rover processing with network rtk and quality...

Presented at ION NTM January 18-20, 2006, Monterey, CA

Rover Processing with Network RTK and Quality Indicators

P. Alves, H. Kotthoff, I. Geisler, O. Zelzer, and H.-J. Euler

Leica Geosystems AG Heerbrugg, Switzerland

BIOGRAPHIES Paul Alves graduated in 2005 with a Ph.D. in Geomatics Engineering form the University of Calgary, Canada. His research interests include network RTK, carrier phase processing, and ambiguity resolution. He is currently working as a software engineer for Leica Geosystems. Holger Kotthoff graduated in geodesy at the Technical University of Bonn, Germany, in 1998. In July of 2000, he joined Leica Geosystems in Switzerland as part of the data processing group, further developing the real-time and post-processing kernel used in various products. Ines Geisler graduated in 2005 with a Diploma in Geodesy from the Dresden University of Technology, Germany. Her research interests are in the field of Network RTK positioning. Ines is currently working as an Application Engineer in the Network Reference Stations team of Leica Geosystems in Switzerland. Oliver Zelzer graduated in Physics at the Technical University of Vienna in 1996 and worked as technical Software engineer in a variety of industries. He joined Leica Geosystems in 2002 and mainly worked on GPS network processing and rover real time GPS data processing. Hans-Juergen Euler studied Surveying at the Technical University of Darmstadt. He graduated in 1990 with a PhD on Fast GPS Integer Resolution in Small-Scale Networks. His research interests are in the field of Network RTK and application of the information in GNSS rovers. He serves in the standardization process of RTCM SC104 as chairman for Network RTK and Galileo. ABSTRACT The multiple reference station approach is widely known as a method for combining the data from a regional reference station network to provide precise measurement

correction to users. This is performed by measuring the regional errors at the reference station locations and interpolating them for the location of the rover. The quality of those corrections is dependent on the reference station spacing, the location of the rover, and the characteristics of the measurement errors. This paper introduces a network RTK quality indicator based on the characteristics of the measurement errors. The indicator assumes that the more linear the regional correlated errors, the better the interpolation methods will perform. The linearity of the network measurement errors is measured and weighted based on the distance to the rover. The quality indicator shown, successfully models the network RTK performance over time in the observation domain. There is an insignificant improvement in the position domain when fixed ambiguities are used however there is a small yet notable improvement in the percentage of fixed ambiguity solutions. INTRODUCTION The quality of Network RTK corrections is a function of the following factors: network geometry, measurement errors, elimination of nuisance parameters (i.e. ambiguities), and the interpolation model that is used. All of these aspects are intermixed, for example, the interpolation model that is used should have the same spatial shape as the measurement errors. Alternatively, if the measurement errors are low then the reference stations can be located further away than if the measurement errors are high. The factors that affect Network RTK performance are generally focused around the qualities and characteristics of the measurement errors. An accurate understanding of the measurement errors leads to an optimal interpolation model and network geometry for a given level of desired rover performance. The following literature focuses on measuring and quantifying the properties of the measurement errors (Alves, 2004; Wanninger, 2004; Wübbena, 2004; Chen, 2003; Raquet, 1998). In some of the more advanced cases the measurement error


properties are extrapolated using network geometry to predict the performance for the rover in addition to the performance of the network. Many quality indicators for network RTK use the residual errors measured by the network reference stations to derive the current error conditions and characteristics (Chen, 2003; Wanninger, 2004). These characteristics are compared against the current interpolation model to determine the model residuals. For example, if the measured network residuals are linear and a linear interpolation model is used then there is a high likelihood that the rover will experience a high level of performance. The model residuals can then be used to predict the model inaccuracies as a function of the distance to the nearby reference stations. For example, if the model residuals are high but the rover is at a reference station then the model errors have no effect. The ability to determine the model residuals is a function of the degrees of freedom of the interpolation model. If there are no degrees of freedom then no residuals can be determined. In this case degrees of freedom can be created by excluding one of the reference stations from the model calculation. NETWORK RTK OVERVIEW Network-Based RTK methods use a network of reference stations to measure the correlated error over a region and to predict their effects spatially and temporally within the network. Although the name suggests that these methods are real-time specific, they can also be used in post-mission analysis. This process can reduce the effects of the correlated errors much better than the single reference station approach, thus allowing for reference stations to be spaced much further apart thereby covering a larger service area than the traditional approach, while still maintaining the same level of rover performance. Network RTK is comprised of six main processes:

