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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What is the use of cross-over points?
Two corrected measurements have to be equal at a cross-over point. // The differences between those measurements cam be used for the orbit determination/correction // the height of the mean sea level can be corrected and improved
What is the meaning of the terms in equation (2.82)? Sketch?
h*: height of the satellite above reference ellipsoid, N: geoidundulation, Zeta_p: permanent part of ocean currents, Zeta_v: variable part of ocean currents, htrp: Delay caused by the tropospheric refration, hion: Delay caused by the ionospheric refraction, hwav: Correction of the wind waves on the sea surface. It is computed based on the “signifant wave height (SWH), that can be reconstructed from the temporal distribution of the reflected pulse (shape of pulse), hsys: instrumental corrections, e: error of the altimeter observation
Planung einer Satellitenmission um Meeresspiegeländerungen von 3mm zu detektieren, mögliche Varianten
Satelliten-Altimetrie, Crossover-Points, Modellierungen von Geoid, Strömungen, etc.
Kalibrierung mit GPS-Bojen / SLR-Stationen. Ev mit InSAR?
Advantages and disadvantages of small and big inclinations? (sat. Altimetry)
Kleine Inklination: grosse Temporale Auflösung, Grosse Inklination: grössere Abdeckung, und vice versa
What does the column "cycles" mean?
Cycles = Repeat Period, Weniger Cross-Overs aber hohe Temporale Auflösung wenn kurze Repeat Periode
Why are the frequencies used in sat. Altimetry mainly 13.9GHz and not less, like GPS?
Viel kleinerer Ionospheren-Effekt, see Page 93.
Sat. Altimetry: for what is the radiometer used?
Wasserdampf in Atmosphäre bestimmen => wet part
Which motions of the stations are modelled?
plate motion // loading effects of the oceans and the atmosphere // solid earth tides and pole tides // variations of the antenna phase center, deformation of the antenna and similar effects (depending on the observation technique)
What is the cause of the station motions?
pole tides: polar motion -> changing centrifugal forces // solid earth tides: gravitational forces of other massive bodies (e.g. moon, sun) // ocean loading: the water mass at high tide presses down the continental plates // atmospheric loading: pressure
Order of magnitude of the different influences on the station coordinates?
pole tides: 1-2cm // solid earth tides: 50cm // ocean loading: vertical: several centimeters (CH:1cm), horizontal: a few millimeters // atmospheric loading: 2cm (vertical)
How is plate motion described?
Rotation of a rigid plate around its pole of rotation on the surface of a spherical earth
What is the meaning of the Love and Shida numbers?
Love and Shida numbers describe the Earth’s response to external forces exerted by celestial bodies due to the elasticity of the Earth // link between the change in the gravitational potential and the resulting change of the station position // Love numbers: for the vertical deformation // Shida numbers: for the horizontal deformation
How are the station coordinates (or other parameters) estimated from the observations of the space geodetic techniques?
by parameter estimation algorithms? (least-square procedures, Kalman filters)
Station Motions, Figure of a Station in Finland. What do you see?
Plate motion in East and North direction, altitude changes (postglacial rebound?) with noise (due to loading effects, tides, groundwatereffects, GPS measurements are weaker in altitude (geometry, atmospherical effects)
Difference between phase and group velocity. When are both velocities the same?
see page 86, Reyleigh-Equation. In the vaccum or non-dispersive area.
In which frequency ranges are there observations of space geodetic techniques (GPS, VLBI, SLR, altimetry)?
GPS: ~1 GHz (1227 and 1575 MHz) //
VLBI: S-band (f~2 GHz),
X-band (f~8 GHz) //
SLR/LLR: ~5*1014 Hz //
Sat. Altimetry: 13.5 - 13.9 GHz
Difference between troposphere and ionosphere
Different height, air pressure, temperature, dispersion, TEC, water vapour, ...
Difference between signal bending and signal delay. What is more important?
delay (up to 100m) more important (beding is important for astronomers, 1-2cm)
Relationship between delay, refractive index and propagation speed
page 90 ff
Altitude range of the ionosphere
50-1000km
How are free electrons generated?
Free electron were thrown out of the gas molecules by the x-ray and ultra-violete radiation from the sun.
On what does the electron density in the ionosphere depend?
a) Solar activity: The intensity of the solar radiation varies with time. There is an 11-year solar cycle _ periodic variations of the number of the Sun spots. b) Altitude: Different gases need different energies to be ionized. As the absorption of the solar radiation is taking place at different altitudes depending on the frequency of the radiation, the individual gases are ionized in different altitude bands
The ionospheric delay depends on ... ?
… the electron density Ne (= number of free electrons per m3)
Order of magnitude of the ionosphere delay for VLBI, GPS, SLR
GPS (f~1 GHz): zenith direction: 1-15 m, low elevations: up to 150 m //
VLBI: S-band (f~2 GHz) _ similar to GPS, X-band (f~8 GHz) _ 4-60 cm // SLR/LLR: (f~5*1014 Hz) _ ionospheric refraction has no influence
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
