Functional Models of Ordinary Kriging for Medium Range Real-time Kinematic Positioning Based on the Virtual Reference Station Technique Al-Shaery, A., Lim, S., Rizos, C., School of Surveying and Spatial Information System, the University of New ABSTRACT This investigates various functional models of the Ordinary Kriging technique, applied to precisely estimate epoch-by-epoch atmospheric corrections for real-time kinematic positioning. A network of Continuously Operating Reference Stations in New South Wales, known as CORSnet-NSW, is utilised to: 1) obtain atmospheric residuals for each reference station, 2) construct an empirical variogram over the network, 3) determine Kriging parameters for three different models: a spherical-, an exponential-, and a gaussian model; and 4) optimise the atmospheric corrections for real-time kinematic positioning. Applying the atmospheric corrections obtained by the Ordinary Kriging functional models, synthetic measurements at a virtual reference station are generated and used for real-time kinematic positioning. Field tests with baselines ranging from 21km to 62km indicate that 1.9cm up to 4.4cm of horizontal accuracy (1 sigma) is achieved. Results show approximately 53% or higher improvement when compared with the results from commercially available software packages. This study has demonstrated the usefulness of a number of Ordinary Kriging functional models where ionospheric and tropospheric delays significantly degrade the positioning quality. INTRODUCTION Real-Time Kinematic (RTK) positioning is an enabling technology that can increase South Wales, Sydney, NSW, Australia productivity of a field GNSS survey. However, RTK can only be reliably used for interreceiver baseline lengths of up to a few tens of kilometres. Such a constraint makes RTK unsuitable for supporting uniform surveys across large areas. The atmospheric effect on GNSS signals is an important bias that needs to be considered for precise positioning, especially for longer baseline length RTK. The ionosphere and troposphere are not alike as the former is a more turbulent medium. In addition, the nature of the effect at the equatorial or tropical regions makes it difficult to model. The troposphere in equatorial and tropical regions is also problematic and should not be ignored when seeking to reliably resolve ambiguities. Although network-based RTK (NRTK) significantly mitigates the distance-dependent errors by estimating the corrections to users within the network, the atmospheric error is still the major challenge for users in equatorial and tropical regions. Ionospheric prediction models can be physical (i.e. derived from physical principles) or empirical (i.e. derived from measured data). Several empirical and physical models have been developed in order to estimate the ionospheric effects on GNSS observations. Single-frequency receiver users are the main beneficiaries of these models as they are unable to exploit the dispersive nature of the ionosphere when a single-frequency receiver is used for positioning. The most commonly used empirical model is the Klobuchar model whose coefficients are transmitted in the navigation message. However, several studies claim that this model only removes approximately 50-60% of the total effect (Camargo et al., 2000;Klobuchar, 1996;Nafisi and Beranvand, 2005).
Users of dual-frequency receivers can combine observations on the L1 and L2 frequencies to generatee an ionosphere-free linear combination to correct for the first-order ionospheric delay. However this observable is not recommended for short baseline RTK as it is noisier than the L1 or L2 observation, and it does not preserve the integer nature of ambiguity parameters. Alternatively, a network of Continuously Operating Reference Stations (Network) can be used to determine the ionospheric delays for each reference station, and allow for the interpolation of the delays for a rover receiver located within the network. For productivity, efficiency and economic reasons, this method is considered to be the best implementation for high precision positioning applications. In the case of the network approach, several studies have been conducted to identify an interpolation technique that provides better results for the ionosphere modelling (non- homogeneous field and multi-scale phenomena) compared to a mathematical functionn or GNSS ionosphere-modelling algorithms using spherical harmonics expansion. Several researchers have indicated that Kriging, mainly used in geostatistics, is a better choice among interpolation methods to model the ionospheric effect obtained from the network because of its capability to take into account spatial and temporal variability of the double-differenced troposphere ionosphere corrections and (Blanch, 2002;Wackernagel, 2003b). 