Titelmaster Geometrical and Kinematical Precise Orbit Determination of GOCE Akbar Shabanloui Institute of Geodesy and Geoinformation, University of Bonn

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Geometrical and Kinematical Precise Orbit Determination of GOCEAkbar ShabanlouiInstitute of Geodesy and Geoinformation, University of Bonn

7th October 2010Cologne, Germany

Titelmaster1Prof. Helfrich Thank you very much for your introduction, Good afternoon ladies and Gentlemen and welcome to my Ph.D presentation with the title of A new approach for a kinematical-dynamical Determination of LEOs based on GNSS observations.Outline2Precise Orbit Determination (POD) principleGOCE Lagrange receiver (clock)Zero difference estimation procedureResultsConclusionsGeometrical Precise Orbit Determination (GPOD)Kinematical Precise Orbit Determination (KPOD)2What is the concept of Precise Orbit Determination (POD) with GNSS observations as main tools. In this case we use the GPS satellite orbits as known parameters from IGS centers, and GPS-SST observations to transfer these positions to LEO satellites. Our final goal in space geodesy is determination of the Earth gravity field is to have a kinematical representation of these LEO satellites. As you see here, CHAMP orbit has been tracked with 4 GPS satellites to determine its point-wise orbit.

Precise Orbit Determination (POD) 3Credit by ESA

3What is the concept of Precise Orbit Determination (POD) with GNSS observations as main tools. In this case we use the GPS satellite orbits as known parameters from IGS centers, and GPS-SST observations to transfer these positions to LEO satellites. Our final goal in space geodesy is determination of the Earth gravity field is to have a kinematical representation of these LEO satellites. As you see here, CHAMP orbit has been tracked with 4 GPS satellites to determine its point-wise orbit.

Precise Orbit Determination (POD) methods4PODGeometricalKinematicalDynamicalPrecise Orbit Determination (or POD) is classified in three main methods namely geometrical, kinematical and dynamical methods. Geometrical-POD is shown with blue colour and kinematical and dynamical with these coloures. The transition from geometrical to kinematical or kinematical to dynamical is possible. 4Geometrical Precise Orbit Determination (GPOD)5

5What are tools to determine geometrical precise orbit? Our observations as I mentioned are Only, just Only geometrical SST observations between the GPSs and LEO satellites, with these SST observations we want to transfer coordinates of GPS satellites to LEO satellite (s). Our model to solve observation equations are, Gauss-Markov model, and our unknowns are LEO satellite (s) positions at the observed epochs namely point-wise epochs.

As you see in the given movies, only Geometrical SST observations have been used, and LEO positions have been estimated in the observed epochs, and representation of orbit isnt continuous, but discrete.

Precise Orbit Determination (POD) methods6PODGeometricalKinematicalDynamicalPrecise Orbit Determination (or POD) is classified in three main methods namely geometrical, kinematical and dynamical methods. Geometrical-POD is shown with blue colour and kinematical and dynamical with these coloures. The transition from geometrical to kinematical or kinematical to dynamical is possible. 6Kinematical Precise Orbit Determination (KPOD)7

7What are tools to determine Kinematical precise orbit? Our observations as I mentioned are geometrical SST observations between GPS and LEO satellites & given representation form of LEO orbit, with these SST observations and given representation form of orbit (short arc) we want to transfer coordinates of GPS satellites to LEO satellite (s). Our model to solve observation equations are, Gauss-Markov model, and our unknowns are LEO satellite (s) representation parameters.

As you see in the given movie, Geometrical SST observations and representation form have been used, and LEO representation parameters (namely short arc parameters) have been estimated therefore LEO positions can been estimated at every epochs, in other word the representation of the LEO orbit is continuous.

Precise Orbit Determination (POD) methods8PODGeometricalKinematicalDynamicalGeometrical POD : point-wise, positionsKinematical POD : continous, positions, velocities and accelerationsDynamical POD : continous, positions, velocities and accelerations based on force function information!= KPOD, e.g. BernPrecise Orbit Determination (or POD) is classified in three main methods namely geometrical, kinematical and dynamical methods. Geometrical-POD is shown with blue colour and kinematical and dynamical with these coloures. The transition from geometrical to kinematical or kinematical to dynamical is possible. 8GPS LAGRANGE receiver onboard GOCE9

Credit by ESAPrecise Orbit Determination (or POD) is classified in three main methods namely geometrical, kinematical and dynamical methods. Geometrical-POD is shown with blue colour and kinematical and dynamical with these coloures. The transition from geometrical to kinematical or kinematical to dynamical is possible. 910GNSS receiver on-board GOCE

Credit by ESA10In simulated case with GPOD to estimate the absolute precise, 15 deg. cut-off angle has been applied to GPS-SST observations. Zero difference carrier phase observations are used in data processing.

