Sep 13, 2006 - master reference A015 to DUNK is 257 km, whereas that to DELF is 413 km. Figure 5 Simulated WARTK CORS stations (red triangles) and.
LAMBDA-Based Ambiguity Resolution for Next-Generation GNSS Wide Area RTK Dennis Odijk, Curtin University of Technology, Australia Sandra Verhagen, Delft University of Technology, The Netherlands Peter Teunissen, Curtin University of Technology, Australia / Delft University of Technology, The Netherlands Manuel Hernández-Pajares, Technical University of Catalonia, Spain J. Miguel Juan, Technical University of Catalonia, Spain Jaume Sanz, Technical University of Catalonia, Spain Jaron Samson, ESA/ESTEC, The Netherlands Michel Tossaint, ESA/ESTEC, The Netherlands
area of GPS ionospheric tomography, GPS data processing algorithms, and radionavigation.
BIOGRAPHY Dennis Odijk is a research fellow in the new GNSS Research Lab of Curtin University of Technology, Australia. Before moving to Australia, he worked as a researcher at Delft University of Technology. His research is focussed on high-precision GNSS, with an emphasis on ionosphere modelling, ambiguity resolution, quality control and prototype software development.
Jaume Sanz is an associate professor of the Department of Applied Mathematics IV at the Polytechnical University of Catalonia (UPC). His current research interest is in the area of GPS data processing Algorithms, GPS ionospheric tomography, Satellite-Based Augmentation Systems (SBAS) and Precise Radio Navigation. Jaron Samson is a Radio Navigation System Engineer at the European Space Agency (ESTEC, The Netherlands). Before joining ESA in 2003, he has worked for Topcon and NLR. Jaron obtained a MSc from Delft University of Technology (Faculty of Geodesy).
Sandra Verhagen obtained her PhD and MSc from Delft University of Technology and is now an assistant professor at the same university. Her research interests are carrier phase ambiguity resolution and quality control for real-time kinematic GNSS applications. Sandra Verhagen is president of IAG Commission 4 “Positioning and Applications”.
Michel Tossaint is a Radio Navigation System Engineer at the European Space Agency (ESTEC, The Netherlands). Before joining ESA in 2001, he has worked for NLR. Michel obtained a MSc from Delft University of Technology (Faculty of Aerospace Engineering).
Peter Teunissen is full professor of Geodesy and Navigation, and Federation Fellow of the Australian Research Council. He is the inventor of the LAMBDA method and has 25 years of GNSS research experience. His current research focuses on investigating the capabilities of the next generation GNSS for relative navigation, attitude determination and formation flying.
ABSTRACT In the present contribution we study how the availability of next-generation multi-constellation Global Navigation Satellite Systems (GNSS), such as GPS and Galileo, will improve Wide Area Real-Time Kinematic (WARTK) ambiguity resolution for users receiving corrections as disseminated by the WARTK service provider. For this purpose, multi-frequency GPS and Galileo data of several permanent and rover stations have been simulated using the Spirent Simulator at ESA/ESTEC. These simulated data have been biased by realistic atmospheric and multipath errors. Two (static) user stations have been simulated, at 257 and 413 km from the master reference station. For the ambiguity resolution of these user
Manuel Hernández-Pajares is an associate professor at the Polytechnical University of Catalonia (UPC) since 1993 and has the full professor habilitation since 2008. He started working on GPS in 1989 for surveying applications, and since 1995 his focus has been in the area of GNSS ionospheric determination and precise radionavigation. He was the Ionosphere WG chairman of the International GPS Service (IGS) from 2002 to 2007. J. Miguel Juan is an associate professor of the Department of Applied Physics at the Polytechnical University of Catalonia (UPC). His current research interest is in the
ION 2010 International Technical Meeting January 25-27, 2010, San Diego, CA
566
and the Fixed Failure-rate (FF) Ratio test (Teunissen and Verhagen, 2009). In the last sections the GNSS data simulation is described and the results of the WARTK user ambiguity resolution and positioning are presented.
baselines the LAMBDA method is used in combination with the Fixed-Failure rate Ratio test. For this paper, the LAMBDA method has been applied to resolve the full vector of ambiguities, without a priori forming of linear combinations, such as the wide-lane or narrow-lane. An important conclusion is that the use of integrated GPSGalileo data will dramatically improve the performance of a WARTK user’s ambiguity resolution as compared to GPS only. In this point the usage of a LAMBDA-based approach helps to ensure the carrier phase ambiguity process.
WIDE AREA RTK The concept of WARTK can be summarized as follows (see Figure 1). A sparse network of Continuously Operating GNSS Reference Stations (CORS) collects and sends their phase and code data to a processing facility, which computes a network solution. The most important product of the network solution is the ionospheric delay experienced by the observed GNSS satellites between the reference stations, since these are the main auxiliary data for WARTK users (or rovers) to speed up the convergence to errors of few centimetres at such distances of hundreds of kilometers. With the usage of a tomographic model of the ionosphere, in conjunction with the geodetic model, the integer phase ambiguities between the network stations resolved, these ionospheric delays become available with very high precision and are input to the prediction of the ionospheric delays at the (approximate) user location. These corrections are then disseminated to the user, in conjunction with other parameters computed from the CORS (such as satellite clocks, orbits and inter-frequency delay code biases) enabling him to precisely correct his observations and to obtain fast integer ambiguity resolution and highprecision positioning results.