1. Processing of the reference station data to resolve the network ambiguities,

2. Selection of the reference stations that will contribute to the corrections for the rover,

3. Generation of the network corrections, 4. Interpolation of the corrections for the rover’s

location, 5. Formatting, and transmission of the corrections,

and 6. Computation of the rover position.

The main task of the network computation is to resolve the ambiguities for all stations in the network to a common ambiguity level, such that the bias caused by the ambiguities is cancelled when double differences are

formed (Euler et al., 2001). The network correction computation uses the ambiguity leveled phase observations from the network reference stations to precisely estimate the differential correlated errors for the region. A subset of stations from the reference network, known as a cell, is selected to generate the correction for the rover based on the rover’s position. One station in the cell, usually the one closest to the rover, is selected as the master station. The correction interpolation process models the network corrections to determine the effects of the correlated errors at the rover’s position. Depending on the correction concept (master-auxiliary, VRS or FKP), the interpolation may be done either by the reference station software or the rover itself. The corrections are formatted in such a way that the rover or standard RTK software can interpret them. Note that until the release of the RTCM 3.0 network messages there has been no official standard for the transmission of network corrections. The interpolated corrections are then applied to the observations of the master station, which are then used by the rover or RTK software to compute its position. METHODOLOGY The proposed method is a geometric approach to determining the linearity of the measurement errors. In single point positioning (no differencing) the absolute magnitudes of all measurement errors affect positioning and navigation performance. When differential methods are used, the difference in measurement errors and not the absolute values of the measurement errors dictate the level of performance. This brought forth the somewhat standard measure of parts per million of the baseline length for describing the magnitude of data set measurement errors. In network RTK the linear trend of the errors is modeled and removed. Consequently, a large parts per million of the baseline length or error gradient value does not dictate the level of performance in network RTK if this gradient is consistent throughout the network. Figure 1 shows an example of this. The red line is the absolute measurement error, which has a high parts per million value (indicated by its slope). The green line indicates the single reference station differential measurement error using the green station as the reference and the blue line shows the differential measurement error using the blue station as the reference. Assuming linear measurement errors and a linear interpolation model for the corrections, the rover would have no measurement errors in network RTK mode. This is because the slope of the measurement errors is modeled in the network RTK interpolation model. Chen (2003) also briefly discusses


this effect when discussing the ionosphere index I95 (Wanninger, 2004).

Abs

olut

e er

ror

Reference stations

Rover

Location Figure 1: Example of the difference between single reference station (shown in blue and green) and multiple reference station measurement error characteristics. The absolute measurement error is shown in red. The method described in this paper uses the change in the slope of the measurement errors as an indicator of the network RTK performance. The linearity of the model is realized in two dimensions by triangulating the network and visualizing the network as a two dimensional surface or triangular facets. If the regional errors are linear then this surface will be flat (although it may not be horizontal) as shown in Figure 2 (top). Non-linearity in this surface appears as joints along the edges of the two dimensional surface (Figure 2, bottom). This method measures the change in the slope of the facets of a 2D triangulated network as an indication of the linearity of the measurement errors.

Figure 2: Examples of the shape of a triangulated network. Top shows a network with only linear measurement errors. Bottom shows a network with non-linear measurement errors.

PROCEDURE The calculation of this quality measure is performed in the following stages:

1. Triangulate the network. 2. Determine the triangle that contains the rover

station and the corresponding three connecting stations.

3. Determine the three triangles that are connected to the edges of the rover’s triangle.

4. Calculate the slopes of the surfaces for the triangle containing the rover and the three connected triangles using the network residuals for the connected stations.

5. Calculate the differences in the slopes along the direction of the intersection of the center surface with the outer surfaces.

