Metaheuristic-based stochastic models for GNSS relative positioning planning

A priori variance–covariance matrix (VCM) estimation of global navigation satellite systems (GNSS) double difference observations in relative positioning is challenging. Existing methods have been limited to estimate variances only or present variables challenging to acquire a priori, unfeasible for observation planning. Ignoring the covariances produces misleading results and compromises reliance on GNSS positioning design. In this study, we propose models to estimate the VCM a priori for planning, based on simple variables accessible to any professional: observation time span, vector length, and ephemeris type. Using a database of over 140,000 GNSS vectors with double difference (DD) observations, we group the data by time span and length range and extract standard deviations and covariances for the linear regression process. The Isolation Forest algorithm is employed to filter outlying observations. Our models provide standard deviations and root square covariances in a local coordinate system, requiring only vector length, observation time span, and ephemeris type as input. Additionally, the equations can be easily implemented in a simple spreadsheet. The results show high coefficients of determination (R2 > 0.8). We tested the models in a simulated GNSS network and verified broadcast ephemeris resulted in 6.5 to 16.7 times larger error ellipsoids compared to the precise ephemeris, indicating higher uncertainty. Ellipsoids differed in flattening and orientation when compared to the null covariance (variance only) approach. Although VCM models better reflect the precision of relative positioning observations, they did not affect the number of non-controllable observations in the observation’s reliability tests.