INTRODUCTION The goal of the concept of Wide-Area Real-Time Kinematic (WARTK) is to provide GNSS users corrections allowing them to extend the current RTK limit of about 10 km to many hundreds of kilometres (Hernández-Pajares et al., 2008). A wide range of highaccuracy (cm-level) applications may benefit from such a WARTK technique, for example precise farming, mining, engineering, transportation systems, meteorology and mapping. The feasibility of the WARTK technique has been demonstrated in several measurement campaigns, both with real GPS data (Hernández-Pajares et al., 2000), as well with simulated Galileo and modernized GPS data (Hernández-Pajares et al., 2003). However, in these feasibility studies no use has been made of the LAMBDA method (Teunissen, 1994) for integer estimation of the carrier phase ambiguities. Since the LAMBDA method is the current state-of-the-art for ambiguity resolution, it is expected that when implemented in the WARTK technique it can significantly improve its performance. In this paper the performance of LAMBDA-based ambiguity resolution is demonstrated for WARTK user’s processing based on Next Generation GNSS signals. Modernized GPS and Galileo data have been simulated using the Spirent simulator at ESA-ESTEC premises. GNSS data have been generated for a simulated European WARTK network, consisting of 8 reference (CORS) stations with inter-station distances up to 1000 km, and two WARTK user stations (rovers), located in the middle of the network. It is mentioned that in (Odijk et al., 2009) comparable WARTK simulations have been carried out, but only Galileo data had been simulated. In the present contribution we demonstrate the feasibility of WARTK based on integrated GPS-Galileo.
Figure 1 Conceptual visualization of Wide Area RTK: ionospheric corrections are determined from a CORS network and disseminated to users (rovers).
At the WARTK processing facility an ionospheric tomography model (see Figure 2) has been computed from the simulated Galileo data of the RIMS reference stations, based on the procedure described in Colombo et al. (1999) and Hernández-Pajares et al. (2000). Next, the ionospheric corrections for the user stations are
This paper is set up as follows. In the following section the principles of WARTK will be briefly summarized. After that, we will address the mathematical model for integrated GPS-Galileo and focus on the method for ambiguity resolution, consisting of the LAMBDA method
ION 2010 International Technical Meeting January 25-27, 2010, San Diego, CA
567
(coordinate) components and zenith tropospheric delay (ZTD) ( g ), the DD ambiguities ( aSYS ) and the DD ionospheric delays ( ıSYS ) for each GNSS. So the only common parameters between both GNSS’s are the position components plus zenith tropospheric delays.
interpolated from the modeled ionospheric delays between the network stations. This interpolation is carried out by using a planar fit of the ionospheric delays for each satellite from the set of reference stations (Colombo et al., 1999).
For each GNSS double differences are formed with respect to its own reference satellite, thus one for GPS and one for Galileo. This is done since each GNSS has its own hardware and clock characteristics. We do not difference between systems, even if the frequencies overlap, as for example done in (Julien et al., 2004). The precision of the GNSS observables is captured in the stochastic model, as follows: Figure 2 Ionospheric tomography using a dual layer ionosphere.
⎛ y ⎞ ⎛ QyGPS D(⎜ GPS ⎟) = ⎜ ⎝ yGAL ⎠ ⎜⎝
GPS-GALILEO AMBIGUITY RESOLUTION
where D(⋅) denotes the mathematical dispersion and QySYS , with SYS ∈ (GPS , GAL) , the variance-covariance
In this section we describe the integrated GPS-Galileo model for the processing of WARTK user’s data. It is assumed that per GNSS there are multi-frequency phase, code and ionosphere observables available. The data collected by the WARTK user’s receiver are processed in single-baseline mode together with the GNSS data of the master reference station of the CORS network, where the ionosphere observables are formed by the WARTK ionospheric corrections disseminated by the processing facility. Alternatively, the user’s processing can be done in undifferenced mode by using the CORS network derived satellite products.
matrix of the DD phase, code and ionosphere observations per GNSS, i.e.: ⎛ QΦ SYS ⎞ ⎜ ⎟ QySYS = ⎜ QPSYS ⎟ ⎜⎜ ⎟ QıSYS ⎟⎠ ⎝ where QΦ SYS , QPSYS and QıSYS denote the variancecovariance matrices of DD phase, code and ionosphere observables. It is assumed that the observations between GPS and Galileo are uncorrelated. The ionosphere-weighted GNSS model The processing of the WARTK user’s data is carried out by means of the ionosphere-weighted GNSS model for relative positioning. The advantage of this model – which was introduced in an early article by (Bock et al., 1986) – is that it allows incorporating the uncertainty of the (WARTK) ionospheric corrections in the stochastic model, in addition to the uncertainty of the multifrequency DD phase and code observations. The uncertainty of the WARTK ionospheric corrections depends on the level of agreement between the interpolated ionospheric delays and the true ionospheric delays between master reference station and rover. This information is provided by the WARTK processing facility.
The integrated GPS-Galileo model The (linearized) model for the integrated processing of GPS-Galileo phase, code and ionosphere data is described in e.g. (Verhagen, 2002):
⎛ y ⎞ ⎛G E (⎜ GPS ⎟) = ⎜ GPS ⎝ yGAL ⎠ ⎝ GGAL where
E (⋅)
AGPS
I GPS AGAL
⎛ g ⎞ ⎜ ⎟ aGPS ⎟ ⎞⎜ ⎟⎜ a ⎟ I GAL ⎠ ⎜ GAL ⎟ ⎜ iGPS ⎟ ⎜i ⎟ ⎝ GAL ⎠
denotes the mathematical expectation,
ySYS = (Φ , P , ıSTYS )T the vector of observed-minuscomputed (multi-frequency) double-differenced (DD) phase ( Φ SYS ), code ( PSYS ) and ionosphere ( ıSYS ) observables of one GNSS, with SYS ∈ (GPS , GAL) . It is assumed that the phase and code observations have been corrected for the major part of the troposphere by applying a standard troposphere model (e.g. Saastamoinen). The unknown parameters of the integrated GNSS model are formed by the relative baseline T SYS
T SYS
ION 2010 International Technical Meeting January 25-27, 2010, San Diego, CA
⎞ ⎟ QyGAL ⎟⎠
In this context, it is mentioned that when the standard deviations of the DD ionospheric observations are set to zero, the ionosphere-weighted model reduces to the ionosphere-fixed model (treating the ionospheric corrections deterministically), while when the ionospheric standard deviation approaches infinity, the ionosphereweighted model reduces to the ionosphere-float model
568
Figure 3 Simulation and data collection test bench at the ESA/ESTEC Radio Navigation Laboratory.
