Pseudo-clock Biases for Precise Point Positioning – The Algebraic Approach
Résumé
As shown in a companion paper devoted to GNSS networks in algebraic graph theory, any (real- or) integer-valued function taking its values on the edges of the GNSS graph can be regarded as the sum of three (real- or) integer-valued functions: a function taking its values on the receiver vertices of this graph, another one on the satellite vertices, and the last one, the closure-delay (CD) function, taking its values on the loop-closure edges. For a given spanning tree, this decomposition is unique. The notion of closure delay generalizes that of double difference (DD). In this framework, particular satellite biases can be estimated and broadcasted to the network users for their precise point positioning (PPP). For example, in the case of large networks, each of these biases includes three (or four) terms: a satellite-clock term, a satellite time-group term, a satellite ionospheric term, and (for the phase) a satellite integer ambiguity mulitplied by the corresponding wavelength. the form of the PPP equations to be solved by the network user is then the same as that of the traditional PPP equations. As soon as the CD ambiguities are fixed and validated, estimates of these float biases can be obtained. the main result of this paper is that no other ambiguity is then to be fixed, hence a better efficiency. In particular, in this approach, it is not necessary to fix the carrier-phase ambiguities, a problem which cannot be easily solved. Indeed, as shown in this paper, when the CD ambiguities are fixed (or when a maximum set of DD ambiguities is fixed), the remaining float problem is not of full rank.