Efficient interpolations to GPS orbits for precise wide area applications

For precise real time or near real time differential GPS positioning in a wide or global area, precise GPS orbits or, alternatively, precise orbital corrections with respect to a reference orbit, such as GPS broadcast ephemerides, must be used. This work tests orbit interpolation methods, in order to represent the GPS orbits and orbital corrections accurately and efficiently for these and other GPS applications. For precise GPS orbits given in the SP3 format at the 15 min interval, numerical tests were conducted using Lagrange and Chebyshev as well as trigonometric polynomial functions. The results have demonstrated that the 19- or 20-term trigonometric function is apparently the most efficient interpolator for a 12 h GPS orbital arc, achieving 1 cm level 3D interpolation accuracy that can meet the requirements of most precise applications. The test results also demonstrated that the 9-term trigonometric function always yields optimal interpolation for a 2 h GPS orbit arc, in terms of interpolation errors, compared to the results when using a different number of terms for the same function or one of the other tested polynomial functions. This is evident from the minimal performance degradation when using the 9-term trigonometric function to interpolate near or at the end of a data interval. By limiting interpolation to the center 15 min to 1.5 h of a 2 h orbit arc, thereby eliminating the need to interpolate near the ends of that interval, users can opt for more terms (11 and 13) or different interpolators to further improve interpolation accuracy. When interpolating the orbital corrections with respect to the GPS broadcast ephemeris, all the tested interpolation functions of 3- to 9-term yield the same suitably accurate results. Therefore, a 3- to 5-term trigonometric function is arguably sufficiently accurate and more efficient for GPS orbital correction messaging in wide area and real time positioning.