New Solutions for the Geoid Potential W0 and the Mean Earth Ellipsoid Dimensions

Journal of Geodetic Science • 3(4) • 2013 • 258-265 DOI: 10.2478/jogs-2013-0031 •

New solutions for the geoid potential W 0 and the Mean Earth Ellipsoid dimensions Research Article

L. E. Sjöberg

Royal Institute of Technology, Stockholm, Sweden

Abstract: Earth Gravitational Models (EGMs) describe the Earth’s gravity field including the geoid, except for its zero-degree harmonic, whichisa scaling parameter that needs a known geometric distance for its calibration. Today this scale can be provided by the absolute geoid height as estimated from satellite altimetry at sea. On the contrary, the above technique cannot be used to determine the geometric parameters of the Mean Earth Ellipsoidal (MEE), as this problem needs global data of both satellite altimetry and gravimetric geoid models, and the standard technique used today leads to a bias for the unknown zero-degree harmonic of the gravimetric geoid height model. Here we present a new method that eliminates this problem and simultaneously determines the potential of the geoid (W0) and the MEE axes. As the resulting equations are non-linear, the linearized observation equations are also presented.

Keywords: Geoid datum • global vertical datum • Mean Earth Ellipsoid • reference ellipsoid © 2013 L. E. Sjöberg, licensee Versita Sp. z o. o. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs license, which means that the text may be used for non-commercial purposes, provided credit is given to the author.

Received 20-06-2013; accepted 16-10-2013

1. Introduction The event of satellite altimetry in the 1970s provided a tool for the realization of a GVD as being the equipotential surface of the Earth’s gravity field that minimizes the sea surface topography The level surface of the Earth’s gravity field defined by the undis- (SST) all over the oceans in a least squares sense (Mather 1978). This turbed sea level is the Gauss-Listing definition of the geoid (Gauss leads to Approach I below, which implies a direct integration of 1828; Listing 1873). Choosing the geoid as the global vertical da- satellite altimetry derived sea surface topography (SST; frequently tum (GVD) implies that the datum is defined by the potential (W ) 0 also denoted Dynamic Ocean Topography) combined with the po- of this particular level surface of the Earth’s gravity field. tential of an Earth Gravitational Model (EGM) all over the oceans. In contrast, Approach II consists in using the same data to first deter- Frequently, the normal potential U1 at the selected reference el- mine the size of the axes of the globally best fitting ellipsoid to the lipsoid; e.g., Geodetic Reference System 1980 (GRS80), is defined geoid surface (called the Mean Earth Ellipsoid; MEE; Heiskanen and to be equal to that of the geoid, which is not well known. Then the Moritz 1967, p. 214), followed by determining W0 from the result. problem is that U1 will not be precise enough for today’s need for A major problem with Approach II is that satellite altimetry is only defining W0, as the data has considerably improved and mean sea successful over the oceans, while the method requires global data. level has been increasing by the order of 1.7 mm/year in the mean Sanchez (2012) reviews the development in the field with lots of during the 1900s and accelerating to more than 3 mm/yr. today references. (e.g., Nicholls and Cazenave 2010). That is, although U1 may be kept fixed to that of GRS80 reference system, the geopotential at In Sections 2 and 3 the two approaches will be reviewed and a short the geoid (W0) frequently needs a realization that better agrees discussion is provided on some of their problems, and in Sect. 4 the with the Gauss-Listing definition. second approach is presented under the consideration that that Journal of Geodetic Science 259 the zero-degree harmonic for the EGM derived geoid model is un- surface spherical harmonic of degree n and order m. One notices known. This implies also that while the above approaches assume that there are no first-degree terms in Eq. (2), which implies that that both the geocentric gravitational constant and the Earth’s the origin of the coordinate system is selected at the Earth’s grav- mean daily angular velocity are known (fixed), we will assume that ity centre. In a similar way a normal gravity field potential can also the former constant is only approximately known. Sect. 5 con- be expressed as a harmonic series of a gravitational potential plus cludes the paper. the rotation potential:

