On the Matern covariance family: a proposal for modeling temporal correlations based on turbulence theory

摘要

Current variance models for GPS carrier phases that take correlation due to tropospheric turbulence into account are mathematically difficult to handle due to numerical integrations. In this paper, a new model for temporal correlations of GPS phase measurements based on turbulence theory is proposed that overcomes this issue. Moreover, we show that the obtained model belongs to the Matern covariance family with a smoothness of 5/6 as well as a correlation time between 125-175 s. For this purpose, the concept of separation distance between two lines-of-sight introduced by Schon and Brunner (J Geod 1:47-57,2008a) is extended. The approximations made are highlighted as well as the turbulence parameters that should be taken into account in our modeling. Subsequently, fully populated covariance matrices are easily computed and integrated in the weighted least-squares model. Batch solutions of coordinates are derived to show the impact of fully populated covariance matrices on the least-squares adjustments as well as to study the influence of the smoothness and correlation time. Results for a specially designed network with weak multipath are presented by means of the coordinate scatter and the a posteriori coordinate precision. It is shown that the known overestimation of the coordinate precision is significantly reduced and the coordinate scatter slightly improved in the sub-millimeter level compared to solutions obtained with diagonal, elevation-dependent covariance matrices. Even if the variations are small, turbulence-based values for the smoothness and correlation time yield best results for the coordinate scatter.