Sensitivity Analysis for GNSS Integer Carrier Phase Ambiguity Validation Test Jinling Wang, Hung Kyu Lee, Steve Hewitson, Chris Rizos, Joel Barnes School of Surveying and Spatial Information Systems, The University of New South Wales, Sydney NSW 2052, Australia, email:
[email protected] Abstract. Global Navigation Satellite Systems (GNSS) have been widely used for many precise positioning and navigation applications. In satellite-based precise positioning, the determination of correct integer carrier phase ambiguities is the key issue. Therefore much effort has been invested in developing robust quality control procedure(s) which can effectively validate the ambiguity resolution results. Such a quality control procedure has been traditionally based on a so-called Fratio. A major shortcoming of this F-ratio is that its probability distribution is still unknown, which precludes the possibility to evaluate the confidence level for the ambiguity validation test. To overcome such a shortcoming an alternative ambiguity validation test based on the so-called W-ratio has been proposed, which allows for a more rigorous quality control procedure (Wang et al., 1998). This paper presents a sensitivity analysis for the new ambiguity validation test based on the W-ratio. The analysis will cover the sensitivities of the W-ratio to undetected gross errors, unmodelled (residual) systematic errors, and geometry strengths relating to a variety of satellite constellations, such as GPS, GPS/pseudolite and GPS/Galileo integration
contain the correct integer ambiguities. The process of searching all possible integer ambiguity combinations within the search window is then performed using a search criterion based on the minimisation of the quadratic form of the least-squares residuals. It is important to note that most ambiguity research techniques provide ambiguity estimation within a few tens of milliseconds, and such a performance normally satisfies most applications. The best integer ambiguity combination from the search that results in the minimum quadratic form of the least-squares residuals will be considered as the most likely (best) solution. The next step for the ambiguity resolution is to apply a so-called discrimination (validation) test to ensure that the most likely integer ambiguity combination is statistically better than the second best combination, as defined by second minimum quadratic form of the least–squares residuals. Traditionally ambiguity validation test procedures have been based on the so-called F-ratio of the second minimum quadratic form of the least-squares residuals and the minimum quadratic form of the least-squares residuals. An alternative ambiguity validation (discrimination) test procedure has been proposed by Wang et al. (1998), in order to overcome the drawback of the validation test using the F-ratio (in which the probability distribution is unknown). This new procedure is based on the ratio (called W-ratio) of the difference between the minimum and second minimum quadratic forms of the least-squares residuals and its standard deviation. A comparative study of the major ambiguity validation procedures has been carried out (Wang et al, 1999), showing that the ambiguity discrimination tests based on F-and W-ratios are generally close to the success probability of ambiguity resolution. However, the results should be different if undetected outlier and/or unmodelled systematic errors are remaining in the raw observations. In addition, a different stochastic modelling method and geometry of the satellite constellation may also affect the ambiguity test. Hence, this contribution will focus on a sensitivity analysis for the ambiguity validation test.
Keywords: GNSS, Ambiguity Validation test, W-ratio 1 Introduction Precise kinematic relative GNSS-based positioning requires the reliable determination of the carrier phase integer ambiguities. (The carrier phase measurements are ambiguous, with the ambiguity - the integer number of signal wavelengths between satellite and antenna - being an unknown value a priori.) Hence the determination of the integer ambiguities, commonly referred to as ambiguity resolution (AR), is the most critical data analysis step for precise GNSS-based positioning. With fixed integer ambiguities, the carrier phases can be used as unambiguous precise range measurements. The ambiguity resolution process consists of two steps, namely: ambiguity estimation and ambiguity validation. In ambiguity search, integer ambiguity parameters are initially treated as real (continuous) parameters. The real-valued (float) ambiguity parameters, together with other unknown parameters such as the coordinates of the roving receiver, can be estimated using a least-squares or Kalman filtering algorithm. The float solution of the real-valued ambiguity estimates and their associated statistics is then used to construct a search window, which is assumed to
2 Ambiguity estimation and validation procedure 2.1 Ambiguity estimation In the case of using least-squares, the so-called Gauss-Markov model for linearised GNSS (single- or double-differenced) measurements is written as:
1
l = Ac xc + Ak xk + v ,
(1)
D = σ 2Q = σ2 P −1 ,
(2)
Ω0 = v$T Pv$ = l T PQv$ Pl = l T Pl − l T PAx$ ,and where f = n−m−t .