2. Approach I: Direct determination of W0 from satellite altimetry and an EGM U (r, θ, λ) = [ ( ) ] GM ∑nmax R n ∑n 2.1. Geometric and gravimetric geoid heights 1 1 + B Y (θ, λ) + Φ (r, θ, λ) , r r nm nm n=2 m=−n By satellite positioning the geodetic height h of the Earth’s surface (3) above the reference ellipsoid can be determined. Assuming also that the orthometric height (H) is known, the geometric determi- whose potential is constant (say, U ) on a chosen (level) reference nation of the geoid height becomes 1 ellipsoid with mass M1. Then one obtains the following series for the disturbing potential: Nh = h − H. (1)

T (r, θ, λ) = W˜ (r, θ, λ) − U (r, θ, λ) For land areas this technique for geoid height determination is usu- ( ) GM ∑∞ R n ∑n ally called GNSS/levelling, where h is determined by GNSS technol- = 1 C Y (θ, λ) , (4) r r nm nm ogy and H is the orthometric height determined by levelling and n=2 m=−n gravity. At sea h is the geodetic height of mean sea level determined from satellite altimetry, while H is the SST, which practically where Cnm = Anm −Bnm. In practice, one tries to choose the con- is either ignored, derived by some oceanographic method or esti- stant GM1 as the best available estimate of the geocentric gravi- mated from satellite altimetry and a preliminary geoid model. Im- tational constant. From now on the disturbing potential estimate portantly, for land applications Eq. 1 suﬀers from inherited system- from the EGM in Eq. (4) will be denoted by T EGM . atic errors, primarily biases, in the levelling networks, which make A direct way to estimate the geoid potential (W0) is to apply Eq. (2) the formula less useful (or even useless) for solving the GVD prob- at the radius vector rg of the geoid (e.g., Dayoub et al. 2012): lem (but useful for transformations from the GVD to local height systems; e.g., Sjöberg 2011). For ocean areas Eq. 1 is most impor- ( ) Wˆ g = Wˆ 0 = W˜ rg, θ, λ , (5) tant despite of the fact that the SST is frequently unknown and simply neglected. This is because, except for the long-wavelength gravity field features as determined by satellite data, there is very where sparse gravity related data available from other sources at sea. rg = rg (θ, λ) = r1 (θ) + N (θ, λ) . (6) As an alternative, the geoid height can also be estimated gravi- metrically from an EGM, such as EGM2008 (Pavlis et al. 2012). Ne- Here r1 (θ) is the geocentric radius vector of the defined reference glecting the atmosphere, the Earth’s gravity potential outside the ellipsoid and N (θ, λ) is the related geoid height (which we fre- topographic masses can be represented as an external type series quently will abbreviate with N), which can be estimated geomet- of spherical harmonics: rically from satellite altimetry in ocean areas according to Eq. (1). Alternatively, one may start from applying Bruns’ formula for the

normal potential at the geoid (Ug; Hesikanen and Moritz 1967, p. W˜ (r, θ, λ) = 84) : [ ( ) ] ∑nmax ∑n n+1 GM1 R T = W − U = W − (U − γ N), (7) 1 + A Y (θ, λ) + Φ (r, θ, λ) , g 0 g 0 1 1 r nm r nm n=2 m=−n which leads to (see also Sacerdote and Sanzó 2004) (2)

W = U − γ N + T , (8) where (r, θ, λ) are the geocentric radius, co-latitude and longitude 0 1 1 g of the computational point, GM1 is an adopted value for the geocentric gravitational constant, Φ is Earth’s rotational potential, each where U1 (= constant) and γ1 are the normal potential and grav- spectral potential component Anm is determined from a global set ity on the reference ellipsoid, respectively. [Actually, γ1 = γ1 (θ), of gravity related data by harmonic analysis up to the chosen de- but we will frequently just use the short notation. In practise, EGM EGM gree and order nmax at the reference radius R and Ynm (θ, λ) is the T = T (r1 (θ) , θ, λ) can be used for representing Tg, 260 Journal of Geodetic Science

