C b φ = B A + C I a (3) I I with covariance matrix Q Q φ (4) QI The weight of the ionosheric observations is governed by their standard deviations. For a standard deviation equal to zero, we arrive at the ionoshere fixed model, for a standard deviation aroaching infinity, we obtain the ionoshere float model. The arameters I can be eliminated from (3), resulting in a measurement model that has a similar structure as the ionoshere fixed model Assuming measurements consist of code and carrier observations, collected at m different frequencies, the GNSS double-difference (DD) baseline measurement model for a single eoch k can be written as CI φ + CI B = B b A a and with covariance matrix (5) B b C = + I B A a C φ () Q C + Q Q C φ I C C T (6) with code observations, φ carrier observations, b baseline arameters, a carrier ambiguities and I ionosheric effects. The corresonding covariance matrix reads Q Q φ () There are several ways to deal with the ionoshere. In the ionoshere fixed model, we assume that ionosheric effects are absent. In that case, the arameters I are elminated from (). In the ionoshere float case, we assume the ionosheric effects are comletely uncorrelated and unknown at both ends of the baseline. The ionoshere fixed and float models are extreme cases. In many cases in ractice baselines are too long to assume there are no residual ionosheric effects in the double difference data, but too short to assume these effects are comletely unknown. We therefore introduce a third model, the ionoshere weighted model. For this model we extend the measurement model () with an additional vector of Note that (5) and (6) are comletely equivalent to (3) and (4). In order to solve for the unknown arameters using least squares, we need the inverse of the covariance matrix of the observations. For the ionoshere float model, comuting the inverse of (6) may be roblematic. However, alying the matrix identity ( X + UYV) = X X U ( Y + VX U ) VX (7) to (6) results in the inverse Q where Q φ Q C D Q C φ T I Q C Q C φ T (8) D = Q + C ( Q + Qφ ) C (9) For the ionoshere float model, the weights of the ionoshere observations become zero and we get Revista Brasileira de Cartografia, N o 54 4

5 T + D = C ( Q Qφ ) C () Note that for the ionoshere fixed model we can simly ignore the second term of (8). Care should be taken to assigne realistic weights to the ionosheric seudo-observations. These observations may, e.g., be derived from the ionosheric estimates of ermanent GPS arrays. In case no such information is available, one may have to revert to samle values of the ionosheric observations that are equal to zero. For this latter case the standard deviations of the ionosheric observations will be larger than for the former. The standard deviations should be chosen sufficiently large not to bias the estimated carrier ambiguities by an amount that may result in a wrong integer ambiguity. The effect of biases on integer carrier ambiguity resolution is described e.g. in [Teunissen, ]. If code and carrier observations are available, both baseline comonents and ambiguities can be resolved using only a single eoch of data (however, as we will see later, the robability of correctly solving the integer ambiguities may be very low). If only carrier observations are used, at least two eochs are required to solve for baseline comonents and ambiguities. The measurement model becomes φk + CI φ j + CI, k, j Bk = B j bk A b j A a () where the subscrits j and k denote the two eochs. Resolution of the integer ambiguities is only ossible for the ionoshere fixed and weighted baselines; for the ionoshere float model, it is not ossible to searate the ambiguities from the ionosheric effects. Not using code observations has the advantage that effects due to multiath do not affect resolution of the integer ambiguities. However, there is also a disadvantage in that the number of satellites required to obtain a solution is much higher: four for a code and carrier solution, but seven for a carrier-only solution (for a moving receiver). This can be seen by eliminating the ambiguities from (). What is left is a trile difference model with two different sets of osition arameters at eochs j and k. There are six unknown arameters; solving for these arameters requires six trile differences, which in turn require seven satellites. If two different satellite systems are available, such as GPS and Galileo, the ambiguities of a mixed air of double differences are in general not integer, due to different hardware and clock characteristics. Therefore the design matrix A and ambiguity vector a should be artitioned as A A = GPS A Gal agps a = () agal This also means that for a carrier-only GPS and Galileo solution, at least eight satellites are required. Float solution Estimate osition and carrier ambiguities LAMBDA method Estimate integer ambiguities 5. AMBIGUITY RESOLUTION Fixed solution Estimate osition (ambiguities fixed) Least squares Integer least squares Least squares Figure 3 The three stes involved in recise GNSS ositioning. Processing GPS data for recise relative ositioning generally consists of the three stes shown