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Satellite Altimetry Ole B. Andersen.

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**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 2

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

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**Sampling the Sea Surface**

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**Sampling the sea surface.**

1 Day 3 Days IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 5

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**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 Time 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 IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 6

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**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. IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 7

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**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). IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 8

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**Ocean Echoes Extreme Terrain**

Different Surfaces – Different Retrackers 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 9

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Benefit of retracking 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. IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 10

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**The Sea surface height mimicks the geoid.**

Altimetry observes the sea surface height (SSH) 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. IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 11

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**Altimetric observations**

The magnitudes of the contributors ranges up to The 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……) 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 12

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**GEOID signal +/- 100 meters**

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 IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 13

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**The Mean Dynamic Topography (MDT)**

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

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**Time Varying Signal Errors**

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 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. IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 15

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**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 16

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**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. 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. IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 17

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**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. IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 18

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

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

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**Data are now ready for computing gravity / geoid.**

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. IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 21

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**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 22

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**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 IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 23

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Geoid to Gravity 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. IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 24

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**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. Ongoing investigating this approach IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 25

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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 IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 26

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**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 IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 27

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**Data and software Satellite Altimetry (Major points).**

Altimetry Pathfinder: RADS/NOAA (Remko): 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) 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 IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 28

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**DNSC Global Marine Gravity Grid**

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**Gravity and Earth Processes**

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

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**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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**North Atlantic Region Gravity diff With KMS02 321.400 obs Mean**

Std Dev. Max abs KMS02 0.44 5.15 49.38 KMS99 0.60 5.69 73.74 45.19 4.79 0.49 DNSC05-DOT36 51.18 5.21 0.50 DNSC05-OC 47.88 4.95 0.48 DNSC05A 92.13 6.10 0.79 NCU01 89.91 6.14 0.68 GSFC00.1 82.20 5.79 0.62 NOAA12 IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 34

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**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 35

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**Sea level change / Climate From Satellite altimetry.**

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**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 37

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**12 Years Sea Level Change (1993-2004) 12 Years Sea Surface**

Temperature change. IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 38

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**Global Sea level change.**

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**OCEAN TIDES From Satellite altimetry.**

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**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 41

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**Tides can be ”dangerous” - BUT TIDES CAN BE PREDICTED.**

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**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. IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 43

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Alias Periods IAG 2006 Geoid School | Copenhagen 30 maj 2006 | OA | side 44

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

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**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 46

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**Land Hydrology – The Amazon**

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

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**Hydrology from GRACE + Satellite Altimetry**

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

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