Satellite Altimetry andSatellite Altimetry andGravimetryGravimetry: : Theory and ApplicationsTheory and Applications

C.K. ShumC.K. Shum1,21,2, Alexander Bruan, Alexander Bruan2,12,1 1,21,2Laboratory for Space Geodesy & Remote SensingLaboratory for Space Geodesy & Remote Sensing

2,12,1Byrd Polar Research CenterByrd Polar Research Center

The Ohio State UniversityThe Ohio State UniversityColumbus, Ohio, USAColumbus, Ohio, USA

ckshum@osu.ckshum@osu.eduedu, , braunbraun.118@.118@osuosu..edueduhttp://geodesy.eng.ohio-state.http://geodesy.eng.ohio-state.eduedu

Norwegian Univ. of Science and TechnologyTrondheimTrondheim, Norway, Norway

2121––25 June, 200425 June, 2004

Satellite Altimetry andSatellite Altimetry and Gravimetry Gravimetry::Theory and ApplicationsTheory and Applications

Tuesday, 22 June 2004Tuesday, 22 June 2004•• Orbital Dynamics & Orbit Determinations II Orbital Dynamics & Orbit Determinations II (AM) By C.K. Shum(AM) By C.K. Shum

–– Nonlinear orbit determination & parameter recoveryNonlinear orbit determination & parameter recovery–– Force, measurement, and Earth orientation modelsForce, measurement, and Earth orientation models

•• Satellite Altimetry II Satellite Altimetry II (AM) By C.K. Shum(AM) By C.K. Shum–– Principles of satellite altimetry, mission design, waveformsPrinciples of satellite altimetry, mission design, waveforms–– Geographically correlated orbit errors and PODGeographically correlated orbit errors and POD–– Instrument, media and geophysical correctionsInstrument, media and geophysical corrections

•• Altimeter Collinear AnalysisAltimeter Collinear Analysis (PM) By Alexander Braun(PM) By Alexander Braun–– Stackfile Stackfile method for oceanography and marine geophysicsmethod for oceanography and marine geophysics–– Mean sea surface, marine gravity field determinationsMean sea surface, marine gravity field determinations–– Models accuracy evaluations and limitationsModels accuracy evaluations and limitations

•• Radar Altimeter Data ProcessingRadar Altimeter Data Processing (PM) By Alexander Braun(PM) By Alexander Braun•• Tutorial onTutorial on iGMT iGMT (continued)(continued) (PM) By Alexander Braun(PM) By Alexander Braun

Background and History:Satellite Altimetry

15 February 2004 C. Shum 9

NASA’S Earth Observing System Satellites: Terra, AquaNASANASA’’S Earth Observing System Satellites: Terra, AquaS Earth Observing System Satellites: Terra, Aqua

Credit: NASA/GSFCCredit: NASA/GSFC

15 February 2004 C. Shum 10

NASA’S Earth Observing System Satellites: Terra, AquaNASANASA’’S Earth Observing System Satellites: Terra, AquaS Earth Observing System Satellites: Terra, Aqua

Credit: NASA/GSFCCredit: NASA/GSFCExample temporal and spatial sampling ofExample temporal and spatial sampling ofsatellite (LEO) measurements from spacesatellite (LEO) measurements from space

SATELLITE ALTIMETRYSATELLITE ALTIMETRYRadar altimetry concept wasRadar altimetry concept wasformulated in the Williamstownformulated in the WilliamstownConference [William Conference [William Kaula Kaula et al.] inet al.] in1969. NASA1969. NASA’’s GEOS-3 is the first radars GEOS-3 is the first radaraltimeter demonstrating thealtimeter demonstrating themeasurement of sea surface heights ofmeasurement of sea surface heights ofthe global ocean.the global ocean.