6. Calculate the shortest distance between the rover position and the interface between the surfaces.

7. Weight the difference in slope values by the inverse of the distance to the surface boundary.

Triangulation of the network can be performed through any of the available methods. In the cases shown, a Delaunay triangulation was used. The double difference slopes are calculated in the north and east directions using a plain fit to the double difference misclosures at the connecting reference stations. The single difference cannot be used because the station clocks will adversely affect the slopes. The difference in the slopes is calculated by using the dot product of a vector of the slopes. The vector perpendicular to the surface is derived from the north and east slopes as follows

=

1en

v (1)

where n is the slope in the north direction and e is the slope in the east direction. The difference in the slopes is calculated by

121

2112 −

⋅⋅

=∆vvvv

sT

. (2)

The absolute value and minus one are added to change the range to a value between 0 and 2. This value represents the difference in the slopes and the linearity of the measurement errors across the local region of the network.


The shortest distance between the rover and the three triangle borders is calculated to weight the three slope differences. The closer the rover is to an edge, the more weight this slope difference should have. The weighting used is the inverse of the distance to the nearest edge. The calculated values are all weighted together to give one value for each double difference as shown:

123

113

112

12323

11313

11212

−−−

−−−

++∆+∆+∆

=rrr

rrrr ddd

dsdsdsδ (3)

where rδ is the rover correction, 12,rd is the shortest

distance between the rover and the edge between slopes 1 and 2. This correction value is, in general, very small because it is a weighted cosine of the angle between the surfaces. To make these values practical for plotting they have all been multiplied by 500. The quality is a relative value that must be calibrated and then fixed. The relative values of the quality indicator are of importance. INTERPOLATION OF CORRECTIONS There are a variety of methods of parameterization of the correction differences for the rover located within or near the network coverage area. All of the methods developed are based on function approximations. The following are examples of investigated interpolation methods:

• Distance-Based Linear Interpolation Method (Dai et al., 2004; Euler et al., 2004),

• Low-Order 2D Surface Model (Dai et al., 2004; Euler et al., 2004; Wanninger, 2000; Fotopoulos & Cannon, 2001), and

• Least-Squares Collocation Method (Raquet, 1998; Marel, 1998; Alves, 2004).

The aim of each of these approaches is to model the distance-dependent errors along the baseline between master and rover station on an epoch-by-epoch and satellite-by-satellite basis. As the input parameters for each interpolation algorithm are between-master and auxiliary single-differenced observables (either carrier-phase residuals for the L1/L2 frequencies or geometric-/ionospheric-free linear combinations), an n-1 independent error vector is generated where n is the number of reference stations involved. In addition to unambiguous (ambiguity leveled) correction differences, coordinates for all reference stations are also needed to either generate coordinate differences (when applying low-order surface models), or baseline lengths between the master and each further

auxiliary station (when processing distance-weighted and least-squares collocation model). The vector of rover corrections is the result of a vector of reference station residuals (estimated errors) and a vector of coefficients (Dai et al., 2004)

V rover V (4)

where V rover is a vector of L1, L2, geometric-free or

ionospheric-free frequencies, is a n-1 vector of coefficients and V is an n-1 vector of errors, Actual GPS measurements have to be observed in order to form the error vector. As a basic premise of all of the interpolation methods the coordinates of stations and satellites have to be known, and the ambiguities have to be correctly solved so that the correction differences share a common integer ambiguity level. All of the coefficients can be calculated from the known coordinates of the reference stations and are therefore derived from the geometry of the rover station and the network (Dai et al., 2004). Since the master-auxiliary concept is applied, the coefficient vector (as well as the error vector) refers to single baselines between a designated master and each auxiliary reference station. Once the correction differences are interpolated for the rover's location they are applied to the raw observations of the master reference station to improve the single baseline positioning between the master and rover station. Single baseline positioning based on network corrections in real-time mode is also referred to as Network RTK positioning. Previous research has shown that the various interpolation methods all produce similar results (Dai et al. 2004). The interpolation method shown in this research will be the least squares collocation method. LEAST-SQUARES COLLOCATION METHOD The least-squares collocation method models the distance-dependent errors based on their spatial correlation. It is assumed that the further apart the reference stations are the less correlated the errors, which leads to a minimization of the errors in a least-squares sense. Least-squares collocation is based on several assumptions (Alves, 2004):

• The generated correction differences are measurements of the true signal s biased by the noise n .


