On User Algorithms for GNSS Precise Point Positioning
Precise Point Positioning (PPP) is a Global Navigation Satellite System (GNSS) processing method with the objective of providing high positioning accuracy without the need for a nearby base station or dense network of reference stations operated by the user. To reach this objective, PPP uses the very precise carrier phase measurements in addition to the coarse pseudorange measurements, and precise satellite orbits and clock offsets. Before the carrier phase measurements can contribute to the position solution, the carrier phase biases must be estimated from the measurements. Additionally, to compute an accurate position, a number of approaches are used in PPP to reduce the error contributions. A sparse global network of reference stations estimates very accurate satellite orbits and clock offsets, and provides these to the user. A priori models are used to correct for the hydrostatic troposphere delay, relativistic effects, carrier phase wind up, phase center offsets and variations. We will mainly consider users on or close to the surface of the Earth, which means that site displacement effects (e.g. ocean loading and solid Earth tides) also should be taken into account. The troposphere wet delay is often estimated from the data, and, in conventional PPP, the ionosphere delays are eliminated by combining dual frequency observations in a reduced number of ionosphere free observations. The accuracy provided by PPP based on GPS is very high; just a few mm for 24h of static data. However, PPP can still be improved in a number of areas which are addressed in this dissertation: The need to both eliminate the ionosphere delays and estimate the carrier phase ambiguities results in a relatively weak model. This leads to long (re)initialization times before an accurate position can be computed. In this dissertation a different approach is proposed, implemented and tested that does not form the ionosphere free combination, but estimates the ionosphere delays from the observations. First the (linearized) GNSS observation equations are introduced and the functional PPP model is derived. From this we determine the estimable parameters, and treat the possibility of constraining the ionosphere estimation by use of external ionosphere data. This is achieved by correcting the observations with ionosphere model values and estimating only the residual ionosphere delay. Finally, a dynamic model for the residual ionosphere delays is determined and positioning results are presented. An accurate stochastic model for PPP is lacking, which leads to a suboptimal weighting of the different observation types, an inadequate quality description of the solution, and unreliable hypothesis testing. In this dissertation the stochastic properties of the PPP model are studied starting with the accuracy of the (real time) satellite orbit and clock offsets, and continuing with the accuracy and correlation of the GNSS observations. For the latter a newly developed method is introduced based on the geometry free model. This analysis also includes an estimation procedure to model pseudorange multipath errors as an auto regressive process, which can also be used to construct a dynamic model for time correlated parameters. Finally, an almost unbiased variance component estimation algorithm is applied to the complete PPP model, and an accurate quality description of the positioning results is obtained. Integrity monitoring, which uses hypothesis testing to detect errors in the model which are not covered by the expected uncertainties, is not well developed for PPP. This subject is related to the previous points, since an accurate stochastic model is required for integrity monitoring and estimating and constraining the ionosphere delay, is key to high integrity performance. The geometry free model is used again in time differenced form to analyze the integrity aspects of (multi frequency) GNSS models and their power to detect errors such as cycle slips on the carrier phase measurements. The null hypothesis and alternative hypotheses are presented and the Minimal Detectable Bias (MDB) is derived for different kinds of errors. Analytical expressions are provided supported by numerical results. The geometry free analysis of pseudorange and carrier phase measurement noise revealed a good agreement with the theoretical expressions linking the measurement noise to the receiver tracking loop parameters and signal to noise ratio. However, strong variations of the pseudorange code measurements over longer time periods were also observed, which dominate the error budget if not accounted for. Modeling of multipath time series as an autoregressive process AR(k) of order k, showed that the estimated AR(1) parameter is almost always significant to a very high level, which indicates strong time correlation of the multipath time series. Indeed the AR(1) model fits the data much better than a white noise or AR(0) model does, significantly reducing the residual variance. Analysis of the real time precise products showed that compared to the satellite orbit prediction, satellite clock prediction is relatively poor and thus dominates the combined error. The quality of newer products with shorter delays is much higher and approaches the post processed products. There also exists strong correlation between the orbit and clock products, resulting from the estimation process, but also due to the use of specific phase center offsets. Users should thus obtain the satellite orbit and clock products from the same provider and not mix products. Analysis of the time differenced geometry free model revealed the strong impact of constraining the ionosphere variations over time on the integrity performance. Instantaneous detection of individual slips in dual and triple frequency data was found to have a high probability of success under moderate ionosphere conditions. Detection of phase slips in single frequency data is more difficult and may lead to a delay in the detection. The size of the smallest pseudorange outliers which can still be detected is, except for single frequency data, insensitive to the ionosphere conditions; the precision of the code measurements is the only determining factor. Testing for a simultaneous phase slip on each frequency leads to a very elongated MDB ellipse or (hyper)ellipsoid if the ionosphere is not constrained, indicating that specific combinations of slips are very difficult to detect. Constraining the ionosphere delay shrinks the MDB ellipse when the precision of the ionospheric pseudo observable increases. The positioning results show that without proper modeling, the pseudorange observations cannot contribute significantly to the carrier phase solution without degrading the accuracy. Furthermore the formal quality description of the solution poorly fits the empirical position errors if time correlated effects are neglected. After solving both these issues, the strength of the combined model was demonstrated by the excellent convergence performance in both the static and kinematic case. Constraining the atmosphere delays to model values further improves the initial convergence of the position estimation.