In the second step, the multi-GNSS float ambiguity solution is input to the LAMBDA method (Teunissen, 1994):
(treating the ionospheric delays as completely unknown parameters and the ionospheric corrections are not used at all), e.g. (Odijk, 2000).
( ⎛ aGPS ⎞ ⎛ aˆGPS ⎞ ⎜ ( ⎟ = S (⎜ ˆ ⎟) ⎝ aGAL ⎠ ⎝ aGAL ⎠
LAMBDA method plus Fixed-Failure-rate Ratio Test To obtain high-precision (sub-dm) user positions, the integrated GPS-Galileo model is solved in three steps: i) float solution, ii) ambiguity resolution and iii) fixed solution.
where S (⋅) denotes the mapping from the real to the ( integer space, and aSYS the estimated integer ambiguities per GNSS. In this context, the use of LAMBDA has some great advantages: • The LAMBDA method is independent of the number of GNSS’s; • The LAMBDA method is independent of the number of frequencies; • The ambiguities are optimally decorrelated, so no predefined linear ambiguity combinations are formed; • The integer least-squares within the method yields the highest possible probability of correct fixing of all ambiguity resolution methods (Teunissen, 1999).
To compute the float solution, the integer property of the DD ambiguities is discarded. The GPS-Galileo float ambiguity solution is denoted as:
⎛ aˆGPS ⎞ ⎜ ˆ ⎟; ⎝ aGAL ⎠
⎛ QaˆGPS ⎜⎜ ⎝ QaˆGAL aˆGPS
QaˆGPS aˆGAL ⎞ ⎟ QaˆGAL ⎟⎠
with QaˆGPS and QaˆGAL the variance-covariance matrices of the float DD ambiguities of GPS and Galileo, respectively, and covariance matrix QaˆGPS aˆGAL accounting for the correlation between the DD ambiguities of both systems.
It is emphasized that we applied the LAMBDA method to the full vector of DD ambiguities. Alternatively, one could try to fix only a subset of ambiguities (e.g. the wide lane); however this has not been done for this paper.
It is remarked that when there would be no common parameters (the baseline coordinates and zenith tropospheric delays) between GPS and Galileo, the correlation between the GPS and Galileo ambiguities would be zero (i.e. QaˆGPS aˆGAL = 0 ) and the success rate of
After integer estimation using LAMBDA, the ratio test with fixed failure rate is executed to decide whether the integers can be reliably accepted or not. The Fixed Failure rate (FF) Ratio test differs from the traditional ratio test used in GNSS processing in the sense that no fixed critical value is used (Teunissen and Verhagen, 2009). However, the critical value is set based on the GNSS model at hand such that the probability of accepting a wrong integer solution (i.e. the failure rate) is below a fixed user-defined threshold (e.g. 0.001). If the integer
integrated ambiguity resolution would be (slightly) lower than the ambiguity success rate based on one of the systems. The presence of the between-system correlation and the fact that the precision of the GPS and Galileo ambiguities from an integrated model is better than based on one system turns out to be very beneficial to ambiguity resolution.
ION 2010 International Technical Meeting January 25-27, 2010, San Diego, CA
569
(static) WARTK rover stations. The distances from master reference A015 to DUNK is 257 km, whereas that to DELF is 413 km.
ambiguities are accepted, the third step is conducted in which the WARTK user’s position is computed with fixed integer ambiguities. GPS-GALILEO DATA SIMULATION Modernized (dual-frequency) GPS and (triple-frequency) Galileo phase and code data have been simulated using the Spirent GNSS Simulators (Spirent GSS7700 and Spirent GSS7800) of the European Navigation Laboratory at ESA/ESTEC in The Netherlands, see Figure 3. GPS and Galileo RF signals were generated and the data were tracked and recorded by a GPS-Galileo multi-frequency receiver and consequently converted to Rinex V3 format. For GPS, data are simulated on the L1C (the modernized civilian L1 frequency, see Betz et al., 2007) and L5 frequencies, but not on the L2 frequency. For Galileo the L1, E5a and E5b frequencies have been assigned. Note that in this set up GPS and Galileo have two overlapping frequencies: L1C-L1 and L5-E5a.
Figure 5 Simulated WARTK CORS stations (red triangles) and rover stations (blue circles).
For the GPS data simulation, a constellation has been used containing 24 satellites divided over six planes and orbiting the Earth at an altitude of approximately 20,000 km and an inclination of 55º. For Galileo a constellation of 27 satellites has been assumed, divided over three planes at an altitude of about 23,200 km, with an orbital inclination of 56º. Figure 4 shows the combined GPSGalileo constellations in one picture.
Two hours of GNSS data have been simulated for 13 September 2006, from 10.10 to 12.10 UTC, at a sampling of 1 Hz. In order to obtain realistic simulations, additional systematic errors for ionosphere, troposphere and multipath have been added to the GNSS phase and code data. Slant ionospheric delays have been simulated using the International Reference Ionosphere (Bilitza and Reinisch, 2008), corresponding to mid-solar cycle conditions in 1993. Tropospheric delays have been simulated using the modified Hopfield model (Goad and Goodman, 1974), considering a constant value for both the dry and wet zenith tropospheric delays and a mapping function. Because of its local properties, multipath has been simulated for the two rover stations only and only for the code data, since code multipath is usually much (about two orders) larger than phase multipath. These code multipath errors have been derived from actual GPS measurements, being compatible with real Galileo (GIOVE) multipath (Simsky et al., 2008).
number of satellites
20
Figure 4 24-satellite GPS (red) and 27-satellite Galileo (blue) constellations (Tiberius et al., 2002).