• The measurements (signal and noise) are zero mean with a normal distribution.

• The true signal ( )ts is unknown, but the

variance-covariance of ( )ts is known.

The signal to be estimated is defined by its covariance, which can be described as the likelihood that two points will have the same values. The distance-dependent errors will be more similar with decreasing baseline length. Raquet (1998) proposes the following least-squares collocation equation for which double-differenced residuals between the master and auxiliary reference stations have to be calculated:

s C slsBT BC ll B

T 1 A x B l (5) where s is a vector of the estimated signals (corrections)

at rover's position, C sl s is the covariance matrix between the signal components of the network observations and the predicted signal (interpolated vector s ), B is the

observation matrix, which transforms C ll into double

difference space, C ll is the variance-covariance matrix of the network observations (measurement vector l), A is the design matrix of the estimated parameters, x is the vector of estimated parameters, l is the vector of network observations. Following the proposed Master-Auxiliary concept Equation (5) can be simplified by using single differences only. Thus the matrix B can be neglected assuming that the correction differences are uncorrelated. The simplified interpolation equation changes to

s C sl sC ll

1 l (6)

By inverting C ll , low variance values become high and high variance values become low, and hence high weight is assigned to precise observations when

multiplying C ll1

by l . A covariance function is used to generate the covariance matrices. It calculates the estimated covariance between the observation's signal components. Depending on the characteristics of a covariance function, each function has a different shape and prediction characteristics. Alves (2004) shows that the greatest difference in the prediction characteristics between several covariance functions will be when the network is sparse or the correlation length is short. The following are the requirements that a

covariance function should fulfill to best suit the error characteristics within a reference station network:

• The covariance function should produce a positive definite variance covariance matrix.

• The prediction should represent the likelihood that the errors measured at the reference stations are the same as the errors measured at the rover station.

• The covariance should converge to zero with increasing distance between stations.

As a consequence, if no reference station is in a position to predict the distance-dependent errors at the rover's location, then the covariance of the signal and the network's observation signal should be zero. Exponential covariance functions have all of the characteristics (described above) for modeling the distance-dependent errors between the stations. In contrast to the proposed covariance function of Raquet (1998) and Marel (1998) which, for example, also takes the satellite's elevation into account. The function applied in this paper is a function of the distance between all reference stations only. There is no need to correct for the elevation of satellites because single differenced observations are introduced in the error modeling process, and the interpolation is done on a satellite-by-satellite basis. The exponential covariance function is

( ) td

=dCF

−

e (7)

where is the correlation length, and d is the 3D-distance between all reference stations. Combining Equations (4) and (6) gives a coefficient vector of

C sl sC ll

1 (8)

DATA SET DESCRIPTION Data from selected reference stations in Lower Saxony, Germany, was used for the evaluation of the interpolation methods and the quality indicator. The geometry of the network is shown in Figure 3.


0640

0663

0657

0681

0650

0651

46.441.6

69.9

57.1

77.3

39.6

38.5

45.4

49.2

29.8

0652

19.2

Figure 3: Reference station network in Lower Saxony, Germany. All distances shown in blue are in km. Data was processed with a one second data rate with an elevation mask of 13 degrees, however results will be shown with a 30 second interval. This data set is from Oct. 31, 2003, which is the day after an ionospheric storm. The ionosphere error for this short baseline reaches up to 2.5 parts per million of the baseline length. All the receivers in this network are Trimble 4700s. The rover station (0652) is located 19 km from the nearest reference station. The reference station baseline lengths range from 30 to 69 km. This is a medium scale network. QUALITY INDEX RESULTS The quality index values can be computed for every double difference satellite pair using only one epoch of data. The quality indicator is always positive and represents the potential magnitude of the network RTK measurement errors. A low value quality index means that the network errors are linear and that interpolation should be able to minimize the errors at the rover. A high value quality index means that the network errors are non-linear relative to the reference station scale and the interpolation may not be able to model the correlated errors. The double difference L1 carrier phase residuals of the single reference station and network RTK solutions are compared to the quality values in Figures 8 to 11. In general, there is a high level of correlation between the level of error in the network RTK solution and the quality value. This shows that this quality indicator is effective in monitoring network RTK performance. Figure 8 shows satellite pair 27 and 10. In this case the double difference residuals are higher at the beginning of the data set and decrease towards the end of the dataset. Once again, there is a high level of correlation between the quality value and the network RTK performance. The

