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					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.

				
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