Relationship between navigation vertical error, VDOP, and pseudorange error in GPS

It is well known that at a given position and time, vertical navigation sensor error is related to the satellite ranging errors by the relationship Z=VDOP*URE, where URE is the user range equivalent error, VDOP is the vertical dilution of precision, and Z is the vertical position error. It is less well known that this relationship is statistically correct even when position and time are variables. This paper develops this and shows that the distribution of Z can be expressed in terms of the distributions of VDOP and URE. Expressions are given for the density, cumulative distribution function, and second order moments of Z. Using the developed formulation, the distribution of Z is investigated for several different scenarios and the results correlated with published work in the field. The paper provides a simple means for estimating vertical accuracy as a function of satellite ranging errors and geometry as an alternative to performing more complex simulations