Inversion of Earth's Changing Shape to Weigh Sea Level in Static

PREPRINT: JOURNAL OF GEOPHYSICAL RESEARCH, DOI: 10.1029/2002JB002290 (IN PRESS 2003)

Inversion of Earth’s changing shape to weigh sea level in static equilibrium with surface mass redistribution

Geoffrey Blewitt1 Nevada Bureau of Mines and Geology & Seismological Laboratory, University of Nevada, Reno, Nevada, USA

Peter Clarke School of Civil Engineering and Geosciences, University of Newcastle, Newcastle upon Tyne, UK

We develop a spectral inversion method for mass redistribution on the Earth’s surface given geodetic measurements of the solid Earth’s geometrical shape, using the elastic load Love numbers. First, spectral coefficients are geodetically estimated to some degree. Spatial inversion then finds the continental surface mass distribution that would force geographic variations in relative sea level such that it is self-consistent with an equipotential top surface and the deformed ocean bottom surface, and such that the total (ocean plus continental mass) load has the same estimated spectral coefficients. Applying this theory, we calculate the contribution of seasonal inter- hemispheric (degree-1) mass transfer to variation in global mean sea level and non- steric static ocean topography, using published GPS results for seasonal degree-1 surface loading from the global IGS network. Our inversion yields ocean-continent mass exchange with annual amplitude (2.92 ± 0.14)×1015 kg and maximum ocean mass on 25 ± 3 August. After correction for the annual variation in global mean vertical deformation of the ocean floor (0.4 mm amplitude), we find geocentric sea level has an amplitude of 7.6 ± 0.4 mm, consistent with TOPEX-Poseidon results (minus steric effects). The seasonal variation in sea level at a point strongly depends on location ranging from 3−19 mm, the largest being around Antarctica in mid-August. Seasonal gradients in static topography have amplitudes of up to 10 mm over 5,000 km, which may be misinterpreted as dynamic topography. Peak continental loads occur at high latitudes in late winter at the water-equivalent level of 100–200 mm.

1. Introduction This development is motivated by recent advances in monitoring the changing shape of the Earth. For exam- We develop a methodology to invert for the redistri- ple, consider the global polyhedron formed by the cur- bution of fluids on the Earth’s surface given precise rent ~250 Global Positioning System (GPS) stations of global geodetic measurements of Earth’s geometrical the International GPS Service. A time series of esti- shape. Specifically we develop (1) inversion of geo- mated polyhedra can be estimated every week [Davies detic station coordinates for a spherical harmonic repre- and Blewitt, 2000], which can then be converted into sentation of Earth’s shape; (2) inversion of Earth’s low-degree spherical harmonic coefficients describing shape for surface mass distribution; (3) inversion for a the shape of the Earth as a function of time [Blewitt et specific surface mass distribution consistent with static al., 2001]. Earlier work by Plag et al. [1996] suggested equilibrium theory on the ocean’s passive response to the possibility of using space geodetic measurements mass redistribution on land. Finally, we demonstrate together with a “known” surface mass distribution to the developed methodology in the simplest possible then invert for Earth’s mechanical properties. However, case, using a published empirical seasonal model of de- more recent evidence suggests that uncertainties in the gree-1 coefficients, and we assess the feasibility of surface load distribution are generally far more signifi- higher resolution inversion. Figure 1 presents an over- cant than differences between Earth mechanical models view of the overall scheme, where here the focus is on [van Dam et al., 1997, van Dam et al., 2001, Tamisiea inversion for the surface load using station positions. et al., 2002]. Hence the focus of this paper is the oppo-

BLEWITT AND CLARKE: EARTH’S SHAPE WEIGHS SEA LEVEL 2 site problem, that is, the estimation of consistent load D(Ω ) = E(Ω ) ˆ + N(Ω )ϕˆ + H (Ω )rˆ (1) distributions, given known mechanical properties of the

Earth. Using the theory developed here, we can in effect use geodetic measurements of land deformation to where (ˆ,ϕˆ ,rˆ) are unit vectors forming a right-handed “weigh” separately the loading of continental water and topocentric coordinate basis pointing respectively in the passive response of sea level. The method also pro- directions east, north, and up. It can be shown [Grafar- duces estimates of variations in the geoid height and the end, 1986] that the lateral component of the displace- ocean’s static topography, which have application to the ment function can also be decomposed into poloidal (or interpretation of satellite altimeter data in terms of the “spheroidal”) and toroidal components, so we can write ocean’s dynamic topography. Expressions are also ob- tained to estimate change in global mean sea level and D(Ω ) = ∇Ψ (Ω) + ∇ × (Γ (Ω )rˆ)+ H (Ω)rˆ (2) the mass of water exchanged between the oceans and the continents. where Ψ (Ω ) is the scalar poloidal surface function, We apply the following development to elastic load- Γ Ω ing theory and equilibrium tidal theory for the investiga- ( ) is the scalar toroidal surface function, and the tion of mass redistribution over timescales of weeks to surface gradient operator is defined by decades. At shorter time periods, ocean currents and ¡ ˆ tidal friction play a dominant role; at longer time peri- ∇ = (1 cosϕ)∂ λ + ϕˆ ∂ϕ (3) ods, mantle viscosity effects start to dominate. As elastic loading is the only deformation process considered Invoking the Love-Shida hypothesis [Love, 1909] that here, it is implicitly assumed that geodetic coordinates no toroidal displacements are forced by surface-normal have been calibrated for other types of solid Earth de- Γ Ω = formations of such as luni-solar tidal deformation, secu- loading, so ( ) 0 , we can express the east and lar motion due to plate tectonics and post-glacial re- north functions in terms of the poloidal component: bound, and transient motions due to the earthquake cycle. It is also extremely important to calibrate for GPS E(Ω) = (1 cosϕ)∂ λΨ (Ω ) station configuration changes, especially antennas. (4) N(Ω ) = ∂ϕΨ (Ω)

2. Spectral Inversion for Earth’s Let us model the height and poloidal functions as Changing Shape spherical harmonic expansions truncated to degree n : Consider a network of geodetic stations located at geographical positions Ω (latitude ϕ longitude λ n n {C,S} i i , i Ω = Φ Φ Ω H ( ) ƒƒ ƒ H nmYnm ( ) for i = 1,2, s ) that provide a time series of station co- = = Φ n 1 m 0 (5) ordinate displacements (e n u ) , corresponding to n n {C,S} i i i Ψ Ω = Ψ Φ Φ Ω ( ) ƒƒ ƒ nmYnm ( ) local east, north, and up in a global terrestrial reference n=1 m=0 Φ frame [Davies and Blewitt, 2000]. As a first step, we develop a purely kinematic model of these vector dis- where our spherical harmonics convention is given in placements as a spectral expansion of appropriate basis Appendix A and truncation issues are discussed later. functions over the Earth’s surface, independent of spe- We define the summation over Φ ∈{C, S} where C cific loading models. This leads to expressions for least squares estimates of the empirical spectral parameters. and S identify sine and cosine components of the expan- This step is therefore independent of the dynamics re- sion. From equations (4) and (5), the east and north sponsible for the deformation (except for consideration functions are therefore modeled by [Grafarend, p.340, of appropriate spatial resolution). 1986]

n n {C ,S} Φ Φ ∂ Y (Ω ) 2.1. Kinematic Spectral Displacement Model Ω = Ψ λ nm E( ) ƒƒ ƒ nm = = Φ cosϕ Consider a vector surface displacement function on a n 1 m 0 (6) sphere decomposed into lateral and height components: n n {C,S} Ω = Ψ Φ ∂ Φ Ω N( ) ƒƒ ƒ nm ϕ Ynm ( ) n=1 m=0 Φ BLEWITT AND CLARKE: EARTH’S SHAPE WEIGHS SEA LEVEL 3

2.2. Degree-0 Considerations ′ Ω = Ω − ∆

D1 ( ) D1 ( ) r ¤ ¤ (9) = = ′ + [( ′ − ′ ) ] The summation for the height function begins at n 1 rˆ H1 1 .rˆ C C 1 because a non-zero H 00Y00 implies an average change in the Earth’s radius, which is theoretically for- where

bidden in a spherically symmetric Earth model where

¤ ¤ the total surface mass is constant. The summation for ′ = − ∆r 1 1 (10) the poloidal function begins at n = 1 simply because ′ = − ∆ H1 H1 r C = Y00 1 and therefore its surface gradient is zero. Hence the degree-0 coefficient is arbitrary as it cannot Clearly the degree-1 vector displacement function de- affect the lateral velocity field. We choose to fix the pends on the arbitrary choice of reference frame origin. gauge of the scalar field Ψ (Ω ) by setting Ψ C = 0 . Equation (10) implies that there exist special refer- 00 ence frames in which either the horizontal displace-

ments or the vertical displacements are zero for degree- 2.3. Degree-1 Considerations 1 deformation [Blewitt et al., 2001]. These are the CL The degree-1 component of displacement is a subtle (center of lateral figure) and CH (center of height fig- problem [Farrell, 1972; Blewitt, 2003] and should not ure) frames, respectively [Blewitt, 2003]. That the verti- mistakenly be represented as a pure translation. Fur- cal degree-1 displacements can be made zero by a sim- thermore, we must carefully consider degree-1 deforma- ple translation proves that the model Earth retains a tion because there are only three (not six) independent shape of constant radius under a degree-1 deformation,

components of the displacement field. Let us start by and hence retains a perfect spherical shape. However, if ¥ writing the degree-1 component of the vector displace- ′ = ′ = ¥ − ≠ H 0 , in general 1 1 H1 0 ; so the surface ment as a function of the six degree-1 parameters from 1 is strained, as detected by GPS [Blewitt et al., 2001] and equations (1), (5), and (6): by very long baseline interferometry [Lavallée and

Φ Blewitt, 2002].

≈ Φ ∂ λY (Ω ) ’ In GPS geodesy, station coordinate time series are in 1 {C ,S}∆ ˆΨ 1m ÷ Ω = 1m ϕ practice often published in an effective center of figure D1 ( ) ƒ ƒ∆ cos ÷ m=0 Φ ∆ Φ Φ Φ Φ ÷ (CF) frame [Dong et al., 1997], defined by no net trans- + ϕˆ Ψ ∂ Ω + ˆ Ω « 1m ϕ Y1m ( ) rH1mY1m ( )◊ lation of the Earth’s surface, and realized through a

rigid-body transformation at every epoch. In the CF = ˆ(−Ψ C sin λ +Ψ S cos λ) 11 11 frame, Blewitt et al., [Fig. 2, 2001] show maximum sea- + ϕ(−Ψ C ϕ λ −Ψ S ϕ λ +Ψ C ϕ ) ˆ 11 sin cos 11 sin sin 10 cos sonal variations in the degree-1 height function of 3 mm. Integrating equation (8) over the sphere, the no net + rˆ(H C cosϕ cos λ + H S cosϕ sin λ + H C sinϕ ) 11 11 10 translation condition is satisfied when (7) ¥ = − 1 H (11) This is equivalent to the vector formula: 1 2 1

¢ ¢ ¡ ¡ which can be derived by substituting the identity D (Ω ) = ˆ( .ˆ) + ϕˆ ( .ϕˆ ) + rˆ(H .rˆ) C S C

1 1 1 1 (8) rˆ = (Y ,Y ,Y ) and using equation (A5). Hence ¡ ¡ 11 11 10 = + ˆ[( − ) ˆ] 1 r H1 1 .r degree-1 is a special case in that there are only 3 (not 6) free parameters. Combining equations (1), (7), and (11), where we define the spatially constant vectors the degree-1 east, north and height surface functions can £ = Ψ C Ψ S Ψ C = C S C be written 1 ( 11 , 11 , 10 ) and H1 (H11 , H11 , H 10 ) . The same degree-one deformation, as observed in a reference frame that has an origin displaced by vector ≈ E (Ω) ’ ≈− 1 0 0’ ≈ H C ’ ∆ 1 ÷ ∆ 2 ÷ ∆ 11 ÷ ∆r = (∆x, ∆y, ∆z) with respect to the original frame, N (Ω) = 0 − 1 0 G(Ω ) H S (12) ∆ 1 ÷ ∆ 2 ÷ ∆ 11 ÷ will have surface displacements (denoted with primes) H (Ω) ∆ 0 0 1÷ ∆ C ÷ « 1 ◊ « ◊ « H10 ◊

BLEWITT AND CLARKE: EARTH’S SHAPE WEIGHS SEA LEVEL 4 where we define C S ≈ ∂λY (Ω) ∂λY (Ω)’ ∆ 0 3 0 20 1 3 nn 1 ÷ ∆ cosϕ cosϕ ÷ ≈ − sin λ cosλ 0 ’ 1 1 ∆ ÷ ∆ ≈− 1 0 0’ ÷ G(Ω ) = − sinϕ cosλ − sinϕ sin λ cosϕ (13) ∆ ∆ 2 ÷ ÷ ∆ ÷ − 1 Ω 3 ∂ C Ω 3∂ S Ω ϕ λ ϕ λ ϕ ∆ ∆ 0 2 0÷G( 1) 0 0 ϕY20( 1) ϕYnn( 1)÷ « cos cos cos sin sin ◊ ∆ ÷ ∆ « 0 0 1◊ ÷ ∆ ÷ YC(Ω)3YS (Ω) 0 3 0 which can be identified as the geocentric to topocentric ∆ 20 1 nn 1 ÷ coordinate rotation matrix. So we can write the degree- A=∆ 4 4 4 ÷ C S 1 geocentric coordinate displacements in the CF frame ∆ ∂λY (Ω ) ∂λY (Ω )÷ ∆ 0 3 0 20 s 3 nn s ÷ as the matrix equation: cosϕ cosϕ ∆ s s ÷ ∆ ≈− 1 0 0’ ÷ Ω = T Ω [− 1 − 1 + ] Ω ∆ 2 ÷ D1 ( ) G ( )diag 2 , 2 , 1 G( )H (14) ∆ − 1 Ω 3 ∂ C Ω 3∂ S Ω ÷ ∆ 0 2 0÷G( s) 0 0 ϕY20( s) ϕYnn( s) ∆ ∆ ÷ ÷ ∆ « 0 0 1◊ ÷ Equation (14) applies quite generally to non-toroidal ∆ C Ω 3 S Ω 3 ÷ deformations. For example, it would not apply to plate « Y20( s) Ynn( s) 0 0 ◊ tectonics, but it would apply to loading models that sat- (18) isfy the Love-Shida hypothesis. Our approach cleanly separates the kinematic spectral inversion problem from and v is column matrix of station coordinate residuals. static/dynamic problems, which require assumptions on Let us assume the stochastic model the spatial and temporal variation of Earth’s mechanical properties. E(vvT ) = C (19)

