Towards a Vertical Datum Standardisation Under the Umbrella of Global Geodetic Observing System Research Article

Journal of Geodetic Science • 2(4) • 2012 • 325-342 DOI: 10.2478/v10156-012-0002-x •

Towards a vertical datum standardisation under the umbrella of Global Geodetic Observing System Research Article

L. Sánchez∗

Deutsches Geodätisches Forschungsinstitut (DGFI), Munich, Germany

Abstract: Most of the existing height systems refer to local sea surface levels, are stationary (do not consider variations in time), and realise diﬀerent physical height types (orthometric, normal, normal-orthometric, etc.). In general, their accuracy is about two orders of magnitude less than that of the realisation of geometric reference systems (sub millimetre level). The Global Geodetic Observing System (GGOS) of the International Association of Geodesy (IAG), taking care of providing a precise geodetic infrastructure for monitoring the system Earth, promotes the standardisation of height systems worldwide. The main objectives are: (1) to provide a reliable frame for consistent analysis and modelling of global phenomena and processes aﬀecting the Earth’s gravity eld and the Earth’s surface geometry; and (2) to support the precise combination of physical and geometric heights in order to exploit at a maximum the advantages of satellite geodesy (e.g. combination of satellite positioning and gravity eld models for worldwide unied precise height determination). According to this, the GGOS Theme 1 ”Unied Height System” was established in February 2010 with the purpose to bring together existing initiatives and to address the activities to be faced. Starting point are the results delivered by the IAG Inter-Commission Project 1.2 ”Vertical Reference Frames” during the period 2003-2011. The present actions related to the vertical datum homogenisation are being coordinated by the working group ”Vertical Datum Standardisation”,which directly depends on the GGOS Theme 1 and is supported by the IAG Commissions 1 (Reference Frames) and 2 (Gravity Field), as well as by the International Gravity Field Service (IGFS). This paper discusses some aspects to take into consideration for the realisation of a standardised globally unied vertical reference system.

Keywords:

GGOS height system • global reference level • global vertical reference system • vertical datum standardisation • W0 reference value © Versita sp. z o.o.

Received 01-10-2012; accepted 18-11-2012

1. Introduction (ITRS, Petit and Luzum 2010) guarantee a globally unied geometric reference frame with reliability at the mm-level. An equivalent high-precise physical reference frame is missing. The existing physical height systems and all the geodetic data depending on their Studying, understanding and modelling global change requires realisation (e.g. gravity anomalies, geoid estimations, digital ter- geodetic reference frames with (1) an order of accuracy higher rain models, etc.) are usable in limited geographical areas only and than the magnitude of the eﬀects we want to study, (2) consis- their combination at regional or global levels presents discrepan- tency and reliability worldwide (the same accuracy everywhere), cies with magnitudes very much higher than the accuracy required and (3) a long-term stability (the same order of accuracy at today. They do not provide a reliable frame for consistent analysis any time). The denition, realisation, maintenance, and wide- and modelling of global phenomena related to the Earth’s gravity utilisation of the International Terrestrial Reference System eld (e.g. sea level variations from local to global scales, redistribu- tion of masses in oceans, continents and the Earth’s interior, etc.), and they are not able to support the precise combination of phys- ∗E-mail: sanchez@dgﬁ.badw.de 326 Journal of Geodetic Science

Table 1. Main characteristics and drawbacks of the existing height systems. Characteristic Drawback The reference level (1) There are so many reference lev- (zero-height surface) is els (vertical datums) as reference tide realised by the mean gauges and (2) they are related to dif- sea level measured at ferent reference epochs. individual tide gauges and averaged during Figure 1. Height datum discrepancies. different time periods. The dynamical ocean (1) Equipotential surfaces passing topography at the local through the different reference tide reference tide gauges gauges realise different (local) geoids, ways the same: to refer all existing physical heights (or geopoten- has not been taken into which are lying very close to sea sur- tial numbers) to one and the same reference surface, which must account. face (< ∼2 m) and are practically par- be realised with high precision globally. The fundamental quanti- allel to each other; but no one coin- ties of interest are the geopotential differences (δW i = W - W i , cides with a global geoid. (2) Relation- 0 0 0 ship between local geoids and between called also height datum discrepancies or vertical datum pa- them and the global one are unknown. rameters, Fig. 1) between a conventional global reference level i The vertical control (1) Vertical networks have been (W0) and the equipotential surface (W 0) realised by the mean has basically been adjusted piece-wisely, (2) systematic sea level estimated at each local tide gauge (i.e. Heck and Rum- extended by means errors significantly growth with the mel 1990, Rapp 1994, Rummel and Teunissen 1988, Sánchez 2007). of spirit levelling distance from reference tide gauge, i during many years and (3) vertical coordinates refer to dif- Most of the proposals oriented to the calculation of δW 0 are and possible vertical ferent epochs. based on the comparison of physical heights (orthometric HO or displacements have normal heights HN ) derived from levelling (reduced by gravity ef- not been taken into fects) with those computed from gravimetric (quasi)geoid models account. (N, ζ) and ellipsoidal heights (h), i.e. (cf. Fig. 1) Different gravity reduc- (1) Vertical coordinates realise dif- tions (sometimes no re- ferent physical height types (orthome- δW ( ) ( ) duction) have been ap- tric, normal, normal-orthometric, etc.) ≈ h − HO − N ≈ h − HN − ζ (1) plied to the measured and (2) the corresponding reference γ level differences. surfaces do not coincide with a proper geoid or quasi-geoid. (3) These sur- see, for instance, Hirt et al. 2011, Kotsakis et al. 2012, Tenzer et faces are height-dependent. al. 2011, Pan and Sjöberg 1998, etc. However, the combination Heights at the border (1) Reference levels and vertical co- of these variables ”as they are” partially reects the inconsisten- between datum zones ordinates are usable only in limited cies included in the determination of the different height types (Ta- present discrepancies geographical areas; (2) their combi- at the m-level. nation at regional or global level is ble 2), misrepresenting the ”best possible” values of the wanted i unsuitable. level discrepancies δW 0. This misrepresentation limits the reliability of the global vertical reference system realisation to the dm- level, being insufficient to support geodetic activities of high pre- ical and geometric heights (i.e. combination of satellite position- cision. In this context, the establishment of a precise global gravity ing with gravity eld models) for worldwide unied precise height eld-related vertical reference system is still an unresolved prob- determination. Table 1 summarises the main drawbacks of the ex- lem. The Global Geodetic Observing System (GGOS) of the Interna- isting height systems. During the last four decades, the unication tional Association of Geodesy (IAG), taking care of providing a pre- of the existing local height systems in a global one has been inten- cise geodetic infrastructure for monitoring the System Earth (Plag sively discussed. Even though different names have been used for and Pearlman 2009), promotes the standardisation of height sys- this theme, e.g. world height system (Rapp and Balasubramania tems worldwide. With the purpose to bring together existing ini- 1992), global or world vertical datum (Rapp 1983a, Balasub- tiatives and to address the activities to be faced, the GGOS Theme ramania 1994, Rapp 1995a), global vertical network (Colombo 1 ”Unied Height System” was established during the GGOS Plan- 1980), global height datum unification (Ardalan and Saffari ning Meeting 2010 (February 1 - 3, Miami/Florida, USA). Starting 2005), global unification of height systems (Rummel 2001), point are the results delivered by the IAG Inter-Commission Project height or vertical datum problem (Heck and Rummel 1990, 1.2 ”Vertical Reference Frames” (IAG ICP1.2, Ihde 2007), which are Sacerdote and Sansò 2001, Sacerdote and Sansò 2004), vertical compiled in the document Conventions for the Definition and datum connection (Xu and Rummel 1991, van Onselen 1997), Realisation of a Conventional Vertical Reference System global unified height reference system (Ihde and Sánchez -CVRS- (Ihde et al. 2007). These CVRS describe the fundamen- 2005, Sánchez 2007, Kutterer et al. 2012), etc., the objective is al- tal aspects to be taken into consideration for the denition and Journal of Geodetic Science 327