i.e. the approximation error of using r1 (θ) for rg is usually negli- where subscript P refers to surface point P and γ¯P is the mean gible.] From Eq. (8) it thus follows that the geoid height above the normal gravity along the normal height at P. reference ellipsoid is given by In a similar way the estimator of Eq. (5) can be averaged over the ocean areas. T EGM ∆W N = − 0 , (9) γ γ 1 1 2.3. Discussion

where ∆W0 = W0 − U1. Eq. (9) shows that in general the geoid If the integration area of Eq. (12) were the whole sphere, the dis- height EGM turbing potential determined by the EGM would vanish and the EGM T N = , (10) estimator would be U1 minus the global average of the geomet- γ1 ric geoid height (timesγ). In all other cases the solution depends determined by the EGM, lacks the unknown correction −∆W /γ , 0 1 on the EGM, which includes both commission and omission errors. which must be determined from geometric data (e.g., by satellite For instance, when σ1 is the area of the oceans, one can expect that altimetry at sea; see Sect. 2.2). For a detailed discussion on the de- the rms geoid error of EGM2008 complete to degree 2159 is about termination of the absolute geoid height from an EGM, see Smith 5-6 cm (Pavlis et al. 2012), corresponding to an uncertainty in W0 (1998). of about 0.5 – 0.6 (m/s)2, and this value is in agreement with nowa- It is important to remember that on the continents Tg is the dis- day’s accuracy in determining W0 (Sanches 2012). turbing potential inside the topographic masses, and its computa-

tion therefore needs a correction for the topography. That is, the 3. Approach II: Joint determination of W0 and the MEE parameters harmonic series for the geoid height in Eq. 2a needs a correction for the analytical downward continuation error or topographic bias 3.1. Introduction EGM of Tg (Sjöberg 1977 and 2007; Martinec 1998, Sects. 7.3-7.4; Ågren 2004), which is -5 cm for the zero-degree harmonic (Sjöberg Here the geometric approach to determine W0 will be presented 2001). under the assumption that the mean angular velocity of the Earth’s daily rotation (ω) is known, and the problem is to estimate both the dimensions [i.e., semi-major and -minor axes a and b (or eccentric- 2.2. Direct estimates of W0 ity e) of the globally best fitting ellipsoid= ( MEE)] as well as the

Equations (5) and (8) are the bases for the direct determination of geoid potential W0 in a joint adjustment from the available geoid the geoid potential. As one estimator Wˆ g of Eq. (5) is directly aver- surface estimates derived from satellite altimetry and an EGM. The

aged over the ocean areas. In two other applications one may take ideal normal potential U0 of the MEE is given by four parameters, advantage of the geodetic height determined by satellite altimetry namely a, b, GM and ω (e.g., Heiskanen and Moritz 1967, Eq. 2- by Eq. (1) and the disturbing potential (determined by an EGM) at 61): the surface point on the undisturbed sea-level (assumed to be the geoid surface), possibly corrected for SST, yielding the following √ GM a2 − b2 ω2a2 result for a first order Taylor expansion (e.g., Sacerdote and Sansó U0 = √ arctan + , (14) a2 − b2 b 3 2004): W = U + T = U − γh + T EGM , (11) 0 g g 1 g g ( √ ) h 2 where hg = h−SST = N is the geoid height determined from or, by using the substitution arctan e/ 1 − e = arcsin(e), satellite altimetry. By taking the mean value over the ocean area one obtains

(σ1) of such point-wise estimates for W0, one obtains the following estimators of the geopotential at the geoid: GM ω2a2 U0 = arcsin (e) + , (15) 1. Averaged geopotential on the ocean ae 3 ∫∫ 1 ( ) ˆ EGM and this potential is also the best choice for W . As will be shown, W0 = U1 + Tg − γ1hg dσ (12) 0 σ1 the geocentric gravitational constant GM is not part of the adjust- σ1 ment, (but it could be indirectly estimated from the adjustment re- and sults). The general computational procedure is outlined below.