Initially designed to measure ocean,Initially designed to measure ocean,radar altimetry has been demonstratedradar altimetry has been demonstratedto be useful in the measurement of landto be useful in the measurement of landand sea ice, land topography, lake andand sea ice, land topography, lake andrivers, etcrivers, etc

15 February 2004 C. Shum 12

MeasurementCoverage:

TOPEX/POSEIDON,JASON:660 latitude coverageERS-1/2, Envisat820 latitude coverageSeasat, Geosat, GFO720 latitude coverageCRYOSAT940 latitude coverageICESAT (Laser)940 latitude coverage

Earth Satellite AltimetersEarth Satellite Altimeters

Altimeter measuresgeocentric sea leveland ice sheetelevation change

Jason

Courtesy: A. Braun

ICESAT

15 February 2004 C. Shum 13CRYOSAT

Courtesy, ESA

Ku-band altimeter (multipleantennas) capable ofnadir, SAR, and InSAR mode.Potential tracking closer tocoastlines. No radiometer.

Satellite Altimetry andSatellite Altimetry and Gravimetry Gravimetry::Theory and ApplicationsTheory and Applications

Tuesday, 22 June 2004Tuesday, 22 June 2004•• Orbital Dynamics & Orbit Determinations II Orbital Dynamics & Orbit Determinations II (AM) By C.K. Shum(AM) By C.K. Shum

–– Nonlinear orbit determination & parameter recoveryNonlinear orbit determination & parameter recovery–– Force, measurement, and Earth orientation modelsForce, measurement, and Earth orientation models

•• Satellite Altimetry II Satellite Altimetry II (AM) By C.K. Shum(AM) By C.K. Shum–– Principles of satellite altimetry, mission design, waveformsPrinciples of satellite altimetry, mission design, waveforms–– Geographically correlated orbit errors and PODGeographically correlated orbit errors and POD–– Instrument, media and geophysical correctionsInstrument, media and geophysical corrections

•• Altimeter Collinear AnalysisAltimeter Collinear Analysis (PM) By Alexander Braun(PM) By Alexander Braun–– Stackfile Stackfile method for oceanography and marine geophysicsmethod for oceanography and marine geophysics–– Mean sea surface, marine gravity field determinationsMean sea surface, marine gravity field determinations–– Models accuracy evaluations and limitationsModels accuracy evaluations and limitations

•• Radar Altimeter Data ProcessingRadar Altimeter Data Processing (PM) By Alexander Braun(PM) By Alexander Braun•• Tutorial onTutorial on iGMT iGMT (continued)(continued) (PM) By Alexander Braun(PM) By Alexander Braun

15 February 2004 C. Shum 16

Earth Satellite Altimetry MissionsEarth Satellite Altimetry Missions

PlannedPlanned:: CRYOSAT (2004), JASON CRYOSAT (2004), JASON or or OSTM (2007)OSTM (2007)ProposedProposed:: ABYSS, NPOESS, GAMBLE ABYSS, NPOESS, GAMBLE

2003ICESAT (laser)

2002ENVISAT

2001JASON

1998GFO

1995ERS-2

1992TOPEX/POSEIDON

1991ERS-1* (Geodetic phase)

1984GEOSAT GM*/ERM

1978SeaSat

1974GEOS 31973Skylab

Launch DateMission

*Non-repeatground tracks

15 February 2004 C. Shum 17

NASA/CNES JASON-1 Altimeter Mission (2001) NASA/CNES JASON-1 Altimeter Mission (2001) NASA/CNES JASON-1 Altimeter Mission (2001)

Credit: NASA/JPLCredit: NASA/JPL

Altitude: 1354 kmAltitude: 1354 km10-day repeat orbit10-day repeat orbit666600 Inclination Inclination

Principle of Satellite Altimetry• Fundamental design• Radar principle

• Temporal-spatial sampling (ground track patterns)

Electromagnetic Spectrum [Source: NASA/JPL]Electromagnetic Spectrum [Source: NASA/JPL]

Radar altimeter operates in Ku-Radar altimeter operates in Ku-band, 13.6 GHz (band, 13.6 GHz (λλ=2.21 cm), C-=2.21 cm), C-band (5.6 GHz), & S-band (4.2 GHz)band (5.6 GHz), & S-band (4.2 GHz)

L-band (1.0L-band (1.0––1.5 GHz), S-band (1.51.5 GHz), S-band (1.5––4.2 GHz), C-band (4.24.2 GHz), C-band (4.2––5.45.4GHz), X-band (5.7GHz), X-band (5.7––10.9 GHz), Ku-band (10.910.9 GHz), Ku-band (10.9––22.0 GHz) 22.0 GHz) [Low [Low ––>high]>high]

