An Analytical Derivation of the Signal-in-Space Root-Mean-Square User Range Error

The concept of the signal-in-space (SIS) root-mean-square (RMS) user range error (URE) is used to evaluate the performance of multiple global navigation satellite systems (GNSSs); however, a complete analytical derivation has not been published. This article describes the instantaneous SIS URE and the instantaneous SIS RMS URE, explains the role of the instantaneous SIS RMS URE in evaluating the statistical accuracy of GNSS signals, and provides an analytical derivation of the instantaneous SIS RMS URE. This derivation is then compared to the equations found in various papers and performance standards to illustrate how the equations, although appearing different, actually measure the same quantity with differing constraints.