# On the estimability of parameters in undifferenced, uncombined GNSS network and PPP-RTK user models by means of \(\mathcal {S}\)-system theory

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## Abstract

The concept of integer ambiguity resolution-enabled Precise Point Positioning (PPP-RTK) relies on appropriate network information for the parameters that are common between the single-receiver user that applies and the network that provides this information. Most of the current methods for PPP-RTK are based on forming the ionosphere-free combination using dual-frequency Global Navigation Satellite System (GNSS) observations. These methods are therefore restrictive in the light of the development of new multi-frequency GNSS constellations, as well as from the point of view that the PPP-RTK user requires ionospheric corrections to obtain integer ambiguity resolution results based on short observation time spans. The method for PPP-RTK that is presented in this article does not have above limitations as it is based on the undifferenced, uncombined GNSS observation equations, thereby keeping all parameters in the model. Working with the undifferenced observation equations implies that the models are rank-deficient; not all parameters are unbiasedly estimable, but only combinations of them. By application of \(\mathcal {S}\)-system theory the model is made of full rank by constraining a minimum set of parameters, or S-basis. The choice of this S-basis determines the estimability and the interpretation of the parameters that are transmitted to the PPP-RTK users. As this choice is not unique, one has to be very careful when comparing network solutions in different \(\mathcal {S}\)-systems; in that case the S-transformation, which is provided by the \(\mathcal {S}\)-system method, should be used to make the comparison. Knowing the estimability and interpretation of the parameters estimated by the network is shown to be crucial for a correct interpretation of the estimable PPP-RTK user parameters, among others the essential ambiguity parameters, which have the integer property which is clearly following from the interpretation of satellite phase biases from the network. The flexibility of the \(\mathcal {S}\)-system method is furthermore demonstrated by the fact that all models in this article are derived in multi-epoch mode, allowing to incorporate dynamic model constraints on all or subsets of parameters.

## Keywords

GNSS Undifferenced model Network PPP-RTK theory \(\mathcal {S}\)-system theory Rank-deficient model Dynamic model## Notes

### Acknowledgments

This work has been done in the context of the Positioning Program Project 1.19 “Multi-GNSS PPP-RTK Network Processing” of the Cooperative Research Centre for Spatial Information (CRC-SI). P.J.G. Teunissen is the recipient of an Australian Research Council (ARC) Federation Fellowship (Project Number FF0883188).

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