Similar as in (Odijk et al., 2009) several locations of the EGNOS Receiver Independent Monitoring Station (RIMS) network have been used to simulate WARTK reference stations, see Figure 5 for their location. Station A015 (Southampton – England) has been assigned as master WARTK reference station. The locations of IGS stations DUNK (Dunkerque – France) and DELF (Delft – The Netherlands) have been used as locations of the two
ION 2010 International Technical Meeting January 25-27, 2010, San Diego, CA
15
GPS Galileo GPS+Galileo
10
5 1000
2000
3000 4000 epoch [1 sec]
5000
6000
7000
Figure 6 Number of GPS and Galileo satellites for the 2 hour time span as visible in DELF above 10 deg elevation.
Table 1 summarizes the simulation scenario. Based on a data cut-off angle of 10 deg, in Figure 6 the number of
570
resolution has been conducted for GPS only, Galileo only and integrated GPS-Galileo. Although triple-frequency Galileo observations are available, we did not include the E5b frequency in the integrated GPS-Galileo computations. The integrated processing is based on dualfrequency GPS (L1C+L5) and dual-frequency Galileo (E1-E5a), and thus the frequencies between the two systems overlap. In (Verhagen et al., 2007) it was by means of ambiguity success rate simulations demonstrated that this is a good combination (with respect to success rate), implying that a third (Galileo) frequency does not add much with respect to this combination, see also (Verhagen, 2002). For the Galileo-only computations we however did include the third frequency.
GPS and Galileo satellites is shown as function of the 2hour time span, as visible for rover DELF. Both number of GPS and Galileo satellites vary between 5 and 8, while the number of combined satellites is 10-16. Table 1 Summary of the WARTK simulation scenarios Date and time 13 September 2006, 10.10-12.10 UTC Data sampling 1 second GNSS signals GPS: L1C (1575.42 MHz) L5 (1176.45 MHz) GAL: L1 (1575.42 MHz) E5a (1176.45 MHz) E5b (1207.14 MHz) CORS EGNOS RIMS stations, stations see Figure 5 for their locations Rover (user) IGS stations DUNK and DELF, stations see Figure 5 for their locations Ionosphere Mid-solar cycle conditions, based on IRI2007 model Troposphere Modified Hopfield model with the following settings: • air pressure: 1014 HPa • temperature: 290 K • relative humidity: 0.31 Multipath Only for code data of rovers, based on actual GPS measurements.
Table 2 Observable-dependent factors needed for computing the standard deviations of the simulated observables GPS GAL Phase factor L1C: 2 mm L1: 2 mm (undifferenced) L5: 2 mm E5a: 2 mm E5b: 2 mm Code factor C1C: 10 cm C1: 8 cm (undifferenced) C5: 10 cm C5a: 5 cm C5b: 8 cm
The stochastic model settings for phase and code applied in all computations are given in Table 2. The values as given in the table are not standard deviations, since an elevation-dependent multiplication factor has been applied as well to each observable: σ ys = [1 + 10 exp(−ε s / 10)]σ y
RESULTS In this section the performance of WARTK ambiguity resolution and positioning is demonstrated for the two rovers DUNK and DELF. To get insight into the quality of the WARTK ionospheric corrections it is first shown to what extent the WARTK corrections reduce the DD ionospheric delays for the two rovers. After that ambiguity resolution results are presented, followed by an evaluation of the positional accuracy.
with σ ys the observable- and satellite-specific standard deviation, ε s the elevation [deg] of satellite s and σ y an observable-dependent factor as given in Table 2. In addition to this, all multi-frequency undifferenced observations are assumed to be uncorrelated, so the only correlation that arises is the correlation due to the double differencing. All phase, code and ionosphere observations are assumed to be uncorrelated in time.
Reduction of ionospheric delays for the WARTK users For the 413-km baseline A015-DELF the DD ionospheric delays based on fixed integer ambiguities have been plotted as function of the time span. At the last page of this paper, in Figure 13 (left) and Figure 14 (left) the DD ionospheric delays as estimated from the GPS and Galileo data are shown, thus without the WARTK corrections, while Figure 13 (right) and Figure 14 (right) depicts the residual DD ionospheric delays (estimated minus WARTK-corrected DD ionospheric delays). The tremendous impact of the corrections is clearly visible from these graphs: while the (absolute) DD ionospheric delays in the GNSS data can reach up to more than 50 cm, the residual (absolute) DD ionospheric delays are all below 4 cm.
To account for the uncertainty in the WARTK ionospheric corrections, an (undifferenced) ionospheric standard deviation of 1 cm has been applied in this case to the 257-km baseline A015-DUNK and a standard deviation of 2 cm to 413-km baseline A015-DELF. Alternatively (though not applied for this paper), the ionospheric standard deviation is applied individually for each observation, as function of the broadcasted ionospheric model computed from the CORS network data, taking into account the different levels of ionospheric activity affecting each satellite (HernándezPajares et al., 2006a-b).