2.4. Inversion Model where E is the expectation operator, and C is the covari- The observation equations for a set of estimated sta- ance matrix associated with the estimated station coor- tion displacement coordinates b = (e n u ) for sta- dinate displacements. The weighted least-squares solu- i i i i tion for the spectral coefficients is = 3 Ω tions i 1, , s at geographic locations i at a given epoch are given by the matrix equation: = T −1 xˆ Cx A C b − (20) = + = ( T −1 ) 1 b Ax v (15) Cx A C A where b is a column matrix of the observed displace- where C is the formal covariance matrix of the esti- ments for the entire network x mated spectral coefficients. Given the estimated spec- Φ Φ T tral coefficients Ψˆ , Hˆ , and matrix C , it is then b = (b b 3 b ) (16) nm nm x 1 2 s straightforward to construct the estimated surface vector displacements, and propagate the errors to compute the x is a column matrix of unknown parameters, which are formal uncertainty in modeled displacement at any loca- the spectral coefficients of the height and poloidal sur- tion on the sphere. face functions

2.5. Truncation Considerations T x = (H C H S H C H C 3H S Ψ C 3Ψ S ) (17) 11 11 10 20 nn 20 nn For a truncation at degree n , there are n(n + 2) spec- Φ Φ Ψ tral coefficients H nm . It is therefore technically possi- Note that terms 1m are not included as free parameters ble to estimate coefficients up to degree n with an ab- due to the no net translation constraint, equation (11). = ( + ) A is the matrix of partial derivatives, which has the fol- solute minimum of s n n 2 stations (for a non- lowing block structure degenerate network configuration). So in principle a 100-station well-distributed network can be used to estimate spectral coefficients up to degree-9. BLEWITT AND CLARKE: EARTH’S SHAPE WEIGHS SEA LEVEL 5

In practice, truncation well below this technical limit pheric mass distribution tends to produce an opposite can have its advantages as an effective spatial filter of change in oceanic mass distribution, thus over the errors in the station heights. Truncation even to a very oceans, geographic variations in T(Ω ) are not strongly low degree might be justified on the grounds that sensitive to the atmospheric component (although spherical harmonic functions are orthogonal over the T(Ω ) would include the variation in atmospheric pres- entire Earth’s surface, and so are effectively orthogonal for a well-distributed network. However, some level of sure averaged over the oceans). On the other hand, if bias will be present for actual network configurations, the application requires that estimates of T(Ω ) be which could be assessed by covariance analysis. In the geometrically interpreted as sea level, then such sea case of low-degree truncation, high degree signal com- level predictions would be strongly sensitive to atmos- ponents would tend to be absorbed into the station dis- pheric pressure (except for global mean sea level, which placement residuals. Conversely, in the case of high- is unaffected). Van Dam et al. [1997] show how to cal- degree truncation, the higher degree estimates would culate the effect of atmospheric pressure on sea level. tend to absorb some of the station position errors. It In this paper, we simplify this aspect of the discussion may be effective to estimate as many degrees as techni- by assuming that the application does not require us to cally possible, but only interpret the results up to a de- consider the atmospheric component of T(Ω ), and so gree less than n ; such considerations would need to be we loosely refer to T(Ω ) over the ocean as sea level, investigated. In principle it should be possible to assess the preci- and over the land as continental water. This does not sion of the spectral coefficients by comparing independ- invalidate the equations, assuming that the variations in ent estimates of the load inversion (described below) the total mass of the atmosphere (due to water vapor) Φ Φ are negligible [Trenberth, 1981]. that use only H or Ψ , and use this to guide the nm nm selection of truncation degree n , or to improve the sto- 3.2. Static Spectral Displacement Model chastic model. The selection of n is not discussed fur- Let us consider the total mass distribution function ther here, except to note that a standard F-test might be as a spherical harmonic expansion: used to assess whether adding an extra degree to the model produces a statistically significant decrease in the ∞ n {C,S} residual variance. Ω = Φ Φ Ω T( ) ƒƒ ƒTnmYnm ( ) (21) n=1 m=0 Φ 3. Spectral Inversion for Mass Distribution 3.1. Surface Mass Redistribution The summation begins at degree n=1 because we assume that, although the surface load can be redistrib- Consider a spherical solid Earth of radius a, plus sur- uted, its total mass is conserved. face mass that is free to redistribute in a thin surface Such a surface load changes the gravitational poten- layer (<

Table 1 shows the geoid height load Love number ∞ l′ ∇Ψ (Ω ) = n ∇[V (Ω )] (1+ k′ ) along with other Love numbers and useful ƒ T n n n=1 g combinations to degree 12, derived from Farrell [1972]. ∞ l′ Ψ (Ω ) = n [ (Ω )] +Ψ For example, the coefficient in equation (24) is shown ƒ VT n 0 = g in Table 1 as ratio “geoid height: load thickness.” n 1 (26) In this paper, we have used Love numbers from Far- ∞ n {C,S} 3ρ = l′ S T Φ Y Φ (Ω ) rell [1972], which refer to a symmetric, no-rotating ƒƒ ƒ n ( + )ρ nm nm elastic, isotropic (SNREI) Earth model based on the n=1 m=0 Φ 2n 1 E Gutenberg-Bullen A model. Farrell's Love numbers are 3ρ Ψ Φ = l′ S T Φ widely used in the geodetic community, for example, in nm n (2n +1)ρ nm elastic Green’s function models of hydrological and at- E mospheric loading [van Dam et al., 2001]. The pre- ′ liminary reference Earth model PREM [Dziewonski and where ln are the lateral load Love numbers, and where Anderson, 1981] yields low-degree load Love numbers Ψ = (as before) we set the integration constant 0 0 almost identical to Farrell’s [Lambeck, 1988; Grafarend without affecting the actual displacements in equation et al., 1997]. More general classes of Earth model have (2). been discussed by Plag et al. [1996]. Mitrovica et al. Table 1 gives the ratio “surface height: load thick- [1994] and Blewitt [2003] have further discussions on ness” up to degree 12. It also gives the ratio “surface model and reference frame considerations. For the in- lateral: surface height,” which is the ratio of lateral to version of seasonal loading, mechanical model consid- height load Love numbers. For the special case of de- erations are relatively minor compared to limitations of gree-1 in the CF frame, equation (11) implies the load model and its spatial resolution. ′ ′ = − for all spherically symmetric models that l1 h1 1 2 BLEWITT AND CLARKE: EARTH’S SHAPE WEIGHS SEA LEVEL 7 obey the Love-Shida hypothesis. With the exception of degree-1, the relative contribution of horizontal data = T −1 yˆ Cy B Cx xˆ suffers from the low surface lateral to surface height (31) − −1 displacement ratio, which for degrees 2 to 9 ranges from C = (B TC 1B) 0.02 to 0.07 (peaking at degree 3). However the simul- y x taneous inversion of poloidal data can be justified because the ratio of horizontal to vertical variance is typi- where Cy is the formal covariance matrix of the esti- cally small for globally referenced coordinates (~0.1) mated mass distribution coefficients. [Davies and Blewitt, 2000], thus the relative information The two-step estimation process prescribed by equa- content of horizontal displacements can be significant, tions (20) and (31) would in principle give an identical despite the low-magnitude Love numbers [Mitrovica et solution as a one-step process. It is convenient to break al., 1994]. the solution into two steps because, as previously discussed, the first step is independent of load Love num- 3.3. Inversion Model bers derived from specific models. More specifically, it From the above development, the spectral coefficients is useful to assess from the first step how well the trun- for the observed surface displacement functions can be cated spectral model fits the original station displace- modeled ment data, and to reject outliers or improve the stochastic model using formal statistical procedures that do not rely on the validity of specific loading models. The xˆ = By + v (27) x second step might be applied separately multiple times using different sets of load Love numbers from various where xˆ , given by equation (20), contains estimated models, or to assess, for example, how consistent are spectral coefficients of displacement according to the the poloidal and height spectral coefficients. In princi- structure of equation (17); y is a column matrix of un- ple the mass distribution spectral coefficients can be known parameters, which are the spectral coefficients of estimated using only height spectral coefficients or only the mass distribution function poloidal coefficients, thus providing an alternative as- sessment of the stochastic model. T y = (T C T S T C T C 3T S ) (28) 11 11 10 20 nn 3.4. Ocean-Land Mass Exchange

and Global Mean Sea Level B is the matrix of partial derivatives which has a diagonal block structure One useful result that can be derived immediately is the total mass exchanged between the oceans and the ′ 3 continents. The effect of ocean-land mass exchange on ≈h1I3 0 0 ’ ∆ ′ ÷ change in global mean sea level is: ∆ 0 h2I 5 ÷ 4 6 3ρ ∆ ÷ Ω Ω B = S ∆ 0 h′I ÷ (29) ——T ( )d ( + )ρ n 2n +1 2n 1 E ∆ ′ ÷ = ocean 0 l2I 5 S ∆ 4 6 ÷ dΩ ∆ ÷ —— (32) ′ ocean « 0 ln I 2n +1 ◊ ∞ n {C,S} CΦ T Φ = nm nm ƒƒ ƒ C Π 2 and vx is column matrix of displacement coefficient re- n=1 m=0 Φ C00 nm siduals. Let us assume the stochastic model Φ where Cnm are spectral coefficients of the ocean T = function C(Ω ), which takes the value 1 over the E(v x v x ) Cx (30) oceans and 0 on land [Munk and MacDonald, 1960], and the normalization coefficient Π is where Cx is given by equation (20). The weighted nm least-squares solution for the mass distribution spectral defined in Appendix A. The ocean function coeffi- coefficients is BLEWITT AND CLARKE: EARTH’S SHAPE WEIGHS SEA LEVEL 8 cients can be computed to some finite degree using where estimates for the mass distribution coefficients coastline data [Balmino et al., 1973] according to are given by equation (31). Ideally, if we knew all the values of the infinite number of total mass distribution Φ 2 Ω Φ Π Φ coefficients Tnm then sea level S( ) would be given C = nm Y dΩ (33) nm 4π —— nm straightforwardly by using the ocean function: ocean S(Ω ) = C(Ω )T (Ω ) Table 2 shows un-normalized ocean function coeffi- ∞ n {C,S} (36) cients up to degree 4, consistent with our convention in = Ω Φ Φ Ω Appendix A. Chao and O’Connor [1988] urge us to be C( )ƒƒ ƒTnmYnm ( ) = = Φ careful due to pervasive “errors which arose from nor- n 1 m 0 malization conventions” in the literature. The coeffi- Ω cients from Balmino et al. [1973] (later reproduced by Similarly, the continental mass distribution L( ) Lambeck [1980]) were chosen because two independent would be given by analysis groups using different conventions corrobo- rated the values. Published coefficients by Munk and L(Ω ) = [1− C(Ω )]T(Ω ) MacDonald [1960] have been questioned by Balmino et ∞ n {C,S} (37) = [ − Ω ] Φ Φ Ω al. [1973], and appear to us to contain errors associated 1 C( ) ƒƒ ƒTnmYnm ( ) with inconsistent application of normalization conven- n=1 m=0 Φ tions. The coefficients by Dickman [1989], while they so that may derive from more accurate satellite data, appear to T(Ω ) = S(Ω ) + L(Ω ) (38) be fundamentally inconsistent with the ocean function’s idempotent nature, and higher order coefficients appear (where L(Ω ) refers to the equivalent height of a col- to be systematically too small (by orders of magnitude), which again suggests normalization problems. umn of seawater on land). Using equations (32) and (33) the mass taken from A problem with the above approach is that the coef- ˆ Φ the ocean and deposited on land is ficients Tnm are in practice only given to a finite degree n so the real discontinuity of mass distribution at the ∆ = −ρ 2 Ω coastlines is poorly represented by a smooth function. A M S Sa ——d ocean more serious objection is that a model of sea level { } Φ Φ (34) formed by truncating equation (36) will not even ap- ∞ n C,S C T = −4πa 2 ρ nm nm proximately conform to an equipotential surface, and so S ƒƒ ƒ Π 2 n=1 m=0 Φ nm is physically unreasonable. Next we shall address this by incorporating the physics of how the oceans respond Global mean sea level is only sensitive to low degree passively to changes in the Earth’s surface potential mass redistribution (because it involves the square of from mass redistribution. spectral coefficients that obey a naturally decreasing power law) and so truncation of equations (32) and (34) 4.2. Relative Sea Level in Static Equilibrium is a relatively minor issue. In hydrostatic equilibrium, the upper surface of the ocean adopts an equipotential surface. Equilibrium 4. Spatial Inversion for Mass Distribution tidal theory neglects ocean currents associated with any 4.1. Model-Free Inversion change in sea level, and so is presumed to be valid for periods longer than the time-constant associated with Without any a priori knowledge of the dynamics of tidal friction, [Proudman, 1960; Munk and MacDonald, mass redistribution, a naïve model-free solution to the 1960] and therefore should be applicable to fortnightly spatial distribution of mass is given by equation (21) periods and beyond [Agnew and Farrell, 1978]. For our intended application of the theory we shall now de- n n {C,S} Φ Φ velop, we are mainly concerned with the forcing of sea Tˆ(Ω ) = Tˆ Y (Ω ) (35) ƒƒ ƒ nm nm level by the seasonal to decadal-scale loading due to n=1 m=0 Φ continental water storage, and so we assume the static equilibrium model. BLEWITT AND CLARKE: EARTH’S SHAPE WEIGHS SEA LEVEL 9

