Geometric Modeling of the Z-Surface and Z-Curve of GNSS Signals and Their Solution Techniques
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1980-2012 IEEE. This paper presents a novel geometric model to characterize the zero-crossing surface (z-surface) and its z-curve for signals emitted by a pair of Global Navigation Satellite System satellites (A, B). The z-surface is the surface of points which have zero difference in pseudorange to (A, B), and the z-curve is the intersection of z-surface with the earth's surface. As a form of time difference of arrival, modeling of the z-surface/ z-curve benefits from the elimination of common error terms (e.g., receiver clock offset) shared by the pair of pseudoranges, so that the resulting model can be used for design of advanced applications, such as timing and positioning integrity monitors and tools for geodetic and atmospheric measurements. The derivation of the z-curve starts with the shape modeling of the terrestrial service volume and the earth's surface. Then, an equi-pseudorange surface of (A, B)'s signals can be placed within these shape models. The z-curve can be derived as a boundary condition of the z-surface. The model can represent the morphing of z-curves with respect to the changes in satellite positions and signal propagation delays. As a result, the z-models can be readily generalized into a k-surface and k-curve, where k represents the pseudorange difference between (A, B). Solutions for modeling and analysis of the z(k)-surface/curve are based on a system of quadratic Cartesian equations. We propose an algebraic method to parameterize them in terms of geocentric latitude and longitude.