satellite altimetry and gravimetry · 2010. 10. 22. · 2. satellite-to-satellite tracking (sst)...

Satellite Altimetry and Satellite Altimetry and GravimetryGravimetry::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

ckshumckshum@@osuosu..eduedu, , [email protected]@osuosu..edueduhttp://geodesy.eng.http://geodesy.eng.ohioohio-state.-state.eduedu

Norwegian Univ. of Science and TechnologyTrondheimTrondheim, Norway, Norway2121––25 June, 200425 June, 2004

GRAVITY MAPPING MISSIONSGRAVITY MAPPING MISSIONS

CHAMP (GFZ), Launched 15 July 2000CHAMP (GFZ), Launched 15 July 2000

GRACE (NASA/GFZ)GRACE (NASA/GFZ)Launched 16 March 2002Launched 16 March 2002Accumulated Accumulated geoid geoid accuracy (150x150): 20 cm (accuracy (150x150): 20 cm (rmsrms))Sensitive to gravity change equivalent to <1 cm Sensitive to gravity change equivalent to <1 cm rmsrmsfluid redistribution at the Earth surfacefluid redistribution at the Earth surface

GOCE (ESA), 2005 LaunchGOCE (ESA), 2005 LaunchAccumulated Accumulated geoid geoid accuracy (250x250): 1 cm (accuracy (250x250): 1 cm (rmsrms))

Atmospheric loading assumed knownAtmospheric loading assumed known

2. Satellite-to-Satellite Tracking (SST)

Range,

(1)

where

Range-rate,

(2)

where

Range-rate rate,

(3)

N.B. All quantities refer to the inertial (non-rotating) frame.

( ) ( ) ,ñ 12121212 exxxxx ⋅=−⋅−=

1212 xxx −≡ ( ) 121212 xxxxe −−≡

1212ñ ex ⋅= &&

1212 xxx &&& −≡

−+⋅= 2

12

2i12

121212 ñ

ñ

1ñ &&&&&& xxe

2. Satellite-to-Satellite Tracking (SST)The linear equation for the gravity recovery from the range-rate and range-rate rate observablecan be derived as follows (Wakker et al, 1989; Seeber, 1993):

"reality = reference (tilde) + residual (delta)"

range-rate:

range-rate rate:

geopotential coefficients:

(4)

(5)

where … relative velocity vector orthogonal to line-of-sight (LOS),

… relative acceleration vector orthogonal to LOS.

ρδρρ &&& +=~

ρδρρ &&&&&& +=~

nmnmnm δβββ +=~

nmnmnm

δββρβ

ρδ ⋅

∂

∂⋅+

∂

∂⋅= 1212

12

xcxe

&&

nmnmnmnm

δββρρ

ρρβρβ

ρδ ⋅

∂

∂⋅

⋅−+−+

∂

∂⋅+

∂

∂⋅= 12

121212

12

212 xecc

dcx

cx

e&&&&

&&

1212 exc ρ&& −=

( ) 12121212 eexxd ⋅−= &&&&


Two alternative approaches for gravity recovery: Energy integral (Jekeli, 1999; Han, 2004): It is derived from considering energy relationshipbetween two satellites (either how-low or low-low).

(6)

(7)

(8)

where T12 is the disturbing potential difference between the two satellites and F1 and F2 are thenon-conservative force vectors such as drag force.

€

T12 = ˙ ˜ x 1 ⋅ ?˙ ? + ˙ ˜ x 2 − ˙ ˜ x 1 e12( ) ⋅ ?˙ x 12 + ˙ ˜ x 12 ⋅ ?˙ x 1 − ˙ ˜ x 1 ˙ ˜ x 12 ⋅ ?e12

+ ?˙ x 1 ⋅ ?˙ x 12 +12

?˙ x 12 ⋅ ?˙ x 12 − ?RE12 −FE12 − ?C

( ) ( )112121122212e112121122212e12 x~

y~y~

x~x~

y~y~

x~ùxyyxxyyxùäRE &&&&&&&& −+−−−+−=

( )∫ ⋅−⋅= dtFE 112212 xFxF &&


LOS acceleration model (Rummel, 1979)

(9)

where is the gravity disturbance vector difference and is the normal gravity vectordifference.