2. Minimized SST (Sacerdote and Sansó 2004): Similar with Sect. 2, first the normal potential U1 of a prelimi- [ ] nary√ reference ellipsoid with geometric parameters a1 and b1 = ∫∫ T EGM −γh 2 P P a1 1 − e and geocentric gravitational constant GM1 is given 2 dσ 1 γ¯P ˆ σ1 by: W0 = U1 + ∫∫ [ ] , (13) 1 2 2 2 dσ GM1 ω a1 γ¯P σ1 U1 = arcsin(e1) + , (16) a1e1 3 Journal of Geodetic Science 261 where all parameters are chosen. The radius vector of the surface as those expressed by an EGM. However, as stated above, satellite of this reference ellipsoid can be written altimetry can provide a geometric estimate of the absolute geoid height over the oceans under the assumption that the SST is known √ 2 2 with suﬃcient accuracy, but such an incomplete integration area r1 (β) = a1 1 − e sin β, (17) 1 for J in Eq. (19a) would only lead to the best fitting reference ellip-

soid and W0 estimated for the ocean areas. Dayoub et al. (2012) where β is the reduced latitude. applied the above technique, and they compared the preliminary According to Heiskanen and Moritz (1967, p. 214), the MEE, is the geoid surfaces given by EGM2008 and a satellite altimetry model ellipsoid, whose mass is the same as that of the real Earth (requires in coastal areas and concluded that the two surfaces agree well that M1 is assumed to be the Earth’s mass), and the axes are such (without specifying the magnitude of the agreement), and they that the global mean square of the geoid height (N) is a minimum: directly filled-in the geoid heights for the land areas by EGM2008 ∫∫ geoid heights. However, the EGM derived geoid height needs the 1 N2dσ = min .(a, e), (18) correction −∆W0/γ1 of Eq. (9), which must be estimated. If the 4π correction is fixed to a preliminary value (which apparently was σ the case in Dayoub 2012, Dayoub private com.), it means that also where σ is the unit sphere. The best known value for GM has a W0 = U1 + ∆W0 has been (more or less) fixed. However, to standard error of the order of 0.8 m3s−2 (Groten 2004), which cor- avoid fixing W0 to an a priory value, the problem can be solved by 2 −2 augmenting the target function of Eq. (19a) by the unknown pa- responds to uncertainties in W0 and the geoid height of 0.1 m s rameter ∆W0. This implies that a, e and ∆W0 are determined in and 1 cm, respectively. As the present uncertainty in W0 is about 0.5 m2s−2 (e.g. Sanchez 2012), the fixing of GM to the best known a combined adjustment. value may improve W0 by adjusting just for the ellipsoidal geome- try parameters by the following geometric approach. This implies 3.2. The combined adjustment approach that once a and e have been fixed, U (and thereby W = U ) 0 0 0 In the previous section the target function J was based on the as- follows from Eq. (15). sumptions that the geocentric gravitational constant is known and However, to be more precise in the approach to follow, it is not agrees with that of the normal potential U1 , and the estimated the integral in Eq. (18) that is to be minimized, but it is the mean geoid surface is continuous and known all over the Earth. In real- square discrepancy between the radius vector of the geoid sur- ity, we have seen that neither of these assumptions is correct. From face estimated by r (β) + N , where r (β) is the radius vector 1 1 satellite altimetry the geoid height is known only over the oceans, of the reference ellipsoid related with the geoid estimate N, and and the EGM geoid height lacks the term∆W0/γ1 as presented in the radius vector r (a, e, β) of a general reference ellipsoid that E Eq. (9). In the approach that follows we are not primarily concerned should be optimized. Hence, mathematically the problem could with the unknown GM, but it is suﬃcient to consider the extra un- be expressed: known x = −∆W0 = U1 − W0. We will assume that the refer- ∫∫ ence ellipsoids for Nh and NEGM are the same (with geometric 1 2 J = [r (β) + N − r (a, e, β)] dσ = min .(a, e), parameters a1 and e1), and radius vector given by Eq. (17). Then 4π 1 E σ the augmented target function reads: (19a) where √ I = pI1(a, e) + (1 − p)I2 (x, a, e) + I3(x, a, e) = min(x, a, e), 2 2 rE (β) = a 1 − e sin β. (19b) (21a) where Once the ellipsoidal parameters a and e of the MEE have been fixed ∫∫ by solving Eq. (19a), the normal potential at the MEE, i.e. U0 of [ h ]2 Eq. (15), can be computed, provided that GM is ( suﬃciently well) I1(a, e) = N + r1 (β) − rE (a, e, β) dσ (21b) known, and this value should also be the estimate for the geopo- σ1 tential value at the geoid, i.e. ∫∫ [ EGM ]2 I2(x, a, e) = x/γ1 + N + r1 (β) − rE (a, e, β) dσ W = U . (20) 0 0 σ1 ∫∫ (21c) [ EGM ]2 However, one problem with this approach is that the present-day I3(x, a, e) = x/γ1 + N + r1 (β) − rE (a, e, β) dσ. uncertainty in GM contributes to about 20% of the uncertainty σ2 in W0 (see Groten 2004). In addition, the main problem to opti- (21d) mize the target function J is that the absolute geoid height is not Here σ1 and σ2 are those parts of the unit sphere that are covered well known globally, but there are only relative geoid models, such by ocean and land, respectively, x = −∆W0 = U1 − W0 (see 262 Journal of Geodetic Science

