Performance Evaluation of Several Interpolation Methods for GPS Satellite Orbit

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Performance Evaluationof Several Interpolation
Methods for GPS Satellite Orbit
Presented by Hamad YousifSupervised by Dr. Ahmed
El-Rabbany
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Presentation Topics

• Introduction
• Errors of Interpolation
• Lagrange Method
• Newton Divided Difference Method
• Trigonometric Method
• Broadcast Ephemeris Method
• Conclusion

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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.

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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.

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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.

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Lagrange Method

• Lagrange Formula

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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
0000
2345
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Newton Divided Difference Interpolation

• Newton Divided Difference Formula

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Trigonometric Interpolation

• The Trigonometric Series

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• 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.

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Comparison between Lagrange and Trigonometric
Interpolation
INERTIAL ORBIT MEAN (cm) MEAN (cm) MEAN (cm) STD (cm) STD (cm) STD (cm) MAX (cm) MAX (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
ECEF ORBIT MEAN (cm) MEAN (cm) MEAN (cm) STD (cm) STD (cm) STD (cm) MAX (cm) MAX (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.
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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.

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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.

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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.