2003;Wielgosz et al., Regarding the tropospheric effect, the delay can be reduced using a priori models such as the Saastamoinen model. However, test results in an equatorial region shows that about ±20cm of double-differencedd tropospheric residualss occur even for short baselines (Musa et al., 2006). Therefore, it is necessary to consider an alternative approach. A network of GNSS reference stations can be used to estimatee tropospheric residuals using the ionosphere-free linear combination. Then such residualss can be precisely interpolated using Ordinary Kriging (OK) to a user receiver within the network. Several performance investigations of interpolation methods have been reported for regional atmospheric error models from Australia where tropical regions and mid-toet al., high latitude areas coexist (Dai 2003;Ouyang et al., 2007;Ouyang et al., 2008;Wu et al., 2008). However no such evaluation of the Kriging interpolationn method in Australia has been reported to date. This paper investigates the performance of Kriging for modelling the atmospheric delay effects (ionosphere and troposphere) from a CORS network in Australia. Firstly, reference stations are used to estimate atmospheric residuals. Secondly, the Kriging method is used to interpolate atmospheric corrections for the location of a user receiver. Thirdly, a virtual reference station (VRS) is established using the interpolated correction. Fourthly, relative positioning is carried out between the VRS and the user receiver. Finally, an accuracy assessment is performed to evaluate the Kriging performance. METHODOL LOGY The main objective of this paper is to assess the improvement in kinematic GPS positioning by using OK functional models. Several modules were developed to achieve this objective (see Figure 1). CORS Netw ork Module DD Module DD Atmsopheric Residuals Short baseli ne DD solution Figure 1: Methodology flowchart CORS-Netwo ork Module Rover DD Atmospheric Corrections VRS generation KRIGING Module VRS Module This module estimates atmospheric residuals at each GPS reference station. Using a network of reference stations with precisely known positions, atmospheric residuals to every observed satellite can be obtained from these network stations. Determining network corrections is carried out using in-house
software developed by the School of Surveying and Spatial Information Systems (SSIS) at the University of New South Wales (UNSW). Six single-difference (between-receivers) ionospheric and tropospheric residuals (QUEN-CWAN, QUEN-MGRV, QUEN- SPWD, QUEN-MENA, QUEN-WFAL, and QUEN-VLWD) are obtained from this module. Such residuals are later used to form the VRS observables in the VRS module. More details concerning this module are given in (Zhang et al., 2009). Kriging Module Ordinary Kriging is used to estimate a value at a location using a neighbouring sample of data whose semi-variogram is known (Wackernagel, 2003). The semi-variogram provides information for interpolation sampling optimisation and for determining spatial patterns. Compared to other interpolation methods such as the inverse distance method, Kriging is a geostatistical method that takes into account the spatial and temporal correlation of data sources using the semi-variogram of the sample data (Wielgosz et al., 2003a). Furthermore, unlike mathematical interpolation methods, this geostatistical method provides an indication of the error in the form of a variance (Wackernagel, 2003). Kriging variance is given by: (1),, 1 where is the Kriging weight parameter for each station (sample point) involved in the Kriging interpolation and is the Lagrange multiplier (LM). The Kriging method can be carried out in three steps: 1. Constructing the experimental variogram, 2. Fitting the experimental variogram to the appropriate theoretical model, and 3. Determining the weight parameters for each reference station. In the second step, the optimum parameters that fit the experimental variogram to an appropriate theoretical variogram model (such as Spherical, Exponential and Gaussian) are determined. The Newton-Raphson method is used to determine the optimum parameters that fit the experimental variogram to a theoretical model by carrying out a least squares solution (Burden and Faires, 2005). A validation technique known as cross-validation is implemented to assist the selection of the appropriate theoretical model. Based on this step, the appropriate model is used in the next step to determine the Kriging weight parameters ( ) for each base station and the LM ( ). In this paper, three functional models were tested: 1) a spherical model, 2) an exponential model, and 3) a gaussian model. Validation Cross-validation is carried out in order to select the theoretical variogram model. Various assumptions about the selected model and the input data are examined to check the size of the data and the presence of any outliers (Wackernagel, 2003). The validation procedure is as follows. Each base station candidate is removed from the data set and treated as a rover. Kriging weight parameters are re-calculated using the n-1 data sample. The removed base station attribute value is ignored and is estimated using the computed weights. There are three statistical parameters which can be used to test the appropriateness of a theoretical model: the mean of normalised residuals (ME), the variance of normalised residuals (VE), and the average squared normalised residuals (ASNR): (2) where: (3) (4) (5) The model having the closest value of ME to zero, and VE and ASNR closest to 1 is the one that is assumed to be best fitting of the experimental variogram (Zhang, 2003).