in Figure 3. In the first ste, known as the float solution, receiver ositions, carrier ambiguities, and ossibly a number of additional nuisance arameters, are determined in a least squares adjustment. The carrier ambiguities are known to be integers. However, after the first ste, only real-valued ambiguities and their associated covariance matrix are available. These estimates and their covariance matrix are the inut for the second ste, in which the integer values of the carrier ambiguities are estimated. Once the integer ambiguities are known, they are alied to the carrier observations, after which these start acting as very recise seudo range measurements. These adated measurements are used as inut for the third ste, known as the fixed solution, in which the final, recise ositions are determined Figure 4 Ambiguity decorrelation by ushing tangents. From left to right: ) ush horizontal tangents of confidence ellise inwards; ) ush vertical tangents inwards; 3) reeat rocedure until resulting ellise resembles a circle as much as ossible, i.e., until no further decorrelation of the ambiguities is ossible. The areas of the original and decorrelated ellise are the same and the integer nature of the ambiguities is reserved. Revista Brasileira de Cartografia, N o 54 5

6 robability density function (df) â robability mass function Success rate = P ( a ( = 4) = df( aˆ) Figute 5 Comutation of success rate, one-dimensional examle (from [Teunissen & de Jong, 999]). The otimal method to resolve the ambiguities is based on integer least squares. An efficient imlementation of this method for ambiguity resolution is the Least squares AMBiguity Decorrelation Adjustment (LAMBDA) method, develoed at Delft University of Technology, [Teunissen, 993], [de Jonge & Tiberius, 996]. This method is based on a decorrelation of the original ambiguities, followed by a search rocedure. The decorrelation is required since the sectrum of the conditional variances of the original ambiguities shows a large discontinuity. As a result, the search sace will be very elongated and the search for the correct set of integer ambiguities will be very inefficient, [Teunissen et al., 994]. After decorrelation of the ambiguities, the sectrum will be much more homogeneous, which results in a more efficient search rocedure. A schematic overview of the decorrelation rocedure is shown in Figure 4. Denoting the real-valued ambiguities and their covariance matrix by â and Qˆ a, resectively, and the decorrelation matrix by Z T, the transformed ambiguities and their corresonding covariance matrix are given by T ˆ T zˆ = Z a Qˆ = Z Q ˆZ (3) z The elements of the matrix Z T are all integers; its determinant is equal to or. After the integer values of the transformed ambiguities, denoted by z (, are determined, the integer values of the original ambiguities are obtained by using the inverse of transformation (3) a ( = Z T z ( (4) Contrary to oular belief, the resolved carrier ambiguities are stochastic rather than deterministic. The integer ambiguities are determined from observations and since these observations carry an inherent uncertainty, the integer ambiguities are also subject to a this uncertainty. Rather than stating that the ambiguities are determined from the observation data, it is more aroriate to say that they are estimated from this data. The ambiguities being estimated have an imortant imlication: they may not always be correct. What we would like to know is, given a articular estimate, the robability that this estimate is indeed the correct integer or integer vector. This robability of correct integer ambiguity estimation will be referred to as success rate, [Teunissen, 998]. For the comutation of the success rate, we need the robability mass function of the integer least squares estimator. This robability mass function can be constructed from the robability density function of the real-valued ambiguity estimates resulting from the float solution, and the integer maing alied in the second ste. The characteristics of the robability density function are catured in the covariance matrix of the real-valued ambiguities. This covariance matrix is a function of the a riori stochastic model of the observations and of the relative receiver-satellite geometry. For its comutation no actual data is required. Consequently, for the comutation of success rates, also no observation data is required. As such, it can be used as a design tool, just like the oular DOP (Dilution of Precision) values. Another alication of the success rate is its use as a system design arameter when looking for the best ossible combination of frequencies for integer ambiguity estimation. Examles of the success rate as a lanning and design tool will be given in Section 6. To illustrate the comutation of success rates, we will consider a simle, one-dimensional examle. The robability density function of the real-valued estimator is shown in Figure 5. This function is continuous because the real-valued estimator can take any real value. The function is centered at the correct integer, due to the unbiasedness of the least squares estimator. For this case, we assume that the correct integer ambiguity is equal to four. In one dimension, Revista Brasileira de Cartografia, N o 54 6

