Generating basis sets of double differences

Abstract.

When a collection of double differences is used to compute global-positioning-system satellite orbits from a permanent network of receiving stations, linear dependence among the double-differenced observations reduces the number of double differences that contribute new information to the computations. A maximal linearly independent subset of a large collection of double differences contains all the information content of the full set. If r is the number of receivers and s is the number of satellites, the original collection of double differences may have size O(r ² s ²), whereas the linearly independent subset has size no greater than O(rs). Only such a smaller independent subset needs to participate in the more expensive double-precision matrix computations to correctly correlate all double differences, detect cycle slips, resolve ambiguities, and compute satellite orbits and station positions and relative velocities. Dependence among double differences is characterized using vector space methods together with geometric characterizations of Boolean matrices. These characterizations lend themselves to fast, robust algorithms for computing maximal linearly independent sets (bases) of double differences. An algorithm is given for constructing a generating independent set of double differences from the Boolean array of receiving-station/satellite connections. Characterizations of generator equivalence allow alternative generating sets to be identified and selected. An updating algorithm to handle local changes in the satellite–receiver connection matrix is also described.