A method of determining a posterior error probability distribution for a parameter measured by a Global Navigation Satellite System (GNSS) receiver. The method comprises receiving a value for each of one or more GNSS measurement quality indicators associated with the GNSS measurement of the parameter. The or each received measurement quality indicator value is provided as an input into a multivariate probability distribution model to determine the posterior error probability distribution for the GNSS measurement, wherein the variates of the multivariate probability distribution model comprise error for said parameter, and the or each measurement quality indicator.

A method of determining a posterior error probability distribution for a parameter measured by a Global Navigation Satellite System (GNSS) receiver. The method comprises receiving a value for each of one or more GNSS measurement quality indicators associated with the GNSS measurement of the parameter. The or each received measurement quality indicator value is provided as an input into a multivariate probability distribution model to determine the posterior error probability distribution for the GNSS measurement, wherein the variates of the multivariate probability distribution model comprise error for said parameter, and the or each measurement quality indicator.

A method for reduced-outlier satellite positioning includes receiving a set of satellite positioning observations at a receiver; generating a first receiver position estimate; generating a set of posterior observation residual values from the set of satellite positioning observations and the first receiver position estimate; based on the set of posterior observation residual values, identifying a subset of the satellite positioning observations as statistical outliers; and after mitigating an effect of the statistical outliers, generating a second receiver position estimate having higher accuracy than the first receiver position estimate.

A method for reduced-outlier satellite positioning includes receiving a set of satellite positioning observations at a receiver; generating a first receiver position estimate; generating a set of posterior observation residual values from the set of satellite positioning observations and the first receiver position estimate; based on the set of posterior observation residual values, identifying a subset of the satellite positioning observations as statistical outliers; and after mitigating an effect of the statistical outliers, generating a second receiver position estimate having higher accuracy than the first receiver position estimate.

The present disclosure provides an authentication method for a satellite-based navigation system that in some implementations includes: obtaining, by an authentication server, a plurality of message authentication bundles from a plurality of reference receiver stations, each message authentication bundle comprising at least one hash for authenticating a corresponding SNS message received by a respective reference receiver station from at least one satellite of the SNS; inserting, by the authentication server at least a subset of the hashes of the obtained message authentication bundles into a probabilistic data structure (PDS), which forms at least a part of a validation filter for authenticating a corresponding subset of SNS messages; and distributing the validation filter to at least one receiver-side device. The disclosure further provides authentication methods performed by a reference receiver station and a receiver-side device, respectively, as well as devices and systems for authenticating SNS messages.

The present disclosure provides an integrity monitoring method of ionosphere gradient based on kinematical to kinematical platform, comprising step 1, constructing geometry-free and ionospheric amplification type detection statistics, based on original triple-frequency carrier phase observations, step 2, adjusting a detection threshold based on a required monitoring false alarm rate, and determining whether the detection statistics are less than the adjusted detection threshold, step 3, comparing a calculated miss-detection rate and a required miss-detection rate, and determining whether the calculated miss-detection rate are less than the required miss-detection rate, and step 4, if the detection statistics are less than the adjusted detection threshold and the calculated miss-detection rate are less than the required miss-detection rate, considering the ionosphere gradient is normal.

An augmentation information adjustment unit (102) reduces an amount of information in augmentation information by combining: update cycle adjustment processing (1021) to set an update cycle of the augmentation information to be an integer multiple of a predetermined update cycle; geographic interval error value adjustment processing (1022) to reduce the number of geographic interval error values by selecting from among a plurality of the geographic interval error values each of which is an error at every predetermined geographic interval out of a plurality of error values, a geographic interval error value at every geographic interval that is an integer multiple of the predetermined geographic interval; and bit count adjustment processing (1023) to reduce a bit count of the error value for each error value. An augmentation information output unit (103) outputs, to an output destination, augmentation information after being reduced in the amount of information by the augmentation information adjustment unit (102).

Methods and apparatus to monitor GPS/GNSS atomic clocks are disclosed. An example method includes establishing a measured difference between an atomic frequency standard (AFS) and a monitoring device. The method also includes modeling an estimated difference model between the AFS and the monitoring device, and computing a residual signal based on the measured difference and the estimated difference model. In addition, the method includes analyzing, by a first detector, the residual signal at multiple thresholds, each of the thresholds having a corresponding persistency defining the number of times a threshold is exceeded before one or more of a phase jump, a rate jump, or an acceleration error is indicated. Furthermore, the method includes analyzing, by a second detector, a parameter of the estimated difference model at multiple thresholds, each of the thresholds having a corresponding persistency defining the number of times a drift threshold is exceeded before a drift is indicated.

Methods and apparatus to monitor GPS/GNSS atomic clocks are disclosed. An example method includes establishing a measured difference between an atomic frequency standard (AFS) and a monitoring device. The method also includes modeling an estimated difference model between the AFS and the monitoring device, and computing a residual signal based on the measured difference and the estimated difference model. In addition, the method includes analyzing, by a first detector, the residual signal at multiple thresholds, each of the thresholds having a corresponding persistency defining the number of times a threshold is exceeded before one or more of a phase jump, a rate jump, or an acceleration error is indicated. Furthermore, the method includes analyzing, by a second detector, a parameter of the estimated difference model at multiple thresholds, each of the thresholds having a corresponding persistency defining the number of times a drift threshold is exceeded before a drift is indicated.

Methods and apparatuses for a receiver of signals from one or more satellite navigational systems to detect and/or eliminate impaired satellites from the set of estimated/acquired satellites in view are described. One method includes acquiring coarse position, time, and frequency values for each of a plurality of satellites from one or more satellite navigational systems, the plurality of satellites being those currently estimated to be in view of the receiver; determining whether one or more of the acquired coarse values are within a minimum range; and if it is determined that the one or more acquired coarse values are within the minimum range: determining a pseudo-true peak of a position domain correlogram comprising Line of Sight (LOS) vectors of each of the plurality of satellites; and identifying any satellite whose cross-correlation peak is beyond a maximum distance from the pseudo-true peak as an impaired satellite.