Let us define “relative sea level” as the column where we define the quasi-spectral sea-level function height of the ocean as measured from the solid Earth surface at the ocean’s bottom to the ocean’s upper sur- n n {C,S}3(1+ k′ − h′ )ρ ∆ ~(Ω) = n n S ˆΦ Φ Ω + V face. The change in relative sea level from an initial S ƒƒ ƒ TnmYnm ( ) (42) = = Φ (2n +1)ρ g equilibrium state to a new equilibrium state can be ex- n 1 m 0 E pressed, to within a constant, as the change in height of the geoid relative to the underlying model of the Earth’s “Quasi-spectral” implies that the coefficients of this surface. In its most general form [Dahlen, 1976], rela- function apply over the oceanic spatial domain rather tive sea level S(Ω ) in static equilibrium is the follow- than the global spatial domain, and it indicates the coef- Ω ficients are intended for spatial interpretation through ing function of geographical position equation (41); expressions for the global spectral coefficients are derived later. The coefficient in equation (42) S(Ω ) = C(Ω )(N(Ω ) − H(Ω ) + ∆V g) (39) is given in Table 1 as “sea level: load thickness” to degree 12. where the geoid height N(Ω ) is given by equation We now evaluate the sea surface geopotential ∆V (24); and C(Ω )H(Ω ) is the height of the deformed subject to the mass exchange condition given by equation (34). Integrating equation (41) over the sphere, we ocean basins, given by equation (25). The constant ∆ find the degree-zero global spectral coefficient is V g is required because, although the sea surface { } Φ Φ will relax to an equipotential surface, it will generally n n C,S 3(1+ k′ − h′ )ρ C Tˆ ∆V have a different potential to its initial potential, depend- Sˆ C = n n S nm nm + C C (43) 00 ƒƒ ƒ ( + )ρ Π 2 00 ing on the total oceanic mass and the irregular ocean- n=1 m=0 Φ 2n 1 E nm g land distribution [Dahlen, 1976]. Therefore ∆V represents the geopotential of the sea surface relative to the Now from our previous discussion on mass exchange, geoid’s potential, and ∆V g represents the height of we must satisfy the condition the sea surface above the deformed geoid. { } Φ Φ Inserting equations (24) and (25) into (39) gives us n n C,S C Tˆ Sˆ C = Tˆ(Ω )dΩ = nm nm = −LˆC (44) “the sea-level equation” appropriate to our problem: 00 —— ƒƒ ƒ 2 00 n=1 m=0 Φ Π ocean nm

» ∞ n {C,S}3(1+k′ −h′ )ρ ∆ ÿ (Ω) = (Ω) n n S Φ Φ Ω + V and so equating (43) and (44), S C …ƒƒƒ TnmYnm( ) Ÿ ( + )ρ n=1 m=0 Φ 2n 1 E g ⁄ { } Φ Φ (40) ∆V n n C,S » 3(1+ k′ − h′ )ρ ÿ C Tˆ = 1− n n S nm nm ƒƒ ƒ … ( + )ρ Ÿ C 2 g n=1 m=0 Φ 2n 1 C Π (45) Φ E ⁄ 00 nm The total load coefficients T themselves include a ~ nm = S C contribution from sea level and so traditionally equa- 00 tions similar to (40) have been either inverted [Dahlen, 1976] or solved iteratively [Wahr, 1982; Mitrovica et where the first term in the square bracket is related to al., 1994], given an applied non-oceanic load (or tidal change in global mean sea level from mass exchange, potential). and the smaller second term (previously derived by Indeed such a calculation would be appropriate if we Dahlen [1976]) is the contribution of average geoid and were to compute how the ocean’s passive response sea floor deformation, which is generally non-zero even modifies the effect of a given land load on the Earth’s in the absence of mass exchange. When using equation shape (something that is generally not accounted for in (41) for interpreting sea level at any point, it is impor- current loading models). However in our application tant to use the spatial rather than spectral representation ˆΦ of the ocean function, because Sˆ(Ω ) must go discon- we have direct access to estimates Tnm , and so have no need to solve the sea level equation in the same sense. tinuously to zero as we cross a coastline toward land. Our estimate for sea level at a given location can therefore be written 4.3. Geocentric Sea Level and Static Topography ~ Sˆ(Ω ) = C(Ω )S (Ω ) (41) We now digress to discuss sea level as inferred by satellite altimetry. Satellite altimetry of the ocean sur- BLEWITT AND CLARKE: EARTH’S SHAPE WEIGHS SEA LEVEL 10 face is inherently sensitive to geocentric sea level rather where explicitly shown is the intuitive notion that mean than relative sea level. Perhaps misleadingly, geocen- geocentric sea level O is the sum of mean relative sea tric sea level has been referred to as “absolute sea level S , given by equation (32), and the mean height level,” when in fact, it is dependent on the choice of reference frame origin; ironically, relative sea level is change of the ocean floor H , where H is given by absolute in this sense, as it is frame-independent. To be equation (25). It can be seen from Table 1 (surface truly self-consistent, the coordinates of the tracking sta- height: load thickness ratio) that geocentric and relative tions used to determine the geocentric satellite position mean sea level only differ by a few percent (at most should be allowed to move with the loading deforma- 11.2% for pure degree 2), which is not surprising since tion, for example using the same epoch coordinates as the sea floor is deformed an order of magnitude less those used to estimate the total load. The ocean’s dy- than the change in sea level. Therefore the cautionary namic topography used to investigate ocean currents can remark above regarding the treatment of station loca- be derived if we know the height of the sea surface tions for satellite altimetry is only important at this above the ocean’s static topography. For purposes of level. our problem, the ocean’s static topography may be defined as the geocentric height of the ocean surface in 4.4. Linearized Sea Level Equation hydrostatic equilibrium if there were no ocean currents Before we solve for the land load, it will be neces- or steric effects. Contributing factors therefore include sary to find the global spectral coefficients of equation the deformation of the geoid and the difference in po- (41). This is the most computationally difficult part of tential between the deformed geoid and the new sea sur- the calculation. We make use of the fact that, as for any face. We formally define the ocean’s static topography function on a sphere, the product of two spherical har- using this new equipotential surface which is given eve- monics must itself be expressible as a spherical har- rywhere as monic expansion. The product-to-sum conversion formula for spherical harmonics is [Balmino, 1994]: ~ O(Ω ) = N(Ω ) + ∆V g (46) ∞ n {C,S} Φ ′ Φ ′′ Φ ,Φ ′,Φ ′′ Φ Y ′ ′ (Ω )Y ′′ ′′ (Ω ) = A ′ ′ ′′ ′′Y (Ω ) (49) ∆ n m n m ƒƒ ƒ nm,n m ,n m nm where V g can be estimated by equation (45) and the n=0 m=0 Φ geoid height can be estimated by where the coefficients can be expressed by applying n n {C,S} 3ρ equation (A7) ˆ (Ω ) = ( + ′ ) S ˆ Φ Φ Ω N ƒƒ ƒ 1 kn TnmYnm ( ) (47) = = Φ (2n +1)ρ n 1 m 0 E Π 2 Φ ,Φ ′,Φ ′′ = nm Φ Ω Φ ′ Ω Φ ′′ Ω Ω Anm,n′m′,n′′m′′ ——Ynm ( )Yn′m′ ( )Yn′′m′′ ( )d (50) Using equation (46) it is straightforward to calculate the 4π mean change in sea level in a geocentric frame (where the degree-1 load Love numbers must be consistent with The integral of triple complex spherical harmonics the type of geocentric frame used to derive the altimeter arises in angular momentum theory of quantum me- coordinates [Blewitt, 2003]), chanics [Wigner, 1959], and can be solved using Clebsch-Gordan coefficients or Wigner 3-j coefficients ~ O(Ω )dΩ [Dahlen, 1976; Balmino, 1994]. Appendix B shows our —— method for the case of classical spherical harmonics. ocean O = Once calculated, these integrals can be used to deter- Ω ——d mine the global spectral coefficients for relative sea ocean level, given the quasi-spectral coefficients. The ocean Φ ~Φ ∞ n {C,S}C O function product-to-sum conversion can be derived us- = nm nm (48) ƒƒ ƒ C 2 ing equation (49) = = Φ C Π n 1 m 0 00 nm Ω ~ Ω = Φ′′ ~Φ ′ Φ′ Ω Φ ′′ Ω C( )S ( ) ƒ ƒCn′′m′′ Sn′m′Yn′m′ ( )Yn′′m′′ ( ) ∞ n {C,S} Φ Φ » 3h′ ρ ÿ C Tˆ n′m′Φ ′ n′′m′′Φ′′ = + n S nm nm …1 Ÿ ′′ ′ ′ ′′ ƒƒ ƒ ( + )ρ C Π 2 = Φ ~Φ Φ ,Φ ,Φ Φ Ω n=1 m=0 Φ 2n 1 E ⁄ C00 nm ƒ ƒ ƒCn′′m′′ Sn′m′ Anm,n′m′,n′′m′′Ynm ( ) ′ ′ ′ ′′ ′′ ′′ = + nmΦ n m Φ n m Φ S H (51) BLEWITT AND CLARKE: EARTH’S SHAPE WEIGHS SEA LEVEL 11 or alternatively in coefficient form as a matrix equation spectral coefficients, which can be used to infer the con- [e.g., equation (14) of Dickman, 1989] tinental load in the spatial domain. Formally, spectral coefficients for the land load up to degree n can be estimated as ~ Φ Φ Φ ′ ~Φ ′ [ Ω Ω ] = Χ , (52) C( )S ( ) nm ƒ nm,n′m′ Sn′m′ Φ Φ Φ ′ ′Φ ′ ˆ = ˆ − ˆ n m Lnm Tnm S nm (54) where we define the coefficients of the product-to-sum Φ where ˆ is given by equation (41). For spatial repre- transformation: Snm ∞ n′′ {C,S} sentation, the quasi-spectral land load function Χ Φ ,Φ ′ = Φ ,Φ ′,Φ ′′ Φ ′′ ~ nm,n′m′ ƒ ƒ ƒ Anm,n′m′,n′′m′′Cn′′m′′ (53) L(Ω ) satisfies the following equation n′′=0 m′′=0 Φ ′′ ~ Lˆ(Ω ) = [1− C(Ω )]L(Ω ) (55) That is, multiplication of a function by the ocean function can be achieved by a linear transformation of that such that the spectral coefficients are the same on both function’s coefficients. Appendix B shows explicitly sides of this equation up to degree n . Equating the how equation (52) is actually computed by a recursive spectral coefficients, we find that algorithm, and Appendix C presents results of this calculation up to degree 2 (as a check for readers wishing n n′ {C,S} ˆΦ = (δ Φ ,Φ ′ − Φ ,Φ ′ )~Φ ′ to reproduce our results or apply our method). Table 3 Lnm ƒ ƒ ƒ nm;n′m′ X nm;n′m′ Ln′m′ (56) shows the product-to-sum transformation coefficients n′=0 m′=0 Φ ′ for spectral interactions to degree 2 × degree 2 assuming the ocean function coefficients of Balmino et al. [1973]. Thus equating (54) with (56) relates the quasi-spectral Due to selection rules on indices that lead to non- ~Φ ˆΦ coefficients Lnm to the estimated load coefficients Tnm . zero integrals, the inner-summation over double-primed ~Φ indices involves only a limited number of contributing To find Lnm up to degree n requires inversion of the ~Φ terms. As Dahlen [1976] points out, one result of this is matrix form of equation (56). Once we find L , we that the ocean function is only required to degree 2n for nm an exact computation of its effect on an arbitrary de- can then construct the land load spatially using equation gree-n function. Thus the spherical harmonic formula- (55). tion is especially well suited for the investigation of low-degree phenomena. 4.6. Reconstruction of the Total Load It should be emphasized that truncation implies that at Higher Degrees a sea level function constructed spatially using global Equation (54) guarantees that our model for land and spectral coefficients will generally be non-zero on land. ocean mass redistribution yields spectral coefficients for This is why the quasi-spectral sea level function is rec- Φ the total load that agree with our original estimates Tˆ ommended for spatial representation of sea level. How- nm ever we need the true spectral coefficients to compute for n = 1,2,n . However, note that equations (54) and the contribution of sea level to the observed total de- (56) can be used to predict higher degree coefficients formation. n > n of the total load, because there is no restriction in these equations on the value of n. 4.5. Inversion for Continental Mass Redistribution ~ ~ The principle behind the following procedure is that Tˆ(Ω) = C(Ω)S (Ω) + [1− C(Ω)]L(Ω) spectral coefficients are applicable to the physics of n n′ {C,S} ˆΦ = [ Φ ,Φ ′ ~Φ′ + (δ Φ ,Φ ′ − Φ ,Φ′ )~Φ′ ] loading, but quasi-spectral coefficients characterize the Tnm ƒƒ ƒ X nm;n′m′ Sn′m′ nm;n′m′ X nm;n′m′ Ln′m′ spatial distribution. Using sea level’s spectral coeffi- n′=0 m′=0 Φ ′ cients, we can subtract the sea level’s contribution from (57) the observed total deformation, and thus infer the continental load’s spectral coefficients. Then the continental Whether these predicted values are physically mean- load’s spectral coefficients can be inverted for quasi- ingful depends on the underlying model assumptions that provide the “extra information”. First consider the extra information from the ocean response model. Our BLEWITT AND CLARKE: EARTH’S SHAPE WEIGHS SEA LEVEL 12 model only considers the passive response of the ocean on sea level or continental water. In effect, we are us- to degree n . This may be a reasonable approximation, ing the solid Earth itself as a weighing machine. because the ocean’s response to higher degree loading is The utility of this demonstration toward answering monotonically decreasing, and at some point becomes specific questions is naturally limited by the hemi- insignificant. (This is ultimately a consequence of spheric-scale spatial resolution implied by degree-1 Newton’s law of gravitation). In any case, the situation truncation. Interpretation of loading at a specific loca- is better than the simpler assumption that mass ex- tion would be suspect for continental water; however it change produces a uniform change in sea level [Chao would be more reasonable for the passive ocean re- and O’Connor, 1988]. sponse, due to its higher sensitivity to low-degree mass Secondly, extra information comes by assuming the redistribution. The main goal of this exercise is to show land load is better characterized by equation (55) rather how physical self-consistency governs the inversion than a simple spherical harmonic expansion truncated to procedure, and into the potential of this technique to the same degree. This assumption is reasonable from investigate the global hydrological cycle. By its nature, the point of view that we do know that the land load is the technique provides valuable integral constraints on exactly zero except on land, and there will generally be the total load, which complements studies using special- a physically meaningful discontinuity in loading at the ized models. The resulting estimates of long- coastlines. wavelength static ocean topography (and geoid defor- Thirdly, by truncating the underlying models at n , mation) might indicate to physical oceanographers the we are implicitly assuming that all total load coeffi- level of errors incurred when interpreting satellite al- cients of degree higher than n are entirely driven by timeter data in terms of dynamic ocean topography. spectral leakage from the lower degree spherical harmonic forcing through the geographical distribution of 5.2. GPS Data and Spectral Inversion continents and oceans. Whether or not this assumption for Earth’s Shape is to be considered reasonable depends on a priori We use the published empirical seasonal model of knowledge. For example, one working hypothesis degree-1 deformation derived from a global GPS net- might be that degree-2 seasonal mass redistribution is work [Blewitt et al., 2001]. This in turn was derived dominated by the interaction of degree-1 (i.e., inter- from weekly estimates of the shape of the global net- hemispheric) loading with the geographical distribution work polyhedron over the period 1996.0-2001.0, pro- of the continents. One indicator to support this hypothe- vided by several IGS Analysis Centers using the fidu- sis is that VLBI baselines have annual signals with a cial-free method [Heflin et al., 1992]. These were then geographic pattern of phase expected of a seasonal de- combined by Blewitt et al. [2001] to produce station gree-1 signal, which is out of phase in opposite hemi- coordinate time series according to the method of Da- spheres [Lavallée and Blewitt, 2002]. vies and Blewitt [2000] and Lavallée [2000]. Residual displacement time series were formed for each station 5. Demonstration by removing estimates of the velocity and initial posi- 5.1. Demonstration Goals tion from the coordinate time series, taking care to re- duce velocity bias by simultaneous estimation of annual We now demonstrate the above method on the sim- and semi-annual displacement signals [Blewitt and plest possible case, where we invert published seasonal Lavallée, 2002]. The degree-1 deformation was then degree-1 loading estimates from the GPS global net- estimated every week, parameterized as a load moment work operated by the IGS. Apart from clarifying the vector in the CF frame. The load moment coordinate steps of the calculation, the demonstration provides ad- time series were then fit to an empirical seasonal model, ditional insight into the relationships between the vari- parameterized by annual and semi-annual amplitudes ous physical parameters, and the results provide an ap- and phases. preciation of the relative magnitudes of the various The load moment data need to be converted into physical quantities discussed. Despite the obvious limi- equivalent height function data before we can start to tation in spatial resolution, it also provides quantitative apply our equations. To do this, we note that equation first-order answers to several research questions. In the (14) is consistent with Blewitt et al. [equation (6), following demonstration, it should be kept in mind that 2001], if we identify although the solutions are constrained by physical self- consistency, they are entirely driven by observations of GPS surface deformation, with no a priori information BLEWITT AND CLARKE: EARTH’S SHAPE WEIGHS SEA LEVEL 13

