It is common practice to use the well-known concept of the minimal detectable bias (MDB) to assess the performance of statistical testing procedures. However, such procedures are usually applied to a null and a set of multiple alternative hypotheses with the aim of selecting the most likely one. Therefore, in the DIA method for the detection, identification and adaptation of model misspecifications, rejection of the null hypothesis is followed by identification of the potential source of the model misspecification. With identification included, the MDBs do not truly reflect the capability of the testing procedure and should therefore be replaced by the minimal identifiable bias (MIB). In this contribution, we analyse the MDB and the MIB, highlight their differences, and describe their impact on the nonlinear DIA-estimator of the model parameters. As the DIA-estimator inherits all the probabilistic properties of the testing procedure, the differences in the MDB and MIB propagation will also reveal the different consequences a detection-only approach has versus a detection+identification approach. Numerical algorithms are presented for computing the MDB and the MIB and also their effect on the DIA-estimator. These algorithms are then applied to a number of examples so as to analyse and illustrate the different concepts.

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An assessment of standalone GLONASS RTK performance is provided using its FDMA and CDMA signals. The new integer-estimable GLONASS FDMA model (Teunissen 2019), which guarantees the integer-estimability of its ambiguities, is employed. We introduce the combined integer-estimable GLONASS FDMA+CDMA model and compare its strength against the FDMA model for instantaneous integer ambiguity resolution and positioning. Various combinations of GLONASS signals are considered including FDMA L1, FDMA+CDMA L1+L3, FDMA L1+L2 and FDMA+CDMA L1+L2+L3. To provide insight into the current RTK performance of GLONASS, we used observations of a short baseline to analyze the integer ambiguity resolution success rate and positioning precision, formally and empirically. To provide insight into the future RTK performance of GLONASS, we present a formal analysis of the integer ambiguity resolution success rate and ADOP, assuming that all the GLONASS satellites transmit FDMA L1, L2 and CDMA L3 signals. A formal analysis of standalone GLONASS ambiguity resolution based on current and future GLONASS constellation is then presented for different locations around the world.

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The goal of this contribution is to assess the impact statistical model selection has on confidence levels of parameter estimators in linear(ized) GNSS models. In the processing of observational data, parameter estimation and statistical testing are often combined. A testing procedure is exercised to select the most likely observational model among the hypothesized ones, which is then followed by the estimation of the identified model parameters. The resulting estimator will inherit the uncertainties involved in both estimation and testing which need to be properly taken into account when computing the corresponding confidence level. The approach that is usually followed in practice to determine the confidence level is to compute the probability of the estimator lying in a region around its true value conditioned on the identified hypothesis. Therefore, use is made of the estimator’s distribution under the identified hypothesis without regard to the conditioning process that led to the decision of accepting this hypothesis. In this contribution, it will be shown that for a proper computation of the confidence level in combined estimation-testing procedures, the associated probability should be conditioned not only on the identified hypothesis, but also on the testing outcome that led to the decision of accepting this hypothesis. Therefore, use need to be made of the conditional distribution of the estimator. We will provide numerical analysis of confidence levels with and without accounting for conditioning on testing decision using a number of examples in the context of GNSS single point positioning. It will be demonstrated that the customary practice which makes use of unconditional distributions to evaluate the confidence level, may give a too optimistic description of the estimator’s quality.@en

The DIA-method, for the detection, identification and adaptation of modeling errors, has been widely used in a broad range of applications including the quality control of geodetic networks and the integrity monitoring of GNSS models. The DIA-method combines two key statistical inference tools, estimation and testing. Through the former, one seeks estimates of the parameters of interest, whereas through the latter, one validates these estimates and corrects them for biases that may be present. As a result of this intimate link between estimation and testing, the quality of the DIA outcome x̄ must also be driven by the probabilistic characteristics of both estimation and testing. In practice however, the evaluation of the quality of x̄ is never carried out as such. Instead, use is made of the probability density function (PDF) of the estimator under the identified hypothesis, say x̂_i, thereby thus neglecting the conditioning process that led to the decision to accept the i^th hypothesis. In this contribution, we conduct a comparative study of the probabilistic properties of x̄ and x̂_i. Our analysis will be carried out in the framework of GNSS-based positioning. We will also elaborate on the circumstances under which the distribution of the estimator x̂_i provides either poor or reasonable approximations to that of the DIA-estimator x̄.

