The use of theThe use of theConsolidated Prediction FormatConsolidated Prediction Format

at Zimmerwaldat Zimmerwald

Werner Gurtner

Astronomical Institute

University of Bern

ILRS Workshop

3-7 October 2005

Eastbourne

CPF Format: SummaryCPF Format: Summary

List of geocentric earth-fixed positions (and velocities) of the satellites, in well-defined coordinate system (ITRS)

Should be suited for satellites, moon, interplanetary ranges (to transponders)

Interval suited for easy interpolation (polynomials, e.g., degree 9)

Auxiliary data in CPF file header Computation of ranges and pointing directions by the

stations (sample software provided) Daily files with a few days worth of positions

Open issuesOpen issues

Do we need velocities? Do we need outbound (and inbound)

geocentric vectors or state vectors? How do we handle general relativistic

corrections?

For satellites up to GPS/Glonass geocentric earth-fixed state vectors alone are OK

Sources of CPF FilesSources of CPF Files

NSGF Starlette, Stella, Topex, Etalon-1/2,

Lageos 1/2, Ajisai, GFO-1, (Envisat)

AIUB/CODE GPS 35/36, Glonass 84/87/89/95

HTSI Beacon-C, Larets

UTX ICESat

Handling of CPF FilesHandling of CPF Files(separately for each satellite)(separately for each satellite)

CPF files are daily received by mail (a few days worth of data)

Add new records to / replace old records in merged file(Take only records in and around predicted pass intervals to keep merged file smaller, delete records older than 10 days)

Generate weekly pass list Compute velocity for late epoch (Lagrange interpolation: Sr HERMITE) Compute osculating elements for late epoch Determine actual pass times (geometry, visibility, illumination)

Example: CPF file for GPS-35Example: CPF file for GPS-35

From: AIUAS3::LASER 19-SEP-2005 04:01:28.92To: LASERSubj: GPS-35 DAILY CPFS CODE

H1 CPF 1 AIUB 2005 9 19 4 262 H2 9305401 3535 22779 2005 9 18 23 59 47 2005 9 23 23 29 47 900 1 1 0 0H910 1 53631 86387.000000 0 -22692683.592 13597988.493 -963615.27310 1 53632 887.000000 0 -22806500.488 13274412.610 1854140.90910 1 53632 1787.000000 0 -22674715.983 12775241.829 4639580.11410 1 53632 2687.000000 0 -22322988.129 12076006.947 7344024.17210 1 53632 3587.000000 0 -21782775.741 11158720.835 9920055.97010 1 53632 4487.000000 0 -21090035.585 10012807.677 12322397.15810 1 53632 5387.000000 0 -20283744.971 8635744.185 14508754.13310 1 53632 6287.000000 0 -19404309.723 7033383.207 16440612.94110 1 53632 7187.000000 0 -18491923.097 5219943.579 18083964.528...99

Prediction GenerationPrediction Generation

Interpolate satellite positions for reflection times in appropriate intervals during satellite pass Interpolation: Sr HERMITE Iteration

Compute Ranges / flight times

(subtract corner cube z positions for GPS, Glonass) Pointing elements (azimuth, elevation) Point-behind angle

Allow for Tropospheric corrections Time biases

Ranges and pointing anglesRanges and pointing angles

tt

tb tr

tr

tbtt

rtb rbr

Station

Satellite

tb-1

point – behind angle

r = rtb + rbr 2 * rbb

rbb

tt: transmit timetb: bounce timetr: receive time

rtt

rrr

Lagrange Polynomial InterpolationLagrange Polynomial Interpolation

Easy to program Use n points to interpolate with polynomial of degree n-1 No need for equally spaced points Apply formula to center interval of given values only First derivative of the formula gives velocities Separate interpolation for x, y, z Does not explicitly give polynomial coefficients Not optimized for speed

1 2 653 4

n=6

Lagrange Polynomial InterpolationLagrange Polynomial Interpolation

Easy to program Use n points to interpolate with polynomial of degree n-1 No need for equally spaced points Apply formula to center interval of given values only First derivative of the formula gives velocities Separate interpolation for x, y, z Does not explicitly give polynomial coefficients Not optimized for speed