≈ H C ’ ≈ m ’ guish how much of this can be attributed spatially to ∆ 11 ÷ h′ ∆ x ÷ loading on land versus the oceans. S = 1 (58) ∆ H11 ÷ ∆my ÷ What we can calculate at this stage is the total ∆ C ÷ M ⊕ ∆ ÷ « H10 ◊ « mz ◊ amount of mass on land (which equals the negative amount of mass in the oceans), by integrating the total where h′ is the height load Love number (CF frame), load over continental areas; thus we can calculate the 1 contribution of mass exchange to mean relative sea and M ⊕ is the mass of the Earth. Table 4 shows the level. Using equation (32), Table 5 shows our results estimated degree-1 seasonal spectral coefficients for the for mean relative sea level, which has annual amplitude height function, derived by equation (58) from the pub- 8.0 ± 0.4 mm peaking toward the end of August. From lished load moment parameters of Blewitt et al. [2001]. equation (34), our result corresponds to ocean-continent 15 Blewitt et al. [Fig. 2, 2001] show the height function mass exchange of annual amplitude (2.92 ± 0.14) × 10 and the gradient of the poloidal function (through their kg, with continental mass peaking toward the end of equation 11) explicitly at two-monthly intervals using a February. stacking technique. The annual degree-1, order-0 (1,0) component dominates, associated with seasonal inter- hemispheric mass exchange, with maximum heights of 5.4. Inversion for Static Ocean Topography 3 mm observed near the North Pole near the end of Au- The sea surface follows the shape of the deformed gust and near the South Pole near the end of February. geoid, thus geocentric sea level and the deformed geoid We scaled by a factor of 1.9 the one-standard devia- are identical for spectral components of degree 1 and tion formal errors in Blewitt et al. [2001], such that the above. Using equation (47) and tabulated “geoid chi-square per degree of freedom equals one when fit- height: load thickness” coefficients in Table 1, Table 4 ting the weekly degree-one deformation solutions to the shows our results for seasonal variation in the deformed empirical seasonal model [D. Lavallée, personal com- geoid. The degree-0 geoid height is constrained to zero munication]. After this scaling, the errors in spectral due to total mass conservation. The annual (1,0) term amplitudes of height are at the level of 0.15 mm, which dominates with amplitude 11.3 ± 0.5 mm, peaking to- is a few percent of the (1,0) annual amplitude. While ward the end of August. The (1,1) terms are both less this appears to reflect formal error propagation and the than half this amplitude. empirical scatter in the data, it would not reflect inter- The degree-1 height of the geoid is reference-frame annual variability in the seasonal cycle, which subjec- dependent. Here it is reckoned with respect to the ori- tively appears to be at the 1-mm level. Therefore the gin of the center of figure frame (CF). Thus the height computed errors might not accurately represent the er- of the degree-1 geoid is simply a consequence of “geo- rors in the seasonal model, however they should indi- center motion,” defined as the motion of CF with re- cate the precision of the geodetic technique, and so sug- spect to the center of mass of the entire Earth system gest the potential of future analyses to resolve extra em- (CM). In the CM frame, for example, there would be pirical parameters for better characterization of the time no degree-1 geoid height variation, and therefore no series. degree-1 component of geocentric sea level. However satellite altimetry measurements are based on the de- 5.3. Inversion for Total Load and rived coordinates of the satellite altimeter, which typi- Mean Relative Sea Level cally do not incorporate seasonal geocenter variations in Season variation in the total load spectral coeffi- the tracking station coordinates, and therefore relate cients was estimated using equation (31) and is shown more closely to CF than CM. in Table 4. The degree-0 term was set to zero to con- However, degree-1 variation in relative sea level is serve mass. The total load follows the same spatial and not frame dependent, since it relates to the sea surface temporal characteristics as the height function, but is height above the deformed ocean floor rather than above opposite in sign (opposite in phase). Therefore the total a frame origin. Table 4 lists our solution for degree-1 load peaks near the North Pole near the end of February. quasi-spectral coefficients of relative sea level, com- The amplitude for the dominant (1,0) spectral compo- puted as the difference between the heights of the de- nent is equivalent to a 59-mm column of seawater; at a formed geoid height and the ocean floor, equation (42). later stage of the calculation we will be able to distin- Table 4 also shows the degree-0 coefficient for geocentric sea level computed as the geopotential height of sea level with respect to the deformed geoid, equation BLEWITT AND CLARKE: EARTH’S SHAPE WEIGHS SEA LEVEL 14

(45). This is identical to the degree-0 quasi-spectral principle, there are contributions from all degrees to relative sea level, because the global average change in infinity). This then enables calculation of the global height of the solid Earth surface is zero (due to mass spectral coefficients for continental water loading using conservation). The annual variation in geopotential equation (54). Due to mass conservation and as Table 6 height of the sea surface is 6.1 ± 0.3 mm, peaking near shows, the degree zero coefficient for the continental the end of August. load, with annual amplitude 5.6 ± 0.3 mm peaking in This is by far the largest annual volumetric compo- late February, is of the same magnitude but opposite nent of mean relative sea level (Table 5), the other two phase as the degree-zero coefficient of relative sea level. volumetric components being the mean marine geoid The annual amplitudes for degree-1 continental load- height with annual amplitude 1.5 ± 0.1 mm, and the ing are approximately 3−5 times larger for the land load mean height of the deformed ocean bottom with annual than for the ocean load; the (1,0) coefficient dominates amplitude 0.40 ± 0.02 mm (of opposite phase). Added at 49.1 ± 2.1 mm, peaking in late February. This di- together appropriately, mean relative sea level has an- rectly relates to the relative contribution of land loading nual amplitude 8.0 ± 0.4 mm (consistent with our previ- and ocean loading to the total degree-1 load (and hence ous calculation using the total load); mean geocentric geocenter variations). sea level (in the CF frame) has annual amplitude 7.6 ± Equation (56) was inverted using the inverse of the 0.4 mm, equation (48), peaking on 25 August. product-to-sum transformation (Table 3) truncated to Seasonal variations in sea level depend strongly on degree 1. Table 6 shows the estimated quasi-spectral location (Figures 2 and 3), with seasonal peak values coefficients of the land load. Figure 2 also shows the ranging from 3 − 19 mm. The smallest variations occur spatial inversion for land load along with sea level. in the northern mid-latitude Pacific; the annual signal Figure 2 shows that the pattern of spatial variation in can become so small that the semi-annual signal domi- oceanic mass redistribution is similar to that on land, nates, which gives rise to two seasonal peak values as although it is approximately a factor of 10 smaller. The low as 3 mm in April and November. The largest varia- land load is dominated by the (1,0) annual coefficient, at tions occur in polar regions. Since the degree-zero 150.1 ± 6.4 mm, peaking at the end of February. How- quasi-spectral coefficient of relative sea level and the ever, the degree-zero coefficient is almost exactly out of (1,0) coefficient are approximately six months out of phase with this (1,0) coefficient, such that it reduces the phase, the geographic pattern of sea level is highly peak load in the Arctic Circle, but enhances it in the asymmetric between the Arctic and Antarctic (Figure 4). Antarctic. Intuitively, the reason for this out-of-phase In the Arctic, these two terms interfere destructively, behavior is related to the larger amount of land in the such that the seasonal variation in sea level peaks at ap- Northern Hemisphere and the need for the total land proximately 9 mm in late March (including the semi- load to be a relatively small mass. Simply put, when the annual terms). However, in the Antarctic, the two coef- (1,0) term is large, it contributes a lot to mass on land. ficients add constructively, such that the seasonal varia- This is partly balanced by a degree zero term of oppo- tion in sea level peaks at approximately 18 mm in mid- site phase such that the total land load is sufficiently August. The intuitive reason for this is that, in the Arc- small so that the land mass budget balances with mean tic, the gravitational attraction of the large amount of relative sea level. water (retained on land in Northern Hemisphere winter) draws sea level higher, however this is partly cancelled 5.6. Discussion of Results by the global reduction in sea level necessary to provide As previously noted, the interpretation of the above mass balance. However, in the Antarctic, these two ef- results should be limited to spatial resolution of hemi- fects add together constructively. The underlying rea- spheric scale. By comparing results with those from son is that far more water is retained on land in the independent techniques, it would be useful to assess the Northern Hemisphere than Southern Hemisphere in extent to which a degree-1 truncated model can predict their respective winter seasons. large-scale phenomena, such as seasonal variation in 5.5. Inversion for Continental Water Topography global mean sea level. First, consider our result on Global spectral coefficients for relative sea level global mean geocentric sea level with annual amplitude were computed from the quasi-spectral coefficients us- 7.6 ± 0.4 mm with maximum sea level occurring on ing the product-to-sum-transformation, equation (52), 25 August. This is to be compared (Fig. 5) with annual where the transformation is given in Table 3. Results of amplitudes from TOPEX/Poseidon, which (after correc- this calculation up to degree 1 are shown in Table 6 (in tion for steric effects) are in the range 7−10 mm, peak- BLEWITT AND CLARKE: EARTH’S SHAPE WEIGHS SEA LEVEL 15 ing during the period 12−24 September [Chen et al., ocean does not change the annual phase significantly 1998, 2002; Minster et al., 1999]. Assuming the large (~1 day). This is in intuitive accord with the above rule steric corrections (of order 5 mm in amplitude, half the of thumb if we multiply 10% by the ratio of ocean to total effect) are correct, this good agreement between land areas. We deduce that models of solid Earth de- our result and that of altimetric methods suggests that formation on the global scale (typically using Green’s seasonal inter-hemispheric mass transfer is the domi- functions) and models of geocenter displacements, nant driving mechanism for seasonal change in sea based on adding together contributions from the land level. In contrast, hydrological models of global mean and oceans, may be biased at this level unless self- sea level differ by up to a factor of three, thus geodesy consistency is rigorously incorporated. provides useful constraints on global hydrological mod- Our spatial inversion for sea level indicates that at- els [Chen et al., 2002]. tention should be paid to the asymmetry in ocean bot- Our result on ocean-continent mass exchange of an- tom pressure in the Arctic versus Antarctic. The dem- nual amplitude (2.92 ± 0.14) × 1015 kg, with continental onstration showed how this is a consequence of a larger mass peaking toward the end of February, implies a seasonal variation in water retained on land in the peak-to-peak seasonal continental mass variation of (5.8 Northern Hemisphere versus the Southern Hemisphere ± 0.3) × 1015 kg. By comparison, satellite radar altim- (thus the geoid height is of opposite phase to global etry (of the surface and base) of the Northern Hemi- mean sea level in the Arctic, and of similar phase in the sphere snowpack have been used to infer a peak-to-peak Antarctic). snow mass of 3 × 1015 kg [Chao et al, 1987; Chang et The potential of this technique is perhaps best illus- al., 1990] peaking during February-March. Taken to- trated by the spatial inversion’s prediction on long- gether with our result, this constrains the planet’s peak- wavelength seasonal gradients in static ocean topogra- to-peak seasonal ground water mass to be < 3 × 1015 kg phy (both relative and geocentric). For example, our (excluding snow), assuming it has a similar phase. This solution predicts a 10 mm east-west sea surface height is consistent with (but stronger than) the order-of- difference spanning the equatorial Pacific Ocean magnitude analysis of Blewitt et al. [2001], which set an (15,000 km) which behaves like an annual see-saw (Fig. upper bound to winter groundwater at < 7 × 1015 kg. 6). The largest see-saw gradients are observed in the Even stronger upper bounds are possible if we assume equatorial Pacific and Atlantic Ocean in the north-south an out of phase contribution to the total mass exchange direction where, at their seasonal peak, sea level can from seasonal snowpack variations in Antarctica (which vary 20 mm over 10,000 km. We caution that such gra- was not included in the satellite radar results). dients in static topography might be misinterpreted in Note in Fig. 5 the asymmetric shape of our estimate terms of basin-scale dynamics when analyzing satellite of seasonal variation in global mean sea level. The altimeter data. maximum slope is 63% greater than the magnitude of the minimum slope. A similar asymmetry is also seen 6. Prospects for Higher Resolution Inversion in the TOPEX-Poseidon results and in the hydrological 6.1. Reconstruction of Total Load and models. This strongly indicates that water runoff in Height Function to Degree 2 Northern Hemisphere spring [Dai and Trenberth, 2002] takes place at a much faster rate than continental water We speculate on prospects for positive identification accumulation toward the end of the year. and interpretation of seasonal signals at higher degrees. While we have shown results of our inversion on Let us first consider degree-2 loading. Using equation seasonal variation in sea level at specific geographic (57), Table 8 shows the degree-2 coefficients of the total locations, it would be difficult to make a meaningful load, assuming that relative sea level and the continental comparison with other techniques, because of the large water load have no significant power above degree 1. uncertainty in the ocean’s dynamic topography. How- We call this the “degree-one dominance hypothesis,” ever one clear conclusion from our results is that physi- and offer no evidence at this stage that it holds, except cal self-consistency forces the ocean to have seasonal to conjecture that seasonal retention of water on land static topography at the level of 10 mm. As a rule of might be dominated by mass exchange between the thumb, sea level variations in offshore regions appeared Northern and Southern Hemispheres. While this hy- to be approximately 10% of the water-equivalent height pothesis has more general implications, our purpose of the adjacent continental load. Table 7 shows that the here is to quantify the magnitude of possible degree-2 passive ocean response to degree-1 loading amplifies signals and thus assess the feasibility of detection. The the annual degree-one land load by 21−27%, but that the solution predicts that degree-2 load would be dominated BLEWITT AND CLARKE: EARTH’S SHAPE WEIGHS SEA LEVEL 16

