Efficient Algorithm for Inverse of the Fully Populated Covariance Matrix in Global Positioning System Relative Positioning

Abstract

Stochastic modeling of the global positioning system (GPS) residual errors yields a fully populated covariance matrix for the GPS carrier phase double difference observations. Implementing this fully populated covariance matrix into a software package usually slows down the numerical computations. This, however, is not the case if an exponential function can be used to approximate the actual covariance function of the GPS residual errors. Using the exponential function results in a block diagonal weight matrix for the double-difference observations. In this paper, an algorithm for the efficient computation of the inverse of this fully populated covariance matrix is developed. Both the ideal case of tracking the same satellites over time as well as the more realistic case of tracking different satellites over time are considered. The storage requirement for the adjustment process is also discussed.