The effect of physical correlations on the ambiguity resolution and accuracy estimation in GPS differential postioning

Description

High accuracy GPS carrier phase differential positioning requires complete modelling of the GPS measurement errors. Generally, in GPS positioning, the mathematical model does not describe the observations perfectly. The main reason for this is the lack of information about the physical phenomena associated with the GPS observation. Therefore, residual error component remains unmodelled. The analysis of many data series representing baselines of different lengths show that, in GPS carrier phase double difference positioning, the residual model errors are positively correlated over a time period of about 10 minutes. Not accounting for this correlation known as physical correlation, usually leads to an overestimation of the accuracy of both the observations and the resulting positions. A simple way of accounting for these correlated residual errors is to model them stochastically through the modification of the observation’s covariance matrix. As the true covariance function is not known, the stochastic modelling of the GPS measurement errors must be achieved by using an empirical covariance function. It is shown that the exponential function is the best approximation for the covariance function of the GPS carrier phase errors in the least squares sense. Although this way of accounting for the unmodelled errors yields a fully populated covariance matrix for the GPS carrier phase double the difference observations, its inverse takes the simple form of a black diagonal matrix. A modified least squares adjustment algorithm incorporating the newly developed, fully populated covariance matrix is derived. The covariance matrix of the ambiguity parameters is used to form a confidence region of a hyperellipsoid around the estimated real values which is then used for searching the likely integer values of the ambiguity parameters. To speed up the searching time, the covariance matrix for the ambiguities is decomposed using Cholesky decomposition. The software DIFGPS is developed to verify the validity of the technique. Real data of several baselines of different lengths observed under different ionospheric activities are used. It is shown that including the physical correlations requires more observations to obtain a unique solution for the ambiguity parameters than when they are neglected. However, for all tested baselines, neglecting the physical correlations leads to an overly optimistic covariance matric for the estimated parameters. Additionally, the use of a scale factor to scale the optimistic covariance matrix was found to be inappropriate. However, without physical correlations, a more realistic covariance matrix is obtained by using data with large sample interval.