Order of Pseudoranges (OOP) of GNSS: Spatial Modeling and Analysis
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This paper presents a temporal-spatial reference model on timing of GNSS signals for receivers based on the order of pseudoranges (OOP), which is an ordered list of visible satellite ID based on their pseudoranges. OOP with respect to a receiver position is essentially a discretized state of GNSS signals for the receiver, and it is a function of time. Points of equal pseudoranges for signals transmitted from two satellites form a zero-crossing surface (z-surface), which sweeps through the co-coverage area of the two satellites as the satellites orbit. We show that the exact form of a z-surface is a hyperboloid, which intersects the Earth surface as a curve segment, i.e. a z-curve. We show that a close form expression for z-curve is inherently difficult to derive but instead a numerical method can be used. Our model takes into account of the error terms such as ionosphere delay, and clock drift, so that one can characterize the width of a z-surface due to uncertainty in estimation. The correctness of the OOP model was tested using the pseudorange data of selected stations in the CORS (Continuously Operating Reference Stations) network of the GPS system. It can precisely predict the zero-crossing time of satellite pairs for all tested cases. An Earth surface area co-covered by a set of satellites can be segmented by z-surfaces into O-zones, each of which represents a particular OOP configuration. A ground area within the coverage of a GNSS system is expected to observe sweeping z-surfaces periodically. As a result, the proposed model offers a natural temporal-spatial reference frame for design of wide area, high confidence applications, such as GNSS timing integrity check points, smart grids, and sensor grids, etc.
