Geometric Modeling of the Z-Surface and Z-Curve of GNSS Signals and Their Solution Techniques

This paper presents a novel geometric model to characterize the zero-crossing surface (<inline-formula> <tex-math notation="LaTeX">$z$ </tex-math></inline-formula>-surface) and its z-curve for signals emitted by a pair of Global Navigation Satellite System satellites (<inline-formula> <tex-math notation="LaTeX">${\mathcal{A}}$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">${\mathcal{B}}$ </tex-math></inline-formula>). The <inline-formula> <tex-math notation="LaTeX">$z$ </tex-math></inline-formula>-surface is the surface of points which have zero difference in pseudorange to (<inline-formula> <tex-math notation="LaTeX">${\mathcal{A}}$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">${\mathcal{B}}$ </tex-math></inline-formula>), and the <inline-formula> <tex-math notation="LaTeX">$z$ </tex-math></inline-formula>-curve is the intersection of <inline-formula> <tex-math notation="LaTeX">$z$ </tex-math></inline-formula>-surface with the earth’s surface. As a form of time difference of arrival, modeling of the <inline-formula> <tex-math notation="LaTeX">$z$ </tex-math></inline-formula>-surface/<inline-formula> <tex-math notation="LaTeX">$z$ </tex-math></inline-formula>-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 <inline-formula> <tex-math notation="LaTeX">$z$ </tex-math></inline-formula>-curve starts with the shape modeling of the terrestrial service volume and the earth’s surface. Then, an equi-pseudorange surface of (<inline-formula> <tex-math notation="LaTeX">${\mathcal{A}}$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">${\mathcal{B}}$ </tex-math></inline-formula>)’s signals can be placed within these shape models. The <inline-formula> <tex-math notation="LaTeX">$z$ </tex-math></inline-formula>-curve can be derived as a boundary condition of the <inline-formula> <tex-math notation="LaTeX">$z$ </tex-math></inline-formula>-surface. The model can represent the morphing of <inline-formula> <tex-math notation="LaTeX">$z$ </tex-math></inline-formula>-curves with respect to the changes in satellite positions and signal propagation delays. As a result, the <inline-formula> <tex-math notation="LaTeX">$z$ </tex-math></inline-formula>-models can be readily generalized into a k-surface and k-curve, where k represents the pseudorange difference between (<inline-formula> <tex-math notation="LaTeX">${\mathcal{A}}$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">${\mathcal{B}}$ </tex-math></inline-formula>). Solutions for modeling and analysis of the <inline-formula> <tex-math notation="LaTeX">$z(k)$ </tex-math></inline-formula>-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.

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