S related for the global IGS network, and how this maps by the T21 amplitude of 11.0 ± 0.9 mm (peaking in late January). into spatial resolution. The results of Table 8 reflect the contribution to degree 2 loading from the interaction of a degree-1 pattern 7. Conclusions of continental water masked by the geographic distribu- We have developed an inversion method for mass tion of the continents. This could lend insight into pos- redistribution on the Earth’s surface given GPS meas- sible mechanisms for polar motion excitation by (2,1) urements of the solid Earth’s varying geometrical shape. coefficients. In particular, we suggest the annual (2,1) The method is based on a load Love number formalism, coefficient as a plausible mechanism to excite the using gravitational self-consistency to infer the relative Chandler wobble, if sufficiently large. contributions of relative sea level and continental water The contribution of degree-1 land loading to degree- loading to the estimated spectral coefficients of the total 2 surface height deformation is largest in the annual co- load responsible for the observed deformation. The C S efficients at H 21 = 0.77 ± 0.07 mm and H 21 = 1.23 method was demonstrated for the simplest possible case ± 0.10 mm, both peaking in the late July to mid-August using published seasonal degree-1 deformations from time frame. Judging by the clear detection of degree-1 the global IGS network [Blewitt et al., 2001]. Inversion height signals of similar magnitude, the annual (2,1) produces results on relative mean sea level in terms of coefficients should be at detectable levels. The magni- the geoid height, the geopotential height of the sea sur- tude of the predicted degree-2 signals is partly due to face (above the geoid), and the height of the deformed the fact that, of all the degrees, degree-2 deformation ocean bottom. The demonstration illustrates how the has the largest surface height to load thickness ratio at relative contribution of the various constituents of sea 11.2% (Table 1), which is more than twice that of de- level is dependent on time of year and geographic loca- gree-1 in the CF frame at 5%. We conclude that degree- tion. 1 land loading alone should be sufficient to create a de- We find the annual amplitude of ocean-continent tectable degree-2 deformation signal. mass exchange is (2.92 ± 0.14)×1015 kg with maximum ocean mass on 25 ± 3 August, corresponding to 6.2. Higher Degrees annual amplitudes for mean relative sea level at Signal detection of higher degrees depends on three 8.0 ± 0.4 mm, and for mean geocentric sea level at things. First, as previously discussed, is the issue of the 7.6 ± 0.4 mm. This result confirms results from number and distribution of GPS stations, which in prin- TOPEX-Poseidon (after steric correction), and places ciple might currently limit detection to around degree-9. strong constraints on physically acceptable hydrological Second, there must be sufficient power in the actual models (which can differ by factors of three). It also loads at higher degrees. Loading models predict sig- confirms that the steric corrections applied to TOPEX- nificant variation in hydrologic loading (at the several- Poseidon data are reasonable as a global average. Tak- mm deformation level) on continental scales [Van Dam ing into account satellite altimeter measurements of the et al., 2001], so this would not appear to be a limiting Northern Hemisphere snowpack (at 3 × 1015 kg, peak- factor, assuming that the goal is inversion in the spatial to-peak), our results strongly constrain global ground domain. water to have a peak-to-peak seasonal variation < 3 × Thirdly, there must be a sufficient deformation re- 1015 kg. sponse of the solid Earth at higher degrees as compared The seasonal variation in sea level at a point strongly measurement errors. As Table 1 shows, beyond degree- depends on location, with typical amplitudes of 10 mm, 2 the surface height to load thickness ratio decreases the largest being ~20 mm around Antarctica in mid- monotonically. By degree-9 the ratio has fallen to 4%. August. Sea level variations are predicted to be smaller This is of similar magnitude to degree-1, which is in the Northern Hemisphere due to the hemispheric clearly detectable. asymmetry in continental area. Sea level is lowered The evidence therefore suggests that spectral inver- everywhere to provide continental water in the Northern sion of loading up to degree-9 and spatial inversion of Hemisphere winter, but at the same time this continental continental-scale loads are feasible. It is recommended water raises sea level in the Northern Hemisphere that covariance analysis be used (in parallel with at- through gravitational attraction. Seasonal gradients in tempts to interpret deformation data) to understand how static topography have amplitudes of up to 10 mm over the estimated spectral coefficients are theoretically cor- 5,000 km, which may be misinterpreted as dynamic topography. Peak continental loads are predicted to occur BLEWITT AND CLARKE: EARTH’S SHAPE WEIGHS SEA LEVEL 17 in polar regions in mid-winter at the water-equivalent where dΩ = d(sinφ)dλ and level of 100-200 mm. An analysis of the potential for this method to esti- 1 » ( − ) ÿ 2 mate the seasonal surface load with higher spatial reso- Π = − δ ( + ) n m ! lution indicates that estimation to degree-9 is feasible. nm …(2 m0 ) 2n 1 Ÿ (A6) (n + m)!⁄ This should allow for spatial resolution of continental- scale hydrology. Finally, we have developed a general scheme (Figure 1), which shows the potential for con- The spherical harmonic coefficients in equation (A1) necting various geodetic data types through models of are therefore the globally-loaded Earth system. For example, it 2 should be possible to jointly invert GPS station position Φ Π Φ f = nm f (Ω )Y dΩ (A7) time series with independent data on seasonal variation nm 4π —— nm in the Earth’s gravity field up to degree and order 4, now available from satellite laser ranging [Nerem et al., It is to be understood that Y S and f S are to be ex- 2000], with potentially much higher spatial resolution n0 n0 predicted from missions like GRACE [Wahr et al., cluded from all equations. 1998]. Appendix B. Product-to-Sum Conversion: Re- Appendix A. Spherical Harmonics Convention cursive Method for Real Spherical Harmonics Recognizing “errors which arose from normalization According to equation (52), the ocean function product- conventions” in the literature [Chao and O’Connor, to-sum conversion formula can be written in coefficient 1988], we choose to use classical, real-valued, unnor- form malized spherical harmonics with the phase convention according to Lambeck [1988]. An arbitrary function Φ » ÿ [ Ω ~ Ω ] = Φ ,Φ ′,Φ′′ Φ′′ ~Φ ′ f (Ω ) defined on a spherical surface as function of po- C( )S ( ) nm ƒ … ƒ Anm,n′m′,n′′m′′Cn′′m′′ ŸSn′m′ (B1) ′ ′Φ′ ′′ ′′Φ′′ ⁄ Ω ϕ λ n m n m sition (latitude , longitude ) can be expanded as: ~ Ω ∞ n {C,S} where S ( ) is the smooth sea level function, and Ω = Φ Φ Ω f ( ) ƒƒ ƒ f nmYnm ( ) C(Ω ) is the ocean function defined to be 1 over the n=0 m=0 Φ oceans and 0 on land. The notation for spherical har- ∞ (A1) » n ÿ monic coefficients is given in Appendix A. Evaluation = f CY C (Ω) + ( f C Y C (Ω) + f S Y S (Ω)) ƒ… 00 00 ƒ nm nm nm nm Ÿ of (B1) first requires a general solution to the following n=0 m=1 ⁄ integral of triple real-valued spherical harmonics de-

(Φ = ) (Φ = ) fined by equation (50): where the cosine C and sine S spherical harmonic basis functions are defined by Π 2 Φ ,Φ′,Φ ′′ = nm Φ Ω Φ′ Ω Φ ′′ Ω Ω Anm,n′m′,n′′m′′ ——Ynm ( )Yn′m′ ( )Yn′′m′′ ( )d (B2) C 4π Y (Ω ) = P (sinφ)cos mλ nm nm (A2) S Ω = φ λ Ynm ( ) Pnm (sin )sin m Typically this integral is discussed in quantum mechanics for the case of complex spherical harmonics. For and where the associated Legendre polynomials are real-valued spherical harmonics, the integral over longitude is different, and there are no negative values of m. First of all, starting with the integral over latitude, = [( − 2 )m 2 n ]( n+m n+m )( 2 − )n Pnm (x) 1 x 2 n! d dx x 1 (A3) our method uses recursion formulae for associated Leg- endre polynomials to derive recursion formulae for the The integral of two spherical harmonics is integrals themselves. The following “selection rules” indicate which combinations of associated Legendre Φ Φ ′ 4π polynomials result in a non-zero integral over latitude: Y (Ω )Y (Ω )dΩ = δ δ δ (A5) —— nm n′m′ nn′ mm′ ΦΦ ′ Π 2 nm

BLEWITT AND CLARKE: EARTH’S SHAPE WEIGHS SEA LEVEL 18

m′′ = m ± m′ from which we can then apply: max(n − n′, m′ ± m′′ ) ≤ n′′ ≤ n + n′ (B3) + + Qn 0;n 0;n + ′ + ′′ = Q = 1 2 3 [(n −n )(n +n +1)+n (n +1)] n n n 2 p n11;n20;n3 2 1 2 1 2 3 3 (B8) + + Q where p is an integer. Let us define the integral over Q = n10;n20;n3 [−(n −n )(n +n +1)+n (n +1)] latitude for those indices that satisfy the above selection n10;n21;n3 2 1 2 1 2 3 3 rules: 1 and then, finally, we can recursively compute: ± = + + Qnm;n′m′;n′′ Pnm (x)Pn′m′ (x)Pn′′ m±m′ (x)dx (B4) = ( + )( − + ) — Qn m ;n m ;n n1 m1 n1 m1 1Qn m −1;n m ;n −1 1 1 2 2 3 1 1 2 2 3 + ( + )( − + ) + The solution to this can always be constructed from the n2 m2 n2 m2 1Qn m ;n m −1;n following general expression: 1 1 2 2 3 + = ( − − )( + + + ) + 1 Qn m ;n m +1;n n3 m1 m2 n3 m1 m2 1 Qn m ;n m ;n + ≡ 1 1 2 2 3 1 1 2 2 3 Qn m ,n m ,n Pn m (x)Pn m (x)Pn (m +m ) (x)dx (B5) + 1 1 2 2 3 — 1 1 2 2 3 1 2 −Q + −1 n1m1 1;n2m2 ;n3 (B9) which can then always be related to (B4) using the following identities derivable from (B5) Note that there are generally many possible pathways to compute a specific coefficient. This is useful for self- ÀQ+ if m ≥ m , (m = m − m ) consistency testing when tabulating the results. − n3m3 ;n2m2 ;n1 1 2 3 1 2 Q = Ã + Equivalently, it can be shown that equation (B5) can n1m1;n2m2 ;n3 Q if m ≥ m , (m = m − m ) Õ n3m3 ;n1m1;n2 2 1 3 2 1 be computed directly by the formula (B6)