Table 2. Inconsistencies making unsuitable the precise combination of physical and geometric heights. Requirement Present status Ellipsoidal heights h and (quasi)geoid heights N must be - Different ellipsoid parameters (a, GM) are applied in given with respect to the same ellipsoid, i.e. the same geometry and gravity. ellipsoid has to be used (1) for the transformation of geo- - h and N are given in different tide systems: centric coordinates into ellipsoidal coordinates, (2) as reference field for the solution of the geodetic boundary value • Oceanography, satellite altimetry, levelling in problem, (3) for scaling global gravity models, etc. mean tide system.

• ITRF positions, GRS80, some geoid models in tide free system.

• Some geoid models, terrestrial gravity data in zero tide system.

Physical heights H and (quasi)geoid undulations N must - Orthometric heights H and geoid height estimations N reflect the same reference surface, i.e. the geoid obtained after the boundary value problem are based on different from subtracting the height of the datum point (H0) from orthometric hypotheses. the height of the other points (Hp) conforming the vertical - H and N refer to different tide systems. reference frame shall be consistent with the geoid derived - Systematic errors over long distances in levelling reduce from gravity (solution of the boundary value problem). the reliability of (Hp)-(H0).

Physical heights H and ellipsoidal heights h must repre- - H and h refer to diﬀerent epochs, and in the most cases, sent the same Earth’s surface. dH/dt is unknown. - Diﬀerent reductions (for Earth-, ocean-, atmospheric tides, ocean and atmospheric loading, post-glacial re- bound, etc.) are applied. 328 Journal of Geodetic Science

realisation of a global vertical reference system that fulls the re- by the IAG Commissions 1 (Reference Frames) and 2 (Grav- quirements mentioned above. The immediate objective of GGOS ity Field), as well as by the IGFS (see Section 7). Further ac- Theme 1 is starting the implementation of these aspects in prac- tivities to be developed in the frame of GGOS Theme 1 in tice, for that the activities to be faced at rst are (cf. GGOS 2020 a mid and long term are (cf. GGOS 2020 Action Plans Action Plans 2011 - 2015, unpublished): 2011 - 2015):

a) Renement of standards and conventions for the global c) Generation of a set of consistent GGOS products for the re- vertical reference system: The document produced by the alisation of the global vertical reference system, including: IAG-ICP1.2 is aligned to the IERS 2003 conventions (Mc- d) A proposal for a global vertical reference frame given by a Carthy and Petit 2004) and is previous to the GOCE mis- set of worldwide homogenously distributed geodetic sta- sion. A review is necessary to take into account updates tions and the corresponding densications at continental included in the IERS 2010 conventions (Petit and Luzum and national levels. 2010) and to consider the new developments offered by GOCE. This also includes the comparison with standards e) Compilation of guidelines for height system unication recently published by other IAG components (e.g. Inter- containing a detailed description about the required input national Gravity Field Service, Bureau Gravimétrique Inter- data, analysis strategy, and expected products. national, International Geoid Service, International Altime- f) Implementation of an information system containing an in- try Service, etc.). Disagreements must be analysed and the ventory of the existing local/regional height systems and corresponding updates (modications) have to be imple- their characteristics (e.g. locations of the reference tide mented in order to achieve a homogenous set of common gauges, period(s) used for computing the mean sea level numerical standards, models, and procedures. This activ- introduced as a zero-height, gravity reductions applied to ity primarily demands the concurrence of GGOS Theme 1, levelling, precision of levelling and gravity data, etc.). the IAG geometric services under the umbrella of IERS (In- ternational Earth Rotation and Reference Systems Service, g) Design of strategies for the appropriate maintenance and www.iers.org ) and the IAG gravity eld services under use in practice of the global vertical reference system con- the umbrella of IGFS (International Gravity Field Service, sidering: www.igfs.net). The work shall be coordinated by the GGOS Bureau for Standards and Conventions (GGOS-BSC), h) Determination of time-dependent changes of the vertical supported by each IAG or GGOS component in its specic reference frame. speciality. i) Alignment of the denition and realisation of the vertical reference system with future improvements in geodetic b) Establishment of a global vertical reference level: Accord- analysis and modelling. ing to Ihde et al. (2007) and Kutterer et al. (2012), the global vertical datum shall correspond to a level surface of j) Interaction with disciplines different to Geodesy requir- the Earth’s gravity eld with a given potential value (W0 ing a vertical reference system; for instance hydrography, = const). In the last years, different W0 estimations have oceanography, etc. been carried out applying a wide range of strategies and models (see Section 6). Although these estimations are Under this umbrella, the present paper discusses some aspects to