The above solution is referred to as the float solution, with the real-valued ambiguity estimates and their associated statistics. The next step is to take into account the fact that the ambiguities should be integer valued. This means that the integer ambiguities can be obtained by applying an appropriate searching method. The best integer ambiguity combination that results in the minimum quadratic form of the least-squares residuals will be considered as the most likely (best) solution. Normally, the first two best ambiguity combinations are identified for validation purposes.
where
l
D Q P
is the n × 1 measurement vector, n is the number of measurements; is the n × 1 vector of the random errors; is the m × 1 double-differenced ambiguity parameter vector, and m is the number of ambiguity parameters; is the t × 1 vector of all other unknown parameters, and t is the number of all other unknowns (except ambiguities); is the design matrix for the ambiguity parameters; is the design matrix for the other unknown parameters; is the covariance matrix; is the cofactor matrix; is the weight matrix; and
σ2
is the a priori variance factor.
v xk xc Ak Ac
Based
on
the
principle
of
2.2 Ambiguity validation with W-ratio test The W-ratio is defined as (Wang et al., 1998):
least-squares
T
study, the a posteriori variance cofactor s$02 is used. In this situation, the W-ratio has a Student’s t-distribution.
(4)
3. Sensitivity analysis for the validation test 3.1 Description of test data sets and data processing To analyse the sensitivity of the carrier phase integer ambiguity validation test based on the W-ratio and Fratio (= Ω s / Ω m ), a variety of numerical tests with simulated measurements were carried out. All of the test measurements (see Table 1) were generated by using the SNAP GNSS simulator (Lee et al., 2002).
(5)
Table 1. Simulated data sets for the tests Data Baseline Num. Of Data Set Length SVs Span A 5 km 6 100 sec B 5 km 5 600 sec
(6)
where Qv$ = Q − AQx$ AT is the cofactor matrix of the residuals. With the estimated residual vector v$ and weight matrix P , the a posteriori variance cofactor can be estimated as:
f
(14)
and δ is the variance factor. Two different variance cofactors can be chosen (Wang et al., 1998). In this
Furthermore, from Equations (1), (3) and (5), the leastsquares residuals are obtained as follows:
Ω s$02 = 0 ,
Var (d ) = δ2Qd k
where x$ = ( x$c , x$k )T and A = ( Ac , Ak ) . Qx$ is the cofactor matrix of the estimated vector x$ , which can be represented by the following partitioned matrices:
v$ = l − Ax$ = Qv$ Pl ,
(13)
2
with:
Qx$c x$k . Qx$k
d = Ω s − Ωm
Qd = 4 ⋅ ( Ks − Km )T Qx−$ 1 ( Ks − Km ) is the cofactor of d ,
x$ = Qx$ AT Pl , (3)
Qx$ c Qx$ = Qx$k x$c
(12)
where
( v Pv = minimum), the estimates of the unknowns x$ in Equation (1) can be obtained:
Qx$ = ( AT PA) −1
d Var (d )
W=
Obs. Type Dual Dual
C
5 km
6
300 sec
Dual
D E
5 km 0.1km
9(4) 13(7)
300 sec 300 sec
Dual Single
Remark Outlier Multipath Stochastic model Pseudolite Galileo
In the simulation of measurements, receiver noise, ionospheric delay, tropospheric delay, and satellite orbit errors were considered, while undetected outliers, unmodelled systematic and multipath errors were added
(7)
2
for the specific datasets depending on the test purpose. Note that temporal correlation between epochs is not considered, and the errors for Galileo are assumed to be similar to those of GPS. Data Set A: The magnitudes of the three outliers are 0.5m, 1.0m, 2.0m for pseudo-ranges and 0.5cm, 1.0cm, and 2.0cm for carrier phases, respectively. Only one outlier is simulated in either pseudo-ranges or carrier phases for each data set. Data Set B: The same multipath pattern was used in all the measurement generation. In one data set, only the measurements (L1/L2 carrier phases) from one satellite were contaminated by multipath. Data set C: Three different patterns of multipath were imposed on the measurements of three satellites, which have a relatively low elevation angle (among the total of five satellites). Data Set D: Dual-frequency data from five GPS satellites and four Pseudolites for analysing an effect of geometry changes. Data Set E: Single-frequency data from six GPS satellites and seven Galileo satellites were used to get an insight of potential benefits of using the upcoming European Global Navigation Satellite System (GNSS) for ambiguity resolution. The data was processed either epoch-by-epoch with dual-frequency measurements, or via accummulated normal equations (multi-epoch) in the case of singlefrequency measurements. The preset standard deviations for both L1 and L2 pseudo-ranges are 0.5m, whereas those for L1 and L2 carrier phase are 0.005m. The best and second best ambiguity vector (ambiguity estimation) was obtained using the Least Squares AMBiguity Decorrelation Adjustment (LAMBDA) method (Teunissen, 1993).