LAMBDA-based WARTK user’s ambiguity resolution With the WARTK ionospheric corrections applied, we will now focus on the performance of the LAMBDA method and FFRatio test for the two rovers. Ambiguity
ION 2010 International Technical Meeting January 25-27, 2010, San Diego, CA
Concerning ambiguity resolution, we applied both instantaneous and multi-epoch ambiguity resolution strategies. Instantaneous or epoch-by-epoch ambiguity resolution is the fastest method, for which LAMBDA and
571
the FFRatio test are based on data of just one observation epoch (no information from previous epochs is used) and thus truly real time. However, for long baselines the strength of the model underlying may not always be sufficient to guarantee successful instantaneous ambiguity resolution, and accumulation of a multiple of epochs (keeping the ambiguities constant) may be needed before the integer ambiguities can be resolved with sufficient reliability.
For sake of comparison, it has been tried to resolve the integer ambiguities without the WARTK ionospheric corrections, thus based on an ionosphere-float ( QıSYS = ∞ ) processing, instead of ionosphere-weighted, as has been done for the instantaneous success rates. A Kalman filter processing is applied with the assumption that the float ambiguities should remain constant in the prediction step (no assumptions on ionosphere and baseline parameters). LAMBDA is still applied on an epoch-by-epoch basis. An ionosphere-float processing without ionospheric corrections is comparable to taking the traditional ionosphere-free combination. Due to the poor precision of the float ambiguities, ambiguity resolution based on the ionosphere-float model is not expected to be fast. Table 4 presents the mean Time-To-Fix-Ambiguities (TTFA) for the GPS-only, Galileo-only and combined GPS-Galileo scenario. The TTFA is defined here as the time needed such that the FFRatio test is accepted and the correct integers are estimated. Note the long time needed in case of GPS only before the ambiguities can be fixed reliably for both baselines. In case of Galileo-only, the needed time is about 50% less compared to GPS only, but still longer than half an hour. The combination of GPS and Galileo data has also in absence of ionospheric corrections an enormous influence: the mean TTFA is only a few minutes. Despite this, applying the WARTK corrections seem to be extremely beneficial for ambiguity resolution.
Table 3 Instantaneous success rates computed including WARTK ionospheric corrections GPS only GAL only GAL only GPS+GAL dual-freq dual-freq triple-freq dual-freq DUNK 92.3% 98.0% 98.6% 100% 257km DELF 52.0% 62.1% 83.0% 99.8% 413km
The success rates of instantaneous ambiguity resolution for the two user baselines are summarized in Table 3. Here the (empirical) success rate is defined as the relative number of epochs for which the correct integers are estimated (the correct values are known from the full 2-hr time span processing) and at the same time for which the FFRatio test is accepted (using a failure rate of 0.1%). As can be seen from the table, the success rate for GPS only of DUNK is already above 90%, which should be attributed to the high-quality WARTK ionospheric corrections. The success rate for DELF with GPS only is much lower (52%), and this should be due to a degradation of the quality of the ionospheric corrections with increasing distance in the adopted equal-ionoweighting scheme. For Galileo only, we considered a dual-frequency and a triple-frequency scenario for both baselines. The dual-frequency success rates are slightly higher than their GPS counterparts, which is likely due to the better precision of the Galileo code data, as compared to GPS (see Table 2). The performance of instantaneous ambiguity resolution relies heavily on the quality of the code data. Adding a third Galileo frequency improves the success rate of DUNK only marginally, but for DELF the improvement is more considerable: from 62% to 83%. The explanation for this is that an additional third frequency becomes more beneficial when the ionospheric corrections have less weight in the adjustment, see also (Odijk et al., 2009). An integrated processing of the simulated GPS and Galileo data yields a 100% success rate for DUNK and almost 100% for DELF. Especially for the longest baseline to DELF the improvement due to two GNSS’s is obvious: even for this baseline length instantaneous ambiguity resolution is almost feasible. When a detailed ionospheric weighting scheme would be applied (see corresponding comments above), the third frequency becomes crucial to guarantee the subdecimeter error level navigation also for receivers placed up to hundreds of kilometers far (see Hernández-Pajares et al., 2004).
ION 2010 International Technical Meeting January 25-27, 2010, San Diego, CA
Table 4 Mean Time-To-Fix-Ambiguities computed excluding WARTK ionospheric corrections (ionosphere-float processing) GPS only GAL only GAL only GPS+GAL dual-freq dual-freq triple-freq dual-freq DUNK 50 min 33 min 33 min 2.5 min 257km DELF 67 min 33 min 33 min 1.7 min 413km
Table 5 presents the overall results based on a Kalman filter processing, again with the ionospheric corrections applied. The empirical probabilities and TTFA are determined based on observation windows of 60 seconds, where the start epoch is shifted by 30 seconds: - Window 1 : 10:30:00 – 10:30:59 - Window 2 : 10:30:30 – 10:31:29 - Window 3 : 10:31:00 – 10:31:59 - etcetera Hence, in total 120 observation windows were considered, and for each window the TTFA and empirical probabilities were determined. The results confirm the tremendous contribution of using a dual-GNSS compared to GPS- or Galileo-only, and that the third Galileo frequency does not contribute much to the ambiguity resolution performance.
572
Table 5 Success rates and TTFA based on Kalman filter processing of 60-second windows (WARTK ionospheric corrections included) GPS GAL GAL GPS+GAL only only only dual-freq dualdualtriplefreq freq freq DUNK mean 99.6% 99.9% 99.9% 100% 257km success rate mean 1.2 1.0 1.0 1 TTFA [s] max 6 3 3 1 TTFA [s] DELF mean 98.0% 98.6% 99.4% 100% 413km success rate mean 2.1 1.7 1.2 1 TTFA [s] 11 9 8 1 max TTFA [s]
0.05
North [m]
North [m]
0.5 0 −0.5 −1 −1
0 East [m]
1
−0.5 0 East [m]
0 −0.05 −0.1 −0.1
1
0 East [m]
0.1
HPE fixed (95%): 0.0071835m 0.1
0.5
0.05
North [m]
North [m]
HPE float (95%): 0.2275m 1
0 −0.5 0 East [m]
0 −0.05 −0.1 −0.1
1
0 East [m]
0.1
HPE fixed (95%): 0.0054245m 0.1
0.5
0.05
North [m]
North [m]
HPE float (95%): 0.16621m 1
0 −0.5 −1 −1
0 East [m]
1
0 −0.05 −0.1 −0.1
0 East [m]
0.1
Figure 9 Float (left) vs. fixed (right) Horizontal Positioning Errors (HPE) for DELF computed using the epoch-by-epoch GPS+Galileo scenario. The 95% HPE circle is depicted in red.