very similar, the discrepancies between the nal W0 val- be considered in the rst two activities, i.e. the standardisation of ues are larger than the expected realisation accuracy, i.e. > the height system components and the adoption of a global verti- 10 cm (or 1 m2/s2). Consequently, the objective is to make cal reference level W 0 . a recommendation about the W0 value to be adopted as 2. Components of a global vertical reference system the conventional reference level for the global vertical ref-

erence system. This W0 value must also be promoted as Any vertical reference system is primarily given by a reference sur- a dening parameter for the computation of an improved face (i.e. the zero-height level) and a vertical coordinate (i.e. a mean Earth ellipsoid and as a reference value for the com- height). If the reference surface and height type depend on the

putation of the constant LG within the IERS conventions. Earth’s gravity eld, we talk about a physical height system; if LG is required for the realisation of the relativistic atomic not, it is a geometrical height system. In the rst case, the typi- time scale, i.e. transformation between Terrestrial Time (TT) cal height types are orthometric and normal heights derived from and Geocentric Coordinate Time (GCT). A formal recom- spirit levelling in combination with gravity reductions. In the sec-

mendation about the W0 value to be adopted within IAG ond case, the most known type is the ellipsoidal height derived and GGOS is a responsibility of the GGOS working group on from satellite positioning techniques (i.e. GNSS: Global and Navi- ”Vertical Datum Standardisation”, which is also supported gation Satellite Systems). Ascertaining ellipsoidal heights presents Journal of Geodetic Science 329 many advantages in comparison with the (levelled) physical ones; and physical) applications. In the last two decades, some discus- for example high accuracy over long distances, quickly and low sions about replacing the GRS80 ellipsoid for a new one were car- cost determination, etc. However, they cannot replace the physi- ried out (e. g. Groten 2002, Hipkin 2002, Grafarend and Ardalan cal heights in many scientic and practical applications because of 1999). Some of these proposals inclusively suggest the replace- their ‘geometrical’ nature, i.e. they do not describe ow of water. ment of a by W0 as denition parameter in the computation of the A modern vertical reference system shall support the precise de- reference ellipsoid (e.g. Burša et al. 2002, Yurkina 1996). Nonethe- termination and combination of physical and geometrical heights less, the main conclusion appointed to keep the GRS80 ellipsoid in such a way, that each of them can alternatively be used in the as the oﬃcial one. Independently of the reference ellipsoid to be appropriate cases. Consequently, this reference system must com- adopted as standard, it has to be a mean Earth ellipsoid, i.e. its prise the respective components: a geometrical one, and a physi- dening parameters M, ω, J2, a or W0 must be close to the real cal one (e.g. Drewes et al. 2002, Ihde and Sánchez 2005, Ihde et al. values of the Earth. Its orientation and position with respect to 2007, Sánchez 2009). It is clear that the reference surface of the ge- the Earth’s body shall be consistent with the denition of the ITRS, ometrical component shall be a level ellipsoid, i.e. an equipotential i.e. its mean point (geometrical centre) has to coincide with the surface of the normal gravity eld (Heiskanen and Moritz 1967, p. ITRS origin of coordinates [x = y = z = 0] (the geo-centre), its 64): minor axis b has to coincide with the ITRS Z-axis, and its reference meridian (zero-longitude meridian) has to coincide with the IERS

