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Carrier phase signals are considered among the key observations in global navigation satellite systems (GNSSs) and several other high-precision inter-ferometric measurement systems. However, these ultra-precise measurements are not fully exploited when the integerness of their inherent ambiguities is discarded during the estimation process. Provided that the integer-estimable functions of their phase ambiguities are properly identified, integer ambiguity resolution (IAR) can be utilized to benefit their parameter solutions. For the GNSS code division multiple access systems with transmitters that broadcast carrier phase signals on identical frequencies, these integer-estimable functions have been characterized and are well-known as double differenced ambiguities. However, this is not the case with "frequency-varying" carrier phase signals that are broadcast by GLONASS satellites, Low-Earth-Orbiting communication satellites, or cellular long-term evolution (LTE) transmitters. This study aims to present full-rank models that can be used to identify integer-estimable ambiguity functions, thereby bringing the observation equations of frequency-varying carrier phase measurements into an IAR-applicable form. Our analytical results are supported by several numerical examples, including GNSS and terrestrial-based IAR as well as a new set of "inter-frequency" integer ambiguities that this study discovers in Galileo multi-frequency carrier phase signals. [ABSTRACT FROM AUTHOR]

To obtain single-receiver Global Navigation Satellite System (GNSS) parameter solutions, the PPP-RTK user-filter combines measurements with time-correlated corrections that are separately computed by the filter of an external provider. The consequence of exercising such double-filtering is that the Kalman filter's standard assumption of having uncorrelated measurements in time becomes violated. This leads the user-filter to lose its 'minimum variance' property, thereby delivering imprecise parameter solutions. The solutions' precision-loss becomes more pronounced when one experiences an increase in the correction latency, i.e., the delay in time after the corrections are estimated and the time they are applied to the user measurements. In this contribution, we propose a new multi-epoch formulation for the PPP-RTK user-filter upon which both the uncertainty and the temporal correlation of the corrections are incorporated. By a proper augmentation of the user-filter state-vector, the corrections are jointly measurement-updated with the user parameter solutions. Supported by numerical results, the proposed formulation is shown to outperform its commonly used counterpart in the minimum-variance sense. [ABSTRACT FROM AUTHOR]

In this contribution, we generalize PPP–RTK theory by allowing the transmitters to transmit on different frequencies. The generalization is based on the integer-estimability theory of Teunissen (A new GLONASS FDMA model. GPS Solutions, 2019). As the theory and associated algorithms provided are generally applicable, they apply to satellite-based carrier-phase positioning as well as to terrestrial interferometric sensory networks. Based on an identification of the constraints imposed on the admissible ambiguity transformations by PPP–RTK, a fundamental network+user condition is found that determines whether PPP–RTK is possible or not. The discriminating contributions of both the network and user observation equations to this PPP–RTK condition are analysed, followed by a description of PPP–RTK enabling classes of measurement scenarios. [ABSTRACT FROM AUTHOR]

Estimability and S -systems are important concepts when dealing with rank-defect models. In this contribution, we generalize the concept of estimability to integer estimability and determine the necessary and sufficient conditions that need to be satisfied for parameter functions to be integer estimable. This is then worked out and applied to the integer estimability analysis of GNSS observation equations. We hereby consider both network ambiguity resolution and single-receiver PPP-RTK ambiguity resolution. In our analyses, use is made of graph theory and properties of the ambiguity incidence matrices of the bipartite and connected network graphs. [ABSTRACT FROM AUTHOR]

The Minimal Detectable Bias (MDB) is an important diagnostic tool in data quality control. The MDB is traditionally computed for the case of testing the null hypothesis against a single alternative hypothesis. In the actual practice of statistical testing and data quality control, however, multiple alternative hypotheses are considered. We show that this has two important consequences for one's interpretation and use of the popular MDB. First, we demonstrate that care should be exercised in using the single-hypothesis-based MDB for the multiple hypotheses case. Second, we show that for identification purposes, not the MDB, but the Minimal Identifiable Bias (MIB) should be used as the proper diagnostic tool. We analyse the circumstances that drive the differences between the MDBs and MIBs, show how they can be computed using Monte Carlo simulation and illustrate by means of examples the significant differences that one can experience between detectability and identifiability. [ABSTRACT FROM AUTHOR]