Elevation weighting strategy has been applied.to weigthing GPS-SST observations.

11GNSS receiver on-board GOCELAGRANGE (Laben GNSS Receiver for Advanced Navigation, Geodesy and Experiments)

12 chanels, dual frequency (L1 and L2) GPS/GLONASSThe clock of the GOCE LAGRANAGE receiver is not steered to integer seconds (free clock system)Interpolation of SST observations (Data Screening with triple differenced method)Interpolation of GPS orbits (Zero differenced)

11In simulated case with GPOD to estimate the absolute precise, 15 deg. cut-off angle has been applied to GPS-SST observations. Zero difference carrier phase observations are used in data processing.


12GNSS receiver on-board GOCE (Rinex SST)

Clock is not steerd to be integer12In simulated case with GPOD to estimate the absolute precise, 15 deg. cut-off angle has been applied to GPS-SST observations. Zero difference carrier phase observations are used in data processing.


13Clock jumps of receiver (Nov. 2009)GPS LAGRANGE receiver clock behavior!



14Clock behavior of receiver (Nov. 2009)Clock jumps of 20 ms at ~27 hours can be seen

20 ms 14In simulated case with GPOD to estimate the absolute precise, 15 deg. cut-off angle has been applied to GPS-SST observations. Zero difference carrier phase observations are used in data processing.


15GPS visibility onboard GOCE (Nov. 2009)7 < Number of GPS satellites (PRNs) < 12



167 < Number of GPS satellites (PRNs) < 12GPS visibility onboard GOCE (Nov. 2009)



17Zero differenced GPOD

Credite by ESA17In simulated case with GPOD to estimate the absolute precise, 15 deg. cut-off angle has been applied to GPS-SST observations. Zero difference carrier phase observations are used in data processing.


Zero DifferenceOnly connection between LEO satellite and GPS satellites,GeometricalOnly pure geometrical relations between LEO and the GPS satellites have to be used, no force models and no constraints,PreciseConsideration all effects on GPS-SST observations and using precise GNSS satellites ephemerides.Zero differenced GPOD1818At the beginning of presentation these three precise orbit determinations methods, three concepts have to be declared.At first Zero difference, Zero difference means that we must use only SST observations between two sender and receiver.

Geometrical observation, is only observation, which it doesnt depend on any dynamical or kinematical model.

What is the meaning of Precise, Precise means that we must consider all systematical error sources on the sender, observations, and on the receiver.Processing conceptGPODZero differencing procedure (ZD)Phase measurementsCode measurements1919The measurements are code (pseudo-range) and carrier phase observations, we use these measurements in Zero difference form, in another words, we use only these observations without differencing in place. Then as I already mentioned, the three main types of orbits can be determined.Precise Orbit Determination (POD) 20

No troposphere effect at GOCE altitude (~250 km)First order ionospheric effect eliminated with Ion-free linear combinationAmbiguity term cannot be solved as Integer (real)!GPS precise orbits (at 15 minutes) and clocks at 30 sec20What is the concept of Precise Orbit Determination (POD) with GNSS observations as main tools. In this case we use the GPS satellite orbits as known parameters from IGS centers, and GPS-SST observations to transfer these positions to LEO satellites. Our final goal in space geodesy is determination of the Earth gravity field is to have a kinematical representation of these LEO satellites. As you see here, CHAMP orbit has been tracked with 4 GPS satellites to determine its point-wise orbit.

21GPS Antenna offsets



22GPS antenna offsets GPS receiver offset with respect to GOCE reference frame is constantOffset with respect to center of mass (COM) is slowly varing!L1 and L2 (L3) Phase Center Offsets (PCO) are derived from IGS ANTEX (ANTenna Exchange format)Phase Center Variation (PCV) can be empirical estimated based on carrier phase residuals! (or ANTEX?)



23Geometrical POD hl-SST

23To show how Geometrical orbit of the LEO is estimated, we assume that 5 GPS satelites are available at every observation epoch. For s1 as a first tracked GPS satellites, carrier phase obseration equation can be written as

An the corresponding linearized model is,

If the same process is repeated for all tracke GPS satellites from s1-s5, then the observation equations at time t can be formulated as,

It is clear for the first epoch, the number of the unknows are more than the observations (the observation system is underdetermined). To solve the problme, the GPS-SST observations at the next epochs must be added to the system. Or in the other words, Geometrical Orbit determination with carrier phase observations must be solved in a Batch processing.

