SatelliteSatellite AltimetryAltimetry andand GravimetryGravimetry:: Theory and Applications C.K. Shum1,2, Alexander Bruan2,1

1,2 Laboratory for Space Geodesy & Remote Sensing 2,1Byrd Polar Research Center

The Ohio State University Columbus, Ohio, USA [email protected], [email protected] http://geodesy.eng.ohio-state.edu Norwegian Univ. of Science and Technology Trondheim, Norway 21–25 June, 2004 GRAVITYGRAVITY MAPPINGMAPPING MISSIONSMISSIONS

CHAMP (GFZ), Launched 15 July 2000

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

GOCE (ESA), 2005 Launch Accumulated geoid accuracy (250x250): 1 cm (rms)

Atmospheric loading assumed known 2. Satellite-to-Satellite Tracking (SST)

Range,

ñ = (x 2 − x1 )⋅ (x 2 − x1 ) = x12 ⋅e12 , (1)

where x12 ≡ x 2 − x1 e12 ≡ (x 2 − x1 ) x 2 − x1

Range-rate,

ñ x e & = & 12 ⋅ 12 (2)

where x& 12 ≡ x& 2 − x& 1

Range-rate rate, 1  i 2 2  &ñ& = e12 ⋅&x&12 +  x& 12 − ñ& 12  ñ12   (3)

N.B. All quantities refer to the inertial (non-rotating) frame. 2. Satellite-to-Satellite Tracking (SST) The linear equation for the gravity recovery from the range-rate and range-rate rate observable can be derived as follows (Wakker et al, 1989; Seeber, 1993):

"reality = reference (tilde) + residual (delta)" ~ range-rate: ρ& = ρ& + δρ& ~ range-rate rate: ρ&& = ρ&& + δρ&& ~ geopotential coefficients: β nm = β nm + δβ nm

 ∂x& 12 c ∂x12  δρ& = e 12 ⋅ + ⋅  ⋅δβ nm (4)  ∂β nm ρ ∂β nm 

 ∂x 2 ∂x 1  2ρ c ⋅c  ∂x  e &&12 c & 12 & c d e 12 δρ&& =  12 ⋅ + ⋅ + − + − 12  ⋅  ⋅δβ nm (5)  ∂β nm ρ ∂β nm ρ  ρ ρ  ∂β nm 

where … relative velocity vector orthogonal to line-of-sight (LOS), c = x& 12 − ρ&e12 d x x e e = & & 12 − ( & & 12 ⋅ 12 ) 12 … relative acceleration vector orthogonal to LOS. 2. Satellite-to-Satellite Tracking (SST)

Two alternative approaches for gravity recovery:

Energy integral (Jekeli, 1999; Han, 2004): It is derived from considering energy relationship between two satellites (either how-low or low-low).

T x˙˜ ??˙ x˙˜ x˙˜ e ?x˙ x˙˜ ?x˙ x˙˜ x˙˜ ?e 12 = 1 ⋅ + ( 2 − 1 12 )⋅ 12 + 12 ⋅ 1 − 1 12 ⋅ 12 1 (6) + ?x˙ ⋅ ?x˙ + ?x˙ ⋅ ?x˙ − ?RE − FE − ?C 1 12 2 12 12 12 12

~ ~ ~ ~ ~ ~ ~ ~ (7) äRE12 = ù e (x12 y& 2 − y 2 x& 12 + x1y& 12 − y12 x& 1 )− ù e (x12 y& 2 − y2 x& 12 + x1 y& 12 − y12 x& 1 ) €

FE F x F x dt 12 = ∫( 2 ⋅ & 2 − 1 ⋅ & 1 ) (8)

where T12 is the disturbing potential difference between the two satellites and F1 and F2 are the non-conservative force vectors such as drag force. 2. Satellite-to-Satellite Tracking (SST)

LOS acceleration model (Rummel, 1979)

1  2 2  &ñ& = e12 ⋅ (äg12 + ã12 + F12 )+  x& 12 − ñ&  ñ   (9)

where ä g 12 is the gravity disturbance vector difference and ã 12 is the normal gravity vector difference.

i i 1  2 2  äg12 ≡ e12 ⋅ äg12 = &ñ& − e12 ⋅ (ã12 + F12 )−  x& 12 − ñ&  ñ   (10)

Using precisely determined orbits and range-rate or range-rate rate observables, the in situ disturbing geopotential difference or acceleration difference can be obtained. The in situ observables are used to model the gravity field globally through fitting spherical harmonic basis functions or locally through the downward-continuation. 3. Satellite Gravity Gradiometry (SGG)

