satellite altimetry and gravimetry · 2010. 10. 22. · 2. satellite-to-satellite tracking (sst)...
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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, , braunbraun.118@.118@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 ⋅−= &&&&
2. Satellite-to-Satellite Tracking (SST)
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 &&
2. Satellite-to-Satellite Tracking (SST)
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
3. Satellite Gravity Gradiometry (SGG)
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
3. Satellite Gravity Gradiometry (SGG)
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=−+∇−∇ &
3. Satellite Gravity Gradiometry (SGG)
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
3. Satellite Gravity Gradiometry (SGG)
… 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
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
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
NOAA WOA-01 [Levitus, 2001]
Dynamic Height Anomaly from WOA-01
Dynamic Topography From WOA-01
NOAA WOA-01 [Levitus, 2001]
0-3000 m
0-1000 m
Oceanic Mass Variation
€
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 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
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