Stochastic Modelling for Real-Time GNSS Positioning
Satellite positioning refers to the process of obtaining positions on or near the Earth‟s surface by measuring ranges to a number of Earth-orbiting satellites. As the positions of the satellites are known at any given time, the observations can be combined in a set of simultaneous equations to determine the coordinates of the receiver, along with some measure of coordinate quality. The most prevalent technique for computing parameters from a set of observations is least squares estimation. Least squares requires a functional model that describes the mathematical relationship between the observations and the unknown parameters and a stochastic model which describes the statistical behaviour of the observations. The estimation process yields both the parameters and their precision estimates. Functional models for GNSS positioning are well known and have remained essentially unchanged for the last two decades. On the other hand, stochastic models are less well understood. Providing a realistic stochastic model in support of GNSS processing is a major challenge. A clear solution is yet to be found and as a result rudimentary models continue to be used in practice.
The stochastic model is represented by the variance-covariance matrix in the least squares algorithm. The diagonal terms of the matrix are variances which describe the precision of individual observations. The off-diagonal terms are the covariances which arise from physical correlations between the observations. Three types of physical correlation have been identified: spatial, temporal, and observation-type. It has been shown by a number of studies that ignoring these correlations will produce unreliable precision estimates of the unknown parameters. However estimating physical correlations, especially for in real-time, has proved an elusive goal. Approaches to model these correlations developed to date are suited only for post-processing applications.
This study proposes a new stochastic model for real-time GNSS processing based on a quantity known as Time Differenced Range Residual (TDRR) that enables empirical noise estimation from raw observations in real-time. The advantages of this approach are that it is based on raw observations, it is computationally efficient and it allows modelling of the main physical correlations.
The TDRR is investigated in this study for its usefulness in empirical noise estimation. It is shown that the TDRR can be used as a tool to investigate the noise characteristics of various GNSS receivers. The stochastic model based on the TDRR is developed in full. This model includes variances for the individual satellite observations as well the spatial correlations. The stochastic model is compared to conventional approaches for processing three short baselines of 3, 9 and 12 km long. Short baselines are chosen for the analysis to minimise the effect of the atmospheric biases on the solution. It is shown that the TDRR-based stochastic model provides more realistic precision estimates for the parameters compared to the conventional approaches, however some limitations in the development are also identified, which require further refinement before the newly developed model can be applied in practical application.