quality indicator follows the behavior of the network RTK performance.

Figure 8: Comparison of the quality value with the double difference residuals L1 carrier phase of the single reference station approach and the 2D interpolation for satellites 27 and 10. Figure 9 shows satellite pair 13 and 4. In this case the quality indicator closely matches the network RTK performance. At the beginning of the data set the network approach provides a high level of improvement over the single reference station approach. During this time the quality indicator shows a relatively low value, which matches the network RTK performance. This shows that the quality indicator is an indication of network RTK quality and not the quality of the single reference station RTK.

Figure 9: Comparison of the quality value with the double difference L1 carrier phase residuals of the single reference station approach and the 2D interpolation for satellites 13 and 4. The next example, shown in Figure 10, is for satellite pair 8 and 13. In this case the quality indicator follows the network RTK performance less closely, however there is


a noticeable change in behavior in the network RTK results, which is shown in the quality value. At the end of the data set the network RTK residuals become larger and more varied. This is shown as an increase in the quality indicator.

Figure 10: Comparison of the quality value with the double difference L1 carrier phase residuals of the single reference station approach and the 2D interpolation for satellites 8 and 13. USE OF THE QUALITY INDEX IN THE POSITION DOMAIN The previous results have shown that the quality indicator is effective in monitoring and tracking the residual measurement errors after applying reference station corrections. This is also an indication of the quality of the reference station corrections and the ability of the network to interpolate measurement errors. An important characteristic of the quality indicator is that an independent value can be calculated for each double difference satellite pair. This property allows for the quality indicator to independently track the measurement quality for each satellite pair. The quality indicator can be used for adjusting the relative weighting of the observations to down-weight observations with expected worse interpolation performance than other measurements. The following results compare the network reference station approach using a standard observation-weighting scheme and the quality indicator as a weighting scheme. The standard observation weighting method is an elevation dependent model where higher elevation satellites are given more weight than lower elevation satellites. Mathematical correlation between the measurements due to base satellite observations being used in multiple double-difference observations is not changed.

The dual frequency data was processed at one Hertz but the quality indicator was computed at 30 seconds and interpolated to the intermediate times. A 15 degree elevation mask was used. Data from the static rover station was processed in kinematic mode to simulate rover performance in the field. POSITION DOMAIN RESULTS The position domain results compare the single reference station approach to the network approach with standard observation weighting and the network approach with the quality indicator weighting. Figure 11 shows the x, y, and z position solutions for the three cases. This data was collected the day after an inospheric storm. The measurement errors are decreasing over time. This explains the improvement of the positioning over time. The single reference station and standard network approach approaches both have small periods where they could not maintain a fixed ambiguity solution. The network solution using the quality indicator was the only solution to maintain a fixed ambiguity solution for the entire dataset.

Figure 11: Position domain results for the single reference station, network approach with and without quality indicator weighting. For further analysis Figure 12 shows the same position solutions as Figure 11 but with a smaller scale. This figure shows the difference between the three solutions. The single reference station approach is clearly worse than the network solutions, especially at the beginning of the dataset when the ionosphere is more active. The difference between the single reference station approach and the network approaches becomes less pronounced towards the end of the dataset. However the z component shows improvement due to the network corrections for nearly the entire length of the dataset.


The difference between the network approach with the standard observation weighting and the network approach with the quality indicator weighting, when only the fixed ambiguity solutions are compared, is insignificant. The two solutions are nearly identical.

