In this article, we compared different approaches for dual-satellite state modelling based on experimental data from different environment types, various combinations of two satellite elevation angles and different azimuth angle separations.

To find an appropriate state model architecture, the first part of this article gives a closed overview on existing state modelling approaches for single-satellite and dual-satellite reception. Three categories of models are presented: first-order Markov models, dynamic Markov models, and semi-Markov models. Based on one measurement example, a detailed analysis of diverse variants of these approaches for dual-satellite state modelling is performed. As evaluation criterion, the single- and dual-satellite state probabilities and the state duration statistics are compared with the measurements. Further on, the practicability of the state modelling approaches in terms of dual-satellite channel models is discussed. It was concluded that, due to the high number of required parameters and the high complexity, dynamic Markov models are not feasible in terms of dual-satellite state modelling.

In the second part of this article, three dual-satellite modelling approaches with low complexity are analysed on a large set of receive scenarios: a first-order Markov model for correlated satellites (Lutz model), a semi-Markov approach assuming a lognormal SDPDF fit, and a semi-Markov approach assuming a piecewise exponential SDPDF fit. For this purpose, the GNSS measurements are separated into various receive scenarios including five environments (urban, suburban, forest, commercial, open), 8 × 8 sections with constant elevation angles of two satellites between 10° and 90°, and seven different intervals of the azimuth angle separation. Parameters for these receive scenarios has been derived for the three selected modelling approaches. Afterwards, the re-modelling results are compared with the measurements in terms of the correlation coefficient, the state probability and the state durations of the critical system state ‘bad bad’ in dependency on the azimuth angle separation and the elevation angles of two satellites for the urban environment. It was shown that the Lutz model accurately re-simulates the correlation coefficient and the state probability, whereas the state duration statistics are insufficiently described. The semi-Markov models describe accurately the state probabilities, the correlation coefficients, and also the state duration statistics. With respect to the number of parameters, the semi-Markov approach using a lognormal fit of the SDPDF is the preferred model for the dual-satellite state modelling and is proposed therefore for the a new dual-satellite channel model for broadcasting applications.

State parameters for the semi-Markov model as well as for the Lutz model for single- and dual-satellite reception are found in the Additional files6,7,8 and9. For the sake of completeness, we also derived Loo parameters from SDARS measurement data describing slow and fast fading effects (cf. Additional file1). By using the two-state model according to Prieto-Cerdeira et al.[4], the parameters enable a simulation of LMS timeseries for single-satellite and dual-satellite reception for different environments, elevation angles, and azimuth separations.

In the near future we will also focus on modelling of the small-scale fading. By analysing the extensive SDARS measurement data in terms of slow- and fast signal variations within MiLADY, some modifications of the two-state model from[4] are indicated. Recently, modifications were proposed in[15]. A validation of these new concepts for a multi-satellite model is topic of ongoing work.

A further task is the state analysis of multi-satellite constellations with three or more satellites. Due to the exponential growth of the model complexity with the number of satellites, new concepts must be investigated. A promising approach would be a Master–Slave concept, where several ‘Slave’ satellites are modelled according to their correlation with one ‘Master’ satellite, while neglecting the correlation between the ‘Slave’ satellites (cf. Section 3). Based on statistical parameters derived from measurement data (such as the joint state probability for ‘bad bad bad’ for three satellites), the Master–Slave concept will be evaluated.

To improve the consistency of a state parameter database, activities are planned to extract state parameters with alternative methods, such as the analysis of environmental images from fish-eye cameras.