Figure 8 and Figure 11 show the position errors based on Galileo-only epoch-by-epoch processing. Compared to GPS only, the float positional accuracy of Galileo is about a factor 3 better than of GPS, which is a result of better code precision (see Table 2) in combination with a better geometry (DOP) of the Galileo satellites. Also the fixed positions obtained with Galileo are more accurate than with GPS.
0.1
Figure 7 depicts the float (left) and fixed (right) horizontal position errors based on GPS only, whereas Figure 10 presents the errors as function of the 2-hr time span. The known position of DELF corresponds to the origin of the North-East-Up system. In the horizontal scatter plots a 95% error circle is plotted as well. From the figures the beneficial effect of ambiguity resolution is visible. However, despite successful ambiguity resolution, especially the fixed vertical positional accuracy may be
ION 2010 International Technical Meeting January 25-27, 2010, San Diego, CA
0
Figure 8 Float (left) vs. fixed (right) Horizontal Positioning Errors (HPE) for DELF computed using the epoch-by-epoch Galileo-only scenario. The 95% HPE circle is depicted in red.
−0.05 0 East [m]
0.05
−1 −1
0
−0.1 −0.1
0.5
Figure 7 Float (left) vs. fixed (right) Horizontal Positioning Errors (HPE) for DELF computed using the epoch-by-epoch GPS-only scenario. The 95% HPE circle is depicted in red.
WARTK user’s positioning accuracy For the epoch-by-epoch scenarios the float and fixed position errors have been computed by subtracting the estimated coordinates with the precisely known coordinates of the rover receivers. In this subsection we will analyze the positional accuracy of rover DELF, since this is the longest user baseline. HPE fixed (95%): 0.012231m 0.1
HPE fixed (95%): 0.012231m 0.1
−1 −1
From Table 3 and Table 5 it should be noted that the performance is quite good. The results in previous experiments (see for instance Hernández-Pajares et al. 2006b), involving actual GPS and simulated GNSS data, confirm the capability of producing ionospheric corrections by the WARTK network of permanent receivers, accurate enough (better than 0.25 TECU of error in differential ionospheric delay) to support a High Precise Positioning service at distances of up to about 400 kilometers.
HPE float (95%): 0.67501m 1
HPE float (95%): 0.67501m 1
North [m]
North [m]
insufficient (i.e. larger than a few dm) for certain times of the day due to a poor geometry (DOP), combined with few satellites and estimation of a ZTD parameter.
The highest positional accuracy is obtained when GPS and Galileo are integrated; see Figure 9 and Figure 12. The 95% float horizontal and vertical positional accuracy are 17 cm and 80 cm, respectively, and this improves to 5 mm and 2.7 cm respectively, with the ambiguities fixed.
573
CONCLUSIONS
Horizontal Position Errors of DELF − GPS ONLY 0.5 FLT HPE (95%): 0.67501 FIX HPE (95%): 0.012231
HPx [m]
0.4
Next-Generation GNSS signals will be very beneficial to WARTK positioning. By means of simulations we have demonstrated that instantaneous dual-frequency ambiguity resolution based on integrated GPS-Galileo is successful for almost 100% of the time for users operating in a sparse WARTK CORS network, having inter-station distances of hundreds of kilometers. Here the user’s ambiguity resolution is based on full integer estimation using the LAMBDA method and integer acceptance by means of the Fixed-failure rate Ratio test. A crucial requirement to this success is the availability of precise ionospheric corrections generated by the WARTK processing facility from the data of the CORS stations. It has also been demonstrated that dual-frequency integrated GPS-Galileo outperforms standalone GPS or Galileo ambiguity resolution, including triple-frequency Galileo only. Besides ambiguity resolution, we have shown that the positional accuracy will improve for integrated GPSGalileo, and this accuracy will be less sensitive to a change in number of satellites and geometry than when using just one system.
0.3 0.2 0.1 0
1000
2000
3000 4000 epoch [1 sec]
5000
6000
7000
Vertical Position Errors of DELF − GPS ONLY 2 FLT VPE (95%): 5.2199 FIX VPE (95%): 0.14579
VPx [m]
1.5
1
0.5
0
1000
2000
3000 4000 epoch [1 sec]
5000
6000
7000
Figure 10 Horizontal (top) and Vertical (bottom) Position Errors as function of time for DELF computed using the epoch-byepoch GPS-only scenario. Horizontal Position Errors of DELF − GAL ONLY 0.5 FLT HPE (95%): 0.2275 FIX HPE (95%): 0.0071835
HPx [m]
0.4 0.3 0.2 0.1 0
ACKNOWLEDGMENTS 1000
2000
3000 4000 epoch [1 sec]
5000
6000
7000
This work has been performed under ESA contract AO-15033/06/NL/HE in the FES-WARTK CCN project, carried out by Delft University of Technology in The Netherlands, in collaboration with and lead by the Research Group of Astronomy and Geomatics of the Technical University of Catalonia (gAGE/UPC) in Spain.
