Functional Models of Ordinary Kriging for Real-time Kinematic Positioning

Ali M. Al-Shaery1, Samsung Lim2 and Chris Rizos3

1,2,3 School of Surveying and Spatial Information System, UNSW, Sydney, NSW 2052, Australia

1ali.al-shaery@student.unsw.edu.au, 2s.lim@unsw.edu.au, 3c.rizos@unsw.edu.au ABSTRACT This paper aims to investigate various functional models of the Ordinary Kriging (OK) technique in order to precisely estimate epoch-by-epoch atmospheric corrections for real-time kinematic (RTK) positioning. A network of Continuously Operating Reference Stations (CORS) in New South Wales (NSW), known as CORSnet-NSW, is utilised to: 1) obtain atmospheric residuals per 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 RTK users. Applying the atmospheric corrections based on the Kriging functional models, “synthetic” measurements at a given virtual reference station (VRS) are generated and used for RTK positioning. A test with a 21km baseline VRS-RTK indicates that less than 2cm of horizontal errors (1 sigma) and 7cm of vertical errors (1 sigma) are achieved. This is approximately a 33% improvement when compared with the results from commercially available software packages. This study has demonstrated the usefulness of a number of OK functional models where ionospheric and tropospheric delays significantly degrade the positioning quality. INTRODUCTION For surveyors, productivity is an issue when a GPS field survey is being carried

out. Real-Time Kinematic (RTK) positioning is a GPS technology known for its productivity. However, RTK has a problem with constraining baseline lengths to typically being below about 20km. Such a constraint makes RTK unsuitable for a uniform survey across large areas. The atmospheric delay effect on GPS signals is an important bias that needs to be considered for precise positioning, especially for longer baseline RTK. The ionosphere and troposphere are not alike as the former is a more turbulent medium, and the absolute value of the delay rapidly varies over time compared to that of troposphere. In addition, the nature of the ionosphere at equatorial or tropical regions makes it difficult to model. Although Network-RTK (N-RTK) effectively removes most of the distance-dependent errors by estimating the corrections to users within the network, the ionospheric effect is still a major challenge for users in tropical regions. The tropospheric delay effect can be reduced by using the International GNSS Service (IGS) ultra-rapid product with a few millimetres precision and 3-4 hours latency(IGS, 2009). Therefore, it is essential for high accuracy users to effectively eliminate or mitigate the ionospheric delay. There are several methods to account for the ionospheric effect on GPS signals: prediction models, dual-frequency observables, relative positioning or using multi-reference stations. Ionospheric prediction models can be physical (i.e. derived from physical principles) or

empirical (i.e. derived from observed data). Several empirical and physical models have been developed. 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, and more accurate, 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 observables made on the L1 and L2 frequencies to generate an ionosphere-free linear combination to correct for the first-order ionospheric delay. However, this observable is not recommended for short baselines as it has more noise than the individual measurements, and it does not preserve the integer nature of ambiguity parameters. For short baselines, relative positioning is the recommended technique to mitigate most of the ionospheric delay effect. Alternatively, a network of Continuously Operating Reference Stations (Network) can be used to determine the ionopsheric delays for each reference station and interpolate the delays for a rover receiver operating within the network without a strong constraint on baseline length. For productivity, efficiency and economic reasons, this method is nowadays considered to be the preferred technique for high precision applications. As for the network approach, several studies have been conducted to identify an interpolation technique that gives better results for modeling ionosphere (non-homogeneous field and multi-scale phenomena) compared to a mathematical function or GPS ionosphere-modeling algorithms using spherical harmonics

expansion (Gao and Liu, 2002;Wielgosz et al., 2003b). Several researchers have suggested that the technique based on Kriging (mainly used in geostatistics) is a better choice among interpolation methods to model the ionospheric effect. This is because of its ability to take into account spatial and temporal variability of the interpolated attributes (i.e. double-differenced ionosphere and troposphere corrections) (Blanch, 2002;Wackernagel, 2003;Wielgosz et al., 2003b). Several investigations of interpolation methods have been reported for regional atmospheric error models from Australia (Dai et al., 2003;Ouyang et al., 2007;Ouyang et al., 2008;Wu et al., 2008). However, no such evaluation for Kriging 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, CORS network 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.

METHODOLOGY The objective of this paper is to assess the improvement in kinematic GPS positioning from using precise CORS network atmospheric corrections interpolated using the ordinary Kriging algorithm. Several modules were developed to address this objective (see Figure 1).

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validation Cross-validation is carried out in order to select the theoretical variogram model better fitting the experimental one. 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. Removed base station original 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 average squared normalised residuals (ASNR):

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The model having the closest value of ME to zero, and VE and ASNR closest to 1 is the one best fitting the experimental variogram (Zhang, 2003). An additional method to examine the appropriateness of theoretical variogram models to fit the experimental variogram is to compare the kriged 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 real reference station as shown in Figure 2). The VRS observables are constructed as follows(Hofmann-Wellenhof et al., 2008):

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, are the carrier phase observable and code observable, respectively, of the VRS based on the code-position of a 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 (r) 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: Δ , ΔT , ΔI , (18) For the code observable, the correction term is: Δ , ΔT , ΔI , (19) It should be mentioned 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 Leica Geo Office (LGO) software was used for this task, processing the generated VRS RINEX file and the rover RINEX file in a kinematic baseline mode. The output of this module is a precise estimate of the rover position with cm-level accuracy. EXPERIMENT Data from five 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 between-receiver differences, as seen in Figure 2.

