Practical multivariate statistical multipath detection methods

Fig. 1 Simulated GPS three-frequency (red: L1, blue: L2, green: L5) multipath error in PRN02 in the LCPC dataset (relative permittivity =3.9)

GPS modernization is being undertaken. The GPS Block IIR-M satellites are already transmitting an unencrypted civil signal (L2C) on L2 frequency and Block III satellites will transmit a new civil signal (L1C) on L1. Moreover, the signal power of L2 will be increased. This will make tracking of L2 much easier and more reliable and will increase the use of L2 in high precision kinematic applications. An additional signal, the so-called L5, will be available on GPS Block IIF satellites scheduled for launch beginning in late 2009. Both the modernised L2 and the new L5 civil signals allow coherent tracking of code and phase and so avoid the losses that occur when tracking the current P(Y) code in L2. This had led to the extensive current interest, e.g. (Hatch et al., 2000) in investigating the potential of three-frequency data for a wide range of applications. On the other hand, the European GNSS, named Galileo, is being developed to provide four carrier frequencies and its Full Operational Capability (FOC) is scheduled to be in 2013. Galileo signals are expected to be available to users in four categories: Open Service (OS), Safety-of-Life (SoL) service, Commercial Service (CS), and Public Regulated Service (PRS).

This paper investigates the application of a sophisticated multiple outlier detection technique, which the author refers to as the cocktail multiple outlier detection algorithm, to multipath contaminated measurements in the three-frequency GPS and Galileo systems (only OS Galileo signals are used in this investigation, i.e., L1, E5a, and E5b) and a combined multiple-frequency GPS and Galileo system. It is tested with simulated data as real GPS L5 (only available in one Block IIR-M satellite now) and Galileo data are not yet available. The results, possibilities and weaknesses of the method are analysed. Note that the results of this investigation are not affected by the movement of the roving receiver because single-epoch data processing is used. Moreover, ambiguity resolution is not the main research interest in this paper, which assumes that ambiguities are fixed before the proposed algorithms are applied therefore no ambiguities are simulated in the carrier-phase data.

Cocktail multiple outlier detection algorithm

For high precision GPS positioning, the observation equations used in parameter estimation are usually linearized as:
E( y) = l + v = Ax (1)
E(v) = 0, l D (v) =D (l) =σ2 Cl (2)
where E( ) denotes the expectation operator, D( ) denotes the dispersion operator, l denotes the vector of double difference carrier phase observations, v denotes the vector of “true” residuals, A denotes the Jacobian matrix, x denotes the vector of unknown parameters, denotes the measurement variance factor, and denotes the cofactor matrix of observations. The unbiased estimate xˆ of x is based on double-difference single-epoch least squares solutions. Such a solution has known statistical properties and an expectation of the residuals of zero. It is given by the following formula.
xˆ = (AT WA)-1 ATWl
with
W = Cl -1     (3)
or in simplified form:
xˆ = N-1b with N = (ATWA),b = ATWl    (4)
where W, the weight matrix, is the inverse of the covariance matrix of observations. Observations contaminated by un-modelled error are generally referred to as outliers and will have the following statistical characteristics:
E=(vl) # 0 and/or D(vi )> σ2 Cl,ii    (5) Since a multipath error is an unmodelled error in the observation model and since more observations will be available in the modernised GPS and/or the new Galileo, the hypothesis that is to be tested here is that the increased redundancy might make multipath detection possible by statistical means. The author proposes to consider a phase observation affected by multipath as an outlier and seek to detect it by statistical testing. Cross (1994), based on Baarda (1968), which first introduced the classical outlier detection method widely used in geodetic network analysis, describes the test statistic ŵi for the ith correlated observation (i.e. where the weight matrix W contains off-diagonal elements) as:
(6)
where Cvˆ = W-1_ AN-1 AT    (7)
ei = [0 0 … 1 … 0 0]T    (8)
vˆ = Axˆ _ l    (9)
The test statistic is tested against a critical threshold c to test if contains a single multipath error whilst all other observations have only random and normally distributed errors
εi : H0 :li = lˉi +εi    (10)
H1: l = lˉ i +εi + Δi    (11)
where lˉ i denotes the true value of the quantities that have been observed. The Tau rejection criterion c described in (Pope, 1976) is used as the critical threshold and the observation is rejected when:

with
c = f (TN, TU,α )    (13) where NT is the number of (non-outlying) observations, NU is the degree of freedom, and α is the desired probability of Type I error. By shifting the position of the unity in the vector ei of Eq. 8, this method can be used to detect outliers in all measurements. The author named this method as MOD in this paper. Investigations described in (Lau, 2005) show that MOD is not sufficiently robust to tackle frequency-dependent multipath errors because it cannot handle the worst case scenario when multiple frequency multipath errors from a particular satellite are all (or any two of the three frequencies) in-phase. Therefore a Cocktail Multiple Outlier Detection (CMOD) algorithm is proposed in this section. The basic principle of this algorithm is based on the MOD method. In the MOD method, the test statistic of each measurement is tested against a critical threshold. However, the CMOD algorithm simultaneously tests all residuals of each satellite against a critical threshold and performs the test for all satellites in sequence. The test statistic (absolute value) of the measurements made on the three frequencies φf1 , φf2 and φf3 of a particular satellite s from a GNSS system is given by:
(14)
where Cvˆ = W-1_ AN-1 AT (7) es = [1 0 … 1 0 … 1 0 … ]T    (15)
in which the measurements of n satellites are arranged as:
[φs1 f1 … φsn f1 φs1 f2 … φsn f2 φs1 f3 … φsn f3 ]T    (16)
and vˆ = Axˆ _ l (9) Although Eq. 14 is theoretically restricted to the detection of only one outlier in the measurements, here the author tests the use of the formula to detect more than one. The author appreciates that this is theoretically incorrect, which is why the author uses the word ‘practical’ in the title of the paper. In order to tackle all possible cases when multipath errors of two or more frequencies from a satellite are in-phase as described in the previous section and shown in Fig. 1, the CMOD algorithm carries out statistical tests for residuals from all possible combinations of frequencies of a satellite. This is done by using different combinations of frequencies in the vector es(with the measurements are arranged as in Eq. 16) as: Test I: This test aims at the detection of the measurements of a satellite contaminated by significant in-phase multipath errors in all the three frequencies at the same time, as illustrated in the two black dotted lines of Fig. 1. es = [1 0 … 1 0 … 1 0 …]T (17) Test II: This test is for significant inphase multipath errors on both f1 and f2. es = [1 0 … 1 0 … 0 0 …]T (18) Test III: This test is for significant inphase multipath errors on both f1 and f3. es = [1 0 … 0 0 … 1 0 …]T (19) Test IV: This test is for significant inphase multipath errors on both f2 and f3. es= [0 0 … 1 0 … 1 0 …]T (20) Tests II to IV are intended to detect and reject serious multipath errors in any two of the three frequencies’ data of a satellite in cases where the multipath error in another frequency is insignificant. An example is shown in the orange dotted line of Fig. 1. The final test is required to detect any multipath error from just one frequency as indicated with the brown line in Fig. 1. This vector ei is the same as the one used in the MOD method: Test V: This test is for significant multipath errors on (f1, f2, or f3). es = [1 0 … 0 0 … 0 0 …]T (8) Tests I to V are performed for all combinations of all satellites at every epoch in order to detect multipath errors on all combinations of frequencies.

Description of testing

Description of simulated test datasets

The first two simulated datasets are referenced to the setup of a real multipath experiment carried out at the Laboratoire Central des Ponts et Chaussées (LCPC) near Nantes in France during May 2002 [6]. The real data was used in the validation of the multipath model used in the simulation of multipath data for the investigations of this paper, the validation results show that the model can generate very realistic phase multipath error (Lau and Cross, 2007). In the experiment, two Leica Geosystems System 530 receivers attached to lightweight AT502 antennas were used and a 5m by 2.5m steel panel

Fig. 2 Diagram showing the geometry of the LCPC test datasets

was constructed and placed about 5m to one of the receiving antennas in order to create a sufficiently large multipath signal. The antenna-reflector geometry is shown in Fig. 2. The geometry of the first two simulated datasets are the same but the reflectors with the assumed relative permittivities of 10 (e.g. flint glass) and 20 are used in multipath simulation. Therefore, the test datasets are denoted as LCPC-10 and LCPC-20 according to the relative permittivity used. The baseline length is about 86 m. The information used in multipath simulation is summarised in Table 1. The sky plot of the satellites-reflector-antenna geometry is shown in Fig. 3. Moreover, details of the satellites whose signals are contaminated by multipath are shown in Table 2.

