Model comparison for GLONASS RTK with low-cost receivers

Abstract

GLONASS ambiguity resolution in differential real-time kinematic (RTK) processing is affected by inter-frequency phase biases (IFPBs). Previous studies empirically determined that IFPBs are linearly dependent on the frequency channel number and calibration values have been derived to mitigate these biases for geodetic receivers. The corresponding IFPB-constrained model is currently the de facto approach in RTK, but the growing market of GNSS receivers, and especially low-cost receivers, makes calibration and proper handling of metadata a complex endeavor. Since IFPBs originate from timing offsets occurring between the carrier phase and the code measurements, we confirm other studies that show that IFPBs are not exactly linearly dependent on the frequency channel number, but rather linearly dependent on the channel wavelength, which calls for a modification in the GLONASS functional model. As an alternative to calibration, we revisit a calibration-free method for GLONASS ambiguity resolution and provide new insights into its applicability. A practical experiment illustrates that the calibration-free approach can offer better ambiguity fixing performance when the uncertainty on the IFPB parameter is large, unless partial ambiguity resolution is performed.

Appendix

The derivations for the calibration-free model presented in previous sections are based on the assumption that IFPBs originate from the DCPB. This appendix presents an alternate derivation showing that the method can also cancel IFPBs present in carrier phase observations, provided that they can be modeled with a partial derivative taking the form of $\lambda^{n} k^{n}$, such that:

$$L_{AB}^{n} = dT_{AB,L} + \lambda^{n} N_{AB}^{n} + \lambda^{n} k^{n} \delta_{AB} .$$

(50)

Selecting a first reference satellite allows for the definition of the biased receiver clock:

$$\overline{dT}_{AB,L} = dT_{AB,L} + \lambda^{1} N_{AB}^{1} + \lambda^{1} k^{1} \delta_{AB} .$$

(51)

The second reference satellite enables the estimation of the biased ambiguity of the reference satellite such that:

$$L_{AB}^{2} = d\bar{T}_{AB,L} - \left( {\lambda^{1} - \lambda^{2} } \right)\bar{N}_{AB}^{1}$$

(52)

with

$$\bar{N}_{AB}^{1} = N_{AB}^{1} - \frac{{\lambda^{2} }}{{\lambda^{1} - \lambda^{2} }}N_{AB}^{12} + \frac{{\left( {k^{1} \lambda^{1} - k^{2} \lambda^{2} } \right)}}{{\lambda^{1} - \lambda^{2} }}\delta_{AB} .$$

(53)

Inserting (51) and (53) into (50) leads to:

$$L_{AB}^{n} = \overline{dT}_{AB,L} - \left( {\lambda^{1} - \lambda^{n} } \right)\left[ {\bar{N}_{AB}^{1} + \omega \delta_{Ab} } \right] + \lambda^{n} \bar{N}_{AB}^{1n} ,$$

(54)

where $\bar{N}_{AB}^{1n}$ has the same definition as (20) and:

$$\omega = \frac{{\lambda^{1} k^{1} - \lambda^{n} k^{n} }}{{\lambda^{1} - \lambda^{n} }} - \frac{{\lambda^{1} k^{1} - \lambda^{2} k^{2} }}{{\lambda^{1} - \lambda^{2} }}.$$

(55)

By recognizing that:

$$\frac{{\lambda^{1} k^{1} - \lambda^{n} k^{n} }}{{\lambda^{1} - \lambda^{n} }} = \frac{{\lambda^{1} k^{1} - \lambda^{2} k^{2} }}{{\lambda^{1} - \lambda^{2} }} = - \frac{f}{\Delta f};$$

(56)

the term $\omega$ cancels from (54) and leads to a consistent system similar to (24)–(26). The only differences are the definition of the estimable receiver clock and reference ambiguity parameters, as shown by (51) and (53), respectively, while the definition of the estimable integer ambiguities remains unchanged.