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

a i a a a a a a a a Ci &x& = &x& + 2Ωia x& + Ω& ia x + Ωia Ωia x (1)

 0 − ω3 ω2  a  0  Ωia =  ω3 − ω1  (2)

− ω2 ω1 0 

where [ ω 1 ω 2 ω 3 ] is three Euler rotation angles of a-frame with respect to i-frame, coordinatized a in a-frame. C i 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. 3. Satellite Gravity Gradiometry (SGG)

The left hand side can be re-written as follows: i i i (3) &x& = ∇ V + F where V is gravitational potential and F is non-conservative acceleration vector. Therefore the following is obtained: a i a a Ci &x& = ∇ V + F (4)

In order to derive the gradiometer tensor equation, consider the following system consisting of pairs of accelerometers. accelerometer

O

Q 1 1 P − Δxa Δxa 2 2

O : Center of gravity P and Q : Locations of proof mass of two accelerometers 3. Satellite Gravity Gradiometry (SGG)

Two equations of motion at P and Q are given by

C a xi = x a + 2Ω a x a + Ω& a x a + Ω a Ω a x a i && P && P ia & P ia P ia ia P (5)

a i a a a a a a a a Ci &x& = &x& + 2Ωia x& + Ω& ia x + Ωia Ωia x Q Q Q Q Q (6)

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

x a = x a = 0 and x a = x a = 0 (7) && P & P && Q & Q

Therefore, ∇ aV + F a = Ω& a + Ω a Ω a ⋅ x a P P [ ia ia ia ] P (8)

a a a a a a ∇ V + F = [Ω& ia + Ωia Ωia ]⋅ x Q Q Q (9)

Taking the difference between them,

∇ aV − ∇ aV + F a − F a = Ω& a + Ω a Ω a ⋅ x a − x a (10) P Q P Q [ ia ia ia ]( P Q ) 3. Satellite Gravity Gradiometry (SGG)

By Taylor linearization,

∇ aV = ∇ aV + M ⋅ Δx a P Q (11)  ∂ 2V ∂ 2V ∂ 2V   x 2 x y x z   ∂ ∂ ∂ ∂ ∂  where ∂ 2V ∂ 2V ∂ 2V M =   ∂x∂y ∂y 2 ∂y∂z   2 2 2   ∂ V ∂ V ∂ V  2 ∂x∂z ∂y∂z ∂z 

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

F a − F a = − M + Ω& a + Ω a Ω a ⋅ Δx a P Q ( [ ia ia ia ]) (12)

Note that the left hand side is the quantities which two accelerometers can measure at P and Q a a a a and Δ x is pre-determined (known) quantity. Therefore, ( − M + [ Ω& ia + Ω ia Ω ia ] ) can be computed and it is denoted by à and is the output o the gradiometer. 3. Satellite Gravity Gradiometry (SGG)

a a a à = −M + [Ω& ia + Ωia Ωia ] … output of gradiometer, "measurement tensor" (13)

In order to extract the pure gravitational tensor from the measurement tensor, we use linear a combination considering the symmetry of M and skew-symmetry of Ω ia as follows:

1 T a (Ã− Ã )= Ω& ia (14) 2 1 T a a ( Ã+ Ã )= −M + Ωia Ωia (15) 2

a The time-integration of the first one can provide Ω ia ( t ) as follows:

t a 1 T a Ω ia (t) = (Ã− Ã )dt + Ωia (t0 ) (16) ∫t0 2

Finally, the gravitational tensor will be computed as follows:

1 T a a M(t) = − (Ã(t) + Ã (t))+ Ωia (t)Ωia (t) (17) 2 Perturbed Satellite Motion

n GM  ∞ n  R   V = 1+ ∑ ∑   (C cosmλ + S sin mλ)P (cosθ ) r  r nm nm nm   n=2 m=0    GM ∞ n GM = + ∑ ∑ Rnm = + R r n=2 m=0 r n GM  R  Rnm =   (Cnm cosmλ + Snm sin mλ)Pnm (cosθ ) r  r  n+1 GM  R  =   (Cnm cosmλ + Snm sin mλ)Pnm (cosθ ) R  r 

Rnm : disturbing function Continue

• where GM is the gravitational constant times the Earth’s mass; R is the Earth’s mean radius; (r,θ,λ) are the coordinates of the satellite; Pnm is the associated Legendre function of degree n and order m; Cnm, Snm are spherical harmonic coefficients. GM/r describes the potential of homogenous sphere; n=1, m=0,1 are zero because the origin of the coordinate system transferred to the center of mass Re-formulated as a function of the orbital elements:

