International Journal of Navigation and Observation

Volume 2015, Article ID 346498, 15 pages

http://dx.doi.org/10.1155/2015/346498

## Next Generation Network Real-Time Kinematic Interpolation Segment to Improve the User Accuracy

^{1}Norwegian Mapping Authority, Geodetic Institute, 3511 Hønefoss, Norway^{2}Department of Mathematical Sciences and Technology, NMBU, 1432 Akershus, Norway^{3}KTH Royal Institute of Technology, 100 44 Stockholm, Sweden^{4}Department of Electronics and Telecommunications, NTNU, 7491 Trondheim, Norway

Received 19 September 2014; Revised 7 December 2014; Accepted 25 December 2014

Academic Editor: Aleksandar Dogandzic

Copyright © 2015 Mohammed Ouassou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper demonstrates that automatic selection of the right interpolation/smoothing method in a GNSS-based network real-time kinematic (NRTK) interpolation segment can improve the accuracy of the rover position estimates and also the processing time in the NRTK processing center. The methods discussed and investigated are inverse distance weighting (IDW); bilinear and bicubic spline interpolation; kriging interpolation; thin-plate splines; and numerical approximation methods for spatial processes. The methods are implemented and tested using GNSS data from reference stations in the Norwegian network RTK service called CPOS. Data sets with an average baseline between reference stations of 60–70 km were selected. 12 prediction locations were used to analyze the performance of the interpolation methods by computing and comparing different measures of the goodness of fit such as the root mean square error (RMSE), mean square error, and mean absolute error, and also the computation time was compared. Results of the tests show that ordinary kriging with the Matérn covariance function clearly provides the best results. The thin-plate spline provides the second best results of the methods selected and with the test data used.

#### 1. Introduction

The use of GNSS and network real-time kinematic positioning to achieve GNSS positions with accuracy at the cm-level is increasing rapidly these years. This is partly due to the development and modernization of the GNSS systems themselves (GPS, GLONASS, Galileo, and Beidou), but it is also caused by a general quest for better position accuracy in many user communities.

High-accuracy GNSS positioning is based on the carrier phase being observable. Using the notation from [1], the basic observation equation that summarizes the relation between observations and error sources is given as follows:
where is the phase observation in cycles, is the wavelength in meters/cycle, is the geometric distance between the receiver and satellite in meters, is the ionospheric signal delay in meters, is the tropospheric signal delay in meters, is the frequency in Hertz, and are the clock errors of, respectively, the receiver and the satellite, is the initial number of cycles at the first observation epoch (the* ambiguity*), and is a noise term given in cycles that mainly accounts for multipath (reflected signals) and receiver noise.

When using the NRTK technique, a network of reference stations is used to estimate the errors in the positioning process, that is, the effects of the ionosphere and troposphere as well as inaccuracies in the satellite position as provided with the broadcast ephemerids from the satellites.

The accuracy of NRTK positioning systems depends on the ability to identify and mitigate the error sources in the system as well as the residual biases. The biases include residual effects from the space segment, signal propagation, environment effects, and receiver noise in the reference network. The mitigation process can be carried out by modeling, estimation, or combinations of observables.

The NRTK processing chain can be summarized as follows: the first step is to collect raw measurements from the network of reference stations, solve for the ambiguities within the reference network, and generate error estimates. The next step is to apply the interpolation/smoothing scheme to generate the RTK corrections for the user location. The RTK corrections are then transmitted to users who can then perform real-time positioning with accuracy at the cm-level.

Figure 1 shows all segments involved in the NRTK processing chain. The figure illustrates the so-called virtual reference station (VRS) concept, which was developed by Landau et al. [2]. Other NRTK standards such as for instance the master auxiliary concept (MAC) also exist [3], but we limit the discussion in this paper to the VRS concept.