(10)

Using precisely determined orbits and range-rate or range-rate rate observables, the in situdisturbing geopotential difference or acceleration difference can be obtained. The in situobservables are used to model the gravity field globally through fitting spherical harmonic basisfunctions or locally through the downward-continuation.

( )

−+++⋅= 22

1212121212 ññ

1äñ &&&& xFãge

12äg 12ã

( )

−−+⋅−=⋅≡ 22

12121212i1212

i12 ñ

ñ

1ñääg &&&& xFãege

3. Satellite Gravity Gradiometry (SGG)

Gradiometer consists of pairs of accelerometers whose outputs are differenced to yield thegradient of acceleration (acceleration difference over the length of separation between twoaccelerometers). For the detail explanation of the principle, we refer Suenkel (Ed.) 1986.Mathematically, we can start from the motion of equation defined as follows:

(1)

(2)

where is three Euler rotation angles of a-frame with respect to i-frame, coordinatizedin a-frame. is a rotation matrix from inertial frame (i-frame) to any rotating frame (a-frame).The superscripts, i and a, indicate the quantities in the inertial and any rotating frames,respectively. The second, third, and fourth terms of the right hand side are the Coriolis acceleration,acceleration due to angular velocity change, and centrifugal acceleration, respectively.

aaia

aia

aaia

aaia

aiaiC xxxxx ΩΩ+Ω+Ω+= &&&&&& 2

−

−

−

=Ω

0

0

0

12

13

23

ωω

ωω

ωωaia

[ ]321 ωωωaiC


The left hand side can be re-written as follows: (3)

where V is gravitational potential and F is non-conservative acceleration vector. Therefore thefollowing is obtained:

(4) In order to derive the gradiometer tensor equation,consider the following system consisting ofpairs of accelerometers.

O : Center of gravityP and Q : Locations of proof mass of two accelerometers

iii V Fx +∇=&&

aaiai VC Fx +∇=&&

PQ axΔ21axΔ−

21

accelerometer

O


Two equations of motion at P and Q are given by

(5)

(6) Due to the feed back mechanism of proof mass (without time delay), we will have

and (7) Therefore,

(8)

(9) Taking the difference between them,

(10)

P

aaia

aiaP

aaiaP

aaiaP

a

P

iaiC xxxxx ΩΩ+Ω+Ω+= &&&&&& 2

Q

aaia

aiaQ

aaiaQ

aaiaQ

a

Q

iaiC xxxxx ΩΩ+Ω+Ω+= &&&&&& 2

0xx ==P

a

P

a &&& 0xx ==Q

a

Q

a &&&

[ ]P

aaia

aia

aiaP

a

P

aV xF ⋅ΩΩ+Ω=+∇ &

[ ]Q

aaia

aia

aiaQ

a

Q

aV xF ⋅ΩΩ+Ω=+∇ &

[ ]( )Q

a

P

aaia

aia

aiaQ

a

P

a

Q

a

P

a VV xxFF −⋅ΩΩ+Ω=−+∇−∇ &


By Taylor linearization,

(11)

where

Note that there are only five independent elements in the matrix, M, because of its harmonic(trace(M)=0) and symmetric characteristics. Finally, we will have

(12) Note that the left hand side is the quantities which two accelerometers can measure at P and Qand is pre-determined (known) quantity. Therefore, can be computed andit is denoted by and is the output o the gradiometer.

a

Q

a

P

a VV xM Δ⋅+∇=∇

∂

∂

∂∂

∂

∂∂

∂∂∂

∂

∂

∂

∂∂

∂∂∂

∂

∂∂

∂

∂

∂

=

2

222

2

2

22

22

2

2

z

V

zy

V

zx

Vzy

V

y

V

yx

Vzx

V

yx

V

x

V

M

[ ]( ) aaia

aia

aiaQ

a

P

a xMFF Δ⋅ΩΩ+Ω+−=− &

axΔ [ ]( )aia

aia

aia ΩΩ+Ω+− &M

Ã


… output of gradiometer, "measurement tensor" (13) In order to extract the pure gravitational tensor from the measurement tensor, we use linearcombination considering the symmetry of M and skew-symmetry of as follows:

(14)

(15) The time-integration of the first one can provide as follows:

(16) Finally, the gravitational tensor will be computed as follows:

(17)

[ ]aiaaia

aia ΩΩ+Ω+−= &MÃ

aiaΩ

( ) aiaΩ=− &T

2

1ÃÃ

( ) aia

aiaΩΩ+−=+ MÃÃ T

2

1

)(taiaΩ

( ) )(2

1)( 0

T

0

tdtt aia

t

t

aia Ω+−=Ω ∫ ÃÃ

( ) )()()()(2

1)( T ttttt a

iaaia ΩΩ++−= ÃÃM

Perturbed Satellite Motion

( ) ( )

( ) ( )

( ) ( )

function disturbingR

PmSmCr

R

R

GM

PmSmCr

R

r

GMR

Rr

GMR

r

GM

PmSmCr

R

r

GMV

nm

nmnmnm

n

nmnmnm

n

nm

n

n

mnm

n

n

mnmnmnm

n

:

cossincos

cossincos

cossincos1

1

2 0

2 0

θλλ

θλλ

θλλ

+

=

+

=

∑ +=∑+=

∑ ∑ +

+=

+

∞

= =

∞

= =

Continue

• where GM is the gravitational constanttimes the Earth’s mass; R is the Earth’smean radius; (r,θ,λ) are the coordinates ofthe satellite; Pnm is the associatedLegendre function of degree n and orderm; Cnm, Snm are spherical harmoniccoefficients. GM/r describes the potentialof homogenous sphere; n=1, m=0,1 arezero because the origin of the coordinatesystem transferred to the center of mass

( ) ( ) ( )θω ,,,0

1Ω∑∑=

∞

−∞==+

MSeGiFa

GMRR nmpq

qnpq

n

pnmpn

n

nm

( ) ( ) ( )( )( ) function tyeccentricieG

function ninclinatioiF

mMqpnpn

C

Scos

S

CS

npq

nmp

nmpq

nmpq

even mn

old mnnm

nmnmpq

even mn

odd mnnm

nmnmpq

=

=

−Ω++−+−=

+

−=

−

−

−

−

θωψ

ψψ

22

sin

Seeber G., Satellite Geodesy, 2003.

Kaula,1966.

Re-formulated as a function of the orbitalelements:

Contunue

• a, b°Gsemi-major, semi-minor axis; f = ν°Gtrue anomaly; E°Geccentricity anomaly; i°Ginclination; _°Gright ascension of theascending node; ω°Gargument of theperigee; ω+ν°Gargument of the latitude; e:eccentricity; M : mean anomaly;

y

xapogee

a

b

perigeeae q1

E fr q2

satellite

, X

Y

Z

Ωω

νi

iperigee

satellite

1.Equation of ellipse

12

2

2

2

=+b

y

a

x

( )

( )

( )2

2

222

2

1

2

22

21

22

11

1

cossin1

tantan

cos1

sin1sin

coscos .2

eapaba

e

eEEe

qq

vf

Eeaqqr

EeaEbqy

eEqEaqaex

−=

−=

−−

===

−=+=

−===

−=⇒=+=

Mass Variations

Atmospheric Mass Variation

Mass Variation of Ocean tide

Mass variation of Continental SurfaceWater

Oceanic Mass Variation

Love Number

h is the ratio of the height of a body tide to thestatic marine tide (introduced by A. E. H. Love).k is ratio of additional potential produced by theredistribution of mass to the deforming potential(introduced by A. E. H. Love). l is the ratio ofhorizontal displacement of the crust to that of theequilibrium fluid tide (introduced by T. Shida).

For a rigid body, h=l=k=0

For a fluid body, the Love number h=l=1

are the spherical harmonic coefficients; kn is theload Love number of degree n that describes theEarth’s elasticity; is the Fully normalizingassociated Legendre function; _E is the averagedensity of the Earth (5517 kg/m3).

