satellite altimetry ole b. andersen.. iag 2006 geoid school | copenhagen 30 maj 2006 | oa | side 2...
TRANSCRIPT
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 2
Content
•The radar altimetric observations (1):•Altimetry data•Contributors to sea level•Retracking•Crossover adjustment
•From altimetry to Gravity and Geoid (2):
•Geodetic theory•FFT for global gravity fields•GRAVSOFT
•Applications. •Accuray Assesment•Applications
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 6
Orbit Parameters
SatelliteRepeat
PeriodTrack
spacingInclination
CoverageTime Span
GEOSAT/GFO 17 days 163 km 108°(+/-72°) 5 years
ERS/ENVISAT 35 days 80 km 98° (+/-82°) 9 years
TOPEX/JASON 10 days 315 km 66.5° 12 years
The coverage of the sea surface depends on the orbit parameters (inclination of the orbit plane and repeat period).
TOPEX/JASON - 10 Days
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 7
ERM – GM data.
ERM DataTOPEX/JASON –
(280 km)
ERS/ENVISAT(80 km)
Geodetic Mission GEOSAT (15 Month)
Drift
ERS-1 (11 Month)2 x 168 days repeat
Equally spacing
GEOSAT+ERS GM data is ESSENTIAL for high resolution Gravity Field mapping.
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 8
Altimetric Observations
Accurate ranging to the sea surface is Based on accurate time-determination
Typical ocean waveformsRegistred at 2000 Hz (every 3 meters) Averaged into 20 Hz.
The 20 Hz height values areAlso noisyAveraged to 1 Hz values (7 km averaging).
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 9
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Extreme Terrain
Different Surfaces – Different Retrackers
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ERS-2 Ocean Waveforms1 6
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NO.
Topex C Sahara 15N000E
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Inland Water Waveforms
Ocean Echoes Extreme Terrain
Desert – Australia Inland Water (River – lake)
R, P. Berry, J. Freeman
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 10
The Sea Ice retracker picks up many data in i.e. polar regions that is not ocean retracket. This retracker also picks up data near coast and in currents (correct echos are similar in shape)
Large distribution of patch waveforms (44%) within 5 km of the coastline, and around 10% from about 25 km from the coast can be seen. At about 50 km from coastline only 60% of waveforms in this region are ocean retracked.
Benefit of retracking
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 11
Satellite AltimetryIf the satellite is accurately
positionedThe orbital height of the
spacecraft minus the altimeter
radarranging to the sea surfacecorrected for path delays
and environmental corrections Yields the sea surface
height: eNh 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.
SSHh
Altimetry observes the sea surface height (SSH)
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 12
Altimetric observations
etNNNh MDTDTMREF )(
etNh )(
The magnitudes of the contributors ranges up toThe geoid NREF +/- 100 meters
Terrain effect NDTM +/- 30 centimetres
Residual geoid N +/- 2 meters
Mean dynamic topography MDT +/- 2 meters
Time varying Dyn topography (t) +/- 5 meters. (Tides + storms + El Nino……)
Nh What We want for Global Gravity is: So we need to ”take out” the rest.
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 13
Remove - Restore.
“ Take These Out”
• Remove-restore technique – enhancing signal to noise.
• “Remove known signals and restore their effect subsequently”
– Remove a global spherical harmonic geoid model (PGM04/06)– Remove terrain effect– Remove Mean Dynamic Topography (MDT)
– Compute Gravity – Restore PGM04/06 global gravity field (Pavlis) – Restore the Terrain effect– No Restoring from Mean Dynamic Topography
')()()( etNetNNNh MDTDTMREF
GEOID signal +/- 100 meters
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 14
The Mean Dynamic Topography (MDT)
-3 30
+/- 2 Meters - Gives up to 3-4 mGal effect
mGal
SSH ~ G + MDT -> So the MDT can be determined like MDT = MSS-N
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 15
Time Varying Signal
noiseretrackrangetidesorbit eeeeee • 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.
)()()()()( ttttt stericpressurewindtides
Tides contribute nearly 80% to sea level variability. Removed using Ocean tide Model (GOT 2000X)Time variable signals are averaged out in ERM data but not in GM data.
Errors
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 16
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
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 17
Crossover – Sea surface Slopes
Enhancing residual geoid height signal for gravity (removing time variability)
• Limit long wavelength contribution (time + error signal FOR GM DATA).
Using crossover adjustment.
MotivationAssumption: the residual geoid signal is stationary at each location so residual geoid
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 will be removed
Using Sea surface Slopes.
MotivationEasier computation (no need for crossover adjustmenstTheoretically straight forward wrt gravity field computation. Time varying signal is not reduced.Transformation from along track to east-west north-south is problematic at the Equator and at
turning lat.
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 18
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 • Problem of Null space – Rank • Bias (rank=1) – mean bias is zero• Bias+Tilt (Rank = 4) Constrain to
prior surf.
dCAccACAx cTT
dT 111 )(
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 21
• Corrected the range for as many known signals as possible. • Retracking enhances amount and quality of data • Removed Long wavelength Geoid part – will be restored. • XOVER: Limited errors + time varying signal (Long wavelength). • Small long wavelength errors can still be seen in sea surface
heights.