Eq. 9) and rE (a, e, β), given by Eq. (20), is the radius vector of the region σ1, these diﬀerences must be weighted in one way or an- MEE , whose parameters a and e are unknown and p is a fixed num- other. The mathematical formulation of the problem is given by ber in the range 0 6 p 6 1 that weighs the contributions from the target function described in Eqs. (21a)-(21d). Nh and NEGM over the ocean areas. Solution: The solution follows from Eqs. (22). If p is set to 0, implying that only EGM data are used, it is shown 4. Approach II: practical solutions by linearization and iteration in Sect. 4.1.1 that the solution is singular. This is obvious, as in this case one tries to solve the problem with only relative geoid heights Below we present two explicit solutions (by linearization and iter- given by the EGM. Alternatively, if p =1 (i.e. only the satellite al- ation) to the system of equations given by Eqs. (22). timetry derived geoid height is employed over the oceans, while the EGM is utilized only over land), the solution discards the infor- 4.1. Solution by linearization mation from the EGM over the oceans. On the contrary, below we suggest that in the application of Eqs. (21a)-(21d) the choice of A first order solution of Eq. (22) is obtained by linearization of the 2 2 radius vector to p should be based on the a priori variances κ1 and κ2 of the satel- rE lite altimetry( and EGM) derived geoid heights, respectively, yielding p = κ2/ κ2 + κ2 . 2 2 1 2 rE (β) ≈ a − g sin β, (25) The least squares condition for the unknowns x, a and e, as speci- fied by Eq. (21a), is satisfied by the three equations where g = ae2/2. (26) ∂I ∂I ∂I = 0, = 0 and 2 = 0, (22) ∂x ∂a ∂e Introducing the abbreviate notations

from which the unknowns can be determined, provided that the R h = r + Nh (27a) equations are independent. 1

If xˆ is the solution for x, and U1 is the a priori value for the ellipsoidal normal potential, the geoid potential estimate finally follows and EGM EGM from R = r1 + N , (27b)

Wˆ 0 = U1 − x,ˆ (23) one obtains the following system of equations (to order e2) by tak-

and by re-inserting the estimates for W0 = U0 , a and e into ing the derivatives of I w.r.t. x, a, and g and equating to zero: Eq. (15), a new estimate for GM becomes