Atmospheric Mass Variation

nmP

w

s

gó

)të,è,(p)të,è,(h =

( ) èdèdësinmë

cosmëèPtë,è,h

1)(2nRó4

)ók3(1

tS

tCnm

E

wn

nm

nm sin)(cos)(

)(

∫∫+

+=

π

h is equivalent water thickness; θ and λ are the latitudeand longitude of surface pressure data, Ps; t is time; g isthe nominal gravity value;_w is the density of water (1000kg/m3).

nmC nmS

Models for Atmosphere

The entire atmosphere is assumed to becondensed onto a very thin layer on theEarth’s surface. The global surfacepressure data are available through :European Center for Medium-range Weather

Forecast (ECMWF)National Centers for Environmental Prediction

(NCEP)

If the vertical structure of the atmosphere shall be taken intoconsideration the vertical integration of the atmospheric masseshas to be performed.

Han(2003)

Ocean tides

{ })(sin)(cos)( tSmëtCmëëè,r,PrR

RGM

t)ë;è,V(r, tnm

tnm

Nmax

0m

Nmax

mnnm

1n

⋅+⋅

= ∑ ∑= =

+

)öùtsin(C)öùtcos(C)t(C 0Snm

0Cnm

tnm +++=

)öùtsin(S)öùtcos(S)t(S 0Snm

0Cnm

tnm +++=

are 4 sets of coefficients of each tidalconstituent; ω is frequency; _0 is initial phase;

CnmC

CnmS

SnmC

SnmS

Ocean Tide Models

The tidal model error represented by thecoefficient difference between CSR4.0 [Eanes and Bettadpur, 1995] and

NAO99[Matsumoto et al., 2000]

Han(2003)

Continental Surface Water

( ) èdèdësinsinmë

cosmë)è(cosPtë,è,h

1)(2nRó4

)ók3(1

)t(S

)t(Cnm

E

wn

nm

nm

+

+=

∫∫π

•Continental water storage data were computed from twolayers (0-10, 10-200 cm) of CDAS-1 soil moisture data andsnow accumulation data. Both data are provided by theNOAA-CIRES Climate Diagnostics Center, Boulder,Colorado, USA, from their web site athttp://www.cdc.noaa.gov/. The global continental data witha spatial resolution of about 2 degrees and a temporalresolution of a day are available in the form of equivalentwater thickness from the web site at the University ofTexas [GGFC, 2002].

Continental Surface Water

Two Models water storage anomaly (WSA)

monthly mean WSA (MWSA)


Oceanic Mass Variation : Seal LevelAnomaly (SLA) - Steric Sea LevelAnomaly Sea level anomaly (SLA) : Observed by

satellite radar altimeters

Steric sea level anomaly: Derived fromtemperature and salinity data according toUNESCO(1981)

TOPEX/Poseidonand Jason

Sea Level Anomaly (SLA)

Monthly sea level anomaly (SLA) fromTOPEX/POSEIDON (T/P); 1 by 1 degreegrids; Instrument, media, andgeophysical corrections are applied;

SLA = Sea Surface Heights (SSH) -Mean Sea Surface (MSS) OSU95MSS is selected.

UncertaintyUncertainty estimated by extending data span to 18 years, estimated by extending data span to 18 years,and based on original 8-yr analysis by [and based on original 8-yr analysis by [Guman Guman et al., 1997]et al., 1997]

DECADAL SEA LEVEL TREND OBSERVED BY ALTIMETERSDECADAL SEA LEVEL TREND OBSERVED BY ALTIMETERS

Estimated Global Sea Level Trend = 2.6Estimated Global Sea Level Trend = 2.6±±0.5 mm/year0.5 mm/year

Geosat, ERS-1/-2 and TOPEX/POSEIDON included

After “geoid” corrections [Peltier, 2003]:Trend = 2.80 mm/yr, ICE-4G model = 2.96 mm/yr, BIFORST model

Dynamic Height Anomaly from WOA-01

• Annual and monthly temperature and salinity data: One-degree objectively analyzed mean. Maximum depth forannual objective analyses reaches 5,500 m (33 layers) andfor monthly objective analyses reaches 1,500 m (24 layers)

