**G1.1 Session: Recent Developments in Geodetic Theory**

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**Estimability in Rank-Defect Mixed-Integer Models: Theory and Applications**

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PJG Teunissen^{1,2}

^{1}GNSS Research Centre, Curtin University, Perth, Australia

^{2}Geoscience and Remote Sensing, Delft University of Technology, The Netherlands

Email: p.teunissen@curtin.edu.au; p.j.g.teunissen@tudelft.nl

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Although estimability is one of the foundational concepts of todays’ estimation theory, we show that the current concept of estimability is not adequately equipped to cover the estimation requirements of mixed-integer models, for instance like those of interferometric models, cellular base transceiver network models or the carrier-phase based models of Global Navigation Satellite Systems (GNSSs). We therefore need to generalize the estimability concept to that of integer-estimability. Next to being integer and estimable in the classical sense, functions of integer parameters then also need to guarantee that their integerness corresponds with integer values of the parameters the function is taken of. This is particularly crucial in the context of integer ambiguity resolution. Would this condition not be met, then the integer fixing of integer functions that are not integer-estimable implies that one can fix the undifferenced integer ambiguities to non-integer values and thus force the model to inconsistent and wrong constraints.

In this paper we present a generalized concept of estimability and one that now also is applicable to mixed-integer models. We thereby provide the operationally verifiable necessary and sufficient conditions that a function of integer parameters needs to satisfy in order to be integer-estimable. As one of the conditions we have that estimable functions become integer-estimable if they can be unimodulair transformed to canonical form. Next to the conditions, we also show how to create integer-estimable functions and how a given design matrix can be expressed in them. We then show how these results are to be applied to interferometric models, cellular base transceiver network models and FDMA GNSS models.

**Keywords: **Estimability, S-system theory, Mixed-integer Models, Integer-Estimability, Admissible Ambiguity Transformation, Interferometry, Global Navigation Satellite Systems (GNSSs)