Statistical characterization of composite protection levels for GPS

Abstract

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 ).

Approximations to composite GPS protection levels for aircraft precision approach and landing

“…The term s 3,i is unit less and is the projection, onto the vertical axis in the locally level coordinate frame, of the unit vector from the user toward the i-th satellite. It is also element (3, 3) of the projection matrix S defined in the previous paper for the x, y, z, and t dimensions in the eastnorth-up locally level coordinate frame (Bruckner et al 2010b). This projection matrix is the same as the LAAS projection matrix except that in LAAS, the locally level coordinate frame is not always aligned with east and north (RTCA 2004(RTCA , 2007.…”

“…The VPL c approximation and one of the two HPL c approximations developed here are derived from PLs described in the literature for a satellite pseudorange bias that has been detected and identified as a system fault. However, we interpret the pseudorange bias as a characteristic of normal GPS operation, and the maximum bound of this bias is incorporated into the PL (Bruckner et al 2010b). Under this new assumption, the two PLs previously described in the literature are shown to be overbounding approximations to exact quantities VPL c and HPL c .…”

“…Improved computational efficiency of PL calculations, relative to the search-based exact VPL c and HPL c algorithms developed in Bruckner et al (2010b), is a motivation for this paper. In the present effort, however, the comparison in computational efficiency remains qualitative, and measuring the improvement is left for future research.…”

“…In Bruckner et al (2010b), a method is also presented by which exact values, i.e. values accurate to a user-defined error tolerance and consistent with statistical assumptions, of VPL c and HPL c are obtained and by which more computationally efficient approximations may be evaluated.…”

“…For further development of these composite PLs, a statistical characterization was previously presented for Global Positioning System (GPS) user range error as a correlated, normally distributed random variable with nonzero mean over the length of the aircraft precision approach operation (Bruckner et al 2010b). This characterization includes modeling GPS error in the position domain as multivariate normal with nonzero mean.…”