PPP–RTK functional models formulated with undifferenced and uncombined GNSS observations

Technique PPP–RTK combines the advantages of both the Precise Point Positioning (PPP) and the Real-Time Kinematic (RTK) positioning. With the emergence of multi-frequency Global Navigation Satellite System (GNSS) observations, it is preferable to formulate PPP–RTK functional models based on original (undifferenced and uncombined) observations. While there exist many variants of the undifferenced and uncombined PPP–RTK models, a unified theoretical framework needs developing to link these variants. In this contribution, we formulate a class of undifferenced and uncombined PPP–RTK functional models in a systematic way and cast them in a unified framework. This framework classifies the models into a code-plus-phase category and a phase-only category. Each category covers a variety of measurement scenarios on the network side, ranging from small-, medium- to large-scale networks. For each scenario, special care has been taken of the distinct ionospheric constraints and the difference between Code Division Multiple Access (CDMA) and Frequency Division Multiple Access (FDMA) signals. The key to systematically formulating these models lies in how to deal with the rank deficiency problems encountered. We opt for the Singularity-basis (S-basis) theory, giving rise to the full-rank observation equations in which the estimable parameters turn out to be the functions of original parameters and those selected as the S-basis. In the sequel, it becomes straightforward to derive for each scenario the user model as it, more or less, amounts to the single-receiver network model. Benefiting from the presented theoretical framework, the relationships and differences between various undifferenced and uncombined PPP–RTK models become clear, which can lead to the better use of these models in a specific situation.