# 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.

## Keywords

Global navigation satellite systems (GNSS) GLONASS Inter-frequency biases Ambiguity resolution## Introduction

With the modernization of the GLONASS constellation, satellites of the GLONASS-M+ and GLONASS-K1 types broadcast signals on a third frequency using code division multiple access (CDMA) (Urlichich et al. 2011). However, the current constellation composed mainly of the GLONASS-M satellites still relies on frequency division multiple access (FDMA). This method of satellite identification ensures that all GLONASS satellites in view transmit signals at a slightly different frequency.

Timing inconsistencies in GNSS signals have long been a concern for GNSS applications. It was recognized early that signals tracked at different frequencies are subject to inter-frequency biases affecting ionospheric studies (Lanyi and Roth 1988). Timing variations between carrier phase and code observations were also found to impact receiver clock estimates for time transfer and cause day-boundary jumps (Defraigne and Bruyninx 2007). By defining clock estimates relying solely on phase measurements, Collins et al. (2010) could demonstrate significant intra-day and longer-term variations in clock estimates involving code measurements. Apart from these timing fluctuations between signals, Sleewaegen et al. (2012) explained that a constant offset between carrier phase and code observations could occur within the digital signal processing section of a GNSS receiver. This timing discrepancy manifests itself as a linear bias in GLONASS carrier phase ambiguities due to the different wavelengths of each satellite and leads to inter-frequency phase biases (IFPBs). Geng et al. (2017) coined the term differential code-phase bias (DCPB) for this timing error and emphasized that the concepts of IFPB and DCPB are not equivalent. This distinction is the key point underlying our derivations of the functional models.

The satellite-dependent wavelengths associated with GLONASS satellites were also found to be problematic for ambiguity resolution (Wang et al. 2001). Due to FDMA, forming double-differenced observations in units of meters between pairs of receivers and satellites does not cancel the ambiguity of the reference satellite. A common approach for eliminating this extra unknown from the system of equations is to compute its value based on a combination of carrier phase and code measurements (Mader et al. 1995). However, due to a misalignment of the phase and code observables within receivers, this procedure leads to IFPBs. From zero-length baseline tests, it was determined empirically that IFPBs between receivers of different manufacturers have a linear dependency with respect to the frequency channel number (Wanninger and Wallstab-Freitag 2007; Al-Shaery et al. 2013). Wanninger (2012) also analyzed a larger sample of receivers running various firmware versions and equipped with different antenna models. That study played a key role in GLONASS RTK by proposing manufacturer-specific calibration values to be used as a priori corrections to the carrier phase observables and recommended estimating a residual IFPB parameter to absorb unit-specific discrepancies from these ensemble means. Providing a priori calibration values for the code-phase bias is now an accepted solution in the high-precision GNSS industry, and the Radio Technical Commission on Maritime (RTCM) Services has defined message type 1230 to exchange such information when available.

With the rapid growth in low-cost GNSS receivers, it is a reasonable assumption that not all GLONASS-enabled receivers have calibrated IFPBs and there is no guarantee that the exchange of metadata will remain consistent. Therefore, Banville et al. (2013) proposed a method for GLONASS ambiguity resolution of mixed receiver types that does not require any external calibration. The approach requires a re-parameterization that allows for the ambiguity of the reference satellite to be explicitly estimated in the positioning filter. This is achieved by selecting two reference satellites that, preferably, have adjacent frequency channel numbers. Recently, Odijk and Wanninger (2017) claimed that this model is only applicable for identical receiver pairs. We will demonstrate that the claim is unfounded in practice.

To demonstrate the applicability of the calibration-free model, the GLONASS functional model for short-baseline RTK processing is first introduced. Two methods for removing the inherent rank deficiency of this system are described: The first one is the IFPB-constrained model with an emphasis on the distinction between the IFPB and DCPB representations. The calibration-free approach of Banville et al. (2013) is then revisited to explicitly show how IFPBs between receivers are absorbed by the model parameters, regardless of receiver type. Numerical and field examples show how each model recovers integer GLONASS ambiguities in practice.