reason would be understood from the estimated a posteriori variance factors shown in the last column, which should be close to unity according to the leastsquares estimation theory (Cross, 1983). However, if the variance factor is significantly different from unity (it may be rejected by an appropriate hypothesis test), it is suspected that outliers (gross errors) exist in the measurements or there is a problem with the fidelity of stochastic and functional model (Ibid, 1983). Since we have already assumed that the outliers are undetected, it can be concluded that the large undetected outliers in the pseudo-range measurements cause the model nonfidelity, and hence lead to unrealistic a posteriori variance estimation, that makes the F- and W-ratios extremely small. Fortunately, such a large error considered in the tests could be detected by the unit variance hypothesis test with critical value 2.7 with 99.0% confidence, although removing the outlier from the specific measurements may not be so easy. Table 2. Averaged statistic values regarding ambiguity validation with respect to undetected pseudo-range outliers (epoch by epoch solution) Success V. Rate (%) Outlier SV F W ˆs 02 (m) F W 13.0 6.8 100.0 100.0 2.675 0.5 3 5.1 3.5 100.0 100.0 7.683 1 1.84 1.8 22.0 0.0 28.145 2 12.6 8.8 100.0 100.0 1.169 0.5 4.9 5.4 100.0 100.0 2.690 1 9 1.2 0.4 38.0 38.0 7.049 2 12.3 5.7 100.0 100.0 3.403 0.5 4.9 2.9 100.0 98.0 10.095 1 17 1.8 1.1 17.0 0.0 36.994 2 12.7 7.2 100.0 100.0 2.299 0.5 4.5 3.4 100.0 99.0 6.014 1 22 1.5 0.84 0.0 0.0 20.363 2 12.0 6.0 100.0 100.0 3.201 0.5 4.6 2.8 100.0 94.0 9.088 1 23 1.7 0.9 9.0 0.0 32.522 2 13.1 8.8 100.0 100.0 1.5487 0.5 5.0 5.3 100.0 100.0 2.973 29 1 1.4 1.0 0.0 0.0 9.380 2
3.2 Impact of undetected outliers Tests were carried out with the simulated Data Set B using the epoch-by-epoch solution mode to study the influence of undetected outliers (gross errors) on the ambiguity validation test. Table 2 shows the averaged Fand W-ratios as well as a posteriori variance values 2
( ˆs 0
),
Outlier FREE
24.4
11.4
100.0
100.0
1.135
Table 3. Averaged statistics values on the ambiguity validation with respect to undetected carrier phase observation outliers (epoch by epoch solution) Success V. Rate (%) Outlier SV F W ˆs 02 (cm) F W
assuming the simulated pseudo-range outliers. In
addition, correct ambiguity validation rates were obtained. Since the best integer ambiguity combinations identified from the search process are identical to the true integer ambiguity, the rates could be easily computed by actual test results using F- and W-ratios with the true values. In the tests, the conventional critical value of 2.0 for F-ratio, and the critical value of 2.998 for the W-ratio with 99% confidence level were used. It can be seen from the results that the performance of the validation tests is getting worse when looking over the test statistics, which, as expected, become smaller in proportion to the increase of the intentionally added outliers. It is very hard to validate the best ambiguity combination with the F- and W-ratio values when an outlier of 2m was added. Actually, the test statistics show that none of the best ambiguity combinations in the worst case (SV22 and 23) can be validated. One possible
3
9
17
22
23
29
0.5 1 2 0.5 1 2 0.5 1 2 0.5 1 2 0.5 1 2 0.5 1 2
Outlier FREE
3
20.6 13.9 5.6 23.1 20.8 16.4 20.8 11.5 3.9 23.5 17.4 7.9 21.4 14.0 5.4 23.7 19.5 11.8 24.4
10.3 9.8 7.9 10.4 9.7 9.4 10.1 8.8 6.2 10.5 9.8 8.4 10.3 10.0 8.4 10.3 10.1 9.3 11.4
100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
1.003 1.031 1.181 1.100 1.124 1.183 1.068 1.100 1.114 1.043 1.123 1.148 1.074 1.094 1.185 1.025 1.121 1.193 1.135
Table 3 is a summary of the ambiguity validation test results with respect to the undetected outliers in carrier phase observations for each satellite. It can be concluded from the results that the correct ambiguity in kinematic positioning can be validated even with a couple of centimetres of outliers in one carrier phase measurement.