Figure 2: Sydney Basin portion of the NSW-CORS network 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 was used as the reference station for the calculations. Atmospheric residuals (ionopsheric and tropospheric) between each station and the reference station were calculated for each satellite. The sample interval was 1 second. Five single-differenced estimated residuals (QUEN-CWAN, QUEN-MGRV, QUEN-SPWD, QUEN-MENA and QUEN-WFAL) were used in the Kriging Module to estimate the rover’s (QUEN-VLWD) 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 DD baseline solution was processed using the LGO software. As the coordinates of VLWD are already precisely known, the performance of the scheme can be directly assessed and validated. RESULTS AND ANALYSIS The network correction module produces ionospheric and tropospheric residuals, and more details on this can be obtained from (Zhang et al., 2009).

Kriging Module Before applying the Kriging algorithm, it is necessary to select the appropriate theoretical model. This module utilises three theoretical models: Spherical, Exponential and Gaussian. Two approaches can be followed to achieve a better selection. In the first approach, kriged atmospheric corrections based on each model can be compared to the estimated corrections from the network correction module. QUEN-VLWD (or VILL as it appears on the map in Figure 2) single-differenced residuals were estimated and used as a reference for the comparison. The second approach uses a cross-validation technique to aid better selection. comparison between network estimated residuals and network krigged residuals This approach is not only used to aid cross-validation but also used to generally validate the Kriging algorithm. It was found that the Spherical model was inappropriate 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 (see Table 1 and Figures 7 and 8). The Exponential model (-0.9mm in the mean and 0.01m in the 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 0.0097m, respectively). In the case of the tropospheric residuals, the Gaussian model performs slightly better than the Exponential, with about the

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same level of difference shown in the case of ionospheric residuals. The Guassian model shows difference mean of -3.5mm to the reference value whereas the exponential has slightly larger difference mean (-3.9mm). This trend is also evident in the standard deviation values (0.005m for Gaussian and 0.0046m for Exponential). 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

Figure 4: Estimated ionospheric residuals (RIon-QV) vs Gaussian Krigged residuals (Ion-Gau)for satellite 25

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 difference 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 7: Statistics of the difference between the estimated and Krigged Ionospheric residuals for both models

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Figure 8: Statistics of the difference between the estimated and Krigged Tropospheric residuals for both models cross-validation results In practice, cross-validation is better than the previous approach to aid 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 ionopsheric 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 noticed 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.

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 11: Average squared of normalised residuals of the Gaussian ionospheric model vs the Exponential model

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Figure 12: Mean error of normalised residuals of the Gaussian tropospheric model vs the Exponential model

Figure 13: Variance error of normalised residuals of the Gaussian tropospheric 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 code-position of the rover receiver (Hofmann-Wellenhof et al., 2008) and the broadcast ephemeris file. The short baseline was processed in kinematic mode using the LGO software. From Figures 15-18 and Table 2, it can be seen that a high accuracy was achieved – less than 2cm horizontal accuracy and 7cm vertical accuracy over 20.7km baseline. Compared to the quoted performance of the VRS concept – 5cm over 35km (Hofmann-Wellenhof et al., 2008), 33% of accuracy improvement has been accomplished. In comparison to the raw baseline (CHIP-VLWD) (see Figures 19-21 and Table 3), statistics show the significance improvement achieved in the horizontal (about  98%) and vertical (about 44%) accuracy when exponentially krigged atmospheric corrections are used. Moreover, the variation in the horizontal coordinates is within a range of less than 2cm (see Figure 15), and about 7cm (see Figure 18) in the vertical component.

Figure 15: Variation in the Easting and Northing coordinate components

Figure 16: Variation in the Easting coordinate component over time

Figure 17: Variation in the Northing coordinate component over time

Figure 18: Variation in the height component over time Table 2: VRS-vlwd Short baseline statistics

E (m) N(m) h(m) rmse 0.014 0.014 0.069 stdev 0.013 0.012 0.037 mean -0.007 -0.006 -0.059

Figure 19: Easting Comparison from corrected and raw baseline

Figure 20: Northing comparison from corrected and raw baseline

Figure 21: Height comparison frm corrected and raw baseline Table 3: chip-vlwd raw baseline (20.69km) statistics

E (m) N(m) h(m) rmse 0.281 0.831 0.123 stdev 0.008 0.011 0.019 mean -0.281 -0.831 -0.122

CONCLUDING REMARKS AND FUTURE WORK 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 is tested and 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 increase the scalability of this optimisation technique. TAKE HOME MESSAGE Network-RTK techniques have been used to improve the GPS surveying productivity by overcoming the limitation of the traditional single-baseline RTK. Network-RTK techniques require an interpolation method in order to provide a user with precise corresponding network corrections. Ordinary Kriging was chosen as an interpolator in this study, and was tested with the GPS data collected from areas where the ionosphere and the troposphere cause significant measurement biases. Three functional models of Ordinary Kriging were investigated: the spherical, the exponential and the Gaussian models. The spherical model was proven to be inappropriate whilst the exponential and the Gaussian models provided promising results. The exponential model performed slightly better than the Gaussian model if the ionosphere effect only was taken into account, however, the Gaussian model performed slightly better than the exponential model if the troposphere only effect was considered.

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 BLANCH, J. 2002 ,an Ionosphere

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