Rank-defect integer estimation and phase-only modernized GPS ambiguity resolution

Carrier-phase ambiguity resolution is usually based on the assumption that the underlying model of observation equations is of full rank. In this contribution the model of observation equations is assumed to be of less than full rank. The well-known three-step procedure of integer least squares is generalized and it is shown how the solution can be affected by the rank deficiency. Although the theory is of interest in its own right, a prime application is found in the problem of phase-only ambiguity resolution in the presence of ionospheric delays. The impact of the third global positioning system (GPS) frequency is therefore studied and it is shown by means of suitable ambiguity transformation which ambiguities are integer estimable and which are not in the case of phase-only modernized GPS. A pitfall when using ionosphere-free linear phase combinations is identified. It is shown that only a particular class of such linear phase combinations permits a parameterization in terms of integer-estimable ambiguities. This pitfall does not manifest itself so clearly with the current dual-frequency GPS system.