U0 = U (X) = const. (2) X-Y plane (Petit and Luzum 2010). In the same way, the normal gravity/potential eld generated by this ellipsoid has to be intro- The corresponding vertical coordinates are then ellipsoidal heights duced as reference eld in the solution of the geodetic bound- and their change with time: ary value problem for the determination of (quasi)geoid models. Complementary, to get a univocally relationship between refer- dh (X) ence surface and vertical coordinates, the ellipsoidal heights and h (X, t) ; (3) dt their change with time have to be obtained from geocentric coordinates [X = (x, y, z)] referring to the realisation of the ITRS, i.e. U0 is univocally estimable as a function of the enclosed mass M the ITRF (International Terrestrial Reference System, Petit (with homogeneous density), angular velocity ω, the dynamical and Luzum 2010). form factor J2, and semi major axis a (see e.g. Heiskanen and Moritz 1967, Eq. 2-61, 2-109 p. 67). At present, the IERS recom- The physical component of the vertical reference system has to be mends the GRS80 ellipsoid (Moritz 2000) for geometrical coordi- also given by a (physical) reference surface and a (physical) ver- nate conversions (Petit and Luzum 2010, p. 40). Within the ‘phys- tical coordinate. Normally, the primary coordinates are level dif- ical community’ different reference ellipsoids are taken into ac- ferences, also called geopotential numbers, which are convention- count. For instance, the ellipsoid dened by a = 6 378 136.3 m and ally transformed into metric quantities like orthometric or normal 1/f = 298.257 is used in the gravity models EGM96 (Lemoine et al. heights. This procedure demands the introduction of the corre- 1998) and EGM2008 (Pavlis et al. 2012) while the GRS80 is utilised in sponding reference surface: the geoid for the orthometric heights, the GOCE data analysis (Gruber et al. 2010 p. 41). Satellite altime- or the quasi-geoid for normal heights. If the denition of the physi- try computations and some mean sea surface models (e.g. CLS01 cal component is based on the geoid (orthometric heights), the re- (Hernandes and Schaefer 2001) or DTU10 (Andersen and Knudsen quired hypotheses about the internal Earth’s mass distribution and 2008)) refer to the ellipsoid dened by a = 6378136.46 m and the vertical gradient of the real gravity must be clearly explained 1/f = 298.25765. Additional examples about the usage of dif- within the denition. If not, there will be as many vertical refer- ferent ellipsoid parameters can be found in recent satellite-based ence systems as applied hypotheses. In the same way, to get a gravity models as EIGEN-GL05C (Förste et al. 2008), ITG-Grace- univocally relationship between reference surface and vertical co- 2010S (Kurtenbach et al. 2009), GOCO01S (Pail et al. 2010), etc. This ordinates both, the geoid and the orthometric heights, have to be kind of inconsistencies is well-known within the geodetic commu- determined using exactly the same hypotheses to reduce the grav- nity and they are usually removed before combining gravity and ity values from the Earth’s surface to the geoid and to estimate the geometrical parameters. Nevertheless, we must keep in mind that mean gravity value along the plumb line. This must also be con- GGOS shall be utilised by other disciplines different to Geodesy sidered for geoid modelling based on satellite gravity data. In ad- and it would be an unreasonable demand to request all the GGOS dition, anytime when the hypotheses change because our knowl- users to be familiarised with basic knowledge about geometrical edge about the internal mass distribution is improved, the deni- and physical parameters of a reference ellipsoid and the transfor- tion of the vertical reference system shall accordingly be modied mation between different ones. In this way, even though the ge- and its realisation has to be re-aligned to the improved denition. ometrical transformation from one ellipsoid to any other is not a Fig. 2 and some publications, e.g. Leismann et al. (1992), Sánchez problem, it would be convenient to introduce a conventional el- (2003), Tenzer et al. (2005), Santos et al. (2006), show the wide lipsoid to be used in a common way for all geodetic (geometrical spectrum of possibilities to estimate orthometric heights and it is 330 Journal of Geodetic Science

surface (W0) and the equipotential surface (WP ) passing through the point (P) of interest:

−∆WP = CP = W0 − WP (6)

This diﬀerence is usually estimated by combining spirit levelling measurements with gravity values (e.g. Heiskanen and Moritz 1967, p. 162, Heck 2003, p. 281):

∫ P ∑P ∼ C (g, dn) = W0 − WP = gδn = gdn (7) 0 0

Figure 2. Magnitude of the gravity reductions (top) for a levelling line being g the mean gravity value along the levelling line between with heights up to 4000 m. Orthometric reductions were calculated after Helmert (Heiskanen and Moritz 1967), two benchmarks, and dn the corresponding measured level diﬀer- Ramsayer (1953, 1954), Baranov (Leismann et al. 1992) ence. The availability of an increasing number of observables de- and Ledersteger (1956). Normal reductions after Moloden- scribing the Earth’s gravity eld and surface enables the utilisation skii (Leismann et al. 1992) are included for comparison. Data taken from Sánchez (2003). of a second approach based on the combination of the disturbing potential (T ) with a reference ellipsoid:

clear that every one of these orthometric height types requires a C (U0,T ) = − (U0 − W0) + γ¯ (ϕ) h − T (ϕ, λ, h) (8) consistent geoid. Only one geoid cannot be the reference surface for all of them. Here, (ϕ, λ, h) are the ellipsoidal coordinates [latitude, longitude, If the vertical reference system denition is based on a quasi-geoid height] of the evaluation point, U is the normal potential at the el- (with normal heights as vertical coordinates), it would be indepen- 0 lipsoid surface (Eq. 2), and γ¯ (ϕ) is the mean normal gravity along dent of hypotheses and worldwide consistent, but the quasi-geoid the theoretical plumb line connecting P (ϕ, λ, h) with the ellip- is not an equipotential surface inside the continental masses, i.e. soid surface (e.g. Hofmann-Wellenhof and Moritz 2005, p. 93). it does not have a physical meaning (Heiskanen and Moritz 1967, Equation (8) is valid only, if U and (ϕ, λ, h) refer to the same p. 109). Thus, in order to formulate a consistent denition, free 0 ellipsoid used for the determination of the disturbing potential of ambiguities, but correct from the theoretical point of view, the T (ϕ, λ, h), which is estimated after the solution of the geodetic physical component of the global vertical reference system shall boundary value problem. The precise combination of the potential be based on geopotential quantities, i.e. the reference level must diﬀerences obtained from Eq. (7) and Eq. (8) conrms the neces- be a given W0 value (Ihde et al. 2007, Sánchez 2009): sity that the vertical reference system includes, together with the usual physical component, a geometrical component supporting W (X) = W0 = const. (4) those observables depending on an ellipsoid. The improvement of the gravity eld modelling thanks to the re- and the vertical coordinates shall be geopotential numbers (and cent gravity-dedicated satellite missions, especially GRACE and their variation with time) referred to this W : 0 GOCE, open the opportunity to derive vertical coordinates by combining these models with an Earth-rotating geocentric reference dC (X) C (X, t) ; (5) system (such as the ITRS), i.e.: dt ( ) The transformation of the geopotential numbers into physi- C C¯nm, S¯nm = W0 − [V (r, θ, λ) + Z (r, θ)] (9) cal heights and the geometrical representation of the surface

W0=const (geoid determination) will then be matter of the realisation. In this manner, inconsistencies (especially due to the ortho- being (V ) the gravitational potential and (Z ) the centrifugal po- metric hypotheses) reducing accuracy and reliability of the vertical tential (e.g. Torge 2001, p. 70): reference system will aﬀect its realisation, but not its denition. n n GM ∑max ∑ [ a ]n 3. Referencing physical heights consistently V (r, θ, λ) = P¯ (cos θ) r r nm n=0 m=0 As already mentioned, the primary physical vertical coordinate is ( ) C¯nm cos mλ + S¯nm sin mλ (10) the potential diﬀerence between an arbitrarily selected reference Journal of Geodetic Science 331 1 Z (r, θ) = ω2r2cos2 (90◦ − θ) (11) 2

