Satellite Geodesy

angular resolution = lambda / D, Here D is the diameter of the telescope. Because the same signal is measured in more than one telescope it’s possible to take the distance between these telescopes as D an not only the real diameter.

large telescope, receiver, high-precision frequency standard, recording device

structure of the radio sources (quasars don’t appear as point sources), antenna structure (deformation, thermal expansion), troposphere (water vapor)

a)“Clock error” (see formula 2.21): The stronger the gravity field is, the more slowly the clocks are running and moving clocks are running more slowly than clocks at rest // b) Signal goes across a Gravity Field (eg. Earth, Sun) (see formula 2.26): a signal is slowed down when propagating through a gravitational field (only relevant when the signal passes very closely to a gravity body)

crosscorelation: shift one data series to the other until the best match is found.

Signal to Noise Ratio very bad => long observation (up to 10’)

Map of known and used (ICRF) Quasars

Minddestens zwei Radioteleskope messen die Signale, welche von eine Quasar ausgesendet werden. Dabei werden die Daten mit einem Zeitstempel versehen um sie danach korrelieren zu können. So können danach die Laufzeitunterschiede vom Quasar zum einen oder anderen Teleskop ermittelt werden.

Daraus können Positionen abgeleitet werden.

Quasars are cores of galaxies. Seen from Earth they have often a complex structure and an extension of up to 60''.

In the unit vector (e) poiting to the Quasars (barrycentric coordinate system)

Static coordinate sysstem with origin in the barycenter of the solar-system (or Earth-Moon-System)

Lorentz-Transformation, Speed is the only parameter

Transformation from Earth-fixed to space-fixed, P: Precession, N: Nutation, U: Rotation matrix with Greenwich sidereal time as argument, X,Y: Polar motion

U = newtonian gravitationl potential at the location of the clock // va = velocity of the atomic clock // c = speed of light // delta_tau_geo = purely geometrical (or newtonian) delay difference between the arrival times of a wavefront at both telescopes. v ist die Geschwindigkeit der Atomuhr verglichen mit dem Massenzentrum des Systems. ca. 4km/s (Speed of a GPS-Satellite with respect to the Earth) ca. 29km/s (Speed of Earth with respect to the sun)

a: 26'600km, Periode: 11h58, I: 55°, 6 Ebenen mit 60° spacing, 24 Satelliten, CDMA (PRN-Codes)

L1: f1 = 1575 MHz, _ = 19cm, C/A-Code, P-Code // L2: f2 = 1227 MHz, _ = 24cm, P-Code

(clear access / coarse acquisition), 1023bits (chips), repeats after 1ms,

chip length (1 bit) = 293m, only modulated on L1, each GPS-Satellite sends own code-sequence.

(protected / precise), repeats after 266.4 days (=2.35*10^14 chips), chip length = 29.3m, transmitted on L1 and L2

on L1 and L2, contains broadcast ephemerides (pseud-keplerian elements), satellite clock corrections (polynomial of 2nd degree), almanach data (approximate orbit info for long-term predictions), information about ionosphere, health status of the satellites

Selective Availability (SA): The accuracy of the positioning is degraded. This works with the so called dithering. There the satellite clock is manipulated in the range of 2 s, witch correspond to 60m. With relative positioning the SA wasn’t a problem. The SA was switched off at May 2, 2000. Anti-Spoofing (AS): The P-Code is encrypted by superposing a additional W-code. => Y-code. The AS was implemented to avoid that position quality can be degraded by a “wrong” artificial GPS-signal. Because of the AS there is an increased noise level in the code measurements ans especially in the carrier phase measurements.

no signal degradation mechanisms

receiver position at reception time, satellite position at emission time, tropospheric delay, ionospheric delay, relativistic correction, influence of multi-path, light travel time, light velocity

re(tE): Receiver position at reception time (tE), rs(tS): =rs(tE-tES): Satellite position at the time of emission tS, trp: signal delay in troposphere, ion, signla delay in ionosphere, rel: relativistic correction, mul: infuence of multi-path

Vorzeichen für Ionosphären-Korrektur (pase advane, group delay), andere Wert für Multipath-Korrektur.

to improve the accuracy: get rid of the satellite clock error, receiver clock error and other effects

2 Reveivers observe simultanous the same satellite // calculate the difference of the two observation equations // eliminated satellite clock error (it’s not totally eliminated - but nearly) // other errors are reduced as well, if baseline is not too long (sallite orbit errors, atmospherical and relativistic effects) // system noise increases by a factor of sqrt(2)

2 receivers, 2 satellites // differences of the observation eqations // eliminates receiver clock error (you still need to know the clock error to calculate the satellite positions in the correct epoch!) // system noise increases by a factor of 2

2 reveivers, 2 satellites measured in two epochs // eliminates the ambiguity // first robust solution for relativ-coordinates // system noise increases by a factor of 2*sqrt(2)

a) multipath-effects b)phasecenter of GPS antenna: actual point of reception. Depending on azimuth and elevation of the GPS satellites. // 1) combining different antenna types: mainly affecting the height component (up to 10cm), horizontal position maximally a few millimeters if tropospheric zenith delays are estimated in the adjustment // 2) long baselines, errors even present if identical antenna types are used! due to different elevtion angle to the same satellite. //

ant(z,a) = -(r0e+(a,z)),

e = unit vector pointing from receiver to satellite,

Corrections are also necessary for the satellite antenna phase center!