+ (n + m )!(n + m )!(n + m +m )! = 2 1 1 2 2 3 1 2 Balmino [1978] presents a method to solve (B4) by de- Qn m ,n m ,n 1 1 2 2 3 2n +1 (n −m )!(n −m )!(n −m −m )! composition of associated Legendre polynomials as an 3 1 1 2 2 3 1 2 × + explicit function of their arguments. For computational n1,m1;n2 ,m2 n3,m1 m2 n1,0;n2 ,0 n3,0 efficiency, and to facilitate control on rationalization of (B10) fractions to mitigate potential overflow problems, we computed and tabulated results for (B5) using recursion + relations for the integrals, which can be derived using where n1 ,m1;n2 ,m2 n3 ,m1 m2 is a Clebsch- recursion relations for associated Legendre polynomials Gordan coefficient, representing a unitary transforma- [Rikitake et al., 1987]. This is a somewhat simpler (but tion between coupled and uncoupled basis states in equivalent) method than suggested by Balmino [1978, quantum angular momentum theory. Clebsch-Gordan 1994]. The recursion algorithm is now summarized. coefficients can be calculated according to the Racah Starting with formulae for order zero, we have: formula [Messiah, 1963] and are tabulated and widely distributed in the particle physics community [Particle δ Data Group, 2002], thus making computer programs + 2 n n Q = 1 2 easier to verify. n10;n2 0;0 + 2n1 1 Now we must compute the integral over longitude. δ ( + )δ + 2n1 n n +1 2 n1 1 n n −1 This vanishes unless the triple product of cosine and Q = 1 2 + 1 2 sine functions involves an odd number of cosines (un- n1 0;n2 0;1 ( − )( + ) ( + )( + ) 2n1 1 2n1 1 2n1 1 2n2 3 derstanding that zero-order sine functions are disal- + 2n −1 lowed). Therefore Φ = F(Φ ,Φ ) is uniquely deter- Q = 3 (B7) 3 1 2 n10;n2 0;n3 ( + ) n3 2n2 1 mined. For these non-zero combinations, it can be + + shown that the longitude integral: ×[(n +1)Q + + + n Q − − ] 2 n10;n2 1,0;n3 1 2 n10;n2 1,0;n3 1 − n3 1 + − Q − n10;n2 ,0;n3 2 n3 BLEWITT AND CLARKE: EARTH’S SHAPE WEIGHS SEA LEVEL 19

π Φ Φ 2 1 2 nominator are divided by their greatest common de- Φ Φ 1 Àsin m λ ¤ Àsin m λ ¤ Λ 1 2 = Ã 1 ‹ Ã 2 ‹ nominator, which can be important to prevent computa- m1m2 π — cos m λ cos m λ 4 0 Õ 1 › Õ 2 › tional problems. (Even doing so, the number of digits Φ Φ ( + )λ F ( 1 , 2 ) required to represent the rational fraction grows quickly, × Àsin m1 m2 ¤ λ Ã ( + )λ‹ d (B11) as large as 11 digits for degree-3 theory, and so at some Õcos m1 m2 › point floating-point calculations become necessary). a(Φ ,Φ ) Our recursive method was amenable to implementa- = 1 2 2(2 − δ )(2 − δ ) tion in a spreadsheet, and the results (for low degree) m1 0 m2 0 are shown in Appendix C. We successfully verified the answers for low-degree expansions using published val- where ues of Clebsch-Gordan coefficients and complex spherical harmonics, which were then converted into results ÀC if Φ = Φ for real-valued spherical harmonics. This was all done F(Φ ,Φ ) = Ã 1 2 1 2 ÕS otherwise by painstaking hand derivation so as to recover the ra- (B12) tional fractions exactly in terms of integer numerators À−1 if Φ = Φ = S a(Φ ,Φ ) = Ã 1 2 and denominators. Finally, we performed a second in- 1 2 Õ+1 otherwise dependent check by writing a FORTRAN program to tabulate the results of equation (B14) by applying the According to identity (B6), we use can use the follow- method of (B10) using a Clebsch-Gordan subroutine. ing selection rules in equation (B5) to compute all al- Since this program has been validated, due to obvious lowed values of the Q coefficients: advantages of speed and accurate bookkeeping, it will be employed for future higher degree calculations. À(m,m′,m′′) if m′′ = m + m′ Œ Appendix C. Product-to-Sum Conversion: (m ,m ,m ) = Ã(m′′,m′,m) if m′′ = m − m′ (B13) 1 2 3 Results for Low-Degrees ÕŒ(m′′,m,m′) if m′′ = m′ − m We provide sample results of equations (52) and and thus compute equation (B4) for any combination of (B1) here as a benchmark to assist those attempting to apply our method. Consider the smooth sea level func- m′′ = m ± m′ . Using all the above tools, the ocean ~ tion S (Ω ) that is defined globally as a spherical har- function product-to-sum coefficients in equation (B1) monic expansion, but is then projected onto the area can be simplified using: covered by the ocean such that the result is exactly ~ ∞ n′′ {C,S} S (Ω ) on the ocean, but zero on land. The spherical Φ ,Φ′,Φ′′ Φ′′ ƒ ƒ ƒ Anm,n′m′,n′′m′′Cn′′m′′ harmonic coefficients of the ocean-projected function n′′=0 m′′=0 Φ′′ are given by n+n′ = Π 2 ΛΦΦ′ + F(Φ ,Φ′) ƒ nm mm′ Qnm;n′m′;n′′Cn′′,m+m′ Φ » ÿ n′′=max( n−n′ ,m+m′) [ Ω ~ Ω ] = Φ ,Φ′,Φ′′ Φ′′ ~Φ′ if n+n′+n′′=even C( )S ( ) nm ƒ … ƒ Anm,n′m′,n′′m′′Cn′′m′′ ŸSn′m′ (C1) ′ ′Φ′ ′′ ′′Φ′′ ⁄ ≥ ′ n m n m À if m m : ¤ n+n′ Œ 2 F(Φ ,Φ′)Φ′ + F (Φ ,Φ′) Œ ~ Œ Π Λ ′ − ′ Q ′′ − ′ ′ ′ C ′′ − ′ Œ Consider that S (Ω ) is given exactly as a degree-2 + ƒ Ã nm m ,m m n ,m m ;n m ;n n ,m m ‹ spherical harmonic expansion. Note that in this case, n′′=max( n−n′ , m−m′ ) Œ if m′ ≥ m : Œ + ′+ ′′= if n n n even ŒΠ 2 ΛF(Φ ,Φ′)Φ Q+ C F(Φ ,Φ′) Œ only ocean function coefficients up to degree 4 are re- Õ nm m′−m,m n′′,m′−m;nm;n′ n′′,m′−m › quired for exact results. Results are now systematically (B14) provided for the projected function up to degree 2 using the method of Appendix B. The numerators and denominators are pure integers Starting with degree 0 of the projected function we (as can be seen by inspection of the above formulae); so have if they are computed separately, the answer can be represented as an exact rational fraction until the final step. At each step in the recursion the numerator and de- BLEWITT AND CLARKE: EARTH’S SHAPE WEIGHS SEA LEVEL 20

~ C ~ 1 ~ 1 ~ ~ S ~ 3 ~ 6 ~ [C(Ω )S (Ω )] = C C S C + C C S C + C C S C [C(Ω )S (Ω )] = C S S C + C S S C + C S S C 00 00 00 3 10 10 3 11 11 11 11 00 5 21 10 5 22 11 1 ~ 1 ~ 3 ~ 3 ~ ≈ 1 6 ’~ + C S S S + C C S C + C C S C + C S S S (C2) + ∆C C − C C − C C ÷S S 3 11 11 5 20 20 5 21 21 5 21 21 « 00 5 20 5 22 ◊ 11 12 ~ 12 ~ + C C + S S ≈ 1 S 18 S ’~ C 18 S ~ C C22 S 22 C22 S22 + ∆− C + C ÷S + C S 5 5 « 5 11 35 31 ◊ 20 7 32 21 (C5) ≈ 3 9 18 ’~ + ∆ C C − C C − C C ÷S S For degree 1 we have « 5 10 35 30 7 32 ◊ 21

≈ 6 S 18 S 108 S ’~ C ~ C ~ ≈ 2 ’~ + ∆− C + C + C ÷S [C(Ω )S (Ω )] = C C S C + ∆C C + C C ÷S C « 5 11 35 31 7 33 ◊ 22 10 10 00 « 00 5 20 ◊ 10 ≈ 6 18 108 ’~ 3 3 + ∆ C − C − C ÷ S + C ~ C + S ~ S C11 C31 C33 S 22 C21S11 C21S11 « 5 35 7 ◊ 5 5 (C3) ≈ 2 9 ’~ ≈ 3 36 ’~ + ∆ C C + C C ÷S C + ∆ C C + C C ÷S C Finally, for degree 2 we have: « 5 10 35 30 ◊ 20 « 5 11 35 31 ◊ 21

~ C C ~ C ≈ 2 C 3 C ’~ C ≈ 3 S 36 C ’~ S 36 C ~ C 36 S ~ S [ Ω Ω ] = + + + ∆ C + C ÷S + C S + C S C( )S ( ) 20 C20 S00 ∆ C10 C30 ÷S10 « 5 11 35 31 ◊ 21 7 32 22 7 32 22 « 3 7 ◊ + ≈− 1 C + 6 C ’~ C + ≈− 1 S + 6 S ’~ S ∆ C11 C31 ÷S11 ∆ C11 C31 ÷S11 [ Ω ~ Ω ]C = C ~ C + 3 C ~ C « 3 7 ◊ « 3 7 ◊ C( )S ( ) 11 C11 S00 C21S10 5 ≈ 2 2 ’~ + ∆ C + C + C ÷ C ≈ 1 6 ’~ 6 ~ C00 C20 C40 S 20 + ∆C C − C C + C C ÷S C + C S S S « 7 7 ◊ « 00 5 20 5 22 ◊ 11 5 22 11 (C6) ≈ 3 10 ’~ ≈ 3 10 ’~ + ∆ C + C ÷ C + ∆ S + S ÷ S ≈ 1 18 ’~ C21 C41 S 21 C21 C41 S 21 + ∆− C C + C C ÷S C « 7 7 ◊ « 7 7 ◊ « 5 11 35 31 ◊ 20 ≈ 24 60 ’~ (C4) + ∆− C + C ÷ C ≈ 3 9 18 ’~ 18 ~ C22 C42 S 22 + ∆ C C − C C + C C ÷S C + C S S S « 7 7 ◊ « 5 10 35 30 7 32 ◊ 21 7 32 21 + ≈− 24 S + 60 S ’~ S ∆ C22 C42 ÷S 22 + ≈ 6 C − 18 C + 108 C ’~ C « 7 7 ◊ ∆ C11 C31 C33 ÷S 22 « 5 35 7 ◊ ≈ 6 18 108 ’~ + ∆ C S − C S + C S ÷S S « 5 11 35 31 7 33 ◊ 22

BLEWITT AND CLARKE: EARTH’S SHAPE WEIGHS SEA LEVEL 21

C ~ C ~ 5 ~ [ Ω ~ Ω ] = C ~ C + ≈ 1 C + 4 C ’~ C [ Ω Ω ] = C C + C C C( )S ( ) C S ∆ C C ÷S C( )S ( ) C22 S00 C32 S10 21 21 00 « 3 11 7 31 ◊ 10 22 7 ≈ 1 1 15 ’~ ≈ 1 C 1 C 10 C ’~ C 10 S ~ S + C − C + C C + ∆ C − C + C ÷S + C S ∆ C11 C31 C33 ÷S11 « 3 10 7 30 7 32 ◊ 11 7 32 11 « 6 14 7 ◊ ≈ 1 1 15 ’~ ≈ 1 C 10 C ’~ C + − S + S + S S + ∆ C + O ÷S ∆ C11 C31 C33 ÷S11 « 7 21 21 41 ◊ 20 « 6 14 7 ◊ ≈ 2 5 ’~ ≈ C 1 C 4 C 6 C 20 C ’~ C + − C + C C + ∆C + C − C + C + C ÷S (C7) ∆ C22 C42 ÷S 20 (C9) « 00 7 20 21 40 7 22 7 42 ◊ 21 « 7 7 ◊ ≈ 3 5 ’~ ≈ 6 S 20 S ’~ S + C − C + C C + ∆ C + C ÷S ∆ C21 C41 5C43 ÷S 21 « 7 22 7 42 ◊ 21 «14 42 ◊ ≈ 3 5 ’~ ~ ≈ 6 C 10 C C ’~ C + − S + S + S S + S S + ∆ C − C + 20C ÷S ∆ C21 C41 5C43 ÷S21 40C44 S 22 « 7 21 21 41 43 ◊ 22 « 14 42 ◊ ≈ 2 1 ’~ ≈ 6 S 10 S S ’~ S + C − C + C + C C + ∆ C − C + 20C ÷S ∆C00 C20 C40 40C44 ÷S 22 « 7 21 21 41 43 ◊ 22 « 7 21 ◊

~ S S ~ C ≈ 1 S 4 S ’~ C [C(Ω )S (Ω )] = C S + ∆ C + C ÷S ~ S ~ 5 ~ 21 21 00 « 3 11 7 31 ◊ 10 [C(Ω )S (Ω )] = C S S + C S S C 22 22 00 7 32 10 10 ~ ≈ 1 1 10 ’~ + S C + C − C − C S ≈ 1 1 15 ’~ C32 S11 ∆ C10 C30 C32 ÷S11 + S − S + S C 7 « 3 7 7 ◊ ∆ C11 C31 C33 ÷S11 « 6 14 7 ◊ ≈ 1 10 ’~ ≈ 6 20 ’~ + ∆ S + S ÷ C + ∆ S + S ÷ C ≈ 1 1 15 ’~ C21 C41 S 20 C22 C42 S 21 + C − C − C S « 7 21 ◊ « 7 7 ◊ ∆ C11 C31 C33 ÷S11 (C8) « 6 14 7 ◊ ≈ 1 4 6 20 ’~ + ∆ C + C − C − C − C ÷ S ≈ 2 5 ’~ C00 C20 C40 C22 C42 S 21 + − S + S C « 7 21 7 7 ◊ ∆ C22 C42 ÷S 20 (C10) « 7 7 ◊ ≈ 6 10 ’~ + − S + S + S C ≈ 3 5 ’~ ∆ C21 C41 20C43 ÷S 22 + S − S + S C « 7 21 ◊ ∆ C21 C41 5C43 ÷S21 «14 42 ◊ ≈ 6 10 ’~ + C − C − C S ≈ 3 5 ’~ ~ ∆ C21 C41 20C43 ÷S 22 + C − C − C S + S C « 7 21 ◊ ∆ C21 C41 5C43 ÷S 21 40C44 S 22 «14 42 ◊ ≈ 2 1 ’~ + ∆C C − C C + C C − 40C C ÷S S « 00 7 20 21 40 44 ◊ 22