(r, θ, λ) are the spherical coordinates [radial distance from geo- centre, co-latitude, longitude]; a, M and ω are the ellipsoid’s semi- major axis, Earth’s mass, and angular velocity, respectively. G is the gravitational constant, P¯nm represents the fully-normalised spherical harmonic functions of degree n and order m, and C¯nm, S¯nm are the fully-normalised spherical harmonic coefficients (e.g. Torge 2001, p. 72). As matter of fact, the reliability (consistency between real and model-derived values) of the model included( in Eq. (10)) ¯ ¯ depends on both, the accuracy of the coefficients Cnm, Snm Figure 3. Ocean levelling. and the maximum degree of the expansion (nmax). For instance, a spherical harmonic expansion up to n, m= 360 allows a spatial resolution of about 70.5 km (e.g. Barthelmes 2009, Tab. 1, p. 20). Inde- coincide with the geoid, Eq. (13) is usually written as: pendently of additional error sources, this implies an omission er- ∫ ror in the geoid undulations of about 0.22 m (e.g. Rapp 1997, Tsch- L 1 erning et al. 1983). To get an accuracy at the mm-level, it is nec- CKL (t, s, p) =WL − WK = δp− L0 ρL (t, s, p) essary an expansion at least up to n, m = 2160, where the omis- ∫ K 1 sion error would be about 0.002 m. The EGM2008 model (Pavlis δp. (14) ρ (t, s, p) et al. 2012) is the only model existing with this high-resolution. K0 K Nevertheless, numerical evaluations show that its accuracy in re- Here (K0, L0) are assumed to be at the level of no-motion while gions with dense terrestrial gravity coverage is about 10 cm while (K,L) are located at the sea surface (Fig. 3). If the geopotential in poor covered areas it is 50 cm only (Pavlis et al. 2012). In this way, difference (i.e. Eq. (13)) between the geoid and the point (K ) is potential differences derived using Eq. (9) would produce physical known, the other levelling points can be referred to W0. The dy- heights with an uncertainty at the dm-level. This is far away of the namic (or geostrophic) levelling assumes that the hydrostatic pres- accuracy obtained from spirit levelling (Eq. 7), and therefore, the sure on the sea surface is balanced by the Coriolis force per unit usage of Eq. (9) is still not suitable. area (i.e. geostrophic equilibrium) and it is given by Another possibility for the determination of potential differences that begins to be considered is the comparison of clock frequencies ∫ K

( f, f0) of high-precision: C (ω, ϕ, υ) = WL − WK = 2ωsin ϕυδl (15) L [ ] f − f0 being ω the angular velocity of the Earth, υ the horizontal wa- C (f) = c2 (12) f0 ter velocity along the levelling (or integration) path (l), and ϕ the latitude where υ was measured. The primary input data in this c is the speed of light. It is expected that a precision of (1×10−17 s) approach are speed measurements of the ocean currents. Equa- measuring time allows geopotential diﬀerences with accuracy of tion (14) is specially applied in open oceans (deep water areas) about 10 cm. At this moment, the required technology is being while Eq. (15) is utilisable in coastal areas to improve (extrapo- improved and Eq. (12) is not applicable in practice yet. late) satellite altimetry measurements. The main drawbacks of the In ocean areas, the determination of potential diﬀerences is based so-called ocean levelling are (1) the poor availability of observa- on the solution of equations of motion of the water masses (hy- tions at high temporal and spatial resolutions, (2) the assumptions drodynamics). There are two basic models (Rummel and Ilk 1995, about the hydrostatic and geostrophic equilibriums, and (3) the Khad 1998): the steric levelling assumes that the water is in hy- certainty with the friction forces are known. Therefore, results pro- drostatic equilibrium above some identiable, isobaric (reference) vided by Eq. (14) and Eq. (15) are compared and improved using surface of no-motion, which coincides with an equipotential sur- the dynamical ocean topography (DOT ) derived from satellite al- face: timetry measurements and a pre-given geoid model. For more de- tails about observing geostrophic currents and deriving steric sea ∫ P 1 surface heights from satellite altimetry see e.g. Le Traon and Mor- C (t, s, p) = W0 − WP = δp. (13) 0 ρ (t, s, p) row (2001). For the purposes of this article, the DOT at any point j (ϕ, λ, h) located at the sea surface can be written as (Fig. 4): The input observables are hydrographical data: temperature (t), [ ] salinity (s) and pressure (p), which permit the estimation of the wa- W0 − Wj DOTi = hs − rj − Nj = (16) ter density, i.e. ρ (t, s, p). Since the no-motion surface does not γj 332 Journal of Geodetic Science

Figure 4. Dynamical ocean topography and local vertical datums. Figure 5. Reference levels on land areas depending on the input data and methods for the determination of vertical coordinates (potential diﬀerences). hs is the height of the satellite with respect to a reference ellipsoid;

rj is the range measurement representing the distance between the satellite and the point j; N , γ and W denote geoid undula- j j j gravity forces, etc.), the interpretation of this denition (and with it tion, normal gravity and gravity potential at j. Equation (16) is only the concept for the realisation of W0) has been changed over the valid, if h and N refer to the same ellipsoid that generates γ . For s j j years depending on the geodetic observations and analysis strate- consistency, it is expected that the values N (ϕ, λ) geometrically j gies available for the geoid determination (e.g. Mather 1978, Heck describe the equipotential surface dened by W . 0 and Rummel 1990, Ekman 1995, Heck 2004). The possible realisa- Independently of the accuracy oﬀered at present by the ap- tions can be summarised as follows (cf. Heck 2004): proaches above mentioned (Eqs. (7), (8), (9), (12), (14), (15) Equipotential surface coinciding with the mean sea level registered and (16)), all of them refer to any arbitrarily selected W value, 0 at an arbitrarily selected tide which in principle realise any equipotential surface. If we are in- i tending to get consistency between these approaches, at least on W0 = W0 (18) land areas, i.e. (cf. Fig. 5): i Since the potential value W0 cannot be precisely determined and