Figure 12: Position domain results for the single reference station, network approach with and without quality indicator weighting (zoomed in). Table 1 shows the position domain statistics for the three solutions. The single reference station approach performs noticeably worse than the network solutions. Comparing all of the solutions (with both float and fixed ambiguities) the network approach improves the positioning solution by 38 percent. When using the quality indicator weighting this improvement increases to 51 percent. Although this is a significant improvement relative to the single reference station approach, the difference between the standard observation weighting and the quality indicator weighting in only in the ability to maintain a fixed ambiguity solution. During observation intervals when both network solutions have fixed ambiguities the difference is negligible. Although negligible the network approach with the quality indicator provides a 0.5% improvement over the standard weighting solution when fixed ambiguity solutions are compared. Table 1: Position domain statistics for the single reference station, the network approach with standard observation weighting for all solutions and only fixed ambiguity solution, and the network approach with quality indicator weighting.

(cm) Mean Standard deviation

RMS

Single RS 3.2 3.2 4.5 Network (all) 1.9 2.0 2.8 Network (fixed) 1.0 2.0 2.2 Network with QI 1.0 2.0 2.2

Table 2 shows the ambiguity domain results for the three solutions. Two ambiguity qualities are shown in the table: The percentage of fixed ambiguities in the position domain (shown in Table 2) relates to the position errors shown in Figures 11 and 12. This is the percentage of fixed ambiguities from the position domain test. There is very little difference between the solutions for this ambiguity resolution statistic. However, the quality indicator network approach is able to resolve all of the ambiguities for the entire length of the dataset. The standard weighting network approach and the single reference station approach are able to fix nearly all of the ambiguities, 99.8 and 99.3 percent, respectively. This is the only notable improvement when using the quality indicator weighting to improve the position domain performance. The second statistic is the percentage of fixed ambiguity attempts, when the ambiguities are searched every ten seconds. The float solution is reset after each search. This means that each ten second dataset is independent. This is an indication of the ability to resolve ambiguities after a reset. This demonstrates the strength of the network approach. The network approaches both fixed 99.4 percent of their ambiguity attempts while the single reference station approach was only able to resolve 67.4 percent. This shows that the network approach is able to better resolve ambiguities after a loss of lock, a system restart or a startup at any given time. There is no difference between the standard observation weighting and the quality indicator weighting for this statistic. Table 2: Ambiguity domain statistics for the single reference station, network approach with and without quality indicator weighting.

% fixed ambiguities in

position domain

% successful fix attempts (every

10 seconds) Single RS 99.3 67.4 Network 99.8 99.4 Network with QI

100.0 99.4

CONCLUSIONS This paper proposes a geometry based quality index to monitor network RTK quality. This quality index can monitor the linearity of the instantaneous network errors in terms of each double difference measurement independently. The quality index was shown over time for three double difference satellite pairs and compared to the L1 carrier phase double difference residuals. In all the cases shown, the quality value closely follows the network RTK results, stating that this geometry-based quality index is a good


indicator for measurement quality after interpolating and applying the corrections. This quality indicator was later used to try and improve the position accuracy by using it as a weighting tool for the measurements. The network approach for both the standard weighting and quality indicator weighting showed a noticeable improvement over the single reference station approach in terms of position accuracy and ambiguity resolution performance, maintaining fixed ambiguities for the entire dataset, while the standard weighting solution fell into float mode for .2% of the dataset. This was the only noticeable difference between the two approaches. When both solutions had fixed ambiguities the difference between the solution’s statistics (mean, standard deviation and RMS) was less than one millimeter. This shows that although the quality indicator is effective in tracking and monitoring network interpolation performance, it had a minimal effect on the position domain solution in this case. The quality indicator weighted method appears to be slightly more robust in terms of maintaining a fixed ambiguity solution, however this was only obvious for 0.2% of the dataset. The rest of the time the quality indicator weighted method provided no noticeable improvement. ACKNOWLEDGMENTS The authors would like to thank Volker Wegener of the Zentrale Stelle SAPOS, LGN, Hannover for kindly supplying us the data used for the numerical analysis. REFERENCES Alves, P. (2004) Development of Two Novel Carrier

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