For example if we add the observation equations from time t2 to time tn, then the Gauss-Markov model for all tracked epochs can be written as

The convergence of the unknowns can be achieved after a few iterartions.

24Ground tracks of GPS, GOCE (2 Nov. 1 Rev.)



25Short arc of GOCE (02 Nov. 2 Rev.)

30 minutes short arc (2009-11-02 02 00 00 - 02 30 00)25In simulated case with GPOD to estimate the absolute precise, 15 deg. cut-off angle has been applied to GPS-SST observations. Zero difference carrier phase observations are used in data processing.


GPOD GOCE results26Estimated geometrical 3D Pos. - PSO

In the GPOD, geometrical configuration of the GPS satellites and the number of them, play an important role in precision of determined point-wise orbits. As you see here, at the top-left figure , the PDOPS and the number of tracked GPS satellites are shown (they are between 5 and 10 satellites), and at top-right GPS satellite elevation angles are shown.

To test the GPOD in error case, the GPS-SST observations are contaminated with white noise, the GPS-SST observations are contaminated with 2 cm white noise. As you see, at the top-left figure, the position differences are in the range of +- 2-3 cm and the corresponding residuals of GPS-SST observations are in the +-5-6 cm. 2627Kinematical POD

27In Kinematical Precise Orbit Determination, we want to connect the geometrical orbit to represent a continous LEO orbit. To do that

Equation of motion w.r.t self-adjoint differential operator can be written as

The corresponding Integral equation of Fredholms type to this differenetial equation can be formulated as

The definition of parameters

If simple newton operator is selected then differential equation can be written as..

The corresponding Fredholms integral equation can be formulated as...With the triangle integral kernel as

The Hammerstein solution of Fredholms intergral equation is shown as.. With linear connection between the short arc boundaries and Bilinear Integral kernel as If we insert the Integral kernel in Freholms intergal equation and compare with Hammerstein solution, then the amplitudes of sine function can be directly estimated.

As you see, the LEO short arc at every epoch is represented through linear connection between the boudaries and the sine function.

28Short arc representationLEO orbit can be represented as:

Gibbs effect!

Precision!

Solution?

fast convergence!or28Now in brief,

As a first method, a satellite shor arc is represented through Fourier series, but Gibbs effect is a major problem in this method.

As a alternative, a short arc is expressed through Euler-Bernoullo polynomials, but precise of this form is the major problem, Now what is the solution. A combination of Hybrid of these two methods seems to be very flexible method to determine short arcs.

Because of smoothness of the difference funtion of Euler-Bernoulli term, convergence of remainder function is very fast, and shows minimum Gibbs effect in the representation step.29

Reference motion Euler-Bernoulli

Determination of Euler-Bernoulli coefficientsA satellite short arc can be represented with the Euler-Bernoulli term up to degree J as:29If we negelct remainder term, then Euler-Bernoulli series can be demonstrated as a summation of Euler and Bernoulli term up to a sufficient degree J.

For example, if a satellite short arc orbit is determined with GPOD (Geometrical Precise Orbit Determination) procedure, then with absolute positions as observations, observation equation is linear with respect to Euler-Bernoulli coefficients. The unknows can be determined in one step.

For example as you see KPOD GOCE results30Estimated kinematical Pos. (J=4) PSO (Fourier index 30 and 40)

In the GPOD, geometrical configuration of the GPS satellites and the number of them, play an important role in precision of determined point-wise orbits. As you see here, at the top-left figure , the PDOPS and the number of tracked GPS satellites are shown (they are between 5 and 10 satellites), and at top-right GPS satellite elevation angles are shown.

To test the GPOD in error case, the GPS-SST observations are contaminated with white noise, the GPS-SST observations are contaminated with 2 cm white noise. As you see, at the top-left figure, the position differences are in the range of +- 2-3 cm and the corresponding residuals of GPS-SST observations are in the +-5-6 cm. 3031Conclusions and recommendationsNo gravity field and no force models have been used in the Geometrical and Kinematical modes (advantage).Kinematical POD can be used to recover the Earths gravity field model based on the hl-SST methods, (GOCE SST model).Empirical PCV results should improve POD of GOCE!GNSS-GOCE satellites configuration and geometrical strength play an important role in POD.Estimated Geometrical Precise Orbit can be used to estimate kinematical POD of GOCE.31Thank youfor your attention!

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Titelmaster Geometrical and Kinematical Precise Orbit Determination of GOCE Akbar Shabanloui Institute of Geodesy and Geoinformation, University of Bonn