F
W
50
Confidence level of W
50 1
3.3 Impact of multipath To investigate how multipath errors in the carrier phase measurements affect the carrier phase integer ambiguity validation test, three tests were performed. A certain type of multipath for L1 and L2 carrier phase was generated, with an assumed standard deviation of approximately 0.013cm, by passing white noise through a first-order Butterworth low-pass filter. Similar to the tests against the undetected outliers, the impact of multipath was just considered in one satellite for each test. Table 4 shows the values of F- and W-ratios and the validation success rates under the simulated multipath environment. Note that under a multipath-free environment, the correct ambiguity rate is 100% of the time when compared with the true values. In this test, two critical values associated with the confidence levels 95% and 99% for the W-ratio were used. In addition, Figure 1 depicts F- and W-ratios as well as the confidence levels of W-ratios. These results reveal that the validation performance is dependent on the location of the multipath errors. Looking over all the results (especially the successful validation rates), when multipath was added to SV21 measurements, the validation results appears to provide the best performance, whereas the validation performances are relatively poor when the multipath was added to SV3 and SV23 measurements. The reason is that the impact of multipath on the float solutions depends on the geometry of the satellite constellation.
0
0
50
50
0
0
50
50
0
0
50
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0
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0 0
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600
0 0
0.9 1
0.5 1
0.5 1
0.5 1
0.5 1
300 Epoch(sec)
600
0.5 0
300 Epoch(sec)
600
Figure 1. F-, W-ratio, and confidence level of the W ratio with respect to the same multipath in different satellite observations
In this example SV3 and SV23 are the worst cases. These satellites actually have the highest elevations. Hence SV23 was used as a reference satellite. However, in practice it is a rare case that satellites with high elevations experience large multipath errors. Table 5. Ambiguity validation test performance under multipath environment SV Free 03 06 17 21 23 SV Free 03 06 17 21 23
Table 4. Averaged F- and W-ratio values as well as successful ambiguity validation rates against simulated multipath in carrier phase measurements (epoch by epoch solution) Successful Validation Rates (%) SV F W F W (95% / 99%) 14.6 6.9 100.0 99.7 96.0 Free 7.1 4.32 79.2 78.6 65.0 3 7.4 5.9 91.2 94.5 83.0 6 6.1 5.3 86.2 93.3 78.3 17 6.8 5.8 94.1 97.5 83.8 21 5.2 4.4 79.5 85.3 66.8 23
F
Accept / Correct W (95% / 99%)
600/600 441/541 544/597 513/596 562/597 446/569
F 0/0 25/59 0/3 0/4 0/3 0/31
598/600 437/541 564/597 556/596 582/597 482/569
576/600 342/541 495/597 466/596 500567 370/569
F
0/600 100/541 53/597 83/596 35/597 123/569
Accept / Incorrect W (95% / 99%) 0/0 24/59 0/3 0/4 0/3 1/31
Reject / Correct Ws (95% / 99%)
0/0 11/59 0/3 0/4 0/3 0/31
F 0/0 34/59 3/3 4/4 3/3 31/31
2/600 104/541 33/597 40/596 15/597 87/569
24/600 199/541 102/597 130/596 97/597 199/569
Reject / Incorrect Ws (95% / 99%) 0/0 35/59 3/3 4/4 3/2 30/31
0/0 48/59 3/3 4/4 3/8 31/31
3.4 Impact of stochastic modeling In Section 3.2 the large undetected outliers brought about the model fidelity problem, and therefore have made it difficult to validate the true ambiguity set. The first test was again performed with Data Set A. However, only measurements with the largest outlier (2m) in the pseudo-range were processed on an epochby-epoch basis, using different weightings (depending on the precision of the measurements). This was possible because the accuracy and precision of the observations is known in the generation procedure. Comparing Table 2 with the Table 6, the validation performance using the Fand W- ratios is significantly improved in terms of the ambiguity validation success rate. In addition, the estimated variances are closer to unity when compared with the results in Section 3.2, even though large pseudo-range errors were included.
Table 5 shows the validation results for the performance of the test with exposure to multipath. The rates of rejecting the correct ambiguities (Type I error) and accepting the wrong ambiguities (Type II error) can be determined. Note that two different confidence levels (95% and 99%) were used for the W-ratio test. The table indicates that multipath makes it more difficult to accept the correct ambiguity combination than to reject the incorrect. The test performance also differs depending on the satellite signal influenced by the multipath effect.