An additional method for examining the appropriateness of theoretical variogram models is to compare the krigged atmospheric residuals to the estimated residuals from the network correction module (assuming that it has the appropriate level of accuracy). VRS Module This module generates VRS observables from real reference station data (here the QUEN CORS station is the reference station used, as shown in Figure 2) and a rover code position. The VRS observables are constructed as follows (Hofmann-Wellenhof et al., 2008):,,,,, (6),,,,, (7) is the carrier phase observable on the L1 or L2 carrier frequency., and, are the carrier phase observable and code observable, respectively, of the VRS based on the code position of the rover receiver., and, are carrier phase observable and code observable of the real reference station (e.g. QUEN station), respectively., is the geometric distance between satellite (s) and receiver located at the VRS, and, represents the geometric distance from a satellite to the real reference station receiver. Δ, is the rover atmospheric (ionospheric and tropospheric) correction obtained from the Kriging Module. For the carrier phase observable, the correction term is:,,, (8) For the code observable, the correction term is:,,, (9) It should be noted that the broadcast ephemeris is used to assess the suitability of the algorithm for RTK applications. Double-Differencing Module In this module a baseline solution is carried out between a rover receiver and the VRS over a short baseline. The LGO software was used for this task, processing the generated VRS RINEX file and the rover RINEX file in kinematic baseline mode. The output of this module is an estimate of the rover position with cm-level accuracy. EXPERIMENT Data from seven stations of the NSW-CORS network located in the Sydney region, Australia, were used to generate network corrections. The stations have a regular distribution with inter-station distance ranging from 20.7km to 62.5km. The in-house network algorithm (Zhang et al., 2009) developed at the SSIS-UNSW was used to generate the network corrections in the form of betweenreceiver differences. 62.5km 49.86km 44.83km 20.69km 33.43km 32.72km Figure 2: Sydney Basin portion of the CORSnet-NSW The study data set used to determine the network corrections is 24 hours in length, from 10 February 2009. A 3hr data portion was used to implement the proposed algorithm. Station QUEN (currently CHIP) was used as the reference station for the calculations. Atmospheric residuals (ionospheric and tropospheric) between each station and the reference station were calculated for each satellite. The sample interval was 1 second. Three tests were carried out using the six single-differenced estimated residuals (QUEN- CWAN, QUEN-MGRV, QUEN-SPWD, QUEN-MENA QUEN-WFAL and QUEN- VLWD). The tests represent different baseline lengths (20.69km, 44.83km and 62.5km) in order to test the proposed algorithm over various baseline lengths. In the first test, VLWD was used as a rover, MGRV in the second, and in the third was SPWD. In each
test, five single-differenced estimated atmospheric residuals, excluding the rover related set, were used in the Kriging Module to estimate the rover s (QUEN-VLWD in the first test, QUEN-MGRV in the second and in the third QUEN-SPWD) corrections. Then, the VRS generation module established a VRS close to the rover station based on the code position and the real reference station data (QUEN). The short double-differenced baseline solution was obtained using the LGO software. As the coordinates of VLWD, MGRV and SPWD are already precisely known, the performance of the algorithm can be directly assessed. RESULTS AND ANALYSIS Before applying the Kriging algorithm it is necessary to select the appropriate theoretical model. Three theoretical models can be used: Spherical, Exponential and Gaussian. Two approaches can be followed to achieve a better selection. In the first approach, krigged atmospheric corrections based on each model can be compared to the estimated corrections from the network correction module. For instance, QUEN-VLWD (or VILL as it appears on the map in Figure 2) singledifferenced residuals were estimated in the KR module and compared to the corrections computed in the CORS-Network module, which is assumed to have enough accuracy that it may be used as a reference for the comparison. The second approach uses a cross-validation technique. Comparison between network estimated residuals and network krigged residuals (see Table 1 and Figures 7 and 8). The Exponential model (-0.9mm mean and 10mm