RTK Fundamentals

Fundamentals
Title	RTK Fundamentals
Author(s)	GMV
Level	Basic
Year of Publication	2011

With origin dating back to the mid-1990s, Real Time Kinematics (RTK) is a differential GNSS technique which provides high positioning performance in the vicinity of a base station. The technique is based on the use of carrier measurements and the transmission of corrections from the base station, whose location is well known, to the rover, so that the main errors that drive the stand-alone positioning cancel out. A RTK base station covers a service area spreading about 10 or 20 kilometres, and a real time communication channel is needed connecting base and rover. RTK, which achieves performances in the range of a few centimetres, is a technique commonly used in surveying applications.

RTK Technique

From an architectural point of view, RTK consists of a base station, one or several rover users, and a communication channel with which the base broadcasts information to the users at real time.

The technique is based on the following high-level principles:

• In the neighbourhood of a clean-sky location, the main errors in the GNSS signal processing are constant, and hence they cancel out when differential processing is used. This includes the error in the satellite clock bias, the satellite orbital error, the ionospheric delay and the tropospheric delay. The main errors left without correction are multipath, interference and receiver thermal noise. Of the errors listed above, the only one which is truly constant with respect to the user location is the satellite clock bias; the rest will show a given dependency with the location as the rover moves away from the base station, being the tropospheric error the first to be fully de-correlated in a few kilometres from the base.
• The noise of carrier measurements is much smaller than the one of the pseudo-code measurements. The typical error of code pseudorange measurements is around 1 m, to compare with 5 mm for carrier phase measurements. However, the processing of carrier measurements is subject to the so-called carrier phase ambiguity, an unknown integer number of times the carrier wave length, that needs to be fixed in order to rebuild full range measurements from carrier ones.
• The phase ambiguities can be fixed using differential measurements between two reference stations. There are different techniques available to fix them, some based on single frequency measurements with long convergence times, other taking benefit of dual frequency observables with shorter convergence. In general, the techniques either depend on a high precision knowledge of the ionosphere, or assume that the two stations are close enough so that the ionospheric differential delay is negligible when compared with the wave-length of the carriers, around 20 cm. The latter is the approached followed in RTK, limiting the service area to 10 or 20 km; the former is used in WARTK to cover big service areas with base stations separated around hundreds of kilometres away. The RTK approach needs continuity in the tracked measurements to avoid re-initialization of the phase-ambiguity filters; this is a severe limitation in urban environments due to the big number of obstructions.

The base station broadcasts its well-known location together with the code and carrier measurements at frequencies L1 and L2 for all in-view satellites. With this information, the rover equipment is able to fix the phase ambiguities and determine its location relative to the base with high precision. By adding up the location of the base, the rover is positioned in a global coordinate framework.

The RTK technique can be used for distances of up to 10 or 20 kilometres, yielding accuracies of a few centimetres in the rover position, to be compared with 1 m that is achieved with code-based differential GPS. Because of its high precision in controlled environments, RTK is extensively used in surveying applications.

RTK Algorithm

As stated in the previous section, one of the main problems in the RTK technique is fixing the phase ambiguities.

The RTK Algorithm is based on double differenced observables that can eliminate selective availability effects as well as other biases. The highlights of the algorithm are described next. At a given epoch, and for a given satellite, the simplified carrier phase observation equation is the following:

[math]\displaystyle{ \qquad \phi =\rho-I+Tr+c(b_{Rx}-b_{Sat} )+〖N\lambda+\varepsilon〗_\phi \qquad \mbox{(1)} }[/math]

Where:

[math]\displaystyle{ I }[/math] is the signal path delay due to the ionosphere;

[math]\displaystyle{ Tr }[/math] is the signal path delay due to the troposphere;

[math]\displaystyle{ b_{Rx} }[/math] is the receiver clock offset from the reference (GPS) time;

[math]\displaystyle{ b_{Sat} }[/math] is the satellite clock offset from the reference (GPS) time;

[math]\displaystyle{ c }[/math] is the vacuum speed of light;

[math]\displaystyle{ \lambda }[/math] is the carrier nominal wavelength;

[math]\displaystyle{ N }[/math] is the ambiguity of the carrier-phase (integer number);

[math]\displaystyle{ \varepsilon_\phi }[/math] are the measurement noise components, including multipath and other effects;

[math]\displaystyle{ \rho }[/math] is the geometrical range between the satellite and the receiver, computed as a function of the satellite [math]\displaystyle{ (x_{Sat}, y_{Sat},z_{Sat}) }[/math] and receiver [math]\displaystyle{ (x_{Rx}, y_{Rx},z_{Rx}) }[/math] coordinates as:

[math]\displaystyle{ \qquad \rho=\sqrt{〖(x_{Sat}-x_{Rx})〗^2+〖(y_{Sat}-y_{Rx})〗^2+〖(z_{Sat}-z_{Rx})〗^2 } \qquad \mbox{(2)} }[/math].

For two receivers a and b making simultaneous measurements at the same nominal time to satellites 1 and 2, the double difference observable is:

[math]\displaystyle{ \qquad \phi_a^{12} - \phi_b^{12} =\rho_a^{12}-\rho_b^{12}-I_a^{12}+I_b^{12}+Tr_a^{12}-Tr_b^{12}+\lambda(N_a^{12}-N_b^{12})+\varepsilon_a^{12}- \varepsilon_b^{12} \qquad \mbox{(3)} }[/math]

In the above equation receiver and satellite clock offsets and hardware biases cancel out. The single difference ambiguities difference [math]\displaystyle{ N_a^{12}-N_b^{12} }[/math] is commonly parameterized as a new ambiguity parameter [math]\displaystyle{ N_{ab}^{12} }[/math]. The advantage of double differencing is that the new ambiguity parameter [math]\displaystyle{ N_{ab}^{12} }[/math] is an integer because the non-integer terms in the GPS carrier phase observation, due to clock and hardware delays in the transmitter and receiver, are eliminated.

Although it would be possible to estimate the double difference ambiguity using a float approach instead of an integer one, this would lead to dm-level accuracy instead of cm-level. Hence, standard RTK fixes the ambiguities to integer figures.

Ambiguity Resolution

As stated in the section above, one of the keys to obtain the best accuracy from RTK is to fix the carrier phase ambiguities to integer numbers. Normally, this is done in three steps:

• The ambiguities are first fixed to float numbers using standard least-square techniques.
• The set of integer ambiguities is set to the one that optimizes the residuals in the surroundings of the float solution.
• The carrier measurements are corrected with the integer ambiguities and they are used to obtain the relative position of the rover to the base station.

Of these three steps, the second one is quite complex, because the float ambiguity covariance ellipsoid in the measurement space is extremely elongated. As a consequence, the brute-force search process is inefficient, normally beyond the computational capabilities of the rover equipment. Several techniques have been developed to deal with this problem; please consult the Carrier Phase Ambiguity Fixing article for more information.