4
Table 6 F- and W-ratios and success validation rates considering realistic stochastic models (epoch by epoch solution) Success V. Rate (%) SV F W ˆs 02 F W 6.2 3.3 100.0 100.0 1.907 3 4.5 3.3 100.0 98.0 1.002 6 6.1 3.2 100.0 100.0 2.019 17 5.5 3.1 100.0 100.0 1.728 21 5.7 3.1 100.0 98.0 2.002 23 5.6 3.7 100.0 100.0 1.195 29
3.5 Satellite configurations In this simulation analysis two different types of satellite configuration will be taken into account. The first one is GPS/pseudolites and the second case is a GPS/Galileo constellation. Pseudolites (PL), also known as "pseudo-satellites", are ground-based transmitters which can be installed wherever they are needed. They therefore offer great flexibility in the augmentation of GPS applications. On the other hand, Galileo is an upcoming European GNSS, which is planned to provide users worldwide with position and velocity capability. Although designed to be a fully independent system, the system is planned to be inter-operable with other systems, such as GPS and GLONASS. The first tests were carried out to explore the impact of pseudolites signals on GPS integer ambiguity validation tests. Observations from five GPS satellites and four pseudolites in different locations were used for the tests. Figure 2 depicts the GPS satellite and pseudolite constellation. Table 9 shows the averaged Fand W- ratio values according to the pseudolite locations (4 cases – see Figure 2) and the number of pseudolites used in the static processing (using only singlefrequency measurements). Comparing GPS/pseudolite results with GPS-only, the validation test statistics were considerably increased for any location of the pseudolites. When looking over the results obtained for the different locations, CASE I location provides the best performance, whereas CASE II and III are a little worse. This is obvious considering the geometry of the GPS satellite constellation. Therefore the pseudolites’ location should be carefully selected to best augment the existing GPS constellation. In addition, another test based on an epoch-by-epoch solution using both GPS and pseudolite dual-frequency measurements was performed, with a different number of pseudolites used. Figure 3 illustrates the F- and W-ratio changes during the test. The results show that one pseudolite can make a big difference, whereas more pseudolites do not significantly increase the test statistics further. On the other hand, the benefits of using pseudolites can be maximised when only 3 or less satellites are available.
Another test was performed with Data Set C simulating multipath errors whose approximate standard deviation is between 0.5cm and 1cm, in three of the satellite measurements for which the elevation angle is relatively low. In order to analyse different scenarios, three types of models were considered namely, the ‘Preset’, ‘Unrealistic’, and ‘Realistic’ models. Both the ‘Preset’ and ‘Unrealistic’ models are based on the known precision. Note that a weight matrix is equally weighted in the ‘Preset’ model. For the ‘Realistic’ model, a realistic measurement noise covariance matrix estimation method (Wang, 1999) was utilised. Table 7 shows the test statistics, ambiguity success rates based on the a posteriori variance factor and the confidence level for the W-test. It is clear that the ‘Unrealistic’ model degrades the validation performance, whereas the ‘Realistic’ model significantly improves the performance when compared with the ‘Preset’ and ‘Unrealistic’ models. Table 7. Averaged statistics values on ambiguity validation according to different stochastic model (epoch by epoch solution) Model Preset Unrealistic Realistic
F-ratio 2.4 1.8 4.6
W-ratio 1.8 1.2 4.2
C. level of W (%) 89.5 78.7 99.3
Data Set B was processed again with the realistic stochastic model (Wang, 1999) to examine its effect on the validation test. Table 8 shows the sensitivity of the test against multipath with an appropriate stochastic model. It is of interest to compare these results with Table 5, in which the ‘Preset’ model was used. Overall the results indicate that the ‘Realistic’ stochastic model does indeed improve the performance of the validation test. Table 8. Ambiguity validation test performance under multipath environment with realistic stochastic models SV
F
Accept / Correct W (95% / 99%)
Reject / Correct Ws (95% / 99%)
F
03 06 17 21
447/554
407/554
271/554
107/554
147/554
271/553
595/600
596/600
530/600
5/600
4/600
70/600
585/600
589/600
521/600
15/600
11/600
79/600
591/600
590/600
552/600
9/600