C (g, dn) ≈ C (U0,T ) ≈ C (Cnm,Snm) ≈ C (f) (17) under the concept that the reference level can arbitrarily be appointed, any geopotential value can be assigned. For instance,

it is necessary, either the utilisation (realisation) of a common W0 in the European Vertical Reference System (EVRS) (Ihde and Au- value (i.e. geoid), or the precise knowledge of the discrepan- gath 2000), the normal potential U0 of the GRS80 ellipsoid was as- i i cies δW0 = W0 − W0 with respect to one and the same signed to the equipotential surface passing through the tide gauge

W0. This is especially important for the combination of physical mark in Amsterdam. Other proposals suggest the determination of i heights derived from levelling and the corresponding obtained W0 taking as a basis a global gravity model (Eq. (10) + Eq. (11)) from GNSS positioning and gravimetric geoid models, methodol- and assuming that the geo-centric coordinates of the tide gauge ogy that nowadays is widely applied (e.g. Kotsakis et al. 2012 and mark are known from GNSS positioning (e.g. Burša et al. 2001, Tenzer et al. 2011). In addition, if the reference level introduced on Ardalan and Safari 2005). These procedures can be adequate for land areas has to be consistent with the reference level supporting local height systems, but their usage in a global vertical reference the estimation of the DOT (as it is desired), both levels shall realise system presents following disadvantages: the same equipotential surface, dened by only one W0 value (cf. a) The DOT at the reference tide gauge is totally neglected Fig. 4). i and the value W0 does not represent a global geoid.

4. Selecting an appropriate W0 realisation b) The global realisation of the corresponding level surface depends on the accuracy of the vertical datum connection, The introduction of a W0 value as reference level is useful to ap- especially of those height systems located at diﬀerent con- point which of the innite equipotential surfaces of the Earth’s tinents. gravity eld is selected as the zero-height surface. In general, the

preferred equipotential surface is those coinciding, in the sense of Another strategy to stimate the W0 reference value is to dene it the least squares, with the global mean sea surface in complete identical with the normal potential generated by a mean Earth el- calm, i.e. the geoid (Gauss 1876, p. 32). Since this condition can- lipsoid: not be satised due to diﬀerent causes (e.g. existence of the continents, oceanic currents, atmospheric pressure eﬀects, external W0 = U0 (19) Journal of Geodetic Science 333

This approximation was especially useful, when the Earth’s grav- Dening the DOT according to Eq. (16), the minimum condition in ity potential eld was not known with the resolution and preci- Eq. (21) can be re-written as (cf. Sacerdote and Sansò 2001): sion as it is today. As already mentioned, a mean Earth ellipsoid is ∫ ∫ [ ] ∂ ∂ W − W 2 computed taking as dening parameters real Earth’s values for the DOT 2ds = 0 j ds = 0; semi-major axis, mass, angular velocity, and dynamical form fac- ∂W0 S ∂W0 S γj ∫ tor. Thus, the reliability of the realisation represented by Eq. (19) Wj S γ 2ds ∫ j depends on the accuracy with those values are determined. The W0 = 1 (22) 2 use of different ellipsoids in practice (see Section 2) proves that S γj ds the GRS80 ellipsoid does not always satisfy the requirements as a Empirical evaluations carried out by e.g. Burša et al. (2002), geodetic reference system in different geodetic applications, par- Sánchez (2007), Dayoub et al. (2012) are based on the computation ticularly those related to the analysis of the Earth’s gravity eld. This of the potential values Wj using a global gravity model (Eq. (10) lets us doubt, if its U0 value is a good approximation for a reference + Eq. (11)) and the ellipsoidal coordinates (ϕ, λ, h) provided by W0 value. mean sea surface models for those points j describing the geome- After the availability of satellite altimetry techniques and the pos- try of the sea surface (cf. Fig. 4). Dayoub et al. (2012) also propose sibility to estimate the DOT, it was suggested that W0 shall corre- the reduction of the sea surface models by a mean dynamic topog- spond to that equipotential surface in relation to which the square raphy model (MDT) in order to get a sea surface closer to the geoid. sum of the DOT estimates at each reference tide gauge around the In this case, the coordinates of the points j are (ϕ, λ,[h − MDT ]). world is a minimum (cf. Eq. (2.4) and Eq. (2.5) in Lelgemann 1977): The applied MDT model is ECCO-2 (Menmenlis et al. 2008), which ∑n is independent of pre-given gravimetric geoid models. In princi- [ ( (i) (i))]2 W0 − W0 + δW = min; ple, DOT and MDT are representing the same, but here they are i=1 distinguish in such a way that, the rst one is derived from satel- 1 ∑n ( ) W = W (i) + δW (i) (20) lite altimetry in combination with a gravimetric geoid, and the sec- 0 n 0 i=1 ond one is obtained from ocean circulation analysis. The numerical evaluation of Eq. (21) or Eq. (22) (including Eq. (10) and Eq. (11)) (i) δW stands for the potential differences generated by the DOT at using either (ϕ, λ, h) or (ϕ, λ,[h − MDT ]) has two remarkable each reference tide gauge i. This approach requires measurements disadvantages (citation from Sacerdote and Sansò 2001, p. 52 and of the sea surface and the gravity eld on ocean and coastal areas 53): in order to precisely determine the DOT around the tide gauges. a) ”... as the function W (ϕ, λ) [equivalent to Eq. (22) using Since a pre-given geoid is necessary, this procedure has to be iter- S Eq. (10) + Eq. (11) over ocean areas only] is not defined ative. The main drawbacks here are: over the whole sphere, the first coefficient of its a) The (marine and terrestrial) gravity data (like gravity expansion is not equal to its mean value...” anomalies or geopotential numbers) necessary for the computation of a high-resolution geoid model are given b) ”... with this procedure a sort of average value of the geopotential over oceans is computed, rather with respect to different reference levels. Thus, W0 depends on the observables included in its computation. than an equipotential surface geometrically averaging the sea surface topography.” b) The reliability of the satellite altimetry in shallow waters is very poor and the minimum condition presented in 5. W0 and the geodetic boundary value problem Eq. (20) cannot be satised with enough accuracy. The introduction of a W0 value either for a local vertical datum c) The addition or omission of any tide gauge demands the or a global one, has been considered until now as irrelevant (e.g. re-denition of the reference level. Heck and Rummel 1990, Rummel and Heck 2000, Heck 2004) because (1) the reference level for the measured potential differences Mather (1978) rened the denition of Lelgemann (1977) and sug- can arbitrarily be appointed, and (2) the direct determination of gests that W0 shall correspond to that equipotential surface in re- an absolute W0 value from observational data is not possible. Re- lation to which the mean DOT at all reference tide gauges is zero. garding to the second point, similarly to the geometrical reference In the same publication, Mather (1978) introduced an ”oceano- system (where coordinates are not directly measurable, but time graphic” interpretation of the geoid extending the condition in intervals, distances, and directions) absolute geopotential values Eq. (20) to all marine areas. With this, W0 represents the level sur- can be precisely estimated by introducing adequate constraints. face with respect to the average of the DOT is zero when sampled The main constraint is vanishing of the gravitational potential (V ) over all the ocean zones (S) worldwide: at innity: ∫ 2 DOT ds = min (21) ∞ S V = 0 (23) 334 Journal of Geodetic Science

