On the sensitivity of the location, size and shape of the GPS ambiguity search space to certain changes in the stochastic model

In this contribution we analyse in a qualitative sense for the geometry-free model the dependency of the location, the size and the shape of the ambiguity search space on di€erent factors of the stochastic model. For this purpose a rather general stochastic model is used. It includes time-correlation, cross-correlation, satellite elevation dependency and the use of an a priori weighted ionospheric model, having the ionosphere®xed model and the ionospherēoat model as special cases. It is shown that the location is invariant for changes in the cofactor matrix of the phase observables. This also holds true for the cofactor matrix of the code observables in the ionospherēoat case. As for timecorrelation and satellite elevation dependency, it is shown that they only a€ect the size of the search space, but not its shape and orientation. It is also shown that the least-squares ambiguities, their variance matrix and its determinant, for, respectively, the ionosphere-®xed model, the ionospherēoat model and the ionosphereweighted model, are all related through the same scalar weighted mean, the weight of which is governed by the variance ratio of the ionospheric delays and the code observables. A closed-form expression is given for the area of the search space in which all contributing factors are easily recognized. From it one can infer by how much the area gets blown up when the ionospheric spatial decorrelation increases. This multiplication factor is largest when one switches from the ionosphere®xed model to the ionospherēoat model, in which case it is approximately equal to the ratio of the standard deviation of phase with that of code. The area gives an indication of the number of grid points inside the search space. Key words. GPS á Ambiguity search space á Stochastic model 1 Introduction Integer ambiguity estimation is a prerequisite for fast, high-precision GPS relative positioning. It is based on the integer least-squares principle min a …âÿ a† Qÿ1 â …âÿ a† ; a integer …1† where â is the vector of real-valued least-squares ambiguities and Qâ its variance matrix. Due to the integer constraints on the double-di€erenced (DD) ambiguities and the fact that the variance matrix is non-diagonal, the solution of Eq. (1) must be obtained by means of a search. An ecient method for computing the integer least-squares ambiguities has been introduced in Teunissen (1993) and reviewed in Kleusberg and Teunissen (1996); it is the least-squares ambiguity decorrelation adjustment (LAMBDA). Implementation aspects of the method can be found in de Jonge and Tiberius (1996) and de Jonge et al. (1996). The method makes use of the ambiguity search space, which when formulated in the space of the original DD ambiguities is de®ned as …âÿ a† Qÿ1 â …âÿ a† v …2† It is located at â, its shape and orientation are governed by Qâ and its size can be controlled by v. In this contribution we will study how the location, size and shape of the ambiguity search space are driven by the stochastic model. For this purpose we will make use of a rather general stochastic model. It includes time-correlation, cross-correlation, satellite elevation dependency and a weighting of the ionospheric delays. Our analysis will be based on the geometry-free model (see e.g. Hatch 1982; Melbourne 1985; WuÈ bbena 1985; Euler and Goad 1991; Dedes and Goad 1994; Euler and Hatch 1994; Teunissen 1996). In order to get a deeper insight as to the way in which the di€erent factors of the stochastic model contribute, our analysis will be of a Journal of Geodesy (1997) 71: 541±551