Volume 10, Issue 4

The present paper investigates accuracy of single and dual-frequency Global Positioning System (GPS) standard point positioning solutions employing different ionosphere error mitigation techniques. The total electron content (TEC) in the ionosphere is the prominent delay error source in GPS positioning, and its elimination is essential for obtaining a relatively precise positioning solution. The estimated delay error from different ionosphere models and maps, such as Klobuchar model, global ionosphere models, and vertical TEC maps are compared with the locally derived ionosphere error following the ion density and frequency dependence with delay error. Finally, the positional accuracy of the single and dual-frequency GPS point positioning solutions are probed through different ionospheric mitigation methods including exploitation of models, maps, and ionosphere-free linear combinations and removal of higher order ionospheric effects. The results suggest the superiority of global ionosphere maps for single-frequency solution, whereas for the dual-frequency measurement the ionosphere-free linear combination with prior removal of higher-order ionosphere effects from global ionosphere maps and geomagnetic reference fields resulted in improved positioning quality among the chosen mitigation techniques. Conspicuously, the susceptibility of height component to different ionospheric mitigation methods are demonstrated in this study which may assist the users in selecting appropriate technique for precise GPS positioning measurements.

This paper introduces a comparison between dual-frequency precise point positioning (PPP) post-processing model, which combines the observations of three different GNSS constellations, namely GPS, Galileo, and BeiDou and real-time PPP model. A drawback of a single GNSS system such as GPS, however, is the availability of sufficient number of visible satellites in urban areas. Combining GNSS observations offers more visible satellites to users, which in turn is expected to enhance the satellite geometry and the overall positioning solution. However, combining several GNSS observables introduces additional biases, which require rigorous modelling, including the GNSS time offsets and hardware delays. In this paper, a GNSS post-processing PPPP model is developed using ionosphere-free linear combination. The additional biases of the GPS, Galileo, and BeiDou combination are accounted for through the introduction of a new unknown parameter, which is identified as the inter-system bias, in the PPP mathematical model. Natural Resources Canada’s GPSPace PPP software is modified to enable a combined GPS / Galileo / BeiDou PPP solution and to handle the newly inter-system bias. A total of four data sets at four IGS stations are processed to verify the developed PPP model. Precise satellite orbit and clock products from the IGS-MGEX network are used to correct of the GPS, Galileo and BeiDou measurements. For the real-time PPP model the corrections of the satellites orbit and clock are obtained through the international GNSS service (IGS) real-time service (RTS). GPS and Galileo Observations are used for the GNSS RTS-IGS PPP model as the RTS-IGS satellite products are not available for BeiDou satellites. This paper provides the GNSS RTS-IGS PPP model using different satellite clock corrections namely: IGS01, IGC01, IGS01, and IGS03. All PPP models results of convergence time and positioning precision are compared to the traditional GPS-only PPP model. It is shown that combining GPS, Galileo, and BeiDou observations in a PPP model reduces the convergence time by 25 % compared with the GPS-only PPP model.

Although Global Positioning System (GPS) primarily provide accurate estimates of position, velocity and time of the receiver, as the signals pass through the atmoshphere carrying its signatures, thus offers opportunities for atmoshpheric applications. Precipitable water vapor (PWV) is a vital component of the atmosphere and significantly influences atmospheric processes like rainfall and atmospheric temperature. The developing networks of continuously operating GPS can be used to efficiently estimate PWV. The Turkish Permanent GPS Network (TPGN) is employed to monitor PWV information in Turkey. This work primarily aims to derive long-term data of PWV by using atmospheric path delays observed through continuously operating TPGN from November 2014 to October 2015. A least square mathematical approach was then applied to establish the relation of the observed PWV to rainfall and temperature. The modeled PWV was correlated with PWV estimated from GPS data, with an average correlation of 67.10 %–88.60 %. The estimated root mean square error (RMSE) varied from 2.840 to 6.380, with an average of 4.697. Finally, data of TPGN, rainfall, and temperature were obtained for less than 2 months (November 2015 to December 2015) and assessed to validate the mathematical model. This study provides a basis for determining PWV by using rainfall and temperature data.

A new method for dealing with systematic effects in laser scanning and visualizing them by confidence regions is derived. The standard deviations of the systematic effects are obtained by repeatedly measuring three-dimensional coordinates by the laser scanner. In addition, autocovariance and cross-covariance functions are computed by the repeated measurements and give the correlations of the systematic effects. The normal distribution for the measurements and the multivariate uniform distribution for the systematic effects are applied to generate random variates for the measurements and random variates for the measurements plus systematic effects. Monte Carlo estimates of the expectations and the covariance matrix of the measurements with systematic effects are computed. The densities for the confidence ellipsoid for the measurements and the confidence region for the measurements with systematic effects are obtained by relative frequencies. They only depend on the size of the rectangular volume elements for which the densities are determined. The problem of sorting the densities is solved by sorting distances together with the densities. This allows a visualization of the confidence ellipsoid for the measurements and the confidence region for the measurements with systematic effects.

Total least squares (TLS) is a technique that solves the traditional least squares (LS) problem for an errors-in-variables (EIV) model, in which both the observation vector and the design matrix are contaminated by random errors. Four- and seven-parameter models of coordinate transformation are typical EIV model. To determine which one of TLS and LS is more effective, taking the four- and seven-parameter models of Global Navigation Satellite System (GNSS) coordinate transformation with different coincidence pointsas examples, the relative effectiveness of the two methods was compared through simulation experiments. The results showed that in the EIV model, the errors-in-variables-only (EIVO) model and the errors-in-observations-only (EIOO) model, TLS is slightly inferior to LS in the four-parameter model coordinate transformation, and TLS is equivalent to LS in the seven-parameter model coordinate transformation. Consequently, in the four- and seven-parameter model coordinate transformation, TLS has no obvious advantage over LS.

There are four recursive algorithms used in the computation of the fully normalized associated Legendre functions (FNALFs): the standard forward column algorithm, the standard forward row algorithm, the recursive algorithm between every other degree, and the Belikov algorithm. These algorithms were evaluated in terms of their first relative numerical accuracy, second relative numerical accuracy, and computation speed and efficiency. The results show that when the degree n reaches 3000, both the recursive algorithm between every other degree and the Belikov algorithm are applicable for | cos  θ  | ∈[0, 1], with the latter better second relative numerical accuracy than the former at a slower computation speed. In terms of | cos  θ  | ∈[0, 1], the standard forward column algorithm, the recursive algorithm between every other degree, and the Belikov algorithm are applicable within degree n of 1900, and the standard forward column algorithm has the highest computation speed. The standard forward column algorithm is applicable for | cos  θ  | ∈[0, 1] within degree n of 1900. This algorithm’s range of applicability decreases as the degree increases beyond 1900; however, it remains applicable within a minute range when | cos  θ  | is approximately equal to 1. The standard forward row algorithm has the smallest range of applicability: it is only applicable within degree n of 100 for | cos  θ  | ∈[0, 1], and its range of applicability decreases rapidly when the degree is greater than 100. The results of this research are expected to be useful to researchers in choosing the best algorithms for use in the computation of the FNALFs.