GMRn n ∞ R F i G e S , M , , nm = n+1 ∑ nmp ( ) ∑ npq ( ) nmpq (ω Ω θ ) a p=0 q=−∞ n−m even n−m even  Cnm  Snm  S = cosψ + sinψ nmpq  S  nmpq C  nmpq − nm n−m odd  nm n−m old

ψ nmpq = (n − 2 p)ω + (n − 2 p + q)M + m(Ω −θ )

Fnmp (i)= inclination function

Gnpq (e)= eccentricity function

Seeber G., Satellite Geodesy, 2003. Kaula,1966. Contunue

• a, b°Gsemi-major, semi-minor axis; f = ν°G true anomaly; E°Geccentricity anomaly; i°G inclination; _°Gright ascension of the ascending node; ω°Gargument of the perigee; ω+ν°Gargument of the latitude; e: eccentricity; M : mean anomaly; 1.Equation of ellipse Z x 2 y 2 2 + 2 = 1 a b satellite i 2. x = ae + q1 = a cos E ⇒ q1 = (cos E − e) ν perigee y = q = bsin E = a 1− e2 sin E i 2 ω 2 2 Ω Y r = q1 + q2 = a(1− ecos E)

2 q2 1− e sin E tan f = tan v = = , X q1 cos E − e 2 2 a − b y e2 = a 2 satellite p = a(1− e2 ) q b r 2 apogee E f x a ae q1 perigee Mass Variations

Atmospheric Mass Variation Mass Variation of Ocean tide Mass variation of Continental Surface Water Oceanic Mass Variation Love Number

h is the ratio of the height of a body tide to the static marine tide (introduced by A. E. H. Love). k is ratio of additional potential produced by the redistribution of mass to the deforming potential (introduced by A. E. H. Love). l is the ratio of horizontal displacement of the crust to that of the equilibrium fluid tide (introduced by T. Shida). For a rigid body, h=l=k=0 For a fluid body, the Love number h=l=1 Atmospheric Mass Variation

ps(è, ë, t) h(è, ë, t) = gó w h is equivalent water thickness; θ and λ are the latitude

and longitude of surface pressure data, Ps; t is time; g is the nominal gravity value;_w is the density of water (1000 kg/m3).

Cnm (t) 3(1+ kn )ów cosmë   = ∫∫ h(è, ë,t)Pnm (cosè ) sinèdèdë 4 Ró (2n 1) sinmë Snm (t)  π E +   Cnm Snmare the spherical harmonic coefficients; kn is the load Love number of degree n that describes the

Earth’s elasticity; P nm is the Fully normalizing associated Legendre function; _E is the average density of the Earth (5517 kg/m3). Models for Atmosphere

The entire atmosphere is assumed to be condensed onto a very thin layer on the Earth’s surface. The global surface pressure 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 into consideration the vertical integration of the atmospheric masses has to be performed. Han(2003) Ocean tides

GM Nmax Nmax R n+1 V(r,è,ë;t)   P (r,è,ë) cosmë C t (t) sin më S t (t) = ∑ ∑   nm { ⋅ nm + ⋅ nm } R m=0 n=m  r  t C 0 S 0 Cnm (t) = Cnm cos(ùt + ö ) + Cnm sin(ùt + ö )

t C 0 S 0 Snm (t) = Snm cos(ùt + ö ) + Snm sin(ùt + ö )

C C S S C nm S nm C nm S nm are 4 sets of coefficients of each tidal constituent; ω is frequency; _0 is initial phase; Ocean Tide Models

The tidal model error represented by the coefficient difference between . CSR4.0 [Eanes and Bettadpur, 1995] and . NAO99[Matsumoto et al., 2000] Han(2003) Continental Surface Water

C (t) 3(1+ k )ó cosmë nm n w h è, ë, t P (cosè) sin èdèdë   = ∫∫ ( ) nm   Snm (t)  4πRó E (2n +1) sinmë  •Continental water storage data were computed from two layers (0-10, 10-200 cm) of CDAS-1 soil moisture data and snow accumulation data. Both data are provided by the NOAA-CIRES Climate Diagnostics Center, Boulder, Colorado, USA, from their web site at http://www.cdc.noaa.gov/. The global continental data with a spatial resolution of about 2 degrees and a temporal resolution of a day are available in the form of equivalent water thickness from the web site at the University of Texas [GGFC, 2002]. Continental Surface Water

Two Models . water storage anomaly (WSA) . monthly mean WSA (MWSA)

