Treatment of Refractivity Fluctuations by Fully Populated Variance-covariance Matrices
Schön, S.¹; Brunner, F.K.^2
1Institut für Erdmessung, Leibniz Universität Hannover; ²Engineering Geodesy and Measurement Systems, Graz University of Technology

The incomplete knowledge of the atmospheric composition and dynamics along the signal path is still an accuracy limitation for GNSS-based parameter estimation. In fact, the dynamic processes in the atmosphere induce correlated wave propagation effects on GNSS signals. Usually, these correlations are not modelled in the variance-covariance-matrix of the observa-tions. Hence, too optimistic uncertainty measures are obtained for GNSS derived parameters like point positions or estimated zenith wet delays.

Based on turbulence theory, we have previously proposed a new variance-covariance-model to account for the physical correlations induced by refractivity fluctuations in the troposphere. Starting with a power-law description of the refractivity fluctuations provided by turbulence theory, the corresponding spectrum is integrated along the lines-of-sight of all satellite-antenna configurations. As a result, expressions for the variances and the covariances of the GNSS phase observations are computed. Subsequently, a fully populated variance-covariance matrix (VCM) is obtained. The integration can be solved analytically for the variances, but for the covariances numerical integration must be used. The magnitudes of the VCM elements depend not only on the satellite-antenna geometry (expressed by the station separation, the azimuths and elevation angles of the satellites) but also on the prevailing atmospheric condi-tions. The latter ones can be parameterised, e.g. by the wind velocity and direction, the so-called structure constant Cn2 of the refractivity, and the outer scale length L0.

In this paper, we analyse the sensitivity of the covariances with respect to the model parame-ter, since the covariances are not given analytically. We will show that the wind velocity and direction are crucial. Finally, we compare the empirical auto- and cross-correlation functions with those predicted by the model for different wind directions and velocities using the dou-ble-differenced phase data from a specially designed test network with station separations between 1 km and 16 km. We show that the geostrophic wind direction and velocity, which can be easily derived from isobaric maps, are appropriate to explain the correlation and decor-relation processes reflected in the computed auto- and cross-correlation functions of the ob-served double-differenced phase data. Consequently this new variance-covariance model con-tributes to model the complex processes in the lower atmosphere and to the adequate treat-ment of physical correlations between GNSS observations in the GNSS data analysis.

The development of new GNSS such as Galileo will on the one hand contribute to enhance the temporal and spatial repeatability of the satellite passes and therefore improve the tropo-sphere sensing and understanding the complex processes in the troposphere. On the over hand for high precision application and for a meaningful interpretation and validation of the esti-mated parameters physical correlations between the GNSS observations cannot be neglected any longer. Consequently, an extended variance-covariance model such as the one proposed in this paper should be implemented.