Performance EvaluationPerformance Evaluationof Severalof Several

Interpolation Methods Interpolation Methods for GPS Satellite Orbitfor GPS Satellite Orbit

Presented by Hamad YousifPresented by Hamad YousifSupervised by Dr. Ahmed El-RabbanySupervised by Dr. Ahmed El-Rabbany

Presentation TopicsPresentation Topics

IntroductionIntroductionErrors of InterpolationErrors of InterpolationLagrange MethodLagrange MethodNewton Divided Difference MethodNewton Divided Difference MethodTrigonometric MethodTrigonometric MethodBroadcast Ephemeris MethodBroadcast Ephemeris MethodConclusionConclusion

IntroductionIntroduction

The IGS have developed three precise GPS The IGS have developed three precise GPS ephemerides:ephemerides:

Ultra rapidUltra rapidRapidRapidFinalFinal

These ephemerides are spaced at 15 minutes These ephemerides are spaced at 15 minutes intervals but many GPS applications require intervals but many GPS applications require

precise ephemeris at higher rates, which is precise ephemeris at higher rates, which is the reason for interpolation.the reason for interpolation.

Interpolation ErrorsInterpolation Errors Function Related Error:Function Related Error:

The amount of this error can be used as a The amount of this error can be used as a measure of how well the interpolating measure of how well the interpolating method approaches the actual value of the method approaches the actual value of the time series.time series.

Computer Generated Error:Computer Generated Error:

This error is the result of computer This error is the result of computer limitations. It depends on the operating limitations. It depends on the operating system, programming language and more system, programming language and more or less on computer hardware.or less on computer hardware.

Interpolation PropertiesInterpolation Properties

Taking too few points produces an Taking too few points produces an unreliable interpolation output.unreliable interpolation output.

Taking a plenty of points is ideally Taking a plenty of points is ideally convenient. However, the computer convenient. However, the computer capability is limited up to a specific capability is limited up to a specific number of points beyond which the number of points beyond which the computer behaves unpredictably.computer behaves unpredictably.

The accuracy degrades noticeably The accuracy degrades noticeably near the end points and tends to near the end points and tends to improve as the interpolator moves improve as the interpolator moves towards the center.towards the center.

Lagrange MethodLagrange Method

0

0 1 1 1

0 1 1 1

, ( ) ( )

( )

:

( )( ) ( )( ) ( )

( )( ) ( )( ) ( )

( )

n

i ii

i i ni

i i i i i i i n

i i

The approximated value of f denoted by p t at any time t

p t a f

where

t t t t t t t t t ta

t t t t t t t t t t

a is referred to as L t which is called the Lagrange Operat

:

1( )

0

:

( )

. .

.

ii

k k

er

and has the following property

for t tL t

otherwise

consequently

p t f

i e Lagrange polynomial returns the exact value

of f at the given collocation points

•Lagrange Formula:

Lagrange InterpolationLagrange 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 InterpolationNewton Divided Difference Interpolation

•Newton Divided Difference Formula:

0 1 0 2 0 1 3 0 1 2 0 1 1

0 1 2

0 0

1 01

1 0

2 0 1 2 02

2 0 2 1

3 0 1 3 03

( ) ( ) ( )( ) ( )( )( ) ( )( ) ( )

, , , , :

( )

( )

( )

( ) ( )

( )( )

( ) ( )

n n

n

p t a a t t a t t t t a t t t t t t a t t t t t t

a a a a can be d et er mi ned as follows

a p t

p t aa

t t

p t a a t ta

t t t t

p t a a t t aa

2 3 0 3 1

3 0 3 1 3 2

0 1 0 2 0 1

0 1 2 1

( )( )

( )( )( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )n n n n

nn n n n n

t t t t

t t t t t t

p t a a t t a t t t ta

t t t t t t t t

Trigonometric InterpolationTrigonometric Interpolation

•The Trigonometric Series:

0

1

1

1

2 2( ) [ cos( ) sin( )]

2

:

2 2cos( )

2 2sin( )

2( )

M

n nn

N

n k kk

N

n k kk

a n np t a t b t

T T

where

ka f t

N T

kb f t

N T

T is the period The fundamental frequencyT

N is the number of collocation points

M is the number of truncated terms

•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

INERTIAL ORBITMEAN (cm) STD (cm) MAX (cm)

dx dy dz dx dy dz dx dy dz

TRIGONOMETRIC0.001

0O.OO3

50.0007 0.0499 0.0841 0.0654

0.3000

0.5000

0.4000

LAGRANGE0.002

50.0067 0.0037 0.0451 0.0756 0.0405

0.4127

0.6374

0.2233

ECEF ORBITMEAN (cm) STD (cm) MAX (cm)

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 MethodBroadcast Ephemeris Method

The direct interpolation of IGS precise The direct interpolation of IGS precise ephemeris has one drawback. The very high ephemeris has one drawback. The very high positive and very low negative values (km) positive and very low negative values (km)

make it difficult to get an accuracy of make it difficult to get an accuracy of millimeter level. As another alternative we millimeter level. As another alternative we

interpolate the residuals of broadcast- interpolate the residuals of broadcast- precise ephemeris whose values are in precise ephemeris whose values are in

meters and therefore it would be easier to meters and therefore it would be easier to get millimeter accuracy.get millimeter accuracy.

ConclusionConclusion Lagrange and Newton Divided Difference demonstrate Lagrange and Newton Divided Difference demonstrate completely identical results in terms of interpolation error.completely identical results in terms of interpolation error.

Excluding the boundaries, the Trigonometric method Excluding the boundaries, the Trigonometric method yielded the best accuracy of all interpolation methods due yielded the best accuracy of all interpolation methods due to the periodic nature of the GPS orbit. This problem can to the periodic nature of the GPS orbit. This problem can be avoided by centering the day to be interpolated among be avoided by centering the day to be interpolated among sufficient data before and after the day. However, in real sufficient data before and after the day. However, in real

time applications no data can be added after the day.time applications no data can be added after the day. Lagrange has a better performance at the boundaries Lagrange has a better performance at the boundaries

which makes it more convenient for real time which makes it more convenient for real time applications.applications.

The interpolation via the broadcast ephemeris has The interpolation via the broadcast ephemeris has produced the best results within the two-hour ephemeris produced the best results within the two-hour ephemeris

period.period.

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.

ReferencesReferences