Vertical Position Errors of DELF − GAL ONLY 2 FLT VPE (95%): 1.406 FIX VPE (95%): 0.036829
VPx [m]
1.5
1
0.5
0
1000
2000
3000 4000 epoch [1 sec]
5000
6000
During the time of the project the first author was working as a researcher at Delft University of Technology. Professor Peter Teunissen is the recipient of an Australian Research Council Federation Fellowship (project number FF0883188). This support is greatly acknowledged. The research of Dr Sandra Verhagen is supported by the Dutch Technology Foundation STW, applied science division of NWO and the Technology Program of the Dutch Ministry of Economic Affairs.
7000
Figure 11 Horizontal (top) and Vertical (bottom) Position Errors as function of time for DELF computed using the epoch-byepoch Galileo-only scenario. Horizontal Position Errors of DELF − GPS & GAL 0.5 FLT HPE (95%): 0.16621 FIX HPE (95%): 0.0054245
HPx [m]
0.4 0.3 0.2 0.1 0
1000
2000
3000 4000 epoch [1 sec]
5000
6000
REFERENCES
7000
Betz JW, Blanco MA, Cahn CR. Dafesh PA, Hegarty CJ, Hudnut KW, Kasemri V, Keegan R, Kovach K, Lenahan LS, Ma HH, Rushanan JJ, Sklar D, Stansell TA, Wang CW, Yi SK (2007) Enhancing the future of civil GPS – Overview of the L1C signal, InsideGNSS, Spring 2007: 42-49.
Vertical Position Errors of DELF − GPS & GAL 2 FLT VPE (95%): 0.80056 FIX VPE (95%): 0.027094
VPx [m]
1.5
1
0.5
0
1000
2000
3000 4000 epoch [1 sec]
5000
6000
7000
Bilitza D and Reinisch BW (2008) International Reference Ionosphere 2007: Improvements and new parameters, Advances in Space Research 42: 599-609.
Figure 12 Horizontal (top) and Vertical (bottom) Position Errors as function of time for DELF computed using the epoch-byepoch GPS+Galileo scenario.
ION 2010 International Technical Meeting January 25-27, 2010, San Diego, CA
574
Bock Y, Gourevitch SA, Counselman CC, King RW and Abbot RI (1986) Interferometric analysis of GPS phase observations. Manuscripta Geodaetica 11: 282-288.
Hernández-Pajares M, Juan JM, Sanz J, Aragon-Angel A, Ramos-Bosch P, Odijk D, De Bakker PF, Van der Marel H, Verhagen S, Fernandez-Hernández I, Toledo M and Samson J (2008) Feasibility study of a European Wide Area Real Time Kinematic system, Proc. NAVITEC ‘08, Noordwijk, The Netherlands, 1012 December 2008, CDROM.
Colombo OL, Hernández-Pajares M, Juan JM, Sanz J and Talaya J (1999) Resolving carrier-phase ambiguities on-the-fly, at more than 100 km from nearest site, with the help of ionospheric tomography, Proc. ION GPS99, Nashville, TN, 14-17 September 1999, pp. 16351642.
Odijk D (2000) Weighting ionospheric corrections to improve fast GPS positioning over medium distances, Proc. ION GPS-2000, Salt Lake City, UT, 19-22 September 2000, pp. 1113-1123.
Goad CC and Goodman L (1974) A modified Hopfield tropospheric refraction correction model, Proc. Fall Annual Meeting of the AGU, San Francisco, CA, 1217 December 1974.
Odijk D, Bakker P de, Verhagen S, Teunissen PJG, Hernández-Pajares M and Samson J (2009) Fast LAMBDA-based ambiguity resolution combined with WARTK technique for Galileo data simulated using ESA’s Galileo Signal Validation Facility, Proc. IGNSS 2009, Surfers Paradise, Australia, 1-3 December 2009, CDROM, 15 pages.
Julien O, Alves P, Cannon ME, Lachapelle G (2004) Improved triple-frequency GPS/GALILEO carrier phase ambiguity resolution using a stochastic ionosphere modeling. Proc. ION-NTM 2004, San Diego, CA, 26-28 January 2004, pp. 441-452.
Simsky A, Mertens D, Sleewaegen JM, De Wilde W, Hollreiser M, Crisci M (2008) MBOC vs. BOC(1,1) Multipath Comparison Based on GIOVE-B Data, InsideGNSS, September/October 2008: 36-40.
Hernández-Pajares M, Juan JM, Sanz J and Colombo OL (2000), Application of ionospheric tomography to real-time GPS carrier-phase ambiguities resolution, at scales of 400-1000 km and with high geomagnetic activity, Geophysical Research Letters, 27(13): 20092012.
Teunissen, PJG (1994) A new method for fast carrier phase ambiguity estimation. Proc. IEEE PLANS ’94, Las Vegas, NV, 11-12 April 1994, pp. 862-873.
Hernández-Pajares M, Juan JM, Sanz J and Colombo OL (2003) Feasibility of wide-area subdecimeter navigation with GALILEO and modernized GPS, IEEE Trans on Geoscience and Remote Sensing, 41(9): 2128-2131.
Teunissen, PJG (1999) A theorem on maximizing the probability of correct integer estimation. Artificial Satellites, 34(1): 3-9. Teunissen PJG, Verhagen S (2009) The GNSS ambiguity ratio-test revisited, Survey Review, 41(312): 138-151.
Hernández-Pajares M, Juan JM, Sanz J, Orús R, GarcíaRodríguez and Colombo O (2004) Wide Area Realtime Kinematics with Galileo and GPS signals, Proc. ION-GNSS 2004, Long Beach, CA, 21-24 September 2004, pp. 2541-2554.
