Analyzing Localizability of LEO/MEO Hybrid Networks: A Stochastic Geometry Approach

With the increase in global positioning service demands and the requirement for more precise positioning, assisting existing medium- and high-orbit satellite-enabled positioning systems with low Earth orbit (LEO) satellites has garnered widespread attention. However, providing low computational complexity performance analysis for hybrid LEO/medium Earth orbit (MEO) massive satellite constellations remains a challenge. In this article, we introduce for the first time the application of stochastic geometry (SG) framework in satellite-enabled positioning performance analysis and provide an analytical expression for the $K$-availiability probability and $K$-localizability probability under bidirectional beam alignment transmissions. The $K$-localizability probability, defined as the probability that at least $K$ satellites can participate in the positioning process, serves as a prerequisite for positioning. Since the modeling of MEO satellite constellations within the SG framework has not yet been studied, we integrate the advantages of Cox point processes and binomial point processes, proposing a doubly stochastic binomial point process for accurate modeling of MEO satellite constellations. Finally, we investigate the impact of constellation configurations and antenna patterns on the localizability performance of LEO, MEO, and hybrid MEO/LEO constellations. We also demonstrate the network performance gains brought to MEO positioning systems by incorporating assistance from LEO satellites.