On the Definition and Realization of a Global Vertical Datum Research Article

Journal of Geodetic Science

• 1(3) • 2011 • 154-157 DOI: 10.2478/v10156-010-0018-z •

L.E. Sjöberg

Royal Institute of Technology, Division of Geodesy, Stockholm, Sweden

Abstract: A Global Vertical Datum (GVD) is naturally defined by the geoid, and there is a well-established consensus to adopt Gauss-Bessel-Listing's definition of the geoid (i.e. as being the level surface of the Earth's gravity field that best fits the undisturbed sea level). The main problem in defining the geoid is therefore to fix its constant geopotential W . Nevertheless, this definition can be interpreted as to fit either ( 0) the geopotential of sea surface to a constant W , or to minimize the height of sea level with respect to the geoid. Although the two ( 0) interpretations lead to apparently different solutions, we show that they are practically the same. To improve the estimation of W , we 0 propose to weight the included data according to their a priori error estimates. Finally we discuss the use of GNSS/levelling data for vertical datum connections, concluding that such data, although indispensable for regional vertical datum connections, in combination with satellite altimetry over the oceans are practically useless for determining the GVD. Also, such a joint adjustment of the GVD and regional vertical datum biases yields inferior local connections vs. a separate adjustment with fixed GVD.

Keywords: Global vertical datum • GVD • geoid • datum uniﬁcation © Versita Warsaw and Springer-Verlag Berlin Heidelberg.

Received 21 January 2011; accepted 8 March 2011

1. Introduction surface becomes just a regional geoid model, which typically differs by 1-2 m from the geoid (e.g., Vanícˇek 1987, p 174). If more than one station is used to define the datum, there will also be The Global Navigation Satellite System (GNSS) is a 3D system coor- inconsistencies/systematic errors in the network. dinated with the high-precision International Terrestrial Reference Similar to the unification of regional/local horizontal datum into Frame (ITRF), whose positioning accuracy of included stations is a global datum, GNSS calls for a consistent, high-accuracy GVD. A at the (several) mm level. Unfortunately ITRF is only a 3D system natural choice of a GVD is the geoid, being a level surface of the in Cartesian coordinates, lacking the connection to actual vertical Earth's gravity field. Although any equipotential surface could be datum. Hence, the vertical component of GNSS is governed by selected as the zero-level for the GVD, the choice of the geoid is various regional and national systems, whose origins typically vary best in concordance with heights above sea level as visualized, e.g. by 1-2 metres from each other. The natural datum for a Global on topographic maps. In this way the problem is mainly reduced Vertical Datum (GVD) is the geoid, which can be defined in various to that of defining the constant potential of the geoid. ways. Historically a geoid model and/or a national/regional height W0 datum is frequently defined by the undisturbed mean sea level However, it is not self-evident which level is the geoid. Already (MSL) at one or several tide gauge stations. The major problem Gauss (1828), Bessel (1837) and Listing (1873) expressed a natural with such a realization of the geoid is that MSL differs systemati- definition of the geoid (Bursa et al. 1999), namely as being the level cally from a level surface of the Earth's gravity field/the geoid; in surface of the Earth's gravity field, which best fits the undisturbed particularly in coastal areas, and therefore the resulting reference mean sea level. Although this definition seems unique, it may be Journal of Geodetic Science 155 interpreted in various ways as we show below. Then we discuss its In Sacerdote and Sansó (2004) and Sanches (2009) mean gravity is realization. approximated by normal gravity on the reference ellipsoid.