7 integer estimation boils down to simle rounding. All real-valued estimates in an interval of +.5 and.5 around an integer value are maed onto that integer. The robability that the integer least squares estimator takes on a articular integer value can therefore be comuted by integrating the robability density function of the real-valued estimator over these maing intervals. Consequently, the success rate is equal to the integral of the robability density function over the maing interval around four, see again Figure 5. Denoting the robability of correctly resolving ( the decorrelated integer ambiguities by P ( z = z), a lower bound of the robability of correct integer ambiguity resolution, i.e., the success rate, is given by ( P( z = z) n (Φ( σ i= zˆ i I ) ) (5) where n is the number of ambiguities, σ z ˆ i I is the conditional variance, resulting from the decomosition T Q zˆ = LDL of the covariance matrix of the transformed ambiguities, with L a lower triangular matrix, D a diagonal matrix and σ z ˆ = D( i, i). The function Φ is i I defined as x Φ( x ) = ex( z ) dz π (6) which is the integral of the Gaussian robability density function. 6. SUCCESS RATES Success rates were comuted for São Paulo (latitude 3 3 S, longitude W), for 3 March 3, using a minimum elevation of. Standard deviations of the undifferenced carrier observations were.3 m, the standard deviations of the code observations are given in Table 4. As exlained in Sections and 3, the recision of the code observations deend on the signals considered for both GPS and Galileo. Unless indicated otherwise, the a riori standard deviation of single-difference ionosheric seudo-observations was chosen as.5 m, corresonding to a baseline length of about 5 km. Table 4: Standard deviations of code observations, used for comutation of success rates. Standard recision L.3 E-L-E.5 L.3 E5b. L5. E6. High recision L.3 E-L-E. L5.5 E5b.5 Success rates are comuted for a number of scenarios. For the first three scenarios, ionoshere fixed (short baselines), ionoshere weighted (medium baselines) and ionoshere float (long baselines) will be considered for single-eoch ambiguity resolution. These scenarios include:. GPS-only, dual- and trile-frequency.. Integrated GPS and Galileo, dual- and trilefrequency. 3. Integrated GPS and Galileo, dual-frequeny, different standard deviations of code observations. For the fourth scenario, only the weighted ionoshere model will be considered. Rather than looking at singleeoch success rates, the number of eochs required to obtain a redefined success rate of 99% are comuted: 4. Integrated GPS and Galileo, dual-frequency, success rate of 99%, different standard deviations of code observations and different standard deviations of ionosheric seudo-observations. The results are shown as 95% ercentiles, which mean that in 95% of all cases (during the day considered), the success rate is greater than the indicated value. Shown in Figure 6 are the number of visible GPS and Galileo satellites for São Paulo on 3 March 3. Visible here means above the cut-off elevation of. As we can see, the total number of satellites ranges from to. It should be noted here that, unlike GPS, the Galileo constellation does not reeat itself every sidereal day (3h56m) but every five days. As a result, the number of satellites, and the results given later in this section, will vary from day to day. Number of satellites ::. ::. ::. 3::. 4::. 5::. 6::. 7::. 8::. 9::. ::. ::. ::. 3::. 4::. 5::. 6::. 7::. 8::. 9::. ::. ::. ::. 3::. GPS Galileo Total Figure 6 Number of visible satellites (above elevation mask) for São Paulo, 3 March 3. Shown in Figure 7 are the success rates for the GPS-only cases. The dual-frequency results are based on the current L and L frequencies, for the trilefrequency results the new L5 signal was added. For short baselines, the dual-frequency success rates are already very high, so adding a third frequency has only a marginal effect. The situation is different for the medium and long baseline cases, although it can be concluded that instantaneous ambiguity resolution is not feasible for most of the time. Revista Brasileira de Cartografia, N o 54 7

8 Success rate [%] % 8% 6% 4% 3% % % 44% eochs are required to attain a success rate of 99%. A second conclusion is that imroved code observations do not contribute much to reducing the number of eochs required to attain a redefined success rate. Formulated differently, code observations do not contribute much when rocessing more than one eoch. % % % % Dual-frequency Trile-frequency % 8% 89% % % 9% Long baseline Medium baseline Short baseline Figure 7 Instantaneous success rates for the GPSonly case (standard recision of code observations). Success rate [%] 6% 4% 49% For an integrated GPS/Galileo solution, the success rates show a significant imrovement, comared to the GPS-only case (see Figure 8). In articular for the weighted ionoshere model (medium baseline length) instantaneous ambiguity resolution becomes feasible