Rather than an absolute potential value, this condition allows the estimation of the potential diﬀerence between a point conven- tionally located at the innity and the equipotential surface passing through the point of interest on or close to the Earth’s surface (Rummel and Heck 2000). Fullling this condition is only possible in the frame of the geodetic boundary value problem and therefore, this frame is the most appropriate for the determina-

tion of ”absolute” potential values like W0. This procedure addi- tionally reduces the ”arbitrariness” of the reference level; then the

obtained W0 value will be in agreement with the geodetic observations included for solving the boundary value problem, i.e. (quasi)geoid determination. The scalar free boundary value problem in spherical and linear approximation (e.g. Sacerdote and Sansò 1986, Heck 1989) is the formulation most applied for the solution of the vertical datum problem. In this case, the so-called vertical datum parameter ∆W0 is included as unknown together Figure 6. Vertical datum parameters by non-connected height sys- with the geometry of the boundary surface Σ and the gravity po- tems. tential W : ∇T = 0, outside Σ (24a) i i ∆W0 were a constant (i.e. ∆W0 = ∆W0) and it appears the new i ∂T 2 2 unknown δW0 , which represents the level diﬀerence between the − − T = gj − ∆W0, on Σ (24b) i ∂r R R local vertical datums (∆W0 ) and that one arbitrarily selected as T →0, at ∞ (24c) the reference level (W0) (Fig. 7):

Function gj represents the observational data (e.g. gravity anoma- i i δW0 = W0 − W0 (28) lies, potential diﬀerences, deections of the vertical, etc.) available

to constitute the boundary conditions. ∆W0 denotes the diﬀer- Compare Eq. (28) with CQi0 in Rummel and Teunissen (1988) and ence between the Earth’s gravity potential W0 and the normal po- follow-on publications, e.g. Heck and Rummel (1990), Xu and Rum- tential U0 introduced for the linearization of the boundary condi- mel (1991), van Onselen (1997). In this way, Eq. (27) would be- tions (observation equations): come:

∆W = W − U (25) ∂T 2 2 2 0 0 0 − − T = g − ∆W + δW i (29) ∂r R j R 0 R 0 Compare Eq. (25) with T in Lehmann (2000), δW in Sacerdote 0 i The unknowns δW0 do not only appear when gravity observables and Sansò (2004), ∆wˆ in Heck and Rummel (1990). W0 is unknown, but it is implicitly included in the observables building referring to different local datums are considered. They also are the boundary conditions, especially in the geopotential numbers present when different kind of observations referring to different and physical heights used for the estimation of gravity anomalies. levels are included, although they are on the same area, for exam- Since these observables do not refer to only one vertical datum, ple in the combination of equations (7), (8) and (9) on land (Fig. 5), i or the combination of equations (14), (15) and (16) on ocean ar- there shall be as many ∆W0 parameters as existing i datums (W0 ) (Fig. 6): eas (Fig. 3 and 4). The use of the boundary conditions given by Eq. (24b), Eq. (27) or Eq. (29) assumes that the horizontal coordinates of the obser- ∆W i = W i − U (26) 0 0 0 vations are given in the conventional terrestrial reference system (i.e. ITRS). If they refer to local geodetic datums, the boundary con- and Eq. (24b) can be written as: ditions must be adequately modied to take into account the ef- fect of the discrepancies between the horizontal datums on the

∂T 2 2 i unknowns (e.g. Sansò and Usai 1995, Sansò and Venuti 2002). − − T = gj − ∆W0 (27) i i ∂r R R The terms ∆W0 (or δW0 if preferred) shall reect the height datum discrepancies only. Therefore, the boundary value prob- Equation (27) means that the boundary surface Σ is split in i un- lem in all vertical datums must be solved by introducing (cf. Ta- connected regions (Sacerdote and Sansò 2004). If they were con- ble 2) (1) the same reference surface and reference gravity poten- nected (for instance by means of spirit or geostrophic levelling), tial eld (preferably those of the conventional ellipsoid, i.e. GRS80) Journal of Geodetic Science 335

and Schaeﬀer 2001) as geometrical representation of the boundary surface Σ. The boundary conditions (gravity disturbances) are derived from the gravity models EGM96 (Lemoine et al. 1998) and EIGEN-GLS04S (Förste et al. 2006) in combination with the GRS80 ellipsoid. Čunderlí and Mikula (2009) apply the boundary element method and include both, ocean and continental areas. On ocean areas, the geometry of the boundary surface is represented by the CLS01 model while on land areas, a combination of the model SRTM_ PLUS V1.0 (Becker and Sandwell 2003) in combination with the geoid undulations derived from the EGM96 model (Lemoine et al. 1998) is included. The gravity disturbances are derived from

