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Satellite Altimetry Ole B. Andersen.

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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

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IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 3

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IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 4 Sampling the Sea Surface

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IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 5 Sampling the sea surface. 1 Day 3 Days

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IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 6 Orbit Parameters Satellite Repeat Period Track spacing Inclination Coverage Time Span GEOSAT/GFO17 days163 km108°(+/-72°)5 years ERS/ENVISAT35 days80 km98° (+/-82°)9 years TOPEX/JASON10 days315 km66.5°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

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IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 7 ERM – GM data. ERM Data TOPEX/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.

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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 waveforms Registred at 2000 Hz (every 3 meters) Averaged into 20 Hz. The 20 Hz height values are Also noisy Averaged to 1 Hz values (7 km averaging).

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IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 9 Different Surfaces – Different Retrackers Ocean Echoes Extreme Terrain Desert – AustraliaInland Water (River – lake) R, P. Berry, J. Freeman

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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

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IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 11 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, eis the error The Sea surface height mimicks the geoid. Altimetry observes the sea surface height (SSH)

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IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 12 Altimetric observations The magnitudes of the contributors ranges up to The geoid N REF +/- 100 meters Terrain effect N DTM +/- 30 centimetres Residual geoid N +/- 2 meters Mean dynamic topography MDT +/- 2 meters Time varying Dyn topography ( t) +/- 5 meters. (Tides + storms + El Nino……) What We want for Global Gravity is: So we need to ”take out” the rest.

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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 GEOID signal +/- 100 meters

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IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 14 The Mean Dynamic Topography (MDT) /- 2 Meters - Gives up to 3-4 mGal effect mGal SSH ~ G + MDT -> So the MDT can be determined like MDT = MSS-N

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IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 15 Time Varying Signal e orbit is the radial orbit error e tides is the errors due to remaining tidal errors e range is the error on the range corrections. e retrak is the errors due to retracking e noise is the measurement noise. 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

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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

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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. Motivation Assumption: 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. Motivation Easier computation (no need for crossover adjustmenst Theoretically 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.

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IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 18 Crossover Adjustment d k =h i ‑h j. 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 c T x=0 Problem of Null space – Rank Bias (rank=1) – mean bias is zero Bias+Tilt (Rank = 4) Constrain to prior surf.

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IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 19 Before - Xover

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IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 20 After Crossover

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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.

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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

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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: P ij 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

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IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 24 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 tranforming the along-track slopes to east-west slopes. Geoid to Gravity

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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.

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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

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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 using Frequency 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

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IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 28 Data and software Satellite Altimetry (Major points). Altimetry Pathfinder: RADS/NOAA (Remko): NASA, ESA (Raw data). DNSC Global Fields (ftp.spacecenter.dk)ftp.spacecenter.dk Marine Gravity (2 min res). (http://www.spacecenter.dk/data)http://www.spacecenter.dk/data Mean Sea Surface (2 min res) Bathymetry (2 min res) Ocean Tide model (30 min res) Software (with geoid school). GRAVSOFT Least Squares collocation: GEOCOL, EMPCOV, COVFIT Interpolation: GEOGRID FFT: GEOFOUR, SPFOUR

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IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 29 DNSC Global Marine Gravity Grid

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IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 30 Gravity and Earth Processes

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IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 31

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IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 32 Marine Gravity Prior to Satellite Altimetry

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IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 33

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IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side DNSC05-DOT DNSC05-OC DNSC05A North Atlantic Region obsMeanStd Dev.Max abs KMS KMS NCU GSFC NOAA12 Gravity diff With KMS02

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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

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Sea level change / Climate From Satellite altimetry.

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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…..

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IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side Years Sea Level Change ( ) 12 Years Sea Surface Temperature change.

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IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 39 Global Sea level change.

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OCEAN TIDES From Satellite altimetry.

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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

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IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 42 Tides can be ”dangerous” - BUT TIDES CAN BE PREDICTED.

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IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 43 Tidal Forces. Difference (P 1 -O) is the Tide generating force = 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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IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 44 Alias Periods

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IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 45 Ocean Tides - M2 loop

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IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 46 GRACE GRACE twin satellites launched March 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.

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IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 47 Land Hydrology – The Amazon Satellite altimetry In rivers Retracking is Essential (P. Berry–De Montford)

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IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 48 Hydrology from GRACE + Satellite Altimetry GRACESatellite Altimetry

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IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 49 Thank You

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