Data are now ready for computing gravity / geoid.
Nh
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 22
Part 2. Geodesy
•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 field determination•DNSC Global Gravity Field.
•Applications:•Accuracy assesment•Applications
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 23
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))
• T Harmonic -> Expand T in spherical harmonic functions:
• Pij are associated Legendre's functions of degree i and order j
• C+D are surface spherical harmonic functions (what you distribute)
• Geoid heights, multip by 1/γ, Gravity anomalies, multip by (i-1)/R
)(sinPsincos ij j D + j C
RM
= T ijij
i
j=02=i
),,(),,(),,( rUrWrT
02
2
2
2
2
2
r
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 24
)2(1
2)(r
N
r
N
r
T
r
TTLg g
Nr
Tr
TL
Nr
Tr
TL
)cos(1
)cos(1
)(
11)(
Geoid N and T (Bruns Formula)
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 tranformingthe along-track slopes to east-west slopes.
T
TLN N )(
Geoid to Gravity
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 25
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.Ongoing investigating this approach
Gravity from altimetry.
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 26
Using FFT
• Flat Earth approx is valid (2-300 km from computation point, (More by Sideris, (1997)).
• The geodetic relations with T are then
• Where F is the 2D planar FFT transform
• DNSC06. • Gravity is estimated in small boxes (3 by 10° ) and pathed together globally
2800 cells• Around 100 million 1 sec ssh observations. • DNSC approach is highly parallel. Computation time is around 1-2 weeks
)()(
)()(
),()/2()( 22
TFkF
TFkF
kkkTFrkgF
x
y
yx
yxykxki
yx
ykxkiyx
dkdkekkFyxfFF
dxdyeyxfkkFfF
yx
yx
)(1
)(
),(),()(
),(),()(
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 27
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 usingFrequency 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 DNSC 12 km is used
22),()/2()( yx kkkTFrkgF
)()( NFGFeeNN
gN
)()()(1
)(4
NFkkNFck
kGF
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 28
Data and software
• Satellite Altimetry (Major points).• Altimetry Pathfinder: http://topex-www.jpl.nasa.gov/• RADS/NOAA (Remko): http://rads.tudelft.nl/rads/rads.shtml• NASA, ESA (Raw data).
• DNSC Global Fields (ftp.spacecenter.dk) • Marine Gravity (2 min res). (http://www.spacecenter.dk/data)• Mean Sea Surface (2 min res) http://www.spacecenter.dk/data• Bathymetry (2 min res) http://www.spacecenter.dk/data• Ocean Tide model (30 min res) http://www.spacecenter.dk/data
• Software (with geoid school). • GRAVSOFT
• Least Squares collocation: GEOCOL, EMPCOV, COVFIT• Interpolation: GEOGRID
• FFT: GEOFOUR, SPFOUR
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 32
Marine Gravity Prior to Satellite Altimetry
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 34
45.194.790.49DNSC05-DOT36
51.185.210.50DNSC05-OC
47.884.950.48DNSC05A
North Atlantic Region
321.400 obs Mean Std Dev. Max abs
KMS02 0.44 5.15 49.38
KMS99 0.60 5.69 73.74
92.136.100.79NCU01
89.916.140.68GSFC00.1
82.205.790.62NOAA12
Gravity diff With KMS02
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 35
Part 3: Applications.
• Sea level Change• Ocean Tides• Land Hydrology
• Sea level changes: – Global coverage – open ocean– Uniform Geocentric reference– About 12 years of T/P time series used
Spatial characteristics
– Calibration needed at tide gauges
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 37
Part 3: Applications.
• Sea level changes: – Global coverage – open ocean– Uniform Geocentric reference– About 12 years of T/P time series used
Spatial characteristics
– Calibration needed at tide gauges
– Sea level change is currently increasing from 2.8 to 3.0 mm / year indicating acceleration…..
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 38
12 Years Sea Level Change (1993-2004)
12 Years Sea SurfaceTemperature change.
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 41
Fascinating Ocean Tides
Tidal Range in Bay of Fundy and English Channel is 15 meters
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 42
Tides can be ”dangerous” -
BUT TIDES CAN BE PREDICTED.
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 43
Tidal Forces.
21
1
)( aR
mmG
21
1
)(R
mmG
31
1
)(2
R
ammGDifference (P1-O) is the Tide generating force =
Force =
31
1
)(2
R
ammGAt 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.
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 46
GRACE
GRACE twin satellites launched March 2002. Status: Mission successfully 4 Years in Space.
40 Monthly Level-2 solutions
Use for climate studies to control global water balance better than hydrological models.
Various corrections for atmosphere+Tides +….. applied leaving
Hydrology as largest contributor to gravity field variations.
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 47
Land Hydrology – The Amazon
High slope
Slope
Water
Land/water
Complex
Very wide
SeaIce-Like
Near specular
PatchOcean
Satellite altimetry In riversRetracking is Essential(P. Berry–De Montford)
IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 48
Hydrology from GRACE + Satellite Altimetry
GRACE Satellite Altimetry