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The Indian Regional Navigation Satellite System (IRNSS) has recently (as of May 2016) become operational. The system has been developed with the objective of offering positioning, navigation, and timing (PNT) to users in its two service areas, covering the Indian landmass and the Indian Ocean, respectively. It is the goal of this contribution to provide further insight into the full-constellation L5 pseudorange single-point positioning (SPP) capabilities of the system. A detailed dilution of precision (DOP) analysis of its two service areas, including the identification, in location and time, of poor receiver-satellite geometries is provided. It is hereby demonstrated how the impact of some of these poor receiver-satellite geometries can be mitigated by means of height-constraining. An overview and analysis of the SPP precision is also provided including easy-to-use representative day-averaged values for a grid of locations covering the two service areas.

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The Indian Regional Navigation Satellite System (IRNSS) has recently (May 2016) reached its full operational capability. In this contribution, we provide the very first L5 attitude determination analyses of the fully operational IRNSS as a standalone system and also in combination with the fully operational GPS Block IIF along with the corresponding ambiguity resolution results. Our analyses are carried out for both a linear array of two antennas and a planar array of three antennas at Curtin University, Perth, Australia. We study the noise characteristics (carrier-to-noise density, measurement precision, time correlation), the integer ambiguity resolution performance (LAMBDA, MC-LAMBDA) and the attitude determination performance (ambiguity float and ambiguity fixed). A prerequisite for precise and fast IRNSS attitude determination is the successful resolution of the double-differenced integer carrier-phase ambiguities. In this contribution, we will compare the performance of the unconstrained and the multivariate-constrained LAMBDA method. It is therefore also shown what improvements are achieved when the known body geometry of the antenna array is rigorously incorporated into the ambiguity objective function. As our ambiguity-fixed outcomes show consistency between empirical and formal results, we also formally assess the precise attitude determination performance for several locations within the IRNSS service area.

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The Indian Regional Navigation Satellite System (IRNSS) has recently (May 2016) become fully operational. In this contribution, for the fully operational IRNSS as a stand-alone system and also in combination with GPS, we provide a first assessment of L5 integer ambiguity resolution and positioning performance. While our empirical analyses are based on the data collected by two JAVAD receivers at Curtin University, Perth, Australia, our formal analyses are carried out for various onshore locations within the IRNSS service area. We study the noise characteristics (carrier-to-noise density, measurement precision, time correlation), the integer ambiguity resolution performance (success rates and ambiguity dilution of precision), and the positioning performance (ambiguity float and ambiguity fixed). The results show that our empirical outcomes are consistent with their formal counterparts and that the GPS L5-data have a lower noise level than that of IRNSS L5-data, particularly in case of the code data. The underlying model in our assessments varies from stand-alone IRNSS (L5) to IRNSS (Formula presented.) GPS (L5), from unconstrained to height-constrained and from kinematic to static. Significant improvements in ambiguity resolution and positioning performance are achievable upon integrating L5-data of IRNSS with GPS.@en