10/600
48/600
23
541/598
564/598
463/598
57/598
34/598
135/598
SV 03 06 17 21 23
F
Accept / Incorrect W (95% / 99%)
F
Reject / Incorrect Ws (95% / 99%)
14/46
8/46
3/46
32/46
8/46
43/46
0/0
0/0
0/0
0/0
0/0
0/0
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0/0
0/0
0/0
0/0
0/2
0/2
0/2
2/2
2/2
2/2
Figure 2. Skyplot for GPS and Pseudolite constellation
5
Table 9. F- and W- ratio according to pseudolites location and the number of used pseudolites in static positioning. PL Num. F W F W location Of PLs I 30.7 94.8 1 99.6 94.8 II 24.2 72.3 2 105.6 104.4 III 23.8 71.2 3 112.4 109.1 IV 31.1 82.3 4 124.0 118.3 GPS only 9.6 35.9
6 Concluding remarks Ambiguity validation is one of the most important steps in ambiguity resolution. In this contribution, we have performed a series of simulations to investigate the sensitivity of the ambiguity validation tests with respect to not only errors remaining in the raw measurements, but different stochastic models and satellite configurations. The numerical results have shown that the validation test statistics, such as F- and W-ratios, are sensitive to the magnitude of remaining measurement errors, and even though the same size or pattern of outliers exists, its effect varies with satellite geometry. It should also be emphasised that existence of outliers in pseudo-range measurements significantly degrades the performance of the validation test. In addition, the results have shown that it was more difficult to avoid Type I errors of the test than Type II errors. It has been shown that appropriate stochastic modelling is critical, not only to improve the estimation precision, but also to enhance the performance of the ambiguity validation test. Finally, it has been demonstrated that the inclusion of pseudolites improves the validation results. However, it is crucial to choose the appropriate pseudolite locations to maximise the benefits of their usage. On the other hand, the analysis with an integrated GPS/Galileo system has shown that OTF ambiguity resolution based on single-frequency measurements could be possible, provided there is a large number of satellite measurements.
The final results shown in Figure 4 were obtained by using single-frequency GPS data only (6 satellites) and integrated GPS/Galileo (6+7 satellites) data, respectively. It is well known that ambiguity resolution with single-frequency GPS measurements is very difficult because the precision of the float solution obtained by least-squares estimation is usually poor and the ambiguities are highly correlated. Hence, dualfrequency measurements are required for ambiguity resolution in the On-The-Fly (OTF) model. As expected, F- and W-ratios in the case of using only GPS are mostly less than their critical values (2.0, and 2.99 respectively with 99% confidence levels). This means that the integer ambiguity cannot be successfully validated. However, the ratios obtained by the integrated GPS/Galileo simulation are sufficient to validate the correct ambiguity for the abovementioned critical values. F-ratio
W-ratio
100 GPS only 75 50 25 0
40 30 20 10 0
100 1 PLs 75 50 25 0
40 30 20 10 0
100 2 PLs 75 50 25 0
40 30 20 10 0
100 3 PLs 75 50 25 0
40 30 20 10 0
100 4 PLs 75 50 25 0 0
40 30 20 10 0 0
100 200 Epoch(sec)
300
Acknowledgements. The second author (HKL) is supported in his PhD research by a Scholarship funded by the Kwanjeong Educational Foundation of Korea. References
100 200 Epoch(sec)
Cross PA (1983) Advanced Least Squares Applied to Position Fixing. Working Paper No. 6, Department of Surveying, Polytechnic of East London, 205 pp. Lee HK, Wang J, Rizos C, Grejner-Brzezinska D, Toth C (2002) GPS/Pseudolite/INS: Concept and first tests. GPS Solutions 6(1-2) : 34-46. Teunissen PJG (1993) A new method for fast carrier phase ambiguity estimation. IEEE Position, Location and Navigation Sysmposium PLANS, Las Vegas, 1115 April, 562-573. Wang J, Stewart MP, Tsakiri M (1998) A discrimination test procedure for ambiguity resolution on-the-fly. J. of Geodesy 72: 644-653. Wang J, Stewart MP, Tsakiri M (1999) A comparative study of the integer ambiguity validation procedures. Proc International Symposium on GPS, Tsukuba, Ibaraki, Japan, 18-22 Oct. Wang J (1999) Modelling and Quality Control for Precise GPS and GLONASS Satellite Positioning. Ph.D thesis, School of Spatial Sciences, Curtin University of Technology, Perth, Australia, 171pp.
300
Figure 3. Impacts of pseudolites on F- and W-ratios
Figure 4. F- and W-ratio change of GPS and integrated GPS/Galileo by single frequency measurements
6