c11x + c12a + c13g = f1 ( ) aêˆ GMˆ = Wˆ − aˆ2ω2/3 c x + c a + c g = f (28) 0 arcsin (eˆ) 21 22 23 2 ( ) c x + c a + c g = f , ( ) eˆ2 31 32 33 3 ≈ Wˆ − aˆ2ω2/3 aˆ 1 − , (24) 0 6 where but this estimate is probably poor compared to estimates based on ∫∫ ∫∫ satellite laser ranging, etc. (e.g. Ries et al. 1992). dσ dσ c11 = (1 − p) 2 + 2 , This concludes the principle of the combined approach. γ1 γ1 σ σ ∫∫1 2∫∫ dσ dσ The assumptions, data, the problem and its solution can be sum- c12 = (1 − p) + p = c21; (29a) γ1 γ1 marized as follows: σ1 σ2 Given (fixed parameters): ω and a reference/level ellipsoid with di- √ 2 ∫∫ ∫∫ mensions a1 and b1 = a1 1 − e1 as well as a normal grav- sin2 βdσ sin2 βdσ ( ) ity field with normal potential U1 = constant and normal gravity c13 = 1 − p + = c31; c22 = 4π; γ1 γ1 γ1 = γ1(β) on the surface of the ellipsoid. σ1 σ2 EGM h (29b) Observations: N (globally) and N (on the oceans; σ1 ); both types of geoid heights refer to the defined reference ellipsoid. ∫∫ Problem: Minimize the global mean square difference between ra- 2 2π c23 = − sin βdσ = − = c32; dius vectors of the surfaces of the geoid and an arbitrary reference 3 ∫∫ σ ellipsoid (with parameters a and e) by varying these parameters 2π c = sin4 βdσ = (29c) and the additional unknown x(= U1 − W0) until the minimum 33 5 is reached. As there are two types of differences available in the σ Journal of Geodetic Science 263 [ ] ∫∫ h EGM √ (1 − p)R + pR ∂rE 2 2 f1 = − dσ; ra = = 1 − E sin β (34c) γ1 ∂a e=E σ1 ∫∫ ∫∫ and [ h EGM ] EGM f2 = (1 − p) R + pR dσ + R dσ [ ] ∂r AE sin2 β σ1 σ2 E re = = − √ . (34d) 2 2 (29d) ∂e a=A,e=E 1 − E sin β

Introducing the residual geoid heights from satellite altimetry and and the EGM: ∫∫ 2 [ h EGM ] f3 = − sin β pR + (1 − p) R dσ. (29e) h h EGM EGM dR = r1 + N − r0 and dR = r1 + N − r0, (35) σ1 respectively, the target function I of Eq. (21a) can be written By solving Eq. (28) the solutions for x, a and e are obtained. How- ever, one should bear in mind that the linear solution will only give I = pI1 (dA, dE) + (1 − p) I2 (x, dA, dE) + I3 (x, dA, dE) a rough solution to the problem. = min (x, dA, dE) , (36a) 4.1.1. Special case where ∫∫ Consider the case with p =0. Then [ h ]2 I1 = dR − radA − redE dσ (36b) ∫∫ σ sin2 β 1 c13 = dσ. (30) ∫∫ [ ]2 γ1 EGM σ I2 = dR + x/γ1 − radA − redE dσ (36c)

σ1

From Heiskanen and Moritz (1967; Eq. 2-126), we realize that 1/γ1 and can be approximated to first order by ∫∫ [ EGM ]2 I3 = dR + x/γ1 − radA − redE dσ. (36d) 1 2 ≈ k0 + k2 sin β, (31) σ2 γ1 The least squares solution for I is obtained by equating its deriva- where k0 and k2 are constants, and therefore tives w.r.t. x, dA and dE to zero. This leads to the following matrix system of equations: c13 ≈ −k0c23 + k2c33. (32)       Fxx Fxa Fxe x fx       Hence, the coeﬃcients for the unknown g of Eq. (28) are lin-  Fxa Faa Fae   dA  =  fa  , (37) early dependent, implying that the system of equations is singular. Fxe Fae Fee dE fe Therefore the system has no solution for p = 0. where ∫∫ ∫∫ 4.2. Solution by iteration dσ dσ F = (1 − p) + ; xx γ2 γ2 The linear solution in Sect. 4.1 can provide the initial values for x, a 1 1 σ1 σ2 and e (denoted x, A and E) in an iterative solution. Alternatively, ∫∫ ∫∫ ra ra the initial values for A and E are taken from a Geodetic Reference Fxa = − dσ − p dσ (38a) γ1 γ1 System, e.g., GRS1980, and x is initially set to zero. Then the MEE σ1 σ1 radius √ 2 2 rE = a 1 − e sin β (33) ∫∫ ∫∫ re re Fxe = − dσ − p dσ; can be expanded to first order as γ1 γ1 σ σ ∫∫ 1 1 F = r2dσ; rE = r0 + radA + redE, (34a) aa a ∫∫σ 2 where √ Fee = re dσ (38b) 2 2 r0 = A 1 − E sin β (34b) σ 264 Journal of Geodetic Science ∫∫ 4π F = r r dσ = − AE (38c) ae a e 3 σ References and Ågren J., 2004, The analytical continuation bias in geoid de- ∫∫ dR EGM termination using potential coeﬃcients and terrestrial gravity fx = − dσ, data. J Geod 78, 314-332 γ1 ∫∫σ ∫∫ [ ] h EGM EGM Bursa M., Kouba J., Kumar M., Mueller A., Radej K., True S. fa = ra pdR + (1 − p) dR dσ + radR dσ C., et al., 1999, Geoidal geopotential and World Height System. σ1 σ2 (38d) Stud. Geophys. Geod., 43, 327-337