• is specific volume; and is specific volume anomaly.is the specific volume of an arbitrary standard sea waterof salinity (S) = 35, temperature (T) = 0 degree andpressure (p) at the depth of the sample.€

α =α35,0,p + δ

α δ

∫∫∫ −==2

1

2

1

2

1,0,3521 ),,(),,(),(

p

p p

p

p

p

pdpdpPTSdpPTSppD ααδ

The last integral is the so-called “standard geopotential distance”

NOAA WOA-01 [Levitus, 2001]

ρα 1=

is density, which can be computed from the equation of State.The equation of state defined by the Joint Panel onOceanographic Tables and Standards (UNESCO,1982) fitsavailable measurements with a standard error of 3.5 ppm forpressure up to 1000 bars, for temperatures between freezing andC, and for salinities between 0 to 42 ( Millero and Poisson, 1981).

ρ

The unit of D is ; 1 dynamic meter =10 ; Dynamic meteris numerically almost equal to the geometric meter. Therefore, D/10(meter) is used to compare with other measurements, such as tidegauge records.

)( 22 sm)( 22 sm


Dynamic Height Anomaly from WOA-01

Dynamic Topography From WOA-01


0-3000 m

0-1000 m


€

C nm (t)S nm (t)

=3?w

4πR?E (2n +1)h φ,λ,t( )∫∫ P nm (cosφ)

cosmλsinmλ

sinϕdφdλ

Have you discover that there is a little bitdifference compared with the formula incontinental Surface Water?

Ans: There is no love number in this equation because water heights derivedfrom altimeters and steric anomaly contain loading effect.

h = Seal Level Anomaly (SLA) - Steric Aea Level Anomaly

1. GRACE monthly gravity field solution (n=120) for eighteen months. Method: Ocean mass variation in term of water heights (WH) are

computed using spherical harmonic coefficients (n = 15) for each month

∑∑= = +

+=Δ

15

0 0 112

)(cos33

),(n

n

m nnm

w

ave

kn

Pa

h θρρ

λθ

))sin()cos(( λλ mSmC nmnm +×is the average density of Earth; is the density of water; areLove numbers;

aveρ wρ nk

Comparison of Altimetric Geoid with MonthlyGRACE Geoid Models

Eighteen-month averaged GRACE geoid is used asreference to compute geoid variations

For observations of altimeter and steric anomaly, lovenumbers are concelled

3. Monthly temperature and salinity data from WOA01 Computation: Averaged monthly dynamic topography (DH) from

temperature and salinity data

Comparison of Altimetric Geoid with MonthlyGRACE Geoid Models

SLA – Steric effect (red curve, i=1,..,18, s=scale)

GRACE (blue curve):

∑=

−−=Δs

ppAvepipii sDHDHSLADH

1,,, )]()[(1

∑ −=Δ=

s

ppAvepii sWHWHDH

1,, )(2

2. TOPEX/POSEIDON(T/P) monthly altimetry sea level. Computation: T/P sea level anomaly (SLA; sea surface heights-mean sea surface); instrument, media, and geophysical corrections;

4. Monthly hydrology data : The land data assimilation system (LDAS) isone of the land surface models developed at NOAA Climate PredictionCenter (CPC).

Mass Variation from GRACE and Alt.- StericAnomaly

∑=

−=s

ppAvepii sWHWHDH

1,, )(

Mass Variation from GRACE and LDAS onLand

TOP: ocean mass variations computed using satellite altimetry and WOA01 dynamicheights and hydrology data from LADSin the month of July 2003 (reference is Feb.2003). Bottom: GRACE observed gravity variations (nmax=15) in the month of July2003 (reference is Feb. 2003).

Left: Mass Variation from GRACERight: Mass variation from Alt., Steric Anomalyand Hydrology Data

Left: Mass Variation from GRACE

Right: Mass variation from Hydrology Data

Left: Mass Variation from GRACERight: Mass variation from Hydrology Data

Top: Massvariation fromHydrology model

Bottom: MassVariation fromGRACE

Top: Massvariation fromaltimetry/thermal

Bottom: MassVariation fromGRACE

satellite altimetry and gravimetry · 2010. 10. 22. · 2. satellite-to-satellite tracking (sst)...

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