Altimeter CrossoverMeasurement Concept:• Active (2-way) nadir pointing microwave (radar) instrument• Accurate clock• Altimeter range (halt)= c(2∆t) where c=speed of light

Implies that the clock needs tobe accurate to < 1 µsec for haltto be accurate to < 1 cm

Radar Altimeter GeometryRadar Altimeter Geometry

• Mean Sea Surface: –100 m to +80 m• Geoid ~ MSS• Ocean topography: ~ several meters• Ellipsoid: ~6378 km• Altimeter altitude: 800 – 1300 km

Radar Altimeter FootprintRadar Altimeter Footprintradius of footprint :R

hcR τ=c – speed of lightτ – pulse width (pulse duration) , actualh – satellite hightGeos-3: h=840, τ = 5.12 ns = 9105.12 −× second , 6.3=RSeasat: h=800, τ =3ns = 9103 −× , ?=R

2

222 2ln16

cH

p += ττ

:pτ radar’s theoretical pulse width:H standard deviation of wave height

Effect of SWH

pulse-length-limitedbeamwidth-limited

1.94 SWH

0.56M SWH

time(gate)

SWH will cause electromagnetic bias (emb) .Thehigher the SWH , the lower received pulse energy

Ocean surface reflectivity and atmosphericOcean surface reflectivity and atmosphericattenuationattenuation

Clear sky attenuation,Clear sky attenuation,radar affected by rain, cloudradar affected by rain, cloudCourtesy: Courtesy: Chelton Chelton et al. [2001]et al. [2001]

Maul [1985]Maul [1985]

Pulse-Limited Radar AltimetryPulse-Limited Radar Altimetry

Courtesy: Courtesy: Chelton Chelton et al. [2001]et al. [2001]

Beam-limited (L)Beam-limited (L) and and pulse-limited (R)pulse-limited (R) altimeter altimeterdesigns. For T/P (1350 km, 13.6 GHz) thedesigns. For T/P (1350 km, 13.6 GHz) theantenna diameter would be antenna diameter would be ~8 m for beam-~8 m for beam-limited altimeter designlimited altimeter design. Pulse-limited altimeters. Pulse-limited altimetersissue many short-pulses and provides anissue many short-pulses and provides anaverage. E.g. average. E.g. antenna width for T/P is ~1.5 mantenna width for T/P is ~1.5 m..

Pulse-Limited Altimeter Footprint andPulse-Limited Altimeter Footprint andoperationsoperations

T/P: bandwidth ~0.3 Ghz (3 ns pulse)

Pulse-Limited Radar AltimetryPulse-Limited Radar Altimetry

Courtesy: Courtesy: Chelton Chelton et al. [2001]et al. [2001]

Averaged waveform returnAveraged waveform returnPlane views of illuminated patternPlane views of illuminated patternof radar with various pulseof radar with various pulseduration for 2 different waveduration for 2 different waveheightsheights

Development at APLof the originalsatellite-based

navigation system(1959-1998, Transit)

Development at APLof the originalsatellite-based

navigation system(1959-1998, Transit)

Courtesy: K. Raney

Pulse-limitedannuli

Pulse-Limited

97/10/13 rkr

Pulse length

SWH > pulse lengthQuasi-flat sea

Track point

Time

Power (F0) Surface response function

Plan view ofilluminationfootprint

(Time delay)

Slope (SWH)

Conventional radar altimetry:

Courtesy: K. Raney

Along track

Relative time delay

0

23

Pulse length Pulse length

Annuli ofequal areas

Pulselimitedfootprint

1

23

Pulse-Dopplerlimitedfootprint

1

0

Altimeters Compared

Two-dimensionalsection of theangular scatteringfunction at eachand everysubsatellite point

Processing: removeextra delay due towavefront curvature,which converts alldata along-track toheight measurements

DDA: More averaging => x2 better precision, x10 better efficiency

Conventional Delay/Doppler

Doppler modulation

Advantage:along-trackincidence andDoppler equivalence(modulo PRF)

Multi-looks at each location

Doppler segmentationpermits closer approach to

land and vegetation

~250 m

Courtesy: K. Raney

Repeat orbits: designed (+/-1 km spacingat equator) for mesoscale oceanographyand sea level, 35-day repeat orbits):optimize temporal sampling andsacrifice spatial coverage

