A few months ago, I highlighted an erratum from the Springer Handbook of Global Navigation Satellite Systems. The chapter on Differential Positioning accused myself and my colleagues at NRCan of making false claims about the applicability of our new approach to GLONASS ambiguity resolution. Contacting the editors did not lead to any hope of having this issue promptly resolved, so we decided that a follow-up paper on this topic was in order to better clarify the intricacies of our approach.

Our paper has just been published in GPS Solutions and is titled: "Model comparison for GLONASS RTK with low-cost receivers”. As part of the Springer Nature SharedIt initiative, you can have access to the paper by clicking here, although you might not be able to download it if your organization does not have proper access rights.

In GLONASS RTK, inter-frequency phase biases (IFPB), if left uncalibrated, contaminate estimated carrier-phase ambiguities and prevent successful ambiguity resolution. For this reason, the Radio Technical Commission on Maritime (RTCM) Services has deﬁned message type 1230 to exchange IFPB calibration values. When this calibration information is available, GLONASS RTK can be performed just like GPS. However, not all reference stations provide the calibration message and newer receiver types, especially low-cost receivers, may not have calibration values available.

When no apriori information on the IFPB is available, a common practice consists of estimating GLONASS ambiguities as float values. Since fixing both GPS and GLONASS ambiguities simultaneously greatly improves ambiguity fixing performance, it is obvious that this solution is suboptimal. Another solution consists of setting up an IFPB parameter in the RTK filter. The resulting ambiguity covariance matrix thus contains the proper correlation information to identify the correct GLONASS integer ambiguities. However, as we show in our paper, this approach can backfire. The following figure shows (in black) that the ratio test used to discriminate the best and second-best integer vectors converges to 1.0 when using this approach, meaning that we cannot perform ambiguity validation reliably.

**Fig 1 Ratio test values with different models for 5-min independent sessions**

Why is the ratio test converging to 1.0? Interestingly, it is possible to form linear combinations of ambiguities that cancel the impact of IFPBs, which is what LAMBDA is typically doing.
Unfortunately, with *n* ambiguities, only *n-1* bias-free linear combinations can be formed. This leaves us with one poorly-defined ambiguity which corrupts the ratio test. A
solution to this problem is to perform partial fixing and fix only the subset of *n-1* bias-free ambiguities.

To avoid performing partial ambiguity resolution, we can use the calibration-free approach initially described by Banville et al. (2013). When selecting two reference satellites, preferably with
adjacent frequency channel numbers, all GLONASS ambiguities naturally converge to integer values. Since we select an additional reference satellite, our ambiguity state vector effectively
contains only the subset of *n-1* bias-free ambiguities. In this case, the ratio test converges appropriately, as shown in red in the above figure. Hence, there is a connection between the
IFPB-constrained approach with partial ambiguity resolution and the calibration-free approach. Both methods can provide identical positioning results with different implementations, i.e. partial
ambiguity resolution vs change in functional model. For those interested, a more in-depth description of the methods is provided in the paper. Numerical examples from a zero-length baseline also
prove the applicability of the models with mixed receiver types.

We hope that this paper will convince the authors of the Differential Positioning chapter that our method is indeed applicable to mixed receivers. In the end, I think that we have been able to turn a lemon into a lemonade, since we have been given the opportunity to explore in more details the connection between GLONASS RTK models.

**References**

Banville S, Collins P, Lahaye F (2013) GLONASS ambiguity resolution of mixed receiver types without external calibration. GPS Solut doi:10.1007/s1029 1-013-0319-7

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2#1Xianglin Liu(Thursday, 28 June 2018 07:58)Without going the detail, I would just like to ask how you exactly define the ratio test here. In the GNSS community, the ratio test value seems always larger than 1.0. (smile)

#2Simon Banville(Thursday, 28 June 2018 21:10)@Xianglin If you flip the numerator and the denominator, you get a value between 0 and 1, and it's much easier to visualize than a value "greater than 1"!