Satellite Altimetry Ole B. Andersen.

DNSC08 Globalt tyngdefelt (ref. O. Andersen) Undervisning i satellit altimetri | OA | side 2

Gravity and Earth Processes Undervisning i satellit altimetri | OA | side 3

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) Undervisning i satellit altimetri | OA | side 4

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Sampling the Sea Surface Undervisning i satellit altimetri | OA | side 6

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). Undervisning i satellit altimetri | OA | side 7

Ocean Echoes Extreme Terrain Different Surfaces – Different Retrackers Ocean Echoes Extreme Terrain Desert – Australia Inland Water (River – lake) R, P. Berry, J. Freeman Undervisning i satellit altimetri | OA | side 8

Sampling the sea surface. 1 Day 3 Days Undervisning i satellit altimetri | OA | side 9

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° Undervisning i satellit altimetri | OA | side 10

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. Undervisning i satellit altimetri | OA | side 11

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. Undervisning i satellit altimetri | OA | side 12

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. Undervisning i satellit altimetri | OA | side 13

Reference ellipsoide Geoiden +/- 100 m Undervisning i satellit altimetri | OA | side 14

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. Undervisning i satellit altimetri | OA | side 15

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 Undervisning i satellit altimetri | OA | side 16

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. Undervisning i satellit altimetri | OA | side 17

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 Undervisning i satellit altimetri | OA | side 18

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 Undervisning i satellit altimetri | OA | side 19

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 Undervisning i satellit altimetri | OA | side 20

Before - Xover Undervisning i satellit altimetri | OA | side 21

After Crossover Undervisning i satellit altimetri | OA | side 22

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. Undervisning i satellit altimetri | OA | side 23

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 Undervisning i satellit altimetri | OA | side 24

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 Undervisning i satellit altimetri | OA | side 25

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. Undervisning i satellit altimetri | OA | side 26

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. Undervisning i satellit altimetri | OA | side 27

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 Undervisning i satellit altimetri | OA | side 28

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. Undervisning i satellit altimetri | OA | side 29

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. Undervisning i satellit altimetri | OA | side 30

KMS Global Marine Gravity Grid (ref. O. Andersen) Undervisning i satellit altimetri | OA | side 31

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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 Undervisning i satellit altimetri | OA | side 33

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2) Gletcher afsmeltning Ændringer skyldes 1) Hav-Temperatur 2) Gletcher afsmeltning Undervisning i satellit altimetri | OA | side 35

OCEAN TIDES From Satellite altimetry.

Tides are Fascinating Undervisning i satellit altimetri | OA | side 37

Tides are Fun Tidal surfing Undervisning i satellit altimetri | OA | side 38

Tides can be ”dangerous” - BUT TIDES CAN BE PREDICTED. Undervisning i satellit altimetri | OA | side 39

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. Undervisning i satellit altimetri | OA | side 40

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.” Undervisning i satellit altimetri | OA | side 41

North Sea – Surges Altimetry versus Tide Gauge Including tides – correlation > 95% Undervisning i satellit altimetri | OA | side 42

Ocean Tides - M2 loop Undervisning i satellit altimetri | OA | side 43

Semi-diurnal Tides. Undervisning i satellit altimetri | OA | side 44

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). Undervisning i satellit altimetri | OA | side 45

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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 Undervisning i satellit altimetri | OA | side 47

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 Undervisning i satellit altimetri | OA | side 48

Seaice-Freeboard.– Arctic Ocean. IceSat (dots) Laser Scanner H. Skourup, DNSC, EGU, 2005 Undervisning i satellit altimetri | OA | side 49