1. Satellite Altimetry
Ole B. Andersen.

2. DNSC08 Globalt tyngdefelt
(ref. O. Andersen)
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3. Gravity and Earth Processes
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4. Content The radar altimetric observations (1):
Altimetry data
Contributors to sea level
Crossover adjustment
From altimetry to Gravity and Geoid (2):
Geodetic theory
FFT for global gravity fields
GRAVSOFT
Applications.
Sea level trend
Ocean tides.
IceSat (Laser altimeter)
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6. Sampling the Sea Surface
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7. Altimetric Observations
Accurate ranging to the sea surface is
Based on accurate time-determination
Typical ocean waveforms
Registred at 20 Hz
The 20 Hz height values are
Too noisy and averaged to give
1 Hz values (7 km averaging).
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8. Ocean Echoes Extreme Terrain
Different Surfaces – Different Retrackers
Ocean Echoes Extreme Terrain
Desert – Australia Inland Water (River – lake)
R, P. Berry, J. Freeman
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9. Sampling the sea surface.
1 Day
3 Days
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10. Orbit Parameters TOPEX/JASON - 10 Days
The coverage of the sea surface depends on the orbit parameters (inclination of the orbit plane and repeat period).
TOPEX/JASON - 10 Days
Satellite
Repeat
Period
Track spacing
Inclination
Coverage
GEOSAT/GFO
17 days
163 km
108°(+/-72°)
ERS/ENVISAT
35 days
80 km
98° (+/-82°)
TOPEX/JASON
10 days
315 km
66.5°
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11. The North Sea – 30 Day coverage
5 ongoing missions
JASON-1
TOPEX TDM
GFO
ERS-2
ENVISAT.
One/two track every day.
Real time altimetry (JASON)
4-6 hours
~5-7 cm accuracy.
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12. GEOSAT / GFO ERS1 / ERS2 / ENVISAT
Altimetry Coverage
GEOSAT / GFO ERS1 / ERS2 / ENVISAT
ERM GM
ERS GM mission 1994
GEOSAT GM mission 1985
GEOSAT+ERS GM data is ESSENTIAL for high resolution Gravity Field mapping.
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13. The Sea surface height mimicks the geoid.
Satellite Altimetry
If the satellite is accurately positioned
The orbital height of the space
craft minus the altimeter radar
ranging to the sea surface
corrected for path delays and
environmental corrections
Yields the sea surface height:
where
N is the geoid height
above the reference ellipsoid,
 is the ocean topography,
e is the error
The Sea surface height mimicks the geoid.
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14. Reference ellipsoide Geoiden +/- 100 m
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15. Altimetric observations
The magnitudes of the contributors ranges up to
The geoid NREF /- 100 meters
Terrain effect NDTM /- 30 centimeters
Residual geoid N /- 2 meters
Mean dynamic topography MDT +/- 1.5 meter
Time varying Dyn topography (t) +/- 5 meters. (Tides + storms + El Nino……)
What We want for Global Gravity is:
So we need to account for the rest.
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16. Remove - Restore.
Remove-restore technique – enhancing signal to noise.
“Remove known signals and restore their effect subsequently”
Remove a global spherical harmonic geoid model (EGM96)
Remove terrain effect
Remove Mean dynamic topography from model.
Compute Gravity
Restore the EGM96 global gravity field
Restore the Terrain effect
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17. Time Varying Signal Errors.
Tides contribute nearly 80% to sea level variability.
removed using Ocean tide Model (AG95, GOT, FES2002, NAO99)
Time variable signals are averaged out in ERM data but not in GM data.
eorbit is the radial orbit error
etides is the errors due to remaining tidal errors
erange is the error on the range corrections.
eretrak is the errors due to retracking
enoise is the measurement noise.
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18. Errors+time varying signals.
ERM data. Most time+error average out.
Geodetic mission data (t) is not reduced
Must limit errors to avoid ”orange skin effect”
NOTICE ERRORS ARE LONG WAVELENGTH
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19. Enhancing residual geoid height signal for gravity
Limit long wavelength (time + error signal).
Using crossover adjustment.
Motivation
The residual geoid signal is stationary at each location.
Consequently the residual geoid observations /sea surface height observations
should be the same on ascending and descending tracks at crossing locations.
Timevarying Dynamic sea level + orbital related signals should not be the same, and should be removed
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20. Crossover Adjustment dk=hi‑hj. d=Ax+v
where x is vector containing the unknown parameters for the
track-related errors.
v is residuals that we wish to minimize
Least Squares Solution to this is
Constraint is needed cTx=0
Case of bias – mean bias is zero
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21. Before - Xover
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22. After Crossover
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23. Data are now ready for computing gravity / geoid.
Corrected the range for as many known signals as possible.
Removed Long wavelength Geoid part – will be restored.
Limited errors + time varying signal (Long wavelength).
Still small long wavelength errors can be seen in sea surface heights. This will be treated in the subsequent Least Squares Collocation interpolation procedure.
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24. Part 2. The radar altimetric observations (1):
Altimetry data
Contributors to sea level
Crossover adjustment
From altimetric heights to Gravity (2):
Geodetic theory
FFT for global gravity fields
Least Squares Collocation
Applications:
Accuracy assesment
Applications
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25. The Anomalous Potential.