standard deviation) has values slightly closer to the reference values in the case of ionospheric residuals compared to the Gaussian values (-1.4mm and about 9.7mm). In the case of the tropospheric residuals, the Gaussian model performs slightly better than the Exponential model, with about the same level of difference in the case of ionospheric residuals. The Guassian model shows difference mean of -3.5mm to the reference value whereas the Exponential model has a slightly larger difference mean (-3.9mm). This trend is also evident in the standard deviation values (5mm for Gaussian and 4.6mm for Exponential). The general conclusion drawn from Figures 7 and 8 is that the ionospheric residuals are well represented by the Exponential model, whereas the Gaussian model better fits the tropospheric residual variogram. Figure 3: Estimated ionospheric residuals (Rion-QV) vs Exponentially Krigged residuals (Ion-Exp) for satellite 25 This approach is not only used to aid crossvalidation but also used to generally validate the Kriging algorithm. It was found that the Spherical model was inappropriate and was therefore not used in the rest of the analysis. No significant differences between the other two models were obtained, as can be noted from the plots of satellite 25 in Figures 3-6. Statistical factors (mean and standard deviation) of the differences between the network estimated residuals and those krigged using both models for both residual types (ionospheric and troposheric) were compared Figure 4: Estimated ionospheric residuals (RIon-QV) vs Gaussian Krigged residuals (Ion-Gau)for satellite 25
Trop_Exp Trop_Gau 0.006 0.004 0.002 0 0.002 0.004 0.006 Mean(m) Stdv(m) Figure 5: Estimated tropospheric residuals (RTro-QV) vs Exponentially Krigged residuals (Tro-Exp) for satellite 25 Figure 6: Estimated tropospheric residuals (RTro-QV) vs Gaussian Krigged residuals (Tro-Gau) for satellite 25 Table 1: Means and standard deviations of the differences between the estimated and Krigged Ionospheric and Tropospheric residuals for both models Mean (m) Stdv (m) Ion_Exp -0.0009 0.01 Ion_Gau -0.0014 0.0097 Trop_Exp -0.0039 0.0046 Trop_Gau -0.0035 0.0050 Figure 8: Statistics of the difference between the estimated and Krigged Tropospheric residuals for both models Cross-validation results In practice, cross-validation is preferred for algorithm validation. The values of ME, VE and ASNR for both models (Exponential and Gaussian) and both residual types (ionosphere and troposphere) are plotted for 100 epochs in Figures 9-14. The preferred value of ME is zero, and for VE and ASNR it is 1. For the ionospheric residuals the average value of the ME of the Exponential model is slightly closer to one than that of the Gaussian model. For the tropospheric residuals no difference can be discerned between the models. For the VE and ASNR, the Gaussian model in the case of the ionospheric residuals shows only an insignificant improvement in results compared to the Exponential model. However, similar results hold for the tropospheric residuals. From both approaches, both models can satisfactorily fit the experimental variograms for both the ionospheric and tropospheric residuals. Hence in this paper the Exponential model is used for the next step. Ion_Exp Ion_Gau 0.012 0.01 0.008 0.006 0.004 0.002 0 0.002 Mean (m) Stdv(m) Figure 7: Statistics of the difference between the estimated and Krigged Ionospheric residuals for both models Figure 9: Mean error of normalised residuals of the Gaussian ionospheric model vs the Exponential model
Figure 10: Variance error of normalised residuals of the Gaussian ionospheric model vs the Exponential model Figure 13: Variance error of normalised residuals of the Gaussian tropospheric model vs the Exponential model Figure 11: Average squared of normalised residuals of the Gaussian ionospheric model vs the Exponential model Figure 14: Average squared of normalised residuals of the Gaussian tropospheric model vs the Exponential model Double-Differenced Short Baseline Module Based on the exponentially krigged ionospheric and tropospheric corrections, a VRS RINEX file was constructed from the real reference station (QUEN) observables file, and the code-derived position of the rover receiver (Hofmann-Wellenhof et al., 2008). The short baseline was processed in kinematic mode using the LGO software. Figure 12: Mean error of normalised residuals of the Gaussian tropospheric model vs the Exponential model From Figures 15-17 and Tables 2 and 3, it can be seen that a high accuracy was achieved less than 2cm, 2.4cm and 4.4cm horizontal accuracy over 20.7km, 44.8km and 62.5km length baseline, respectively. Compared to the claimed performance of the VRS concept, i.e. 5cm horizontal over 35km (Hofmann- Wellenhof et al., 2008), these test results indicate that approximately 53%, 171% and
