Statistical characterization of composite protection levels for GPS

A statistical characterization is presented of Global Positioning System (GPS) user range error as a normally distributed random variable with non-zero mean over the length of the aircraft precision approach operation, correlated from one GPS measurement epoch to another and from one satellite to another. This leads directly to modeling GPS error in the position domain as multivariate normal with non-zero mean. Based on this model, a vertical composite protection level VPL_c and a horizontal composite protection level HPL_c are each calculated as scalar values from a univariate normal distribution displaced from the origin by the worst-case position domain bias combination possible, given the maximum possible individual satellite bias magnitudes and the satellite geometry. A method is then presented by which exact values—that is, values accurate to a user-defined error tolerance and consistent with statistical assumptions—of VPL_c and HPL_c are obtained, and by which computationally efficient approximations may be evaluated. A statistical quadratic form under the multivariate normal distribution is used to derive a new class of protection levels based on the probability enclosed within a radius defined in two or more dimensions. A central Chi-square representation of this quadratic form is also presented and is incorporated into a six-step computational procedure for the two-dimensional composite radial protection level RPL_c. This procedure is extended to the spherical protection level (SPL_c) and the ellipsoidal protection level (EPL_c).