## GLONASS functional model

*L*) and code (

*C*) observations to satellite

*j*reduces to a simple timing-equation form only and reads:

*L*and

*C*, respectively. This distinction is necessary primarily to model the DCPB, i.e., the timing delays induced by the digital signal processing (DSP) component of GNSS receivers (Sleewaegen et al. 2012). Different timing references for the carrier and code observable can, therefore, be modeled as:

## The IFPB-constrained model

The a priori values for \(\delta_{AB}^{\prime }\) derived from a dense network of receivers (Wanninger 2012), zero-length baselines (Al-Shaery et al. 2013), or even global networks of receivers (Tian et al. 2015; Geng et al. 2017) serve the purpose of removing the bulk of the IFPB error. Even though this quantity is quite repeatable from unit to unit of the same manufacturer, variations in the order of a few millimeters per channel were noticed. These slight deviations do not prevent the tightly constraining of this parameter in the positioning filter and achieving fast ambiguity resolution.

If no a priori calibration values are available for a given receiver type, a looser initial constraint must be applied to the IFPB parameter. Since this parameter is not directly observable without fixing at least one double-differenced ambiguity, the IFPBs will propagate into the estimated ambiguities and will corrupt their integer nature (Takac 2009). When the rover position is precisely determined, as in case of long observation sessions, Habrich et al. (1999) showed that it is often possible to fix a first GLONASS ambiguity whose channel separation with respect to the reference satellite \(\left( {k^{n} - k^{1} } \right)\) is small, preferably ± 1. In this case, the standard deviation of \(\delta_{AB}^{\prime }\) drops significantly and the integer nature of other ambiguities is revealed. The same principle can be applied when the precise position is obtained from a GPS ambiguity-fixed solution (Yao et al. 2017). Since the uncertainty in the IFPB parameter is reflected in the ambiguity covariance matrix, fixing ambiguities using integer least squares with an ambiguity decorrelation procedure, e.g., LAMBDA (Teunissen 1993), still allows for fast ambiguity resolution when the IFPB is loosely constrained. However, we will show that this approach is not necessarily optimal.

## The calibration-free model

It is important to stress that the derivations for the calibration-free model do not require code measurements to remove the rank deficiency of the system. Therefore, contrary to the IFPB-constrained model, the calibration-free approach is not affected by IFPBs originating from the DCPB. While inter-frequency carrier phase biases from other sources could impact the validity of the approach, no study has demonstrated their existence so far. For the sake of completeness, “Appendix” shows that the calibration-free model can still provide integer ambiguities in the presence of phase biases as long as they can be modeled as a function of \(\lambda^{n} k^{n}.\)

## Connection between models

Since the ambiguity covariance matrix contains the correlation information associated with the uncertainty of the IFPB parameter, the decorrelation procedure of LAMBDA typically forms IFPB-canceling linear combinations. However, since only \(n - 1\) independent bias-free ambiguities can be formed, partial ambiguity resolution should be performed when the uncertainty on the IFPB parameter is large.

## Numerical examples

The application of the two models introduced above is presented using GLONASS data from both geodetic-quality receivers and low-cost single-frequency receivers. Data from geodetic receivers are analyzed first. A dataset collected on January 1, 2012, at the University of New Brunswick (UNB) campus in Fredericton, Canada, used NovAtel OEMV3 (UNBN) and Trimble NetR5 (UNB3) receivers connected to the same antenna. Therefore, differencing measurements between the two receivers cancel all error sources, except timing offsets (clocks and biases) and carrier phase ambiguities. From previous calibration sessions, the inter-frequency phase biases between these receiver types have been determined to be 30 mm/channel (Wanninger 2012).

