Satellite Geodesy
Exam questions of the lecture «Satellite Geodesy» by Prof. Dr. Rothacher, D-BAUG, ETH Zürich
Exam questions of the lecture «Satellite Geodesy» by Prof. Dr. Rothacher, D-BAUG, ETH Zürich
How is the ionospheric delay corrected for in VLBI, GPS, SLR?
The ionospheric refraction is computed by an integration of the refractive index n along the path s of the signal (neglecting S - G). // Ne… electron density // S … lengths of the actual, real signal path // G … straight lined geometric distance
Altitude range of the troposphere
0-10km
On what does the refractivity depend in the case of the troposphere?
on the refractive index n=c/v (ratio of speed of light c in the vacuum and the speed of light v in the atmosphere
Difference between dry and wet part of the delay
The dry part accounts for about 90% of the total delay and is mainly depending on the pressure of the dry air and can be modelled quite well based on pressure measurements on the ground. In general the air pressure and so the dry part of the delay is changing rather slowly with time (2 cm/12 hours) // The wet part (caused by water vapor) only amounts to a maximum about 40 cm but is highly variable in time and space and therefore very difficult to model because of the unknown distribution of water vapor and water in the atmosphere.
How is the tropospheric delay corrected for in VLBI, GPS, and SLR?
Mapping functions (simpelst with 1/cos(z))
Difference in the influence of the troposphere on VLBI, GPS, SLR?
For radio waves in the frequency range from 100 MHz to about 15 GHz (as in case of VLBI and GPS) the refractive index n is independent of the frequency (--> identical delay for all frequencies) // For optical wavelengths (SLR/LLR with a frequency of around 5*1014Hz ) the tropospheric refraction is dispersive
Order of magnitude of the tropospheric delay for VLBI, GPS, SLR?
The total delay (dry and wet part) amounts to about 2.3 m in the zenith direction (~8 ns in time units) and about 25 m at an elevation of 5°.
The wet part is maximal 40 cm.
What is a mapping function? Expression for the simplest mapping function
Often the TEC value in zenith direction (VTEC = Vertical Total Electron Content) is modelled and not directly the TEC in a given direction. With the mapping function mion(z) and the VTEC EV as input parameter it is possible to compute the TEC E for an arbitrary angle z.
Difference between the mapping function for the ionosphere (single layer) and for the troposphere
(z … zenith angle) ionosphere: mion(z)=1/cos(z') and sin(z')=R/(R+H)*sin(z) // troposphere: mtrp(z)=1/cos(z)
How are troposphere parameters estimated? What are the difficulties?
GPS (estimation) and Meteo-Measurements // big local and temporal differences
Formula 4.58. What is that? P, T, e?
Saastamoinen-Modell, Modellierung der tropospährischen Refraktion. Erster Term in runder Klammer: dry Part. Zweiter Term in runder Klammer: wet Part. Dritter Term in runder Klammer: Erdkrümmung. P: Luftdruck, T: Temperatur, e: Wasserdampfdruck , berechnbar aus relativer Luftfeuchtigkeit
Formula 4.58, Was von den dreien (P,T,e) hat den grössten Einfluss?
Temperatur (3-27 mm / 1°C), Druck (2mm / mbar), relative Luftfeuchte (0.6 - 4 mm / 1%)
Formula 4.58, Weshalb 1/cos(z)? Wie heisst dieser Teil? Warum tan^2(z)?
Einfachste Mapping-Function <= Mapping Function // Erdkrümmung
Wie gross ist der Einfluss der Troposphäre bei GPS, VLBI, Altimetrie?
GPS: (non-dispersive) 2.3m (Zenith) (wetpart: 5-40cm), 25m (5° elev) // VLBI: (non-dispersive) dito // Altimetrie: (non-dispersive) dito // SLR/LLR: (dispersive) 2.45 (Zenith) (wetpart: 0.1-0.6cm), 7m (20° elev)
Wie gross ist der Einfluss der Ionosphäre bei GPS, VLBI, Altimetrie?