Acknowledgments. We thank H.-P. Plag, an anonymous re- viewer, and T. Herring for their constructive reviews. We gratefully acknowledge contributions by: T. van Dam, H.-P. Plag, and P. Ge- gout of the IERS Global Geophysical Fluids Center’s “Special Bu- reau for Loading” for discussions on the need for self-consistent integration of models; G. Balmino for correspondence on Wigner 3-j theory and product-to-sum conversion; and D. Lavallée for providing scaled formal errors of the load moment data and for producing Figure 2. This work was supported in the U.S. by the National Sci- ence Foundation, Geophysics Program, grant EAR-0125575, and in the U.K. by the Natural Environment Research Council, grant NER/A/S/2001/01166. BLEWITT AND CLARKE: EARTH’S SHAPE WEIGHS SEA LEVEL 22

References Lambeck, K., The Earth’s Variable Rotation, Cambridge University Press, New York, 1980. Lambeck, K., Geophysical Geodesy: The Slow Deformations of the Agnew. D.C., and W.E. Farrell, Self-consistent equilibrium ocean Earth, Oxford University Press, New York, 1988. tides, Geophys. Journ. Roy. Astron. Soc. 55, 171-182, 1978. Lavallée, D., and G. Blewitt, Degree-1 Earth deformation from very Balmino, G., K. Lambeck, and W.M. Kaula, A spherical harmonic long baseline interferometry, Geophys. Res. Lett., 29(20), 1967, analysis of Earth’s topography, Journ. Geophys. Res., 78, 478-481, doi:10.1029/2002GL015883, 2002. 1973. Lavallée, D., Plate Tectonic Motions from Global GPS Measure- Balmino, G., On the product of Legendre functions as encountered ments, Ph.D. Thesis, University of Newcastle upon Tyne, UK, in geodynamics, Studia Geophys. et Geod., 22, 107-118, 1978. 2000. Balmino, G., Gravitational potential harmonics from the shape of an Love, A.E.H., The yielding of the Earth to disturbing forces, Proc. homogeneous body, Cel. Mech. and Dyn. Astron., 60, 331-364, R. Soc. London A, 82, 73-88, 1909. 1994. Messiah, A., Quantum Mechanics, Volume II, North Holland, Am- Blewitt, G., Self-consistency in reference frames, geocenter defini- sterdam, 1963. tion, and surface loading of the solid Earth, Journ. Geophys. Res., Minster, J.F., A. Cazenave, Y.V. Serafini, F. Mercier, M.C. Gen- doi:10.1029/2002JB002082, 2003. nero, and P. Rogel, Annual cycle in mean sea level from TOPEX- Blewitt, G., and D. Lavallée, Effect of annual signals on geodetic Poseidon and ERS-1: inference on the global hydrological cycle, velocity, Journ. Geophys. Res., 107(B7), Global Planet. Change, 20, 57-66, 1999. doi:10.1029/2001JB000570, 2002. Mitrovica, J.X., J.L. Davis, and I.I. Shapiro, A spectral formalism Blewitt, G., D. Lavallée, P. Clarke, and K. Nurutdinov, A new for computing three-dimensional deformations due to surface global mode of Earth deformation: Seasonal cycle detected, Sci- loads 1. Theory, Journ. Geophys. Res., 99, 7057-7073, 1994. ence, 294, 2,342-2,345, 2001. Munk W.H. and G.J.F. MacDonald, The Rotation of the Earth, Chang, A.T.C., J.L. Foster, and D.K. Hall, Satellite estimates of Cambridge University Press, New York, 1960. Northern Hemisphere snow volume, Remote Sens. Lett. Int. Journ. Nerem, R.S., R.J. Eanes, P.F. Thompson, and J.L. Chen, Observa- Remote Sens., 11, 167-172, 1990. tions of annual variations of the Earth’s gravitational field using Chao, B.F., W.P. O’Connor, A.T.C. Chang, D.K. Hall, and J.L. Fos- satellite laser ranging and geophysical models, Geophys. Res. ter, Snow-load effect on the Earth’s rotation and gravitational Lett., 27, 1783-1786, 2000. field, 1979-1985, Journ. Geophys. Res., 92, 9,415-9,422, 1987. Particle Data Group (Hagiwara K., et al.), 2002 review of particle Chao, B.F., and W.P. O’Connor, Effect of a uniform sea-level physics, Phys. Rev. D 66, 010001, 2002. change on the Earth’s rotation and gravitational field, Geophys. Plag, H.-P., H.-U. Juttner, and V. Rautenberg, On the possibility of Journ., 93, 191-193, 1988. global and regional inversion of exogenic deformations for me- Chen, J.L., C.R. Wilson, D.P. Chambers, R.S. Nerem, and B.D. Ta- chanical properties of the Earth’s interior, Journ. Geodynamics., pley, Seasonal global water mass balance and mean sea level 21, 287-309, 1996. variations, Geophys. Res. Lett., 25(19), 3555-3558, 1998. Proudman, J., The condition that a long-period tide shall follow the Chen, J.L., C.R. Wilson, B.D. Tapley, D.P. Chambers, and T. Pek- equilibrium law, Geophys. Journ. Roy. Astron. Soc., 3, 9,423- ker, Hydrological impacts on seasonal sea level change, Global 9,430, 1960. and Planetary Change, 32, 25-32, 2002. Rikitake, T., R. Sato, and Y Hagiwara, Applied Mathematics for Dahlen, F.A., The passive influence of the oceans upon the rotation Earth Scientists, D. Reidel Publishing Company, Boston, 1987. of the Earth, Geophys. Journ. Roy. Astron. Soc., 46, 363-406, Tamisiea, M.E., Present-day secular variations in the low-degree 1976. harmonics of the geopotential: Sensitivity analysis on spherically Dai, A. and K.E. Trenberth, Estimates of freshwater discharge from symmetric Earth models, Journ. Geophys. Res., 107(B12) 2378, continents: Latitudinal and seasonal variations, Journ. Hydrome- doi:10.1029/2001JB000696, 2002. teorology, 3, 660-687, 2002. Trenberth, K., Seasonal variations in global sea level pressure and Davies, P., and G. Blewitt, Methodology for global geodetic time the total mass of the atmosphere, Journ. Geophys. Res., 86, 5,238- series estimation: A new tool for geodynamics, Journ. Geophys. 5,246, 1981. Res., 105, 11,083-11,100, 2000. van Dam, T.M., J. Wahr, Y. Chao, and E. Leuliette, Predictions of Dickman, S.R., A complete spherical harmonic approach to luni- crustal deformation and of geoid and sea-level variability caused solar tides, Geophys. Journ. Int, 99, 457-468, 1989. by oceanic and atmospheric loading, Geophys. Journ. Int., 129, Dong, D. J. Dickey, Y. Chao, and M. K. Cheng, Geocenter varia- 507-517, 1997. tions caused by atmosphere, ocean and surface ground water, van Dam, T.M., J. Wahr, P.C.D. Milly, A.B. Shmakin, G. Blewitt, Geophys. Res. Lett., 24(15), 1867-1870, 1997. D. Lavallée, and K.M. Larson, Crustal displacements due to conti- Dziewonski A.M. and D.L. Anderson, Preliminary reference Earth nental water loading: Geophys. Res. Lett., 28, 651–654, 2001. model, Phys. Earth Planet. Int. 25, 297-356, 1981. Wahr, J.M., The effects of the atmosphere and oceans on the Earth’s Farrell, W.E., Deformation of the Earth by surface loads, Rev. Geo- wobble. 1. Theory, Geophys. Journ. Roy. Astron. Soc., 70, 349- phys. and Space Phys., 10, 761-797, 1972. 372, 1982. Grafarend, E.W., Three-dimensional deformation analysis: global Wahr, J.M., M. Molenaar, and F. Bryan, Time variability of the vector spherical harmonic and local finite element representation, Earth's gravity field: Hydrological and oceanic effects and their Tectonophysics, 130, 337-359, 1986. possible detection using GRACE, Journ. Geophys. Res., 103(B12), Grafarend, E.W., J. Engels, and P. Varga, The spacetime gravita- 30,205–30,229, 1998. tional field of a deforming body, Journ. Geodesy, 72, 11-30, 1997. Wigner, E.P., Group Theory and its Applications to Quantum Me- Heflin, M.B., W.I. Bertiger, G. Blewitt, A.P. Freedman, K.J. Hurst, chanics of Atomic Spectra, Academic Press, New York, 1959. S.M. Lichten, U.J. Lindqwister, Y. Vigue, F.H. Webb, T.P. Yunck, ______and J.F. Zumberge, Global geodesy using GPS without fiducial G. Blewitt, University of Nevada, Reno, Mail Stop 178, sites, Geophys. Res. Lett., 19, 131-134, 1992. Reno, Nevada 89557 ([email protected]) PREPRINT: JOURNAL OF GEOPHYSICAL RESEARCH, DOI: 10.1029/2002JB002290 (IN PRESS 2003)

Table 1: Load Love Numbers and Combinations to Degree 12. Surface Geoid Surface Surface Lateral: Surface Height: Geoid Height: Sea Level: Degree Height Height Lateral Surface Height Load Thickness Load Thickness Load Thickness n h′ 1+ k′ l′ l′ 3ρ h′ 3ρ (1+ k′ ) 3ρ (1+ k′ − h′ ) n n n n S n S n S n n h′ ( + )ρ ( + )ρ ( + )ρ n 2n 1 E 2n 1 E 2n 1 E 1* -0.269 1.021 0.134 0.500 -0.050 0.190 0.240 2 -1.001 0.693 0.030 0.029 -0.112 0.077 0.189 3 -1.052 0.805 0.074 0.071 -0.084 0.064 0.148 4 -1.053 0.868 0.062 0.059 -0.065 0.054 0.119 5 -1.088 0.897 0.049 0.045 -0.055 0.045 0.101 6 -1.147 0.911 0.041 0.036 -0.049 0.039 0.088 7 -1.224 0.918 0.037 0.030 -0.046 0.034 0.080 8 -1.291 0.925 0.034 0.026 -0.042 0.030 0.073 9 -1.366 0.928 0.032 0.023 -0.040 0.027 0.067 10 -1.433 0.932 0.030 0.021 -0.038 0.025 0.063 11 -1.508 0.934 0.030 0.020 -0.037 0.023 0.059 12 -1.576 0.936 0.029 0.018 -0.035 0.021 0.056 First three columns computed from Farrell [1972] (interpolated for degrees 7, 9, 11, and 12). Load thickness defined as the height of a column of seawater, specific density 1.025. *Frame-dependent degree-1 numbers computed in center of figure (CF) frame [Blewitt et al., 2001], except for last column, which is frame invariant. Degree-1 numbers should be transformed into the isomorphic frame adopted by a specific investigation [Blewitt, 2003].

Table 2: Ocean Function Spectral Coefficients Used in this Analysis Coefficient Degree-0 Degree-1 Degree-2 Degree-3 Degree-4 Φ Φ Φ Φ Φ Φ Cnm C0m C1m C2m C3m C4m C Cn0 0.69700 -0.21824 -0.13416 0.11906 -0.07200 C Cn1 -0.18706 -0.05164 0.04753 0.03415 S Cn1 -0.09699 -0.06584 -0.03456 0.02846 C Cn2 0.02582 0.02391 0.02012 S Cn2 0.00129 -0.03040 -0.00470 C Cn3 -0.00223 -0.00317 S Cn3 -0.01241 0.00030 C Cn4 0.00030 S Cn4 -0.00213 Normalization removed from coefficients of Balmino et al. [1973]. Convention for un-normalized coefficients in Appendix A.

BLEWITT AND CLARKE: EARTH’S SHAPE WEIGHS SEA LEVEL 24

Table 3. Product-to-Sum Transformation Coefficients. Χ Φ ,Φ ′ nm,n′m′ n’,m’,Φ’ n,m,Φ 0,0,C 1,0,C 1,1,C 1,1,S 2,0,C 2,1,C 2,1,S 2,2,C 2,2,S 0,0,C 0.697 -0.073 -0.062 -0.032 -0.027 -0.031 -0.040 0.062 0.003 1,0,C -0.218 0.643 -0.031 -0.040 -0.057 -0.063 -0.094 0.123 -0.156 1,1,C -0.187 -0.031 0.755 0.002 0.062 -0.100 -0.078 -0.283 -0.290 1,1,S -0.097 -0.040 0.002 0.693 0.002 -0.078 -0.223 -0.093 -0.214 2,0,C -0.134 -0.094 0.103 0.003 0.638 0.027 0.012 0.084 -0.045 2,1,C -0.052 -0.035 -0.056 -0.043 0.009 0.771 -0.012 -0.124 -0.064 2,1,S -0.066 -0.052 -0.043 -0.124 0.004 -0.012 0.612 0.076 0.003 2,2,C 0.026 0.017 -0.039 -0.013 0.007 -0.031 0.019 0.744 -0.085 2,2,S 0.001 -0.022 -0.040 -0.030 -0.004 -0.016 0.001 -0.085 0.720 Transformation defined by equation (53), with computational method shown in Appendices B and C. Calculation uses ocean function coefficients from Table 2. The strong diagonal nature is evident.

