The Fixed Point Theorem of Ambiguity Resolution for Precise Point Positioning of GPS Networks: Theory and Applications

The properties of ambiguity resolution are explored both theoretically and by experiment when applied to GPS network solutions that have first been derived by precise point positioning (PPP). Since its invention by Zumberge et al. [1997], PPP has become popular for regional network processing, because formal covariances between stations are zero, and so processing time scales linearly with the number of stations (unlike traditional processing models that scale quadratically). PPP network solutions can be further improved by the application of ambiguity resolution, however, the processing time for this step generally scales quadratically [Blewitt et al., 1989]. Furthermore, ambiguity resolution introduces inter-station correlations that cause subsequent network kinematic analysis to scale quadratically rather than linearly. Thus some of the practical advantages of PPP are subsequently lost. A theoretical understanding of how PPP networks respond to ambiguity resolution may point the way to faster, linear processing schemes, and may help to assess from a theoretical perspective current ad hoc schemes for producing solutions that are experimentally almost identical to optimal solutions. A reasonable condition for implementing such schemes is that the differences between optimal and sub-optimal solutions should be statistically insignificant ("near-optimal"). Here a fixed point theorem is derived, which identifies linear combinations of network parameters that are theoretically invariant under ambiguity resolution. This theorem is useful to assist the design and justification of near-optimal network processing schemes that scale linearly with the number of stations. In addition to reducing processing time, linear schemes readily lend themselves to parallel processor implementation, thus there is the potential to reduce real processing time by several of orders of magnitude for extremely large networks. Such schemes would allow for very rapid, multiple reanalysis of extremely large networks to assess various models.