In this contribution, we present and analyze datasnooping in the context of the DIA method. As the DIA method for the detection, identification and adaptation of mismodelling errors is concerned with estimation and testing, it is the combination of both that needs to be considered. This combination is rigorously captured by the DIA estimator. We discuss and analyze the DIA-datasnooping decision probabilities and the construction of the corresponding partitioning of misclosure space. We also investigate the circumstances under which two or more hypotheses are nonseparable in the identification step. By means of a theorem on the equivalence between the nonseparability of hypotheses and the inestimability of parameters, we demonstrate that one can forget about adapting the parameter vector for hypotheses that are nonseparable. However, as this concerns the complete vector and not necessarily functions of it, we also show that parameter functions may exist for which adaptation is still possible. It is shown how this adaptation looks like and how it changes the structure of the DIA estimator. To demonstrate the performance of the various elements of DIA-datasnooping, we apply the theory to some selected examples. We analyze how geometry changes in the measurement setup affect the testing procedure, by studying their partitioning of misclosure space, the decision probabilities and the minimal detectable and identifiable biases. The difference between these two minimal biases is highlighted by showing the difference between their corresponding contributing factors. We also show that if two alternative hypotheses, say Hi and Hj, are nonseparable, the testing procedure may have different levels of sensitivity to Hi-biases compared to the same Hj-biases. [ABSTRACT FROM AUTHOR]

In the past few decades the Global Positioning System (GPS) has been the number one positioning tool in a range of Geodesy and Geophysics applications. GPS modernisation from dual-frequency to triple-frequency signals has lately also been complemented by global GLObal' naya NAvigatsionnaya Sputnikovaya Sistema (GLONASS), Galileo and BeiDou Navigation Satellite System (BDS), as well as the regional Quasi-Zenith Satellite system (QZSS) and Navigation with Indian Constellation (NavIC). Single-frequency receivers, in particular, need ionospheric models to be able to precisely determine positions, and by making use of ionospheric models one can also speed up the parameter convergence for dual- and multi-frequency GNSS applications. [Extracted from the article]

Integer ambiguity resolution (IAR) is the key to fast and precise GNSS positioning and navigation. Next to the positioning parameters, however, there are several other types of GNSS parameters that are of importance for a range of different applications like atmospheric sounding, instrumental calibrations or time transfer. As some of these parameters may still require pseudo-range data for their estimation, their response to IAR may differ significantly. To infer the impact of ambiguity resolution on the parameters, we show how the ambiguity-resolved double-differenced phase data propagate into the GNSS parameter solutions. For that purpose, we introduce a canonical decomposition of the GNSS network model that, through its decoupled and decorrelated nature, provides direct insight into which parameters, or functions thereof, gain from IAR and which do not. Next to this qualitative analysis, we present for the GNSS estimable parameters of geometry, ionosphere, timing and instrumental biases closed-form expressions of their IAR precision gains together with supporting numerical examples. [ABSTRACT FROM AUTHOR]

Single-receiver integer ambiguity resolution (IAR) is a measurement concept that makes use of network-derived non-integer satellite phase biases (SPBs), among other corrections, to recover and resolve the integer ambiguities of the carrier-phase data of a single GNSS receiver. If it is realized, the very precise integer ambiguity-resolved carrier-phase data would then contribute to the estimation of the receiver’s position, thus making (near) real-time precise point positioning feasible. Proper definition and determination of the SPBs take a leading part in developing the idea of single-receiver IAR. In this contribution, the concept of array-based between-satellite single-differenced (SD) SPB determination is introduced, which is aimed to reduce the code-dominated precision of the SD-SPB corrections. The underlying model is realized by giving the role of the local reference network to an array of antennas, mounted on rigid platforms, that are separated by short distances so that the same ionospheric delay is assumed to be experienced by all the antennas. To that end, a closed-form expression of the array-aided SD-SPB corrections is presented, thereby proposing a simple strategy to compute the SD-SPBs. After resolving double-differenced ambiguities of the array’s data, the variance of the SD-SPB corrections is shown to be reduced by a factor equal to the number of antennas. This improvement in precision is also affirmed by numerical results of the three GNSSs GPS, BeiDou and QZSS. Experimental results demonstrate that the integer-recovered ambiguities converge to integers faster, upon increasing the number of antennas aiding the SD-SPB corrections. [ABSTRACT FROM AUTHOR]