Primal and dual mixed-integer models for Global Navigation Satellite Systems

Mixed-integer models arise in several geodetic problems, including precise positioning and remote sensing in Global Navigation Satellite Systems (GNSS), as well as deformation monitoring through Interferometric Synthetic Aperture Radar (InSAR) or fringe phase observations from Very Long Baseline Interferometry (VLBI). These problems generally involve two types of unknowns: integer ambiguities a ∈ Z^n and real-valued parameters b ∈ R^p, whose accuracy can be significantly improved by correctly resolving the ambiguities. However, in some cases, a large number of ambiguity components are involved and need to be correctly resolved; therefore the ambiguity resolution process becomes a bottleneck for the computations.

One research question is therefore how to effectively tackle the challenges of high-dimensional ambiguity resolution and its computational complexity, while ensuring a successful resolution of the ambiguities. Additionally, a second question arises regarding whether it is possible to solve this challenging problem in the domain of real-valued parameters, e.g. positioning coordinates, given that those are usually the parameters of interest for the user. In response to such questions, this doctoral dissertation is structured in two main parts: the first one looks in the integer ambiguity domain, presenting new flexible estimators and algorithms, ultimately merged into the new Least-squares AMBiguity Decorrelation Adjustment (LAMBDA) 4.0 toolbox; the second one focuses on the real-valued parameter domain where the integerness of ambiguities is still taken into account, where novel dual estimators are presented and an optimal, globally convergent solution is constructed using the branch-and-bound method.