Non-repeat (Geodetic) orbits: designedfor fine-spatial sampling, suffers fromtemporal sampling (Geosat GM, ERS-1Geodetic phase, proposed ABYSSmission)

10-day Repeat

35-day Repeat

17-day Repeat

GEOSAT GEODETIC MISSION GROUND TRACK PATTERNGEOSAT GEODETIC MISSION GROUND TRACK PATTERNGEOSAT GEODETIC MISSION GROUND TRACK PATTERN

Orbit Determination:Dynamic, reduced

dynamic, kinematic

€

˙ ̇ r = −µ r r3← vector← scalar

+ ∇U + F

Equation of Motion:

U - conservative (gravitational) forcesF - Non-conservative forces

PRECISION ORBIT DETERMINATION METHODS

Dynamical Equations of Motion:

( )tcvrfr

rGMr ,,,

3∑+=&&

vr , - Position and Velocity Vectors

( )tcvrf ,,,∑ - Perturbation Forces

Gravitational:

• Non-spherical Earth• Luni-solar and planetary• Solid Earth tides• Ocean tides• General relativity

Nongravitational:

• Atmospheric drag• Direct solar radiation pressure• Earth albedo radiation pressure• Empirical forces

c - Constant Parameters• Dynamical• Kinematical

DOMINANT PERTURBATIONS ONDOMINANT PERTURBATIONS ONNEAR-EARTH ORBITING SATELLITESNEAR-EARTH ORBITING SATELLITES

• Gravitational– Geopotential, N-body, solid Earth and ocean tides (astronomical)– Cryospheric, oceanic, hydrological, atmospheric mass variations*– Secular mass variations due to postglacial rebound, sea level, etc.*– General relativity

• Nongravitational *Currently not modeledCurrently not modeled– Atmospheric drag– Solar radiation pressure (includes Earth eclipsing)– Earth radiation pressure (optical and infrared)

• Non-rotating (Inertial) and Terrestrial reference frames– Station positions, horizontal velocities, vertical motion*– Precession, nutation, Earth rotation, polar motion– Geocenter motion* and loading (tidal, atmospheric*, hydrological*)

• Satellite thrust/thermal radiation models• S/C attitude (CM motion wrt tracking sensors and instrument)

Accelerations on Satellite Orbits

Chelton et al. [2001]

SLR Tracking System

Chelton et al. [2001]

DORIS Tracking System

Chelton et al. [2001]

Global Positioning System SatellitesGlobal Positioning System Satellites

Geosat Geosat Orbit Error Spectra: height vs SlopeOrbit Error Spectra: height vs Slope

OO

Sandwell Sandwell and Zhang, JGR [1989]and Zhang, JGR [1989]

Radial Orbit Error of Radial Orbit Error of ~5 m~5 mat 40,000 km scale (onceat 40,000 km scale (onceper revolution), is aboutper revolution), is about~0.8 ~0.8 µµradrad

After After crossover adjustmentcrossover adjustmentof orbits, the once per revof orbits, the once per reverror reduces to error reduces to ~0.15 ~0.15 µµradrad

SPATIAL REPRESENTATION OF THE RADIAL ORBIT ERRORDUE TO GEOPOTENTIAL PERTURBATION

For 0=q , radial orbit error [Tapley and Rosborough, 1985]

( )λλ mmr SCD lmlmc

lmplmp

l

p

l

ml

sincos001

)0( −=Δ Φ∑∑∑==

∞

=

&

( )λλ mm SCD lmlms

lmplmp

l

p

l

ml

cossin001

−± Φ∑∑∑==

∞

=

&

where

Dlmp - function of satellite altitude and inclination

Φ&c

lmp and Φ&

s

lmp - latitude functions

+ sign denotes satellite is on ascending pass

- sign denotes satellite is on descending pass

Geographical mean radial orbit error:

( )λλγ mm SCD lmlmc

lmplmp

l

p

l

ml

sincos001

+=Δ Φ∑∑∑==

∞

=

&

Geographical variability error about the mean:

( )λλ mmv SCD lmlms

lmplmp

l

p

l

ml

cossin001

−±=Δ Φ∑∑∑==

∞

=

&

SPATIAL REPRESENTATION OF ALTIMETER CROSSOVERERROR DUE TO GEOPOTENTIAL PERTURBATION

Single satellite crossovers:

νΔ=Δ 2x

)cossin(2001

λλ mSmC lmlms

lmplmp

l

p

l

mlD −= Φ∑∑∑

==

∞

=

• Zonals unobservable (to this level of approximation)

Dual satellite crossovers:

jiji vvyyx Δ−Δ+Δ−Δ=Δ

( ) ( )λλ mSmCD lmlmiclmplmpl

p

l

ml

sincos~

001

+Φ= ∑∑∑==

∞

=

( ) ( )λλ mSmCD lmlmiclmplmpl

p

l

ml

sincos~

001

+Φ− ∑∑∑==

∞

=

( ) ( )λλ mSmCD lmlmislmplmpl

p

l

ml

cossin~

001

−Φ∑∑∑==

∞

=

m

( ) ( )λλ mSmCD lmlmislmplmpl

p

l

ml

cossin~

001

−Φ± ∑∑∑==

∞

=

for satellites i and j

Predicted T/P ErrorDue to Gravity

Courtesy: John Ries

Predicted JASON Orbit Error Due to Gravity

Courtesy: John Ries

Mean rms = 22.4 cmVariability rms = 21.7 cmTotal radial orbit error(EGM96, 50x50) = 31.2 cmEstimated error (150x150)= ~50 cm rms

Note: Geopotential covariance computedto only 50x50, ISS sensitive to ~130x130

International Space Station (ISS)

Error SourceError SourceERS-1/-2ERS-1/-2 Orbit Orbit (cm) (cm)

T/P T/P Geosat Geosat GFO GFO Cryosat Cryosat ISSISS Orbit Orbit Orbit Orbit Orbit Orbit Orbit Orbit Orbit Orbit (cm) (cm) (cm) (cm) (cm) (cm) (cm) (cm) (cm) (cm)

Constant gravity 2 1 3 3 15 50Constant gravity 2 1 3 3 15 50Radiation forces 2 2 3 3 2Radiation forces 2 2 3 3 2 10 10Atmospheric dragAtmospheric drag 3

15 February 2004 C. Shum 98

Inferred Sea SurfaceHeights from Altimetry

15 February 2004 C. Shum 101Courtesy: Courtesy: Chelton Chelton et al. [2001]et al. [2001]

15 February 2004 C. Shum 102

Sea Surface Height

wherehorbit the altitude of altimeter orbit;halt the raw altimeter range;hinsru the total of the instrument corrections;hssb the sea state bias correction;hdry the dry troposphere correction;hwet the wet troposphere correction;hion the ionosphere correction;htides the ocean tide correction, solid Earth tide correction and

the pole tide correction;hib the inverted barometer correction;b the altimeter bias;e the contribution of random and systematic errors.

€

hssh = (horbit − halt − hinsru − hssb − hdry − hwet − hiono

€

−htides − hib ) + b + e

Instrument Corrections• Acceleration error

• Doppler-shift error• Oscillator-drift error• Pointing-angle & sea state corrections (altimeter dependent)

• Other drift corrections (Internal calibration, point target response, etc.,

altimeter dependent)• Time tag biases

GFO Timing Stability ComparisonsGFO Timing Stability Comparisons

GFOGFO GeosatGeosat

USO Height Correction ComparisonsUSO Height Correction Comparisons

GFOGFO GeosatGeosat

Timing Bias Estimates - Laser OrbitsTiming Bias Estimates - Laser Orbits

OSU Time Tag BiasOSU Time Tag BiasEstimates(11/00 - 2/01):Estimates(11/00 - 2/01):~1.5 ~1.5 msecmsec

Internal Calibrationcorrections and tidegauge calibrations(RA bias and drift)

Courtesy: G. Hayne and D. Hancock

-20.0

-15.0

-10.0

-5.0

0.0

5.0

10.0

15.0

20.0

50000000 100000000 150000000 200000000

SPTR Range Corrections to ERS-1 Radar Altimeter

SPTR

Ran

ge C

orre

ction

s

Seconds Past 1990

-40.0

-30.0

-20.0

-10.0

0.0

10.0

20.0

160000000 170000000 180000000 190000000 200000000 210000000 220000000 230000000 240000000