Tiberius CCJM, Pany T, Eissfeller B, Jong CD de, Joosten P and Verhagen S (2002) Integral GPS-Galileo ambiguity resolution. Proc. ENC-GNSS ’02, Copenhagen, Denmark, 27-30 May 2002, CD ROM.
Hernández-Pajares M, Juan JM and Sanz J (2006a) Medium-scale traveling ionospheric disturbances affecting GPS measurements: Spatial and temporal analysis, Journal of Geophysical Research, Vol. 111, A07S11, doi: 10.1029 / 2005JA011474.
Verhagen S (2002) Performance analysis of GPS, Galileo and integrated GPS-Galileo, Proc. ION GPS 2002, Portland, OR, 24-27 September 2002, pp. 2208-2215. Verhagen S, Teunissen P, Odijk D (2007) Carrier-phase ambiguity success rates for integrated GPS-Galileo satellite navigation, Proc. SANE2007, Perth, Australia, 15-18 April 2007, Vol. 107, No. 2, pp. 139-144.
Hernández-Pajares M, Juan JM, Sanz J (2006b) Real time MSTIDs modelling and application to improve the precise GPS and GALILEO navigation, Proc. ION GNSS-2006, Forth Worth, TX, 26-29 September 2006, pp. 1358-1368.
ION 2010 International Technical Meeting January 25-27, 2010, San Diego, CA
575
90
60 0
0
6000 epoch
7000
0
0
−1
90
1
0
1000 epoch
0
−1
90
1
0
0
0
−1
90
1
0 30004000500060007000 epoch
0
−1
5000 epoch
0
5000 epoch
• ambiguity−fixed
6000 epoch
7000
−0.05
90
0.05
1000 epoch
90
0.05
5000
90
0.05
0
500 epoch
−0.05
90
0.05
0 30004000500060007000 epoch
5000
6000 epoch
30 −0.05
0
− satellite elevation
5000 epoch
0
PRN 28 90
0.05
90
60 0
epoch
60 0
PRN 26
60 0
30 −0.05
0
90
30
0
0 7000
5000 epoch
60
30 −0.05
0
PRN 18
60
0
−0.05
0
0
0
0
30
0
PRN 17
30 −1
0
60
−0.05
60 0
PRN 10 0.05
0
90
30
30
0
0
0
−0.05 5000
0.05
60
30
60
30 0
90
PRN 22
1
60
−1
0
0
0
PRN 9
0.05
60
PRN 28 90
30 0 7000
90
30
PRN 26
60
6000 epoch
60
30
PRN 22 1
5000
90
60 0
500 epoch
5000 epoch
0
PRN 18
30
−1
0
PRN 17
60
0
30
PRN 8
0.05
60
30
PRN 10 1
−1
90
60
30 −1 5000
PRN 5
1
Satellite elevation [deg]
DD ionospheric delay [m]
PRN 9
1
Satellite elevation [deg]
PRN 8 90
residual DD ionospheric delay [m]
PRN 5 1
0
0
5000 epoch
• ambiguity−fixed
30 −0.05
0
5000
0
epoch
− satellite elevation
Figure 13 Estimated ambiguity-fixed DD ionospheric delays vs. residual ambiguity-fixed DD ionospheric delays for A015-DELF (GPS). PRN 29 was reference satellite for the whole time span and is not plotted.
0
60 0
−1
90
1
0
5000 epoch
PRN 157 1
0
1
0
1000 epoch
0
−1
90
1
4000 epoch
0 6000
1000
PRN 163
2000 epoch
3000
1
90
0.05
30 0
• ambiguity−fixed
1000 2000 epoch
0.05
0
1000 epoch
−0.05
0
0.05
0
−0.05
90
0.05
0
200 epoch
4000 epoch
0 6000
60 30 −0.05 6600
6800 7000 epoch
0
PRN 162 90
0.05
90
60
60 0
30 −0.05
1000
2000 epoch
3000
0
30 −0.05 4000
6000 epoch
0
− satellite elevation
0.05
90
60 0
60 0
30 5000 epoch
90
PRN 164 90
0
0
0
PRN 162
0
5000 epoch
30
0
0.05
− satellite elevation
90
0
60
• ambiguity−fixed
−0.05
−0.05
PRN 161
30
0
30
0
PRN 161
0
6000 epoch
5000 epoch
60
60
−1
90
0
PRN 163
0
0
−0.05
0
30 −1 4000
0
PRN 162
90
30 5000 epoch
−0.05
60 0
60
0
90
30
PRN 157
− satellite elevation
60 0
4000 6000 epoch
0.05
PRN 164 90
0
0
−0.05 2000
0.05
60
30
0
0
90
30
60
30
• ambiguity−fixed
1
1
60
−1
0
0
6800 7000 epoch
PRN 156
0.05
60
30 −1 6600
PRN 155 90
PRN 162 90
0
200 epoch
90
0
30 0
0
60
PRN 162
0
5000 epoch
30
60
0
0
60
PRN 162
DD ionospheric delay [m]
90
0
1
−1
−1
PRN 161
30
−1
0
PRN 161
60
−1
30
Satellite elevation [deg]
4000 6000 epoch
30
0
0.05
60 0
30 −1 2000
PRN 154 90
Satellite elevation [deg]
60
1
Satellite elevation [deg]
90
0
30 −0.05
0
• ambiguity−fixed
1000 2000 epoch
0
Satellite elevation [deg]
PRN 156
1
residual DD ionospheric delay [m]
DD ionospheric delay [m]
PRN 155 90
residual DD ionospheric delay [m]
PRN 154 1
− satellite elevation
Figure 14 Estimated ambiguity-fixed DD ionospheric delays vs. residual ambiguity-fixed DD ionospheric delays for A015-DELF (GAL).
ION 2010 International Technical Meeting January 25-27, 2010, San Diego, CA
576