GPS: (dispersive) 1-15m (Zenith), 150m (low elev) // VLBI: (dispersive) S-Band (2GHz), wie GPS, X-Band (8GHz) 4-60cm // Altimetrie: (dispersive) 2-20cm // SLR/LLR: (non-dispersive) ionospheric refraction doesn’t play any role!4
Rotation der Erde: Ist das etwas konstantes?
Nein. Die Erdrotation wird durch die Gezeitenreibung verlangsamt. Die Erde hat das gleiche schon mit dem Mond “gemacht”
Weshalb besteht ein Zusammenhang zwischen Verlangsamung der Erdrotation und Entfernung des Mondes?
Es gibt drei Erhaltungssätze in der Physik, hier: Drehimpuls muss erhalten bleiben
Was für Effekte bewirkt der Mond sonst noch? (abgesehen von der Gezeitenreibung)
Feste Erdgezeiten, Schwankungen in der Erdrotationsachse, davon abgeleitet auch Polgezeiten, er wollte noch auf generelle Relativität bei Mondnahen VLBI-Messungen hinaus.
Wie kann man die Erdrotation und ihre Änderung messen?
GNSS, LLR, VLBI
Wo in der ausgleichungsrechnung werden die parameter für die
troposphärische refraktion eingesetzt? Was kommt raus, etc?
Page 101, In the Observation Equation
=> The Partial derivative of the Obs. Equ. gives the Mapping Function.
Which parameter can, which cannot be determined by VLBI?
Which parameter can, which cannot be determined by VLBI?
Which parameter can, which cannot be determined by VLBI?
Which parameter can, which cannot be determined by GNSS?
Which parameter can, which cannot be determined by GNSS?
Which parameter can, which cannot be determined by GNSS?
Which parameter can, which cannot be determined by SLR?
Which parameter can, which cannot be determined by SLR?
Which parameter can, which cannot be determined by SLR?
Which parameter can, which cannot be determined by LLR?
Which parameter can, which cannot be determined by LLR?
Which parameter can, which cannot be determined by LLR?
How can the high angular resolution of VLBI be explained?
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.
Which instruments/sensors are required for a VLBI station?
large telescope, receiver, high-precision frequency standard, recording device
Which are the most important error sources for the VLBI of today?
structure of the radio sources (quasars don’t appear as point sources), antenna structure (deformation, thermal expansion), troposphere (water vapor)
Which relativitic effects have to be considered in VLBI analyses (withhout formulas)?
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)
Principle of the correlation of the two signals from the two telescopes
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’)
What do you see? (Fig. 2.3)
Map of known and used (ICRF) Quasars
Wie funktioniert VLBI?
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.
How big are Quasars?
Quasars are cores of galaxies. Seen from Earth they have often a complex structure and an extension of up to 60''.
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Principle of the correlation of the two signals from the two telescopes
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’)
What do you see? (Fig. 2.3)
Map of known and used (ICRF) Quasars
Wie funktioniert VLBI?
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.
How big are Quasars?
Quasars are cores of galaxies. Seen from Earth they have often a complex structure and an extension of up to 60''.
Where in de Observation Equation of VLBI do you find the coordinates of the Quasars?
In the unit vector (e) poiting to the Quasars (barrycentric coordinate system)
What is a barycentric intertial system?
Static coordinate sysstem with origin in the barycenter of the solar-system (or Earth-Moon-System)
Transformation from barycentric to earth fixed system? Parameters?
Lorentz-Transformation, Speed is the only parameter
What is that strange PNUXY?
Transformation from Earth-fixed to space-fixed, P: Precession, N: Nutation, U: Rotation matrix with Greenwich sidereal time as argument, X,Y: Polar motion
Formular 2.21, discuss, what speed is v? For GPS?
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)
Charactericits of the GPS satellite constellation?
a: 26'600km, Periode: 11h58, I: 55°, 6 Ebenen mit 60° spacing, 24 Satelliten, CDMA (PRN-Codes)
Structure of the GPS signals?
L1: f1 = 1575 MHz, _ = 19cm, C/A-Code, P-Code // L2: f2 = 1227 MHz, _ = 24cm, P-Code
C/A-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.
P-Code?
