Receiver-satellite Geometry Research Articles

Automatic Calibration of Process Noise Matrix and Measurement Noise Covariance for Multi-GNSS Precise Point Positioning

The Expectation-Maximization algorithm is adapted to the extended Kalman filter to multiple GNSS Precise Point Positioning (PPP), named EM-PPP. EM-PPP considers better the compatibility of multiple GNSS data processing and characteristics of receiver motion, targeting to calibrate the process noise matrix Qt and observation matrix Rt, having influence on PPP convergence time and precision, with other parameters. It is possibly a feasible way to estimate a large number of parameters to a certain extent for its simplicity and easy implementation. We also compare EM-algorithm with other methods like least-squares (co)variance component estimation (LS-VCE), maximum likelihood estimation (MLE), showing that EM-algorithm from restricted maximum likelihood (REML) will be identical to LS-VCE if certain weight matrix is chosen for LS-VCE. To assess the performance of the approach, daily observations from a network of 14 globally distributed International GNSS Service (IGS) multi-GNSS stations were processed using ionosphere-free combinations. The stations were assumed to be in kinematic motion with initial random walk noise of 1 mm every 30 s. The initial standard deviations for ionosphere-free code and carrier phase measurements are set to 3 m and 0.03 m, respectively, independent of the satellite elevation angle. It is shown that the calibrated Rt agrees well with observation residuals, reflecting effects of the accuracy of different satellite precise product and receiver-satellite geometry variations, and effectively resisting outliers. The calibrated Qt converges to its true value after about 50 iterations in our case. A kinematic test was also performed to derive 1 Hz GPS displacements, showing the RMSs and STDs w.r.t. real-time kinematic (RTK) are improved and the proper Qt is found out at the same time. According to our analysis despite the criticism that EM-PPP is very time-consuming because a large number of parameters are calculated and the first-order convergence of EM-algorithm, it is a numerically stable and simple approach to consider the temporal nature of state-space model of PPP, in particular when Qt and Rt are not known well, its performance without fixing ambiguities can even parallel to traditional PPP-RTK.

The ADOP and PDOP: Two Complementary Diagnostics for GNSS Positioning

Ambiguity dilution of precision (ADOP) and position dilution of precision (PDOP) are two popular scalar-diagnostics used in Global Navigation Satellite System (GNSS) positioning. Where the ADOP is a predictor for carrier-phase ambiguity resolution performance, the PDOP is meant to predict the receiver-satellite geometry’s capability for precise positioning. We will show, however, that although the PDOP works well for code-based positioning, one has to exercise great care in using the PDOP for real-time kinematic (RTK) positioning. We show that the ADOP and PDOP have distinct behaviors, an important consequence of which is that one can have time periods with small PDOPs, and thus seemingly good geometry for precise positioning, but at the same time large ADOPs, thus showing that successful ambiguity resolution and therefore precise positioning will not be possible. Also, the reverse situation may occur, i.e., having large PDOPs with small ADOPs. In such a situation, the large PDOPs should not automatically lead to the conclusion of poor position performance, because the large gain that ambiguity resolution brings will often still make precise positioning possible. We will analyze and explain this complementary behavior of the PDOP and ADOP and demonstrate this both analytically and empirically. For this analysis we use real Global Positioning System (GPS) single- and multifrequency signals and GPS/Quasi-Zenith Satellite System (QZSS), GPS/Navigation with Indian Constellation (NAVIC) L5 signals of two baselines located in Perth, Australia.

Fast Phase-Only Positioning with Triple-Frequency GPS.

In this contribution, we study the phase-only ambiguity resolution and positioning performance of GPS for short baselines. It is well known that instantaneous (single-epoch) ambiguity resolution is possible when both phase and code (pseudorange) data are used. This requires, however, a benign multipath environment due to the severe effects multipath has on the code measurements. With phase-only processing, one would be free from such severe effects, be it that phase-only processing requires a change in receiver-satellite geometry, as a consequence of which it cannot be done instantaneously. It is thus of interest to know how much change in the relative receiver-satellite geometry is needed to achieve successful phase-only ambiguity resolution with correspondingly high precision baseline solutions. In this contribution, we study the two-epoch phase-only performance of single-, dual-, and triple-frequency GPS for varying time spans from 60 s down to 1 s. We demonstrate, empirically as well as formally, that fast phase-only very-precise positioning is indeed possible, and we explain the circumstances that make this possible. The formal analyses are also performed for a large area including Australia, a part of Asia, the Indian Ocean, and the Pacific Ocean. We remark that in this contribution "phase-only" refers to phase-only measurements in the observation model, while the code data are thus only used to compute the approximate values needed for linearizing the observation equations.

Impacts of Ionospheric Scintillations on Dilution of Precision and Positioning Errors

Equatorial scintillation impacts negatively on GPS navigation measurements by reducing the number of satellites that are readily available for a GPS receiver to calculate a navigation solution. Ideally, a minimum of four satellites is required to calculate a valid navigation solution. We analyzed GPS data from Ascension Island during the Air Force Research Laboratory (AFRL) campaign of the solar maximum year of 2002 (5–19 March 2002). The study showed that strong scintillations impacted the receiver-satellite geometry, with attendant poor dilution of precisions and positioning errors. On the non-scintillation nights, the GPS receiver showed good tracking capability, as it consistently maintained lock on six to eight satellites, and the GDOP and PDOP values were generally less than 2.2 with lesser positioning errors. In contrast, on the scintillation nights, between 2200 and 2300 UT, the number of satellites that the receiver maintained lock on were consistently reducing to four and in some cases fewer than four. Around this period, the GDOP or PDOP values were higher than 6.0, and significant vertical and horizontal positioning errors were recorded. Equatorial scintillation is capable of adding about 3 metres error to the overall error budget of a GNSS system.