Information regarding selected GLONASS satellites from the UNBN-UNB3 baseline on January 1, 2012

ID | PRN | \(k^{j}\) | \(\lambda^{j}\) | \(L_{AB}^{j}\) | \(C_{AB}^{j}\) |
---|---|---|---|---|---|

1 | R11 | 0 | 0.18714 | 14.400 | 15.185 |

2 | R12 | − 1 | 0.18720 | 18.300 | 14.802 |

3 | R13 | − 2 | 0.18727 | − 10.939 | 14.733 |

4 | R01 | 1 | 0.18707 | 185.412 | 15.154 |

### The IFPB-constrained model

### The calibration-free model

### Connection between models

## Field experiment

The ambiguity fixing performance of the IFPB-constrained and calibration-free models are assessed using data from a u-blox M8T EVK receiver and antenna collected on top of the Netherlands Measurement Institute (NMi) building in Delft (de Bakker 2017; de Bakker and Tiberius 2017). IGS station DLF1 is located approximately 13 m away and is used as the base station. DLF1 runs a Trimble NetR9 receiver and supplies data at 1 Hz sampling interval allowing between-station single-differenced measurements with the u-blox receiver. For the purpose of our demonstration, 1 h of GPS + GLONASS data were selected starting at 00:00:00 GPST on August 10, 2016, and this data set was divided into 12 independent 5-min sessions.

Two post-processed kinematic (PPK) solutions were obtained with identical processing settings, except for the handling of the IFPBs. The first solution implements the IFPB-constrained model of (15): Since no IFPB calibration values are available for the u-blox receiver, an a priori constraint of zero with an uncertainty of 5 cm was applied to the \(\delta_{AB}^{'}\) parameter. The second solution is based on the calibration-free model of (24)–(26). Since the system is of full rank, no additional constraint is needed. In both of these models, ambiguity resolution was attempted for both GPS and GLONASS simultaneously. Code measurements were included in the system with a system-specific code-clock parameter as in (2), and satellite-specific IFCBs were estimated for GLONASS with a priori constraints of 10 m.

## Conclusion

Our derivations confirmed the discussion of Geng et al. (2017) showing that IFPBs originating from timing offsets between GLONASS carrier phase and code measurements do not follow the empirically derived linear relationship with respect to the frequency channel number. This concept has been extended by proposing a more rigorous partial derivative for the IFPB-constrained model which considers the product of the wavelength and channel number. It was also emphasized that IFPBs are not inherent to the phase observables, but only appear in the carrier phase functional model when introducing code observations.

The calibration-free model of Banville et al. (2013) removes singularities in the GLONASS functional model by selecting two reference satellites allowing for an explicit estimation of the reference satellite ambiguity. Such a formulation remains unaffected by IFPBs originating from the DCPB since it is independent from code observations. While inter-frequency carrier phase biases from other sources could impact the validity of the approach, no study has demonstrated their existence so far. A numerical example and a field experiment further confirm the validity of the model and demonstrate that the claim made by Odijk and Wanninger (2017) that the calibration-free model is not applicable to different receiver types is unfounded.

The calibration-free model requires two reference satellites to define a system of full rank. Extending the model of Banville et al. (2013), the constraint that these two reference satellites require adjacent frequency channel numbers can be relaxed, although the wavelength of the estimated ambiguities is reduced by their channel spacing. When using this model, the estimated ambiguities are shown to be linear combinations canceling IFPBs, thereby preserving their integer nature. A connection between the IFPB-constrained and calibration-free models can, therefore, be defined by an appropriate, unidirectional, transformation. Similarly, the decorrelation procedure of LAMBDA cannot cancel IFPB effects in all linear combinations, and partial ambiguity resolution is recommended for successful ambiguity validation when the uncertainty on the IFPB parameter is large.

Both the IFPB-constrained and calibration-free models can achieve fast ambiguity resolution. However, GLONASS ambiguities only maintain their integer nature with the full-rank calibration-free model and, in practice, this provides for a stronger and more robust solution, especially where IFPB calibration values are unknown or of poor quality as for low-cost receivers.

## Notes

### Acknowledgements

The authors would like to acknowledge the Geodetic Research Laboratory at UNB for sharing GNSS data from their continuously operating receivers. The initiative of Peter F. de Bakker to openly share his u-blox data is greatly appreciated. The constructive comments of an anonymous reviewer and the editor contributed in improving the original manuscript. This paper is published under the auspices of the NRCan Earth Sciences Sector as contribution number 20170171.

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