GDOP and the Cramer-Rao bound

Abstract

The GDOP is frequently thought of as a number signifying the effect of satellite geometry on computed position. More generally, it is well known that the GDOP matrix is the covariance of the linearized least squares errors in estimating position and bias from pseudoranges with unit variances. However a much stronger statistical interpretation is possible. In this paper we explore the relationship between the GDOP matrix and the Cramer-Rao bound of classical statistical point estimation. We also detail an interpretation of GDOP made in earlier work. In the first part of the paper we show that the GDOP matrix is actually the Cramer-Rao lower bound on estimates of position and bias given that the pseudorange errors are Gaussian distributed. In light of recent work indicating that pseudoranges not affected by SA are likely not Gaussian, we discuss generalization of this result to symmetric non-Gaussian pseudorange errors. In the second part of the paper, the GDOP is interpreted in terms of covariance about the average line of sight vector. This interpretation is used to compare ranging systems with pseudoranging systems. It is demonstrated that ranging systems are inherently more accurate, given the same geometry and measurement variances. Conditions under which the two systems are equivalent are derived. The role of the clock bias in this relationship is detailed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>