Another virtual test site is at the IGS global tracking station LBCH in Long Beach, United States. In the LBCH test dataset, a concrete wall (reflector) is set 5m to the north of the antenna and a reference station 100m to its south. A relative permittivity of 7 (see Table 1) was used in the simulation of multipath in order to create strong multipath. This value is based on Stavrou and Saunders (2003), which found that the real part of the complex permittivity of concrete varied from 6.2 to 7 (for signals in the range of 1 to 95.9 GHz). Table 2 summarises the multipath simulation.

Also a kinematic data set, denoted as K-HK7-300, was simulated based on a 720m trajectory along a real street in Hong Kong. The roving antenna was assumed to set about 29cm (measured to L1 phase centre) above a 1m by 1m steel carry platform that travel at 1ms-1. The buildings were assumed to be made of concrete with a relative permittivity of 7. The reference station was assumed to be about 500 m to the north of the road. Table 3 summarises the simulation – note that some satellites are blocked for some of the time.

In the all of the test datasets, Leica System 530 receivers and AT502 antennas were assumed to have been used in data collection. Moreover, normal distributed random measurement noise with the standard deviation of 1 mm was generated in each

Fig. 3 Sky plot of Galileo (underlined) and GPS satellites in the LCPC datasets; the green area represents the refl ector; the grey area indicates that the data from the satellites in this area are contaminated by multipath

phase measurement. It should be pointed out that it was necessary to assume the use of a particular receiver and antenna because factors such as receiver correlator spacing and antenna gain pattern have an impact on the simulated multipath. The author does not believe the choices the author has made affect the overall conclusions of this research. Testing methodology Five scenarios with different GNSSs or combinations of frequencies have been tested and results compared with the known positions in all cases:

• Scenario 1: the current single

frequency GPS data;

• Scenario 2: the modernised

dual-frequency GPS data,

• Scenario 3: the future threefrequency

GPS data,

• Scenario 4: the future OS threefrequency

Galileo data, and

Table 1 Information for multipath simulation in the static test datasets

Table 2 Multipathing satellite information in the static test datasets

Table 3 Multipathing satellite information in the kinematic test datasets

• Scenario 5: the future OS Galileo
+ GPS multiple-frequency data.

Note that the reason for Scenario 1 is that
many surveying and geodetic applications only use L1 data in fixed solutions. Although L2 may have been used at the ambiguity fixing stage it may be too noisy to contribute usefully to the final solution.

Results of the cocktail multiple outlier detection (CMOD) algorithm for multipath error detection

Summaries of the positioning errors after rejection of measurements with detected ultipath by CMOD in the test scenarios 3 to 5 for LCPC-10, LCPC-20, LBCH-7 and K-HK7-300 datasets are shown in Tables 4, 6, 8, and 10 respectively. The percentage improvements (or deteriorations) when compared with single-frequency GPS and the same scenarios using MOD are also shown in the table. The results of scenarios 1 to 5 without using MOD and CMOD (i.e., standard least squares) are also shown in the table for comparison. Tables 5, 7 and 9 show the successful rates for detection of multipath errors in scenarios 3, 4 and 5 for LCPC-10, LCPC-20, and LBCH-7 datasets respectively. In the results of K-HK7-300 dataset, CMOD shows deteriorations in 3D position accuracy in scenarios 3 to 5 (see the right-most column in Table 10) when compared with the current singlefrequency GPS system or the multiplefrequency least squares only solutions. This is because the multipath errors on GPS L2 and L5 and/or Galileo E5a and E5b frequencies are highly correlated (their frequencies are very similar) when reflections occur at the very close carrying platform. The introduction of another highly correlated multipath error clearly further drags the estimated position away from the true position. Table 11 shows the coherence of phase multipath errors against the differential path delays and Table 12 shows the correlation functions between frequencies and differential path delays. The orange highlighted values in Table 12 show that the high correlations (correlation functions are greater than 0.95) of some frequency pairs occur when the differential path delays (also antenna-reflector distance) are very short (about less than or equal to one metre). The performance of CMOD is worse than that of MOD in this dataset. There are two potential reasons. Firstly, the relatively small multipath errors may not be detected by MOD but they may be detected by CMOD, which should have a positive impact. However, since there are many multipath contaminated measurements at each epoch in this dataset as shown in Table 3, many measurements are rejected, which then weakens the satellite geometry. Secondly, the highly correlated low-frequency multipath errors from the very close reflector (the carrying platform) may lead to false detection of measurements without multipath error. In addition to the problem of correlation of multipath errors, the number of multipath contaminated measurements in this dataset is much more than the number of good measurements. This in itself leads to difficulty in the detection of multipath errors.