SPTR Range Corrections for ERS-2 RA

SPTR

Ran

ge C

orrec

tion (

mm)

Seconds Past 1990

ERS-1 and ERS-2 (Old) SPTR Range Corrections

Credit: ESA/ESRIN

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

0 5 10 15 20 25 30 35 40

GEOSAT Altimeter Calibration Averaged Per Cycle

GEOSAT Internal Altimeter Height Calibration (cm)GEOSAT Internal Clock Drift Calibration (cm)GEOSAT Total Calibration (cm)

Jan 1, 1987 Jan 1, 1988

Delta

Ran

ge (c

m)

Cycle

GEOSAT Internal Calibration and Oscillator Drift Corrections

Media and Geophysical Corrections• “Correction” is defined as physical

or instrument phenomena that we “understand” and could quantify with specified accuracy • Otherwise, these phenomena are signals

Atmospheric Attenuation of RadarAtmospheric Attenuation of RadarTropsphere Tropsphere (Dry and Wet) and (Dry and Wet) and Ionospere Ionospere Delays [Source: NASA/JPL]Delays [Source: NASA/JPL]

Atmospheric Refractions on RadarAtmospheric Refractions on Radar

R _universal gas constant (8.317 11 −− ⋅⋅ kmolJ )

waterρ _density of water vapor(5.7) Can be written as_( assuming =g constant , =T constant = aT )

as TwPh 723.11027.25 +×=Δ − (6.8)

dry component wet component

wetdry hh Δ+Δ=

dzzP airs ∫∞

=0

)(ρ (6.9)

∫∞

→=0

)( dzzw waterρ difficult to model

=aT average temperature

cmhcm

meterh

wet

ary

306

31.2

Atmospheric Refractions on RadarAtmospheric Refractions on Radar index of the ionosphere_

22

1f

Nn α+=

N = number of free elections per unit volumeα = 80.5 23 −smf = radio frequency in Hertz

Error in range dzNf

dzn ∫∫∞∞

=−=020 2

)1(α

2

2.40fE

= (6.4)

∫∞

→=0NdzE columnar value of free elections (6.5)

1816 1010

ATMOSPHERE ATTENUATIONSATMOSPHERE ATTENUATIONS

Chelton Chelton et al. [2001]et al. [2001]

15 February 2004 C. Shum 116

CODE GIM-TOPEX TEC (mean and rms)

1995–2001

COMPARISON OF NCEP(GFO) AND GFO MWR WET DELAY

COMPARISON OF GFO MWR AND ERS-2 MWR (ATSR) WET DELAY

Revised NOAA IGDR Data (Dec 6-22, 1999)

SWH BUOY CALIBRATION (D. Cotton)SWH BUOY CALIBRATION (D. Cotton)Buoy data fit: Buoy data fit: 12 cm 12 cm rms rms (26 cm for TOPEX; 32 cm for ERS-2)(26 cm for TOPEX; 32 cm for ERS-2)Preliminary results (limited calibration data used)Preliminary results (limited calibration data used)

σσ00 BUOY CALIBRATION (D. Cotton) BUOY CALIBRATION (D. Cotton)Buoy data fit : Buoy data fit : 1.28 m/s (1.27 m/s for TOPEX; 1.23 m/s for ERS-2)1.28 m/s (1.27 m/s for TOPEX; 1.23 m/s for ERS-2)Preliminary results (limited calibration data used)Preliminary results (limited calibration data used)

COMPARISONS WITH TOPEX SWH/COMPARISONS WITH TOPEX SWH/σσ0010-day Averages within 66S-66N10-day Averages within 66S-66N

Preliminary results indicate GFO offsetsPreliminary results indicate GFO offsetswith TOPEX SWH and with TOPEX SWH and σσ00 values, confirmingvalues, confirmingD. HancockD. Hancock’’s calibration resultss calibration results

Pressure Field and Inverted Barometer

Chelton et al. [2001]

Tides: Solid Earth tides, (geocentric) ocean tides, pole tides

ASSESSMENT OF TIDE ERROR USINGASSESSMENT OF TIDE ERROR USINGMODEL COMPARISONSMODEL COMPARISONS

Yu et al. [2000]Yu et al. [2000]