The Indian Regional Navigation Satellite System (IRNSS) has recently (May 2016) become fully-operational and has been provided with the operational name of NavIC (Navigation with Indian Constellation). It has been developed by the Indian Space Research Organization (ISRO) with the objective of offering positioning, navigation and timing (PNT) to the users in its service area. This contribution provides for the first time an assessment of the IRNSS L5-signal capability to achieve instantaneous attitude determination on the basis of data collected in Perth, Australia. Our evaluations are conducted for both a linear array of two antennas and a planar array of three antennas. A pre-requisite for precise and fast IRNSS attitude determination is the successful resolution of the double-differenced (DD) integer carrier-phase ambiguities. In this contribution, we will compare the performances of different such methods, amongst which the unconstrained and the multivariate-constrained LAMBDA method for both linear and planar arrays. It is demonstrated that the instantaneous ambiguity success rates increase from 15% to 90% for the linear array and from 5% to close to 100% for the planar array, thus showing that standalone IRNSS can realize 24-h almost instantaneous precise attitude determination with heading and elevation standard deviations of 0.05 and 0.10 degrees, respectively.

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The current Galileo constellation in April 2017 comprises both in-orbit validation and full operational capability satellites transmitting signals on five frequencies, i.e., E1, E5a, E5b, E5, and E6. We analyze the power, multipath and noise of these signals using the data collected by four short baselines of various lengths and receiver/antenna types in Perth, Australia, as well as the Netherlands. In our analysis, the Galileo signals, except E5, show different relative noise and multipath performance for different receiver/antenna types. The E5 signal, with a weak dependency on the type of receiver/antenna, shows a significantly lower level of multipath and noise with respect to the other signals. Estimations of the E5 code standard deviation based on the data of each of the mentioned baselines gives a value of about 6 cm, which is further reduced to about 1 cm once the data are corrected for multipath. Due to the superior stochastic properties of E5 signal compared to the other Galileo signals, we further analyze the short-baseline real-time kinematic performance of the Galileo standalone E5 observations. Our findings confirm that the Galileo E5 data, if corrected for the multipath effect, can make (almost) instantaneous ambiguity resolution feasible already based on the current constellation.

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A first assessment of GLONASS CDMA L3 ambiguity resolution and positioning performance is provided. Our analyses are based on GLONASS L3 data from the satellite pair SVNs 755-801, received by two JAVAD receivers at Curtin University, Perth, Australia. In our analyses, four different versions of the two-satellite model are applied: the geometry-free model, the geometry-based model , the height-constrained geometry-based model, and the geometry-fixed model. We study the noise characteristics (carrier-to-noise density, measurement precision), the integer ambiguity resolution performance (success rates and distribution of the ambiguity residuals), and the positioning performance (ambiguity float and ambiguity fixed). The results show that our empirical outcomes are consistent with their formal counterparts and that the GLONASS data have a lower noise level than that of GPS, particularly in case of the code data. This difference is not only seen in the noise levels but also in their onward propagation to the ambiguity time series and ambiguity residuals distribution.

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Knowledge of inter-system biases (ISBs) is essential to combine observations of multiple global and regional navigation satellite systems (GNSS/RNSS) in an optimal way. Earlier studies based on GPS, Galileo, BDS and QZSS have demonstrated that the performance of multi-GNSS real-time kinematic positioning is improved when the differential ISBs (DISBs) corresponding to signals of different constellations but transmitted at identical frequencies can be calibrated, such that only one common pivot satellite is sufficient for inter-system ambiguity resolution at that particular frequency. Recently, many new GNSS satellites have been launched. At the beginning of 2016, there were 12 Galileo IOV/FOC satellites and 12 GPS Block IIF satellites in orbit, while the Indian Regional Navigation Satellite System (IRNSS) had five satellites launched of which four are operational. More launches are scheduled for the coming years. As a continuation of the earlier studies, we analyze the magnitude and stability of the DISBs corresponding to these new satellites. For IRNSS this article presents for the first time DISBs with respect to the L5/E5a signals of GPS, Galileo and QZSS for a mixed-receiver baseline. It is furthermore demonstrated that single-frequency (L5/E5a) ambiguity resolution is tremendously improved when the multi-GNSS observations are all differenced with respect to a common pivot satellite, compared to classical differencing for which a pivot satellite is selected for each constellation.

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