Conclusion

A practical cocktail multiple outlier detection algorithm, called CMOD, is proposed to tackle the undetected outlier problem in classical multiple outlier detection method (MOD) when phase multipath errors in two or more frequencies of a satellite are in-phase. Tests with static test datasets showed that using CMOD with GPS or Galileo three-frequency data may not improve positioning accuracy when compared with MOD results. The author believes that this is due to the fact that there is insufficient redundancy to take advantage of the multiple outlier detection process used in CMOD. The result is the rejection of too many measurements/ satellites, which then weakens the satellite geometry. However, using CMOD with combined GPS and Galileo multiplefrequency data (scenario 5) shows a substantial increase in correct detection of multipath errors and significant reduction in false detection. Only one false detection occurred in only one of the static test datasets that were tested.

Table 4 RMS positioning errors in millimetres and percentage improvement of using CMOD for the test scenarios in LCPC-10 dataset

Table 5 Approximate percentages of correct and wrong detections of using CMOD for scenarios 3 to 5 in the LCPC-10 dataset

Table 6 RMS positioning errors in millimetres and percentage improvement of using CMOD for the test scenarios in LCPC-20 dataset

Table 7 Approx. percentages of correct and wrong detections of using CMOD for scenarios 3 to 5 in the LCPC-20 dataset

Table 8 RMS positioning errors in millimetres and percentage improvement of using CMOD for the test scenarios in LBCH-7

Table 9 Approx. percentages of correct and wrong detections of using CMOD for scenarios 3 to 5 in the LBCH-7 dataset

Table 10 RMS positioning errors in millimetres and percentage improvement of the test scenarios in K-HK7-300 dataset

However CMOD shows a significant deterioration when compared with MOD and the current single-frequency GPS system when dealing with our kinematic data set. The reason has been identified as being due to a very close reflector, which led to a large number of highly correlated multipath contaminated observations (almost all signals are affected by multipath). Actually the author believe that any very close reflector (less than one metre) has the potential to lessen the advantage of using multiple-frequency GNSS data, this is because the frequencies of GPS L2 and L5, and Galileo E5a and E5b are extremely close and hence the magnitude and phase of carrier-phase multipath errors are extremely close. Note that the results of this paper are not affected by the movement (static/kinematic) of the roving receiver because single-epoch data processing is used.

Table 11 Phase multipath errors of the GPS and Galileo frequencies for various differential path delay (DPD)

Table 12 Correlation functions of phase multipath errors among different GPS and Galileo frequencies in Table 11 against the differential path delay (DPD)

Finally the author remarks that, when combining GPS and Galileo, if the number of measurements contaminated by multipath is small compared to the total number of measurements at any epoch, and if there are no very close reflectors (or at least no reflectors that lead to small additional path lengths), then the author believes that the performance of CMOD will be better than that of MOD. Moreover in this case CMOD will be an extremely effective way to reduce the impact of multipath by 3-12% when comparing with MOD and by 65-73% when comparing with the standard least squares solution using the current reliable GPS L1 data. This in turns leads to the recommendation to avoid, as far as possible, locating GNSS antennas close to reflecting surfaces even for the future multiple-frequency GNSS.

References

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[2] Cross PA (1994) Advanced Least Squares Applied to Position-Fixing. Working Paper No. 6, Department of Land Surveying, University of East London. ISSN 0260-9142

[3] Baarda,W (1968) A Testing Procedure for Use in Geodetic Networks. Publication on Geodesy, New Series, 2(5), Netherlands Geodetic Commission

[4] Pope AJ (1976) The Statistics of Residuals and The Detection of Outliers. NOAA Technical Report NOS 65 NGS 1

[5] Lau, L. (2005) Phase Multipath Modelling and Mitigation in Multiple Frequency GPS and Galileo Positioning. PhD Dissertation, University of London.

[6] Bétaille D, Maenpa J, Euler HJ, Cross PA (2003) New Approach to GPS Phase Multipath Mitigation. In: Proceedings of ION National Technical Meeting 2003

[7] Lau L, Cross P (2007) Development and Testing of a New Rigorous Ray-Tracing Approach to GNSS Carrier-Phase Multipath Modelling. J. Geodesy, 81(11):713-732. DOI 10.1007/s00190-007-0139-z

[8] Stavrou S, Saunders S (2003) Review of Constitutive Parameters of Building Materials. In: Proceedings of the 12th International Conference on Antennas and Propagation, Exeter, 31st March – 3rd April 2003, pp 211-215