The anomalous potential T is the difference between the actual gravity potential W and the normal potential U
T is a harmonic function outside the masses of the Earth satisfying
(²T = 0) Laplace (outside the masses)
(²T = -4) Poisson (inside the masses ( is density))
Expanding T in spherical harmonic functions:
Pij are associated Legendre's functions of degree i and order j
Geoid heights, multiplying the coefficients by 1/γ.
Gravity anomalies, multiplying the coefficients by (i-1)/R
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26. Bruns formula links N with T N can be expressed in terms of a linear functional applied on T (γ is the normal gravity) Gravity and T Deflection of the vertical (n,e) Deflection of the vertical is related to geoid slope Geoid slopes (east, west) can be obtained from altimetry by tranforming the along-track slopes to east-west slopes.
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27. Gravity from altimetry.
Three feasable ways
1) Integral formulas (Stokes + Vening Meinesz + Inverse)
Requires extensive computations over the whole earth.
Replace analytical integrals with grids and is combined with FFT
2) Fast Fourier Techniques.
Requires gridded data (will return to that).
Very fast computation.
Presently the most widely used method.
3) Collocation.
Requires big computers.
Do not require gridding.
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28. 2D FFT – Flat Earth approximation
Flat Earth approx is valid (2-300 km from computation point, Sideris, 1997).
The geodetic relations with T are then
Where F is the 2D planar FFT transform
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29. From height to gravity using 2D FFT
An Inverse Stokes problem
High Pass filter operation enhance high frequency.
Optimal filter was designed to handle white noise + power spectral decay obtained using
Frequecy domain LSC with a Wiener Filter (Forsberg and Solheim, 1997)
Power spectral decay follows Kaulas rule (k-4)
Resolution is where wavenumber k yields (k) = 0.5
For KMS 12.5 km is used.
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30. GRAVSOFT and excercises.
Summary of steps:
1) Extract data from data base (full geoid signal).
2) Remove long wavelength field (get unknown geoid residuals+ noise)
3) Perform crossover adjustment (geoid residuals)
4) compute gravity anomalies (get gravity residuals from geoid residuals)
Use GEOCOL on data points or FFT on GRID of DATA
5) Restore grid of gravity residuals (get total gravity field)
Least Squares collocation:
GEOCOL - least-squares collocation and comp.of reference fields.
EMPCOV, COVFIT empirical covariance function estimation and fitting.
Interpolation
GEOGRID fast and reliable program for handling the interpolation of the observations onto a regular grid is the collocation based algorithm
FFT
The GRAVSOFT software library has facilities to do two-dimensional planar FFT via the routine geofour
Multiband spherical 2D FFT can be handled using the routine spfour.
Finally software for carrying out the one-dimensional FFT is available via the routine sp1d.
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31. KMS Global Marine Gravity Grid
(ref. O. Andersen)
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33. Applications. Sea level changes: Global coverage – open ocean
Uniform Geocentric reference
About 15 years of data
Spatial characteristics
Calibration needed at tide gauges
Dynamic Phaenomenas – El Nino
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35. 2) Gletcher afsmeltning
Ændringer skyldes
1) Hav-Temperatur
2) Gletcher afsmeltning
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36. OCEAN TIDES From Satellite altimetry.

37. Tides are Fascinating
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38. Tides are Fun Tidal surfing
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39. Tides can be ”dangerous” - BUT TIDES CAN BE PREDICTED.
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40. Tidal Forces. Force = Difference (P1-O) is the Tide generating force =
At P2 the force away from the moon is =
At P3 the force is directed towards O
In Addition we have the centrifugal force which must be added.
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41. Tidal Forces P
Water surface P balances FEarth = Fcentrifugal To this the tides are added.
Force by the Moon is 1.4 Force by the Sun because Moon is much closer.
Major tidal constituent M2 is created by the attraction of the Moon.
Major solar tidal constituent is called S2
Daily rotation of the Earth creates Semi-daily or semi-diurnal tides.
Diurnal tides are created because the semi-diurnal are not ”identical.”
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42. North Sea – Surges Altimetry versus Tide Gauge
Including tides – correlation > 95%
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43. Ocean Tides - M2 loop
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44. Semi-diurnal Tides.
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45. Applications - IceSat – Laser Altimetry
Laser Altimetry - ICESAT.
Difference with radar altimetry
High resolution Ice-rifs observations
from ICESAT.
Sea – Ice Freeboard in the Arctic
comparison With laser scanner.
Partly provided by Helen Fricker (SCRIPPS, California).
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47. Difference - Radar and Laser Altimetry
GLAS has much smaller footprint than radar altimeter instruments such as ERS and ENVISAT’s RA-2 (3-10 km)
Small footprint enables GLAS to measure small-scale features on the ice sheet, previously unresolved in radar altimetry (65-70 meters)
Icesat will give unprecedented elevation information containing exquisite detail across ice sheet features such as: Ice shelf rifs/edges etc (examples).
Radar altimeter pulse (frequency 13.8 GHz) penetrates the surface of the ice, leading to volume scattering within the snow-pack. Effect increases in the dry snow zone and high accumulation areas
Observations at 40 Ghz corresponding to 150 meters distance between individual observations
GLAS ~65m
ERS = 3 –10 km
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48. Region correspond to one radar altimetric obs
MODIS: 19-Oct-2003
GLAS Laser 2A
mélange L1 T1 T2 T2
GLAS footprints
Region correspond to one radar altimetric obs T2 L1
Fricker, 2004, AGU
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49. Seaice-Freeboard.– Arctic Ocean.
IceSat (dots)
Laser
Scanner
H. Skourup, DNSC, EGU, 2005
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