Satellite Altimetry Ole B. Andersen.. IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 2 Content The radar altimetric observations (1): Altimetry.

Satellite Altimetry

Ole B. Andersen.

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



Sampling the Sea Surface


Sampling the sea surface.

1 Day

3 Days


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


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.


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).


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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


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


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)


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.


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

http://www.gfz-potsdam.de/pb1/op/grace/results/grav/g003_eigen-cg01c.html


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


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


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


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.


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 )(


Before - Xover


After Crossover


• 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


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


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


)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


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.


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

)(

),(),()(

),(),()(


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


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

ftp://ftp.spacecenter.dk/

http://www.spacecenter.dk/data













DNSC Global Marine Gravity Grid


Gravity and Earth Processes



Marine Gravity Prior to Satellite Altimetry



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


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

Sea level change / Climate

From Satellite altimetry.


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…..


12 Years Sea Level Change (1993-2004)

12 Years Sea SurfaceTemperature change.


Global Sea level change.

OCEAN TIDES

From Satellite altimetry.


Fascinating Ocean Tides

Tidal Range in Bay of Fundy and English Channel is 15 meters


Tides can be ”dangerous” -

BUT TIDES CAN BE PREDICTED.


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.


Alias Periods


Ocean Tides - M2 loop


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.


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)


Hydrology from GRACE + Satellite Altimetry

GRACE Satellite Altimetry


Thank You

Satellite Altimetry Ole B. Andersen.. IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 2 Content The radar altimetric observations (1): Altimetry.

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