It should be mentioned that for the evaluation of the geopotential WP can be estimated from satellite altimetry and an Earth Gravi- there are three additional fundamental constants, namely the tational Model (EGM): geocentric gravitational constant, the mean angular velocity of the Earth's rotation and the second zonal Stokesian harmonic W U − γ h T ; (5) P = 0 0 P + P coefficient, but they are well determined and adopted, e.g. by GRS80, and they will not be further discussed in this paper. where and are normal potential and gravity at the reference U0 γ0 ellipsoid, while hP and TP are geodetic height and disturbing 2. Two interpretations of the geoid potential at P: Here hP is determined by satellite altimetry, while T is obtained from an EGM. W deviates from the ideal Let the gravity potential of the sea level at point P be WP . Due P P equipotential surface W W mainly for two reasons: a) MSL to various static and dynamical physical effects (caused by tides, = 0 salinity, currents, dominating winds and air pressure, etc.) the deviates systematically from the geoid, both in space and time, potential of the Sea Surface Topography (SST, i.e. the orthometric due to the dynamic effects and eustatic sea level rise, and b) there height of the sea surface) is not a level surface of the Earth's gravity are erroneous data in both satellite altimetry and the EGM. The systematic errors of type a) are of the order of m in field, but differs from point to point over the sea. The gravity ±(1 − 2) potential of the geoid W is then defined by the integral (e.g. coastal regions but much smaller in the open sea, while the errors ( 0) Bursa et al. 1999) of type b) are only of the order of a few centimetres in satellite altimetry data and of the order of a few decimetres in the EGM ZZ (dependent on the choice of EGM). Also, as MSL is not fixed in time, W − W 2dσ :; (1) ( 0 P ) = min but SST increases by some 2-3 mm/yr, the geoid, and thereby the Ω GVD, defined by the constant , should be fixed in time, or, at W0 where and are the unit sphere and its part covered by the sea, least, time-tagged. σ Ω respectively, yielding the estimate If there were no systematic effects in WP , MSL would coincide with the geoid. In this case Eq. (3) would practically imply that W0 ZZ is defined by minimizing the MSE of the geoid height over the sea. W 1 W dσ (2a) ˆ 0 = P Ω Ω 3. Further comparisons of the two interpretations with the Mean Square Error (MSE) The mean gravity at point , deviates less than 1 from P; g¯P ‰ normal gravity on the reference ellipsoid γ , so that Eq. (4a) can ZZ ( 0) 2 be approximated by (cf. Sacerdote and Sansó 2004) s2 1 W − W dσ: (2b) = ˆ 0 P 1 Ω Ω RR WP dσ γ2 0 Alternatively (cf. Sacerdote and Sansó 2004 and Sánchez 2009), Wˆ ≈ Ω : (6) 0 RR 1 dσ W may be defined as the level surface that minimizes the SST, γ2 0 0 yielding the condition Ω As increases by approximately 5 from the equator to the γ0 ‰ ZZ W − W 2 pole, Eq. (6) implies that W is down-weighted by up to 1% ( 0 P ) dσ :; (3) P 2 = min towards the pole. However, this weighting is only theoretical and g¯P Ω has no practical meaning. This conclusion can be drawn from the where is mean gravity between the geoid and sea surface at following reasoning. g¯P Using the simplified notation −2, the difference between point P. Then the solution becomes q = g¯p the two interpretations of the GVD of Eqs. (2a) and (4a) can be RR WP dσ written 2 g¯P Wˆ Ω ; (4a) 0 = RR 1 dσ RR 2 q WP − Wˆ dσ g¯P 0 Ω dW Wˆ − Wˆ Ω ; (7) 0 = 0 0 = RR qdσ with the weighted MSE Ω

2 and by applying Schwarz' inequality one obtains Wˆ −W RR 0 P dσ 2 g¯P 2 Ω s : (4b) 2 P(2) 2 = RR 1 dσ |dW | s2 Ω ; (8a) g2 0 6 1 ¯P P(1) 2 Ω [ ] 156 Journal of Geodetic Science

where One main argument against such an approach has been that the ZZ gravimetric geoid (which is also needed in this method) cannot P(k) qk dσ: (8b) = be sufficiently well determined over the continents due to the Ω uncertainty of the topographic mass distribution. However, as As q ≈ γ−2 ≈ constant, it holds with only minor approximation shownbySjöberg(2004), eveniftheuncertaintyofthetopographic ¯0 that density is of the order of 20%, the propagated error to the gravimetric geoid will not exceed 2.2 cm for elevations within |dW0| 6 s1; (9) 1 km. Hence this problem is not a serious one. i.e. the difference between the two interpretations does not exceed However, the contribution of GNSS/leveling data in determining the standard error of . Hence, the two interpretations are Wˆ 0 the global datum is little or none. This conclusion is verified practically the same, and therefore there is no practical advantage by a simple example in the Appendix, where it is assumed that in estimating by the more complicated Eq. (4a) vs. Eq. (2a). W0 the geoid height needed in fixing the by GNSS/leveling is W0 Again, if there were no systematic effects in WP (implying that MSL purely provided by a satellite-only EGM (not deteriorated by the coincides with the geoid), then s2 and s2 would be the variances regional datum biases). Also, if the gravimetric geoid height is 1 2 of the two estimates of . W0 determined by the modification of Stokes' formula by an EGM (e.g., Sjöberg 2003), Stokes' integral is limited to a small cap around 4. A weighted mean estimator the computation point with little contribution to the gravimetric The estimation of the GVD according to Gauss-Bessel-Listing's geoid height estimate, practically implying that the above con- definition, when considering that the available geopotential at clusion holds also for this case. On the other hand, important to sea level suffers from both systematic and random errors, is say, regional datum connection can hardly be provided without (WP ) achieved in an optimum way by a weighed least squares solution, satellite positioning/leveling.

which considers the various qualities of the individual WP . Using the notation p for the weight function, the estimator and its MSE 6. Concluding remarks follow from There is a rather common agreement in the geodetic community to RR pW dσ P define the potential of the GVD according to Gauss-Bessel-Listing's W Ω ; (10a) ˆ 0 = RR geoid definition. So far it is best realized by considering data pdΩ Ω from satellite altimetry data and an EGM over the oceans. The and two versions of interpretation applied today, Eqs (2a) and (4a), RR p2s2 dσ are practically the same. By considering the errors of W in each W P MSE Wˆ Ω : (10b) point, a better estimate can be achieved by weighting the data 0 = 2 RR pdσ according to Eq. (10a). Ω Finally, GNSS/leveling data are most useful in connecting regional Eq. (10b) assumes that there are no correlations among the obser- datum or a regional vertical datum to a fixed GVD, but it will not improve the determination of the GVD. vation errors, and s2 is the MSE of WP . If p σ 2/s2 , where σ 2 W = 0 W 0 is a constant, Eq. (10b) attains its minimum: References  −1 ZZ dσ MSE Wˆ   : (10c) 0 = 2 Bessel F.W. (1837): Über den Einfluss den Unregelmäigkeiten der sW Ω Figur der Erde auf Geodätische Arbeiten und ihre Vergleichung mit den Astronomischen Beschtimmungen. Astronomische Nachrichten, T. 14, 269: 329-331. 5. On the use of data over continental areas