Dayoub N., Edwards S.J. and More P., 2012, The Gauss- ∫∫ ∫∫ [ h EGM ] EGM Listing potential value W0 and its rate from altimetric mean fe = re pdR + (1 − p) dR dσ + redR dσ. sea evel and GRACE. J Geod 86, 681-694 σ1 σ2 (38e) Gauss F.W., 1828, Bestimmung des Breitenunterschiedes After each iteration, A and E are updated to the next step by zwischen den Sternwarten von Göttingen und Altona durch Beobachtungen am Ramsdenschen Zenithsector, Vander- A = A + dA and E = E + dE. (39) schoeck und Ruprecht, Göttingen, 48-50.

Groten E., 2004, Fundamental parameters and current After the iteration is stopped, the final estimate for W0 is given by (2004) best estimates of the parameters of common relevance Eq. (24), i.e. to astronomy, geodesy and geodynamics. J Geod 77, 724-731 Wˆ0 = U1 − xˆ (40) Heiskanen W.H. and Moritz H., 1967, Physical Geodesy, W with xˆ being the final estimate for x. H Freeman and Co., San Francisco and London 5. Conclusions Listing J.B., 1873, Über unsere jetzige Kenntnis der Gestalt The geoid potential W0 can be directly estimated as a correction und Grösse der Erde. Nachr. d Kgl Gesellschaft d Wiss und der to the normal potential of the reference ellipsoid and the oceanic Georg-August-Univ, Göttingen, 33-98. average of the diﬀerence between geoid heights from satellite altimetry and an EGM. Systematic errors in the data propagate into Martinec Z. (1998): Boundary-value problems for gravi- biased solutions, which are therefore sensitive to the chosen data. metric determination of a precise geoid. Lecture Notes in Earth Typical systematic error sources are lacking SST information for the Sciences 73, Springer satellite altimetry (which locally may reach several decimetres to metre) and truncation error in the EGM derived geoid height. The Mather R. S., 1978, The role of the geoid in four-dimensional eﬀects of these error types will change with the area of integration. geodesy. Marine Geodesy 1, 217-252 The latter error source would vanish, if the averaging is extended to the whole Earth, but that would require geometric geoid heights Nicolls R. J. and Cazenave A., 2010, Sea-level rise and its in continental regions, which are prone to other types of biases. impact on coastal zones. Science 328, 1517-1520 The above technique cannot be used to estimate the axes of the

MEE, but their relations can be conditioned by Eq. (16) once U0 = Pavlis N. A., Simon A. H., Kenyon S. C.and Factor J. K., 2012, The

W0 has been fixed. Development and Evaluation of the Earth Gravitational Model

The alternative technique to solve for W0 is to first determine the 2008 (EGM2008). JGR 117, B04406 axes of the MEE and then find the geoid potential. The main problem with this technique is that it propagates a bias from the un- Ries J.C., Eans R.J., Shum C.K. and Watkins M.M., 1992, known zero-degree harmonic of the gravimetric geoid height into Progress in the determination of the gravitational coeﬃcients the solution. This problem is solved by the new technique dis- of the Earth, GRL 19(6), 529-531

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