Performance Evaluation of Several Interpolation Methods for GPS Satellite Orbit Presented by Hamad Yousif Supervised by Dr. Ahmed El-Rabbany Presentation Topics Introduction Errors of Interpolation Lagrange Method Newton Divided Difference Method Trigonometric Method Broadcast Ephemeris Method Conclusion Introduction The IGS have developed three precise GPS ephemerides: Ultra rapid Rapid Final These ephemerides are spaced at 15 minutes intervals but many GPS applications require precise ephemeris at higher rates, which is the reason for interpolation. Interpolation Errors Function Related Error: The amount of this error can be used as a measure of how well the interpolating method approaches the actual value of the time series. Computer Generated Error: This error is the result of computer limitations. It depends on the operating system, programming language and more or less on computer hardware. Interpolation Properties Taking too few points produces an unreliable interpolation output. Taking a plenty of points is ideally convenient. However, the computer capability is limited up to a specific number of points beyond which the computer behaves unpredictably. The accuracy degrades noticeably near the end points and tends to improve as the interpolator moves towards the center. Lagrange Method •Lagrange Formula: Lagrange Interpolation •INTERPOLATION ALGORITHM: The 24-hour data is divided into 23 overlapping segments each of 9 terms as shown below: SEGMENT 1 SEGMENT 2 SEGMENT 22 SEGMENT 23 00:00 23:45 Newton Divided Difference Interpolation •Newton Divided Difference Formula: Trigonometric Interpolation •The Trigonometric Series: •This method is suggested by [Mark Schenewerk, A brief review of basic GPS orbit interpolation strategies, 2002]. •The code is taken from: http://www.noaa.gov/gps-toolbox/sp3intrp •The Trigonometric coefficients are computed using an algorithm called Singular Value Decomposition. Comparison between Lagrange and Trigonometric* Interpolation MEAN (cm) STD (cm) MAX (cm) INERTIAL ORBIT dx dy dz dx dy dz dx dy dz TRIGONOMETRIC 0.0010 O.OO35 0.0007 0.0499 0.0841 0.0654 0.3000 0.5000 0.4000 LAGRANGE 0.0025 0.0067 0.0037 0.0451 0.0756 0.0405 0.4127 0.6374 0.2233 MEAN (cm) STD (cm) MAX (cm) ECEF ORBIT dx dy dz dx dy dz dx dy dz TRIGONOMETRIC 0.0126 0.0007 0.0021 0.1032 0.0580 0.0696 1.2 0.3000 0.2000 LAGRANGE 0.0016 0.0120 0.0034 0.1548 0.2501 0.0623 1.5216 3.3753 0.4276 * The boundaries of the Trigonometric are not included. According to Schenewerk (2003) the error at the boundaries is 8.2 cm for INERTIAL and 10.3 cm for ECEF. Broadcast Ephemeris Method The direct interpolation of IGS precise ephemeris has one drawback. The very high positive and very low negative values (km) make it difficult to get an accuracy of millimeter level. As another alternative we interpolate the residuals of broadcast- precise ephemeris whose values are in meters and therefore it would be easier to get millimeter accuracy. Conclusion Lagrange and Newton Divided Difference demonstrate completely identical results in terms of interpolation error. Excluding the boundaries, the Trigonometric method yielded the best accuracy of all interpolation methods due to the periodic nature of the GPS orbit. This problem can be avoided by centering the day to be interpolated among sufficient data before and after the day. However, in real time applications no data can be added after the day. Lagrange has a better performance at the boundaries which makes it more convenient for real time applications. The interpolation via the broadcast ephemeris has produced the best results within the two-hour ephemeris period. References Press, W.H., S.A. Teukolosky, W.T. Vetterling, B.P. Flannery (2002). Numerical Recipes in C++: The Art of Scientific Computing. Cambridge University Press. Schenewerk, M. (2003). “A Brief Review Of Basic GPS Orbit Interpolation Strategies.” GPS Solutions, Vol. 6, No. 4, pp. 265-267. Spiegel, M.R. (1999). Mathematical Handbook of Formulas and Tables. McGraw Hill. Armed Forced, Munich.