As 2/3 of the Earth's surface is ocean, the above techniques to fix Bursa M., Kouba J., Kumar M., Mueller A., Radej K., True S. C., et the geoid should be quite accurate. The question is whether data al. (1999): Geoidal geopotential and World Height System. Stud. over the continents could improve the result. Geophys. Geod. 43: 327-337. Since long it has been proposed to include also GNSS/leveling data in fixing (e.g., Rummel and Teunissen 1988). The advantage Bursa M., Groten E., Kouba J., Radej K., Vatrt V., Vojtísková M. W0 of this technique would be not only to use data (mainly satellite (2002): Earth's dimension specified by Geoidal Geopotential. Stud. altimetry) over the oceans, but also independent data from the Geophys. Geod. 46: 1-8. continents to determine both the global datum and regional datum biases. Bursa M., Kenyon S., Kouba J., Sima Z., Vatrt V., Vitek V., et al. Journal of Geodetic Science 157

(2007): The geopotential value for specifying the relativistic Case b) Data both on the oceans and continents W0 atomic time scale and a global vertical reference system. J. Geod. 81: 103-110. In addition to the above observations on the oceans we now assume GNSS/leveling observations ( ) on the continents, and m L2 Gauss F.W. (1828): Bestimmung des Breitenunterschiedes that there is a datum parameter y that connects the ``regional'', zwischen den Sternwarten von Göttingen und Altona durch continental datum to the global datum. The geoid height is Beobachtungen am Ramsdenschen Zenithsector, Vanderschoeck given directly from an EGM (i.e. with no contribution from Stokes' und Ruprecht, Göttingen, 48-50. formula). Then the observation equations can be written:

Listing J.B. (1873): Über unsere jetzige Kenntnis der Gestalt ! ! ! ! e 0 x L ε und Grösse der Erde. Nachr. d Kgl Gesellschaft d Wiss und der n 1 − 1 ; (A2) e e y = L ε Georg-August-Univ, Göttingen: 33-98. m m 2 2

Rummel R., Teunissen P. (1988): Height datum definition, height where ek is a k-dimensional column vector with only units, and datum connection and the role of the geodetic boundary value εk are independent observation error vectors. For simplicity we problem. Bull. Geod. 62: 477-498. assume that all observations have equal weight, implying that the normal equations for Eqs. (A2) can be written Sacerdote F., Sansó F. (2004): Geodetic boundary value problems and the height datum problem. IAG Symposia 127: 174-178, ! ! ! n m m x S S Springer. + 1 + 2 = ; (A3) m m y S2 Sánchez L. (2009): Strategy to establish a global vertical reference system. In H. Drewes (Ed.) Geodetic Reference Frames, IAG where S is the sum of observations of vector L . Hence, the least Symposia 134, 273-278. i i squares solution of the system becomes

Sjöberg L.E. (2003): A general model of modifying Stokes' formula and its least-squares solution. J Geod 77: 459-464. nS − mS x S /n and y 2 1 (A4) ˆ = 1 ˆ = nm Sjöberg L.E. (2004): The effect on the geoid of lateral density variations. J. Geod. 78: 34-39. with the variances

Vanícˇek P. (Ed.) (1987): Four-dimensional geodetic positioning - σ 2 n m σ 2 m Report of the IAG SSG 4.96. Manuscr Geod 12(4). σ 2 0 and σ 2 σ 2 + 0 : (A5) x = n y = 0 nm = m 1 + n Appendix

Here we will compare two scenarios for GVD determination: Case Case c) Regional datum connection to the ﬁxed GVD a) with n satellite altimetry observations on the oceans and Case b) Here we assume that is fixed and it is requested to estimate with m additional GNSS/leveling observations on the continents. x y from only. The least squares solution and its variance become Finally, in Case c) it is assumed that the GVD has been fixed and L2 the regional datum bias is to be fixed from GNSS/leveling.

y S /m and σ 2 σ 2/m : (A6) ˆ = 2 y = 0 Case a) Data on the oceans (only) Assuming n observations of equal weight, contained in the vector From these results we conclude that L , the least squares solution for x W and its variance become 1 = 0 · the standard error of x is the same in Cases a) and b). x S /n and σ 2 σ 2/n ; (A1) ˆ = 1 x = 0 · the regional datum connection is better in Case c) where σ 2 is the variance of unit weight, S1 is the sum of terms in vs. Case b). 0 vector L1.