101% of accuracy improvement has been achieved. As expected, the vertical accuracy is lower than that of the horizontal accuracy, but it follows the same trend of declining accuracy with increasing baseline length. Over a 20.7km baseline, 8.4cm vertical accuracy was achieved. Over the double baseline length (44.8km), the accuracy fell to more than double the value (18.1cm). Similarly, for three times the first baseline length (62.5km), the height accuracy (32.7cm) was three times worse than that of the first baseline. Figure 17: Variation in the Height component over time Table 2: VRS-rover horizontal (Hz), Short baseline statistics station rms (m) stdv (m) mean (m) VLWD 0.019 0.018 0.007 (20.69km) MGRV 0.024 0.023 0.008 (44.83km) SPWD (62.50km) 0.044 0.038 0.023 Figure 15: Variation in the Easting coordinate component over time Table 3: VRS-rover vertical (Hofmann-Wellenhof et al.), Short baseline statistics station rms (m) stdv (m) mean (m) VLWD 0.084 0.042-0.073 (20.69km) MGRV 0.181 0.045-0.176 (44.83km) SPWD (62.50km) 0.327 0.092 0.313 0.050 0.040 0.030 Hz_rms (m) Hz_stdv (m) Hz_mean (m) Figure 16: Variation in the Northing coordinate component over time 0.020 0.010 0.000 VLWD MGRV SPWD Figure 18: VRS-rover Horizontal statistics results
0.400 0.300 0.200 0.100 0.000 0.100 0.200 0.300 ht_rms (m) ht_stdv (m) ht_mean (m) Figure 19: VRS-rover Height statistics results CONCLUDING REMARKS This study has demonstrated the capability of Ordinary Kriging to improve real-time kinematic GPS positioning when a CORS network is available, especially in the areas where the ionosphere and troposphere cause significant measurement biases. It was demonstrated that Ordinary Kriging has successfully aided the VRS technique to achieve a high accuracy, even with the broadcast ephemeris. However, it should be noted that further investigations with a large number of reference stations (e.g. 50+ stations) are required to justify this claim. Moreover, further rigorous tests should be carried out using longer baselines between the rover and the reference stations in order to investigate the scalability of this optimisation technique. ACKNOWLEDGMENT The first author would like to thank Mr Shaocheng Zhang for the use of his data. He is also grateful to the scholarship provider, the Saudi Higher Education Ministry, and especially the University of Umm Al-Qura. REFERENCES VLWD MGRV SPWD BLANCH, J. (2002) an Ionosphere Estimation Algorithm for WAAS based on Kriging. 15th International Technical Meeting of the Satellite Division of the U.S. Institution of Navigation, Portland OR,24-27 September, 816-823. BURDEN, R.R. and J.D. FAIRES (2005) Numerical Analysis, Belmont: Thomson Books. CAMARGO, P.D.O., J.F.G. MONICO and L.D.D. FERREIRA (2000) Application of Ionospheric Correction in the Equatorial Region for L1 GPS users. Earth Planets and Space, 52(11), 1083-1089. DAI, L., S. HAN, J. WANG and C. RIZOS (2003) Comparison of Interpolation Algorithms in Network-based GPS Techniques. The Journal of Navigation, 50(4), 277-293. HOFMANN-WELLENHOF, B., H. LITCHTENEGGER and E. WASLE (2008) GNSS - Global Navigation Satellite Systems: GPS, GLONASS, Galileo and more, Vienna: Springer- Verlag. KLOBUCHAR, J.A. (1996) Ionospheric Effects on GPS. IN AXELRAD, P., et al. (Eds.) Global Positioning System: Theory and Applications Washington: American Institute of Aeronautics and Astronautics. MUSA, T., S. LIM, T. YAN and C. RIZOS (2006) Mitigation of Distance- Dependent Errors fro GPS Network Positioning. IGNSS Symposium, Gold Coast,17-12 July, CD-ROM procs. NAFISI, V. and S. BERANVAND (2005) Estimation of Total Electron Content Using Single Frequency Weighted Observations for Esfahan Province. The Journal of Surveying Engineering, 131(2), 60-66. NETWORK, S.A.C. (2010), Network RTK. Accessed 22/02/2010, <http://www.cors.com.au/technicalinfo/network-rtk>. OUYANG, G., J. WANG and J.L. WANG (2007) Generating a 3D TEC Model for Australian with Combined LEO Satellite and Ground Base GPS Data. 20th International Technical Meeting of the Satellite Division of the U.S. Institution of Navigation, Fort Worth, TX,25-28 September, 2285-2290. OUYANG, G., J. WANG, J.L. WANG and D. COLE (2008) Analysis on Temporal- Spatial Variations of Australian TEC. IN DEHANT, V., et al. (Eds.) International Association of Geodesy
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