Location determination using WiFi fingerprinting versus WiFi trilateration

Many applications in the area of location-based services and personal navigation require nowadays the location determination of a user not only in an outdoor environment but also an indoor. Typical applications of location-based services (LBS) mainly in outdoor environments are fleet management, travel aids, location identification, emergency services and vehicle navigation. LBS applications can be further extended if reliable and reasonably accurate three-dimensional positional information of a mobile device can be determined seamlessly in both indoor and outdoor environments. Current geolocation methods for LBS may be classified as GNSS-based, cellular network-based or their combinations. GNSS-based methods rely very much on the satellite visibility and the receiver-satellite geometry. This can be very problematic in dense high-rise urban environments and when transferring to an indoor environment. Especially, in cities with many high-rise buildings, the urban canyon will greatly affect the reception of the GNSS signals. Moreover, positioning in the indoor/outdoor transition areas would experience signal quality and signal reception problems, if GNSS systems alone are employed. The authors have proposed the integration of GNSS with wireless positioning techniques such as WiFi and UWB. In the case of WiFi positioning, the so-called fingerprinting method based on WiFi signal strength observations is usually employed. In this article, the underlying technology is briefly reviewed, followed by an investigation of two WiFi-positioning systems. Testing of the system is performed in two localisation test beds, one at the Vienna University of Technology and another one at the Hong Kong Polytechnic University. The first test showed that the trajectory of a moving user could be obtained with a standard deviation of about ±3–5 m. The main disadvantage of WiFi fingerprinting, however, is the required time consuming and costly signal strength system calibration in the beginning. Therefore, the authors have investigated if the measured signal strength values can be converted to the corresponding range to the access point. A new approach for this conversion is presented and analysed in typical test scenarios.

A proof of Nielsen's conjecture on the GPS dilution of precision

The dilution of precision terms for relative positioning are bounded from above by the corresponding dilution of precision terms for point positioning. This result is proven for the case of four satellites and conjectured to be valid for the case of more than four satellites. A proof of this conjecture is given. We also extend the result by giving two different lower bounds for the dilution of precision terms. The first lower bound depends on the receiver-satellite geometry, whereas the second does not. The proof of the bounds is based on the solution of a generalized eigenvalue problem.

A canonical theory for short GPS baselines.

This contribution is the third of four parts. Based on the gain-number concept, canonical forms will be presented of the ambiguity search spaces of the geometry-based model, the time-averaged model, and the geometry-free model. These forms reveal the intrinsic geometry of the search spaces and allow one to study their size, shape, and orientation as function of data precision, sampling rate, satellite redundancy, and change in receiver-satellite geometry. The canonical forms are also used to address the problem of search halting. The phenomenon of search halting is explained and it is shown how decorrelating ambiguity transformations can largely eliminate this computational burden.

A canonical theory for short GPS baselines.

The present contribution is the second of four parts. It considers the precision and correlation of the least-squares estimators of the carrier phase ambiguities. It is shown how the precision and correlation of the double-differenced ambiguities as well as of the widelane ambiguities are effected by the observation weights, by the number of satellites tracked, by the number of observation epochs used, and by the change over time of the relative receiver-satellite geometry. Also the ability of the widelane transformation to decorrelate and to improve the precision is investigated.

A canonical theory for short GPS baselines.

The present contribution is the first of four parts. It considers the precision of the floated and the fixed baseline. A measure is introduced for the gain in baseline precision which is experienced when the carrier phase double-differenced ambiguities are treated as integers instead of as reals. The properties of this measure are analyzed, and it is shown by means of principal angles how it relates to the change over time of the relative receiver-satellite geometry. We also present canonical forms of the baseline variance matrices for different measurement scenarios. These canonical forms make the relation between the various variance matrices transparent and thus present a simple way of studying their relative merits.

GPS double difference statistics: with and without using satellite geometry

In this contribution GPS statistics are presented for the case that the relative receiver-satellite geometry is included in the single baseline model and for the case that the relative receiver-satellite geometry is excluded. It is shown that the statistics are linked through a particular form of a phased adjustment. Based on the stepwise approach of a phased adjustment, the impact of using satellite geometry or dispensing with it, on the least-squares estimators, on the teststatistics and their associated reliability, and on the integer ambiguity estimation, is presented and analyzed.

An analytical study of ambiguity decorrelation using dual frequency code and carrier phase

In this contribution the integer least-squares estimation of the double differencedL1 andL2 ambiguities is analyzed, under the provision that the relative receiver-satellite geometry is dispensed with. The variance-covariance matrix of the ambiguities is instrumental for gaining insight into the characteristics of the ambiguity fixing process. A qualitative geometric description in detail is therefore given of the ambiguity search space. The elongation, the correlation coefficient and the areas of the ambiguity search space and its enclosing boxes are all given as function of the ratio or product of the carrier phase and code standard deviations. It is shown that the ambiguity search space is very elongated and that the ambiguities are highly correlated. It is also shown how the high correlation between the ambiguities can be used to ones advantage for the transformation to new ambiguities. This is done by means of the least-squares ambiguity decorrelation adjustment method. The improvements in terms of decorrelation, elongation and precision are shown and the corresponding optimal time-invariant ambiguity transformations are given for a practical range of code and measurement precisions.