during most of the day. The dualfrequency Galileo system considered here was assumed to use the E-L-E frequency of MHz and the E5b frequency of 7.4 MHz. Success rate [%] % 8% 6% 4% % % % 85% Dual-frequency % % In the third scenario, only dual-frequency GPS (L, L5) and Galileo (E-L-E, E5b) were considered for different recisions of the code observations, see Table 4. As can be seen from Figure 9, the long baseline (ionoshere float model) benefited the most from the imroved recision of the code observations. High recision code observations are even more beneficial than standard recision observations for the trilefrequency case. However, instantaneous ambiguity resolution is ossible for only a limited eriod of the day. For the last scenario again the integrated dualfrequency GPS and Galileo system from scenario 3 were considered, but this time only for the medium baseline case. Instead of looking at single-eoch ambiguity resolution, it was now investigated how many eochs were required for a success rate of 99%. Results are shown in Figure for the standard- and highrecision code measurements for a standard deviation of.5 m for the ionosheric seudo-observations; in the same figure similar results are shown for a standard deviation of. m for these seudo-observations. It can be concluded that only a limited (-4) number of 35% 9% Trile-frequency Long baseline Medium baseline Short baseline Figure 8 Instantaneous success rates for integrated GPS/Galileo ositioning (standard recision code observations). % % % Standard recision High recision Long baseline Medium baseline Short baseline Figure 9 Instantaneous success-rates for integrated dual-frequency GPS/Galileo and different recisions for the code observations (see Table 4). Number of eochs Number of eochs ::. ::. ::. 3::. 4::. 5::. 6::. 7::. 8::. 9::. ::. ::. ::. 3::. 4::. 5::. 6::. 7::. 8::. 9::. ::. ::. ::. 3::. Satellites High recision Standard recision ::. ::. ::. 3::. 4::. 5::. 6::. 7::. 8::. 9::. ::. ::. ::. 3::. 4::. 5::. 6::. 7::. 8::. 9::. ::. ::. ::. 3::. Satellites High recision Standard recision Figure Number of eochs, required to attain a success rate of 99% for dual-frequency GPS/Galileo and standard and high recision code observations and for a ionosheric seudo-observations standard deviation of.5 m (to) and. m. The carrier observations in this case seem to rovide enough geometrical strength by themselves to warrant reliable ambiguity resolution. The additional advantage is that it is not necessary to use code observations, which often suffer from multiath, which in turn affect the resolution of the carrier ambiguities, see [Joosten et al., ] Number of satellites Number of satellites Revista Brasileira de Cartografia, N o 54 8

9 7. CONCLUSIONS For integrated GPS/Galileo ositioning ambiguity success rates are significantly higher than for GPS-only, in articular for medium and long baselines. However, for very long baselines, instantaneous ambiguity resolution is not feasible, due to ionosheric effects. Fortunately, the influence of the ionoshere is smaller for shorter baselines. For medium baselines (u to km), ambiguities can be resolved nearly instantaneously, using only a limited number of observation eochs. 8. REFERENCES FONTANA, R.D., W. CHEUNG, P.M. NOVAK, T.A. STANSELL, JR.,. The new L civil signal. Proceedings ION GPS, Salt Lake City, USA, HEIN, G.W., J. GODET, J.L. ISSLER, J.C. MARTIN, R. LUCAS-RODRIGUEZ, T. PRATT,. The Galileo frequency structure and signal design. Proceedings ION GPS, Salt Lake City, USA, TEUNISSEN, P.J.G., 993. Least-squares estimation of the integer GPS ambiguities. Invited lecture, Section IV Theory and Methodology, IAG General Meeting, Beijing; also in Delft Geodetic Comuting Centre, LGR series No. 6, Delft University of Technology, Delft, 6. TEUNISSEN, P.J.G., 998. Success robability of integer GPS ambiguity rounding and bootstraing. Journal of Geodesy, 7, TEUNISSEN, P.J.G.,. Integer estimation in the resence of biases. Journal of Geodesy, 75, TEUNISSEN, P.J.G., P.J. DE JONGE, C.C.J.M. TIBERIUS, 994. On the sectrum of GPS DD ambiguities. Proceedings ION GPS-94, Salt Lake City, USA, TEUNISSEN, P.J.G., C.D. DE JONG, 999. Predicting the success of ambiguity resolution. Presented at 34 th CGSIC meeting, Nashville, USA. htt:// (accessed 3 March 3). HEIN, G.W., J. GODET, J.L. ISSLER, J.C. MARTIN, P. ERHARD, R. LUCAS-RODRIGUEZ, T. PRATT,. Status of Galileo frequency and signal design. Proceedings ION GPS, Portland, USA, DE JONGE, P., C. TIBERIUS, 996. The LAMBDA method for integer ambiguity estimation: imlementation asects. Delft Geodetic Comuting Centre, LGR series No., Delft University of Technology, Delft, v+49. VAN DIERENDONCK, E.J., C. HEGARTY,. The new L5 civil GPS signal. GPS World, Vol., No. 9, JOOSTEN, P., T. PANY, J. WINKEL,. The imact of unmodelled multiath on ambiguity resolution. In: Proceedings ION GPS, Portland, USA, NTIA, 3. US frequency allocation chart. htt:// (accessed 3 March 3). SALGADO, S., S. ABBONDANZA, R. BLONDEL, S. LANNELONGUE,. Constellation availability concets for Galileo. Proceedings ION NTM, Long Beach, USA, Revista Brasileira de Cartografia, N o 54 9