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**Integer Ambiguity Resolution for Frequency-Varying Carrier Phase Signals: Theory and Numerical Results**

*Amir Khodabandeh, Department of Infrastructure Engineering, The University of Melbourne, and Peter J.G. Teunissen, Department of Geoscience and Remote Sensing, Delft University of Technology*

Peer Reviewed

Background and motivation: The carrier phase signals have served as key observations in various high-precision measurement systems and remote-sensing applications such as Global Navigation Satellite Systems (GNSS) (Teunissen and Montenbruck 2017), Very Long Baseline Interferometry (VLBI) (Hobiger et al. 2009), Interferometric Synthetic Aperture Radar (Kampes and Hanssen 2004), underwater acoustic positioning (Viegas and Cunha 2007), Radio Interferometric Positioning Systems (RIPS) (Maroti et al. 2005), and Opportunistic Navigation with non-conventional sensors (Shamaei and Kassas 2019). It is, for instance, the provision of GNSS carrier phase signals that leads to millimeter-level positioning parameter solutions. GNSS carrier phase measurements have also been shown to improve the precision of several other important model parameters such as, e.g., those for atmospheric sounding, instrumental calibration and time transfer (Khodabandeh and Teunissen 2018). The starring role of carrier phase measurements is particularly pronounced in case the integerness of the phase ambiguous cycles, i.e., the so-called ambiguities, can be fully exploited in the estimation process. Integer ambiguity resolution (IAR) is therefore often applied to integer-resolve the phase ambiguities. Ideally, one aims at resolving all available ambiguities of the measurement model. However, due to the lack of information content in the underlying system of observation equations, only certain functions of the ambiguities can be considered integer-estimable. If the integer-estimability condition is not met, IAR may fail to deliver correct solutions to the original integer ambiguities (Teunissen 2019). For the code division multiple access (CDMA) based navigation systems whose transmitters broadcast carrier phase signals on identical frequencies, such integer-estimable functions are the well-known double-differenced (DD) ambiguities (Khodabandeh and Teunissen 2019). This is, however, not the case with frequency-varying carrier phase signals as broadcast by, e.g., the GLONASS satellites, Low-Earth-Orbiting (LEO) communication satellites (Khalife et al. 2020), or cellular long-term evolution (LTE) transmitters. Since the corresponding wavelengths of such signals are transmitter-dependent, the classical double-differencing technique produces non-integer combinations of the phase ambiguities which clearly cannot serve as input to IAR. The goal of the present contribution is therefore to introduce new algorithmic tools that bring the observation equations of such frequency-varying carrier phase signals into the form to which IAR is applicable.

Methodology: Following the integer-estimability theory as developed by Teunissen (2019), we present an integer-sweeping algorithm to reparametrize mixed-integer rank-deficient models into their integer-estimable parametrized versions. . A prime example of such mixed-integer models is given by the linearized model of undifferenced GNSS observation equations having carrier phase and pseudorange observables that are linked to integer-valued phase ambiguities and real-valued unknown parameters, such as e.g., position coordinates, atmosphere parameters, receiver and satellite clock parameters, and instrumental biases. Our proposed integer-sweeping algorithm makes use of operations that consist of adding or subtracting integer multiples of a column to another column, and of reordering or sign-changing columns. Careful execution of such elementary operations is shown to deliver an admissible integer transformation with which one can form full-rank measurement models and identify the corresponding integer-estimable ambiguities.

Results and conclusions: We discuss the integer-estimability theory, presenting a theorem that provides the necessary and sufficient conditions for phase ambiguity functions to be integer-estimable. It shows that the conditions of unbiased estimability and integerness, although necessary, are not sufficient to guarantee that the unbiased integer estimation of these functions will do proper justice to the intrinsic integerness of carrier phase measurement models. Through a number of real-world GLONASS and simulated non-GNSS datasets, we show the feasibility of successful integer ambiguity-fixing in the measurement models formed by frequency-varying carrier phase signals. The measurement models considered range from a simple GLONASS frequency division multiple access (FDMA) single-baseline to a more involved network of non-GNSS sensors.

References:

- Hobiger T., Sekido M., Koyama Y., and Kondo T. (2009). Integer phase ambiguity estimation in next-generation geodetic Very Long Baseline Interferometry. Advances in Space Research, 43(1): 187–192

- Kampes B.M., and Hanssen R.F (2004). Ambiguity resolution for permanent scatterer interferometry. IEEE Transactions on Geoscience and Remote Sensing, 42(11): 2446–2453

- Khalife J., M. Neinavaie and Z. M. Kassas (2020). Navigation with Differential Carrier Phase Measurements from Megaconstellation LEO Satellites. IEEE/ION Position, Location and Navigation Symposium (PLANS): 1393-1404

- Khodabandeh A., and Teunissen P.J.G. (2018), On the impact of GNSS ambiguity resolution: geometry, ionosphere, time and biases. Journal of Geodesy, 92(6): 637–658

- Khodabandeh A., and Teunissen P.J.G. (2019) Integer estimability in GNSS networks. Journal of Geodesy 93(9):1805–1819

- Maroti M., Volgyesi P., Dora S., Kusy B., Nadas A., Ledeczi A., Balogh G., and Molnar K. (2005). Radio Interferometric Geolocation. Proceedings of the 3rd international conference on Embedded networked sensor systems, pp. 1–12.

- Shamaei K., and Kassas Z.M. (2019). Sub-Meter Accurate UAV Navigation and Cycle Slip Detection with LTE Carrier Phase Measurements. ION GNSS+ Conference Miami, FL, September 16–20, 2019.

- Teunissen P.J.G., Montenbruck O. (eds). (2017). Handbook of global navigation satellite systems. Springer, Berlin

- Teunissen P.J.G. (2019). A new GLONASS FDMA model. GPS Solutions, https://doi.org/10.1007/s10291-019-0889-0

- Viegas D.C.d.N., and Cunha S.R. (2007). Precise positioning by phase processing of sound waves. IEEE Transactions on Signal Processing, 55(12): 5731-5738

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