1 2 653 4

n=6

Software: Interpolation SubroutineSoftware: Interpolation Subroutine

SUBROUTINE HERMITE(ITYP,X,Y,Z,NMAX,NVAL,XP,YP,ZP,IRCODE) --------------------------------------------------------

Interpolation by a polynomial using NVAL out of NMAX given data points

Input : ITYP : 1: use Lagrange polynomial of degree NVAL-1 2: use Hermite formula: Polynomial of degree 2*NVAL-1

NVAL : number of points to use for interpolation NMAX : number of given points in list X(I) : arguments of given values (I=1,...,NMAX) Y(I) : functional values Y=f(X) Z(I) : derivatives Z=f’(X) XP : interpolation argument

Output: YP : interpolated value at XP ZP : first derivative of YP (ITYP=1 only) IRCODE: return code (0=ok, 2=error)

The function selects the NVAL values to be used for interpolation such that the interpolated data point is located in the center interval. (Works best for NVAL = even number, of course).

Subroutine CPF_INTERSubroutine CPF_INTERSubroutine CPF_INTER********************

INPUT:

TT : EPOCH (TRANSMIT=FIRING TIME) MODE : 0: NO FLIGHT TIME APPLIED FLIGHT TIME = 2 * INSTANT RANGE AT EPOCH TT 1: OUTBOUND VECTOR (AZIMUTH, ELEVATION) FLIGHT TIME = 2 * INSTANT RANGE AT BOUNCE TIME 2: OUTBOUND AND INBOUND VECTOR FLIGHT TIME = OUTBOUND + INBOUND

STAGEO() : GEOCENTRIC STATION COORDINATES XYZSTALON : STATION LONGITUDE (EAST > 0, RADIANS) STALAT : STATION LATITUDE (NORTH > 0, RADIANS)

TTAB() : TABULATED EPOCHSOBJTABX(): TABULATED X-COORD. OF OBJECT (ITRF)OBJTABY(): TABULATED Y-COORD. OF OBJECT (ITRF)OBJTABZ(): TABULATED Z-COORD. OF OBJECT (ITRF)NTAB : NUMBER OF TABULATED EPOCHS/COORDINATESNINT : NUMBER OF TABULATED VALUES TO USE

OUTPUT:

RANGE : ONE-WAY RANGE (M)FLTIME : TWO-WAY FLIGHT TIME (SEC)AZIOUT : OUTBOUND AZIMUTH (DEG)ELEOUT : OUTBOUND ELEVATION (DEG)DIFAZI : POINT-BEHIND (AZIMUTH, ARC SECONDS)DIFELE : POINT-BEHIND (ELEVATION, ARC SECONDS) DIFFERENCE INBOUND MINUS OUTBOUND AT EPOCH TT

Flight time difference CPF-IRV (Lageos-2)Flight time difference CPF-IRV (Lageos-2)

NSGF

Flight time difference CPF-IRV (GPS-36)Flight time difference CPF-IRV (GPS-36)

/ NSGF (6 hrs)

Flight time difference IRV-CPF (ICESat)Flight time difference IRV-CPF (ICESat)

Lageos-1: Observed - predictedLageos-1: Observed - predicted

Lageos-2: Observed - predictedLageos-2: Observed - predicted

Glonass-84: Observed - predictedGlonass-84: Observed - predicted

GPS-36: Observed - predictedGPS-36: Observed - predicted

ICESat: Observed - predictedICESat: Observed - predicted

ICESat: Observed - predictedICESat: Observed - predicted

Earth TidesEarth Tides

not applied

applied

PerformancePerformance

GPS CODE < 2 ns (!) Glonass CODE < 4 ns Ajisai NSGF < 10 ns BE-C NSGF < 200 ns (occ. up to 500 ns)

Envisat NSGF < 20 ns (IRV ESOC up to500 ns!)

ICESat UTX < 200 ns (occ. up to 1000 ns)

Lageos NSGF < 7 ns LaretsHTSI < 100 ns (recently worse)

ConclusionsConclusions

Relatively easy to implement Core computation easy Handling of CPF files: Merge daily mails or ftp-ed

files, keep them limited in size Significant improvement of prediction accuracy Approach all prediction providers to generate CPF

in parallel with IRV Encourage stations to switch to CPF predictions

(allow for both, CPF and IRV, in the transition phase)