(protected / precise), repeats after 266.4 days (=2.35*10^14 chips), chip length = 29.3m, transmitted on L1 and L2
Nav-Message of GPS Signal?
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
Which mechanisms exist for the artificial degradation of the GPS signals?
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.
Which mechanisms exist for the artificial degradation of the GLNASS signals?
no signal degradation mechanisms
Which terms are present in the observation equation of GPS?
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
What is the meaning of the terms in equation (2.43)
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
Differences between the phase and code observation equations
Vorzeichen für Ionosphären-Korrektur (pase advane, group delay), andere Wert für Multipath-Korrektur.
Why are often differences formed in GPS analyses (especially double-differences)?
to improve the accuracy: get rid of the satellite clock error, receiver clock error and other effects
Explain single Diff in GPS measurements?
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)
Explain double Diff in GPS measurements?
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
Explain tripple Diff in GPS measurements?
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)
What type of instrumental effects have to be considered in GPS? What can be done to reduce their adverse effects?
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!
What differences exist between GLONASS and GPS?
(GPS/GLONASS), altitude (26'600km, 25'500km), Period (11h58', 11h16'), Inclination (55°, 65°), orbital planes (6, 60°spacing, 3, 120° spacing), CodeDivisionMultipleAccess <=> FrequencyDMA
Which Meteo-Parameters have to be taken in account when you are measuring with GPS?
Air pressure, temperature, humidity
What is the influence of water vapour to the accuracy of GPS?
5-40cm (wet part)
Formular 4.23, explain. How is it for GPS? Other Effects in the Atmosphere?
Rayleigh-Equation. GPS: Ionosphäre Dispersiv (=> 2 Frequenzen, Einfluss 1 - 15m), Troposphäre non-dispersiv,
v_ph: kann grösser als c sein und ist grösser als v_gr,
phase advance und group delay,
Ionosphäre ist Orts und Zeitabhängig (TEC)
Tell something about the Era before GPS?
Optical measurements (with Photos) to the satellites (strobe-light and balloon-satellites) with relative coordinates to the stars
Compare the relativistic effects of GPS and VLBI?
Lorentz-Transformation nur bei VLBI, Unterschiede der Uhren bei VLBI viel kleiner (vernachlässigbar), bei GPS nicht (Sender und Empfänger), Signale im vergleichbaren Wellenlägenbereich, relativistische Effekte (Allgemeine Relativität) auf Signale gleich.
Was entspricht bei GPS der Quellstruktur von VLBI?
Variationen des Antennenphasezentrums, Ungenauigkeit der GPS-Bahnen
Why aren't there multi-path effects for VLBI?
GPS antennas are omnidirectional, VLBI antennas are directed.
Simplest form of the observation equation of SLR/LLR?
rho = c * delta_t / 2
Advantages of SLR/LLR?
favorable propagation characteristics of visible light in the atmopshere (water vapor has no influence) // direct distance measurement are performed // satellites with a very long life-time // simple instrumentation on the satellite (nothing but a retro-reflector)
Disadvantages of SLR/LLR?
depending on weather conditions (clouds, fog) // expensive infrastructure and instrumentation at the ground station // very heavy equipment (transport)
Main components of an SLR/LLR system
retro-reflectors // generator of the laser pulses // optical telescope // receiving system: detector and analyzer of the returning pulse (echo) // timing equipment
Which parameters can be determined by SLR, which ones by LLR?
Nutation (LLR), Polar Motion (both), Length of Day (both), Ocean tide amplitude (SLR), Coord. + Velocities (both), Geocenter (SLR), etc. see Table 2.1
Simplest form ot the observtion eqation of satellite altimetry?
rho = h* - h, with rho = c * delta_t / 2
What are altimetry data used for? What can be determined with altimetry data?
sea level also possible for bigger lakes and Iceshields. // Geoid over the oceans very accurately and with hight spatial resolution // gravitiy anomalies (computed out of the altimeter geoid) => new earth models // information about the dynamics of the oceans (currents, ocean tides)
What is a cross-over point?
A Point where a descending and a ascending groundtrack cross-over eachother