BLEWITT AND CLARKE: EARTH’S SHAPE WEIGHS SEA LEVEL 25

Table 4. Seasonal Degree-1 Inversion for Static Ocean Topography Annual Semi-Annual Parameter Amplitude Phase* Amplitude Phase* (mm) (deg) (mm) (deg) Height Function Spectral Coefficients from GPS Inversion1 ˆ C 2.97 ± 0.12 236 ± 2 0.67 ± 0.12 27 ± 10 H10 ˆ C 0.90 ± 0.15 266 ± 9 0.31 ± 0.15 249 ± 26 H11 ˆ S 1.30 ± 0.13 165 ± 6 0.27 ± 0.12 121 ± 25 H11 Total Load Spectral Coefficients2 ˆ C 0.0 ± 0 — 0.0 ± 0 — T00 ˆ C 59.4 ± 2.5 56 ± 2 13.5 ± 2.4 207 ± 10 T10 ˆ C 18.0 ± 3.0 86 ± 9 6.3 ± 2.9 69 ± 26 T11 ˆ S 26.1 ± 2.6 345 ± 6 5.4 ± 2.4 301 ± 25 T11 Geocentric Sea Level and Geoid Height Spectral Coefficients3 ˆ C 0.0 ± 0 — 0.0 ± 0 — N 00 ~ C = ∆ 6.1 ± 0.3 234 ± 3 0.7 ± 0.3 20 ± 22 O00 V g ~ C = ˆ C 11.3 ± 0.5 56 ± 2 2.6 ± 0.5 207 ± 10 O10 N10 ~ C = ˆ C 3.4 ± 0.6 86 ± 9 1.2 ± 0.6 69 ± 26 O11 N11 ~ S = ˆ S 5.0 ± 0.5 345 ± 6 1.0 ± 0.5 301 ± 25 O11 N11 Relative Sea Level Quasi-Spectral Coefficients4 ~ C = ∆ 6.1 ± 0.3 234 ± 3 0.7 ± 0.3 20 ± 22 S00 V g ~ C = ~ C − ˆ C 14.3 ± 0.6 56 ± 2 3.2 ± 0.6 207 ± 10 S10 O10 H10 ~ C = ~ C − ˆ C 4.3 ± 0.7 86 ± 9 1.5 ± 0.7 69 ± 26 S11 O11 H11 ~ S = ~ S − ˆ S 6.3 ± 0.6 345 ± 6 1.3 ± 0.6 301 ± 25 S11 O11 H11 * π − −φ Uses the convention cos[2 f (t t0 ) f ] where f is frequency, t is time, t0 is φ π 1 January; so f 2 f is time when f-harmonic is maximum. 1Derived from equation (58) using data from 1996.0-2001.0 in center of figure frame [Blewitt et al., 2001]. One-standard deviation formal errors are scaled here by factor 1.9 to normalize the chi-square per degree of freedom for the seasonal model. 2Degree-0 total load constrained to zero due to conservation of mass. All other coefficients use surface height to load thickness ratio in Table 1. 3Degree-0 geocentric sea level computed by equation (45). All other coefficients use geoid height to load thickness ratio in Table 1. 4Can be computed equivalently using sea level to load thickness ratio in Table 1. BLEWITT AND CLARKE: EARTH’S SHAPE WEIGHS SEA LEVEL 26

Table 5. Seasonal Inversion for Global Mean Oceanographic Parameters Annual Semi-Annual Parameter Amplitude (mm) Phase Amplitude Phase (deg) (mm) (deg) Mean Relative Sea Level* S 8.0 ± 0.4 234 ± 3 1.0 ± 0.7 20 ± 22 Mean Marine Geoid Height N 1.5 ± 0.1 234 ± 3 0.2 ± 0.1 20 ± 22 Mean Ocean Floor Height H 0.40 ± 0.02 54 ± 3 0.05 ± 0.02 200 ± 22 Mean Geocentric Sea Level O = S + H 7.6 ± 0.4 234 ± 3 0.9 ± 0.4 20 ± 22 Sea Surface Geopotential Height (above deformed geoid) ∆V g =O − N 6.1 ± 0.3 234 ± 3 0.7 ± 0.3 20 ± 22 Mean values over oceanic areas computed in spectral domain using ocean function coefficients of Table 1. *Mean relative sea level computed equivalently using either total load spectral coefficients or the quasi-spectral sea level coefficients.

BLEWITT AND CLARKE: EARTH’S SHAPE WEIGHS SEA LEVEL 27

Table 6. Seasonal Inversion for Continental Water Topography Annual Semi-Annual Parameter Amplitude Phase Amplitude Phase (mm) (deg) (mm) (deg) ~ Relative Sea Level Global Spectral Coefficients1 Sˆ(Ω ) = C(Ω )S (Ω ) ˆ C 5.6 ± 0.3 234 ± 3 0.7 ± 0.3 20 ± 22 S00 ˆ C 10.3 ± 0.4 57 ± 2 2.3 ± 0.4 206 ± 10 S10 ˆ C 3.9 ± 0.6 80 ± 9 1.1 ± 0.6 71 ± 29 S11 ˆ S 4.4 ± 0.4 345 ± 6 0.9 ± 0.4 305 ± 26 S11 Continental Water Global Spectral Coefficients2 ˆC =− ˆ C 5.6 ± 0.3 54 ± 3 0.7 ± 0.3 200 ± 22 L00 S00 ˆC = ˆ C − ˆ C 49.1 ± 2.1 56 ± 2 11.2 ± 2.0 207 ± 10 L10 T10 S10 ˆC = ˆ C − ˆ C 14.2 ± 2.4 88 ± 10 5.2 ± 2.3 69 ± 26 L11 T11 S11 ˆS = ˆ S − ˆ S 21.7 ± 2.1 345 ± 6 4.5 ± 2.0 300 ± 26 L11 T11 S11 ~ Continental Water Quasi-Spectral Coefficients3 Lˆ(Ω ) = [1− C(Ω )]L(Ω ) ~C 30.1 ± 2.0 238 ± 4 2.3 ± 1.9 345 ± 18 L00 ~C 150.1 ± 6.4 58 ± 2 34.7 ± 6.1 205 ± 10 L10 ~C 61.2 ± 11.2 85 ± 10 24.1 ± 10.9 65 ± 26 L11 ~S 68.5 ± 7.2 337 ± 6 15.1 ± 6.8 316 ± 26 L11 Errors propagated formally without a priori uncertainty on total load coefficients of degree 2 and higher (which may have spectral leakage). 1Computed in the spectral domain by product-to-sum transformation (Table 3) according to equation (52). Full spectrum exists in principle. 2Given to the degree to which total load has been estimated (degree 1). 3Computed using inverse of product-to-sum matrix truncated at degree-1.

Table 7. Amplification of Degree-1 Loading by Passive Ocean Response Annual Semi-Annual Parameter Comparison Amplitude Phase Shift2 Amplitude Phase Shift2 Ratio1 (deg) Ratio1 (deg) ˆ C ˆC 1.21 0.1 1.20 -0.2 T10 relative to L10 ˆ C ˆC 1.27 -1.6 1.22 0.4 T11 relative to L11 ˆ S ˆS 1.20 0.1 1.20 0.7 T11 relative to L11 1Defined as the ratio of spectral amplitudes of the total (continental plus ocean) load to the continental load. 2Defined as the difference in phase between the total load and continental load. BLEWITT AND CLARKE: EARTH’S SHAPE WEIGHS SEA LEVEL 28

Table 8. Reconstruction of Total Load and Height Function Annual Semi-Annual Parameter Amplitude Phase Amplitude Phase (mm) (deg) (mm) (deg) Consistency Check, Degree 1 Total Load1 ˆ C 0.0 ± 0 — 0.0 ± 0 — T00 ˆ C 59.4 ± 2.5 56 ± 2 13.5 ± 2.4 207 ± 10 T10 ˆ C 18.0 ± 3.0 86 ± 9 6.3 ± 2.9 69 ± 26 T11 ˆ S 26.1 ± 2.6 345 ± 6 5.4 ± 2.4 301 ± 26 T11 Predicted Degree-2 Total Load2 ˆ C 5.1 ± 1.4 28 ± 16 4.9 ± 1.3 225 ± 16 T20 * ˆ C 6.9 ± 0.6 48 ± 5 0.2 ± 0.6 79 ± 148 T21 * ˆ S 11.0 ± 0.9 22 ± 5 1.1 ± 0.9 286 ± 46 T21 ˆ C 0.5 ± 0.5 88 ± 59 1.3 ± 0.5 44 ± 20 T22 ˆ S 5.3 ± 0.5 50 ± 5 0.3 ± 0.5 76 ± 102 T22 Predicted Degree-2 Height Function2 ˆ C 0.57 ± 0.15 208 ± 16 0.55 ± 0.15 45 ± 16 H20 * ˆ C 0.77 ± 0.07 228 ± 5 0.03 ± 0.07 259 ± 148 H 21 * ˆ S 1.23 ± 0.10 202 ± 5 0.12 ± 0.10 106 ± 46 H 21 ˆ C 0.05 ± 0.05 268 ± 59 0.15 ± 0.05 224 ± 20 H 22 ˆ S 0.60 ± 0.05 230 ± 5 0.03 ± 0.05 256 ± 102 H 22 Computed only using estimates and full covariance matrix of quasi-spectral coefficients (to degree 1) for relative sea level and continental water by applying equation (57) up to degree 2. 1Coefficients to degree 1 should (and do) agree with initial values in Table 3. 2Degree-2 predictions only represent the contribution of loading given quasi-spectrally to degree 1, by its spectral interaction with the ocean function. Actual degree-2 load will also include higher quasi-spectral degree loading, which may contribute constructively or destructively. *Seasonal degree-1 annual loading interacting with the ocean function leaks strongly into annual degree-2, order-1 terms, indicating a potential mechanism to excite the Chan- dler Wobble.

BLEWITT AND CLARKE: EARTH’S SHAPE WEIGHS SEA LEVEL 29

Land Load Hydrology Surface Density

Satellite Altim- etry Tide Gauges & Mass Exchange Bottom Pressure

Geocentric Relative Sea Level + Sea Level −

Global Load Surface Density

Equipotential Gravitation Momentum Sea Floor Sea Surface Deformation

Load Po- Geocenter Dis- tential placement

LLN Theory Frame Theory

Geoid Solid Earth Deformation Deformation

Satellite Gra- Station Station vimetry Gravimetry Positioning

Figure 1. The basic elements of our analytical integrated loading model, building on a figure from Blewitt [2003], which incorporated self-consistency of the reference frame and loading dynamics. Here we incorporate self- consistency in the static, passive response of ocean loading. Phenomena are in round boxes, measurement types are in rectangular boxes, and physical principles are attached to the connecting arrows. The arrows indicate the direction leading toward the computation of measurement models, which this paper inverts for the case of measured station positions. Although the diagram suggests an iterative forward modeling solution [e.g., Wahr, 1982], we develop a closed-form solution that allows for true inversion. BLEWITT AND CLARKE: EARTH’S SHAPE WEIGHS SEA LEVEL 30

Figure 2. Monthly snapshots of mass distribution on land and in the oceans derived from observed degree-1 deformation. The left panel shows (top to bottom) 1 January through 1 June, and the right panel from 1 July through 1 December. Different scales are used for land and ocean distributions, as there is a factor of ten more load variation on land than in the oceans.

BLEWITT AND CLARKE: EARTH’S SHAPE WEIGHS SEA LEVEL 31

25 Geoid Height 20 (a) North Atlantic Relative Sea Level

) 15 Geocentric Sea Level

m Ocean Bottom

m 10 (

l Geopotential Height e 5 v e L 0 a e

-5 e v

t -10 a l e R -15 -20 -25 25 0 30 60 90 120 150 180 210 240 270 300 330 360 20 (b) South Atlantic Phase of the Year (deg) 15 )

m 10 m (

l e 5 v

e L 0 a

e S

-5 e v i t -10 a

l e

R -15 -20 -25 25 0 30 60 90 120 150 180 210 240 270 300 330 360

20 (c) North Pacific Phase of the Year (deg)

15 ) m 10 m

(

e 5

v e L 0 a e

S -5 e v

i t -10 a l

e R -15 -20 -25 0 30 60 90 120 150 180 210 240 270 300 330 360 Phase of the Year (deg)

Figure 3. Seasonal variation in the constituents of sea level at three sample locations showing very different behavior: (a) North Atlantic (45°N, -30°E), where sea level variation is smaller than geoid variation due to less total oceanic water (which controls the sea surface geopotential) in Northern Hemisphere winter when gravitational attraction from land is largest; (b) South Atlantic (-45°N, -10°E), where geoid variation and oceanic water are in phase, causing larger variations in sea level; (c) North Pacific (45°N, -170°E) where competing effects approximately cancel the annual variation, so semi-annual variation dominates. BLEWITT AND CLARKE: EARTH’S SHAPE WEIGHS SEA LEVEL 32

25 Arctic 20 Antarctic 15 ) Global Mean m 10 m (

e 5 v e L 0 a e S

-5 e v i t -10 a l e

R -15 -20 -25 0 30 60 90 120 150 180 210 240 270 300 330 360 Phase of the Year (deg)

Figure 4. Seasonal variation in relative sea level compared between the Arctic (75°N, 0°E) and Antarctic (-75°N, 180°E). As would be expected for seasonal forcing, the phase is opposite between these locations. The difference in amplitude can be explained almost entirely by the annual variation in mean sea level, which in turn is caused by the asymmetry in continental area between the Northern and Southern Hemispheres.

BLEWITT AND CLARKE: EARTH’S SHAPE WEIGHS SEA LEVEL 33

15 This study T/P-Steric (1) T/P-Steric (2) 10 T/P-Steric (3) )

m GEOS-1

m CDAS-1 (

l 5 e v e L

e 0 S

n a e -5 M

l a b o l

G -10

-15 0 30 60 90 120 150 180 210 240 270 300 330 360 Phase of the Year (deg)

Figure 5. Seasonal variation in geocentric global mean sea level, comparing the results of this study to those derived from TOPEX/ Poseidon data minus a steric model: (1) Chen et al. [2002], (2) Chen et al. [1998], (3) Minster et al. [1999]. Our geocentric results are with respect to the center of figure (CF) frame [Blewitt, 2003]. Also shown for comparison are two hydrological predictions of relative global mean sea level calculated from the NCAR CDAS-1 model [Chen et al., 2002] and the NASA GEOS-1 model [Chen et al., 1998], although other hydrological predictions not shown here can differ in amplitude by a factor of three [Chen et al., 2002].

BLEWITT AND CLARKE: EARTH’S SHAPE WEIGHS SEA LEVEL 34

15 1-Mar 1-Jun 10 1-Sep

) 1-Dec m m

( 5

l e v e L

a 0 e S

c i r t n

e -5 c o e G -10

-15 120 150 180 210 240 270 Longitude (deg E)

Figure 6: Snapshots of the longitudinal profile in equatorial sea level across the Pacific Ocean, showing a seasonal “see-saw” effect.