Oceanic Mass Variation

Oceanic Mass Variation : Seal Level Anomaly (SLA) - Steric Sea Level Anomaly . Sea level anomaly (SLA) : Observed by satellite radar altimeters . Steric sea level anomaly: Derived from temperature and salinity data according to UNESCO(1981) TOPEX/Poseidon and Jason Sea Level Anomaly (SLA)

Monthly sea level anomaly (SLA) from TOPEX/POSEIDON (T/P); 1 by 1 degree grids; Instrument, media, and geophysical corrections are applied; SLA = Sea Surface Heights (SSH) - Mean Sea Surface (MSS) . OSU95MSS is selected. DECADAL SEA LEVEL TREND OBSERVED BY ALTIMETERS

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

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

Estimated Global Sea Level Trend = 2.6±0.5 mm/year Uncertainty estimated by extending data span to 18 years, and based on original 8-yr analysis by [Guman et al., 1997] Dynamic Height Anomaly from WOA-01

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

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

p2 p2 p2 D( p1, p2 ) = δ (S,T, P)dp = α(S,T, P)dp − α35,0, pdp ∫p1 ∫p1 ∫p1 The last integral is the so-called “standard geopotential distance”

NOAA WOA-01 [Levitus, 2001] Dynamic Height Anomaly from WOA-01

α = 1 NOAA WOA-01 [Levitus, 2001] ρ ρ is density, which can be computed from the equation of State. The equation of state defined by the Joint Panel on Oceanographic Tables and Standards (UNESCO,1982) fits available measurements with a standard error of 3.5 ppm for pressure up to 1000 bars, for temperatures between freezing and C, and for salinities between 0 to 42 ( Millero and Poisson, 1981).

The unit of D is ( m 2 s 2 ) ; 1 dynamic meter =10 ( m 2 s 2 ) ; Dynamic meter is numerically almost equal to the geometric meter. Therefore, D/10 (meter) is used to compare with other measurements, such as tide gauge records. Dynamic Topography From WOA-01

0-3000 m

0-1000 m

NOAA WOA-01 [Levitus, 2001] Oceanic Mass Variation

    Cnm (t) 3?w cosmλ   = ∫∫ h(φ,λ,t)Pnm (cosφ)  sinϕdφdλ  Snm (t)  4πR?E (2n +1) sinm λ 

h = Seal Level Anomaly (SLA) - Steric Aea Level Anomaly € Have you discover that there is a little bit difference compared with the formula in continental Surface Water? Ans: There is no love number in this equation because water heights derived from altimeters and steric anomaly contain loading effect.

Comparison of Altimetric Geoid with Monthly GRACE Geoid Models 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 3aρ 15 n 2n +1 h( , ) ave P (cos ) Δ θ λ = ∑∑ nm θ 3ρw n=0 m=0 1+ kn

× (Cnm cos(mλ) + Snm sin(mλ))

ρave is the average density of Earth; ρ w is the density of water; k n are Love numbers; . Eighteen-month averaged GRACE geoid is used as reference to compute geoid variations . For observations of altimeter and steric anomaly, love numbers are concelled Comparison of Altimetric Geoid with Monthly GRACE Geoid Models 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; 3. Monthly temperature and salinity data from WOA01 Computation: Averaged monthly dynamic topography (DH) from temperature and salinity data

4. Monthly hydrology data : The land data assimilation system (LDAS) is one of the land surface models developed at NOAA Climate Prediction Center (CPC). SLA – Steric effect (red curve, i=1,..,18, s=scale) s DH1 [(SLA ) (DH DH )] s Δ i = ∑ i, p − i, p − Ave, p p=1 GRACE (blue curve): s ΔDH 2i = ∑ (WHi, p −WH Ave, p ) s p=1 Mass Variation from GRACE and Alt.- Steric Anomaly Mass Variation from GRACE and LDAS on Land

s DH (WH WH ) s i = ∑ i, p − Ave, p p=1 TOP: ocean mass variations computed using satellite altimetry and WOA01 dynamic heights 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 July 2003 (reference is Feb. 2003). Left: Mass Variation from GRACE Right: Mass variation from Alt., Steric Anomaly and Hydrology Data Left: Mass Variation from GRACE Right: Mass variation from Alt., Steric Anomaly and Hydrology Data Left: Mass Variation from GRACE Right: Mass variation from Hydrology Data Left: Mass Variation from GRACE Right: Mass variation from Hydrology Data Top: Mass variation from Hydrology model

Bottom: Mass Variation from GRACE Top: Mass variation from altimetry/thermal

Bottom: Mass Variation from GRACE