EIGEN-GLS04C. ∆W0 is called perturbation of the Dirichlet boundary condition (cf. δW Eq. (8), Čunderlí and Mikula 2009, p. 231) and it takes different values depending on the continent; therefore, the authors suggest as nal value those determined over Figure 7. Vertical datum parameters by connected height systems. the ocean areas only. From this section, we can conclude that there are two basic for- mulations for the introduction of a unied reference level: the rst in the linearization, and (2) the same global gravity model de- one recommends the adoption of an existing reference level, i.e. rived from satellite-only observations for modelling the long wave- the vertical datum of any already established local height system; length component of the Earth’s gravity eld. With the same pur- while the second one proposes the determination of an ”absolute” pose, the solution of the boundary value problem must follow reference level, i.e. independent of the existing local vertical da- the Molodenskii’s theory, concretely in continental areas; other- tums. In the rst case, denition and realisation of a ”absolute” wise, uncertainties in the required assumptions for the classical ap- vertical datum is considered not important, since the fundamen- proach (the geoid determination) can be misinterpreted as verti- tal vertical coordinates are level differences and the starting value cal datum inconsistencies. Once the quasi-geoid is properly de- to convert these differences in ”absolute” values can arbitrarily be termined, it can be transformed into a geoid (if it is wanted) by selected (e.g. Heck and Rummel 1990, Rummel and Heck 2000, introducing the desired hypothesis. On ocean areas, the multiple Heck 2004). Here it is assumed that the reference level is already vertical datum dependence of the boundary value problem can be realised and the most important task is the connection of the exist- avoided, if only one kind of data is used to formulate the boundary ing height systems with that selected as absolute reference, espe- condition on Σ (Eq. (24b), Eq. (27) or Eq. (29)). For instance, by ap- cially height systems located in different continents (e.g. Rummel plying exclusively satellite altimetry data and satellite-only global and Teunissen 1988, Xu and Rummel 1991, Rapp and Balasubrama- i gravity models, we would have only one ∆W0 (i = 1) and the cor- nia 1992, Rummel and Ilk 1995). In this formulation, the scalar free responding W0 obtained from Eq. (26) can be assumed as a the boundary value problem is widely applied. In the case of an abso- global reference level (Sánchez 2008). In this case, the function gj lute vertical datum, the primary step is to dene a global reference in Eq. (24b) corresponds to the gravity disturbance surface (datum) assumed to be available over the world. The next task is its realisation, which includes the connection of the exist-

δgP = gP − γP (30) ing local height systems (e.g. Balasubramania 1994, Rapp 1995a). Sánchez (2008, 2009) proposed to reach the rst objective (realisation of the global reference level) by applying the ﬁxed boundary at the sea surface, and the boundary condition takes the form value problem on ocean areas and the second one (connection of the local height datums to the global one) by solving the scalar ∂T − = δg, (31) free boundary value problem. ∂r

6. Current W0 estimates which represents, together with Eq. (24a) and Eq. (24c) a ﬁxed gravimetric boundary value problem (e.g. Lehmann 2000, Again, we have to mention that any W0 reference value can be ap- Sacerdote and Sansò 2004, Heck 2011); i.e. the geometry of the pointed for the determination of vertical coordinates (cf. Heck and boundary surface is known and the only one unknown to be de- Rummel 1990, Rummel and Heck 2000, Heck 2004). However, to termined is the gravity potential. Sánchez (2008) presents some get the worldwide consistency desired within a global vertical ref- computations applying the analytical solution of the xed bound- erence system, the selected W0 value must be realisable with high- ary value problem summarised in Hofmann-Wellenhof and Moritz precision at any time and everywhere around the world. Since W0 (2005, p. 303) and taking the sea surface model CLS01 (Hernandes represents only one quantity and it is not suﬃcient to estimate 336 Journal of Geodetic Science

position and geometry of the equipotential surface it is dening, of the IAG Special Commission SC3, Fundamental Con-

the main problem to solve here is not the determination of the stants, XXII IAG General Assembly). At present, LG is con-

W0 per se, but its realisation. Therefore, we insist on the neces- sidered as a ”dening” constant (free of uncertainty) and its value 2 −2 sity to estimate it from real observations of the Earth’s gravity eld is based on W0= 62 636 856.0 ± 0.5 m s (see IERS Convention and surface. If those observations are also applied for modelling a 2010, Petit and Luzum 2010).

global geoid (zero-height surface), a geometrical representation of Per denition, W0 is the geopotential value of the geoid, and the the W0 equipotential surface is obtained and the requirements for geoid shall be the equipotential surface that best approximates the its realisation are achieved. Nevertheless, the empirical estimation global mean sea surface when totally calm (i.e. if no external forces

of W0 shall follow specic standards and conventions. Otherwise, would act on the oceans). Recent empirical estimations (see Ta- 2 −2 there would be given as many W0 reference values (i.e. zero-height ble 3) demonstrate that this value W0= 62 636 856.0 ± 0.5 m s surfaces) as methods of computation. As an example Table 3 sum- corresponds to an equipotential surface located about 20 cm be-

marises diﬀerent global W0 values computed after the publication neath the sea surface scanned by satellite altimetry. Although this of the GRS80 ellipsoid (local estimations, i.e. based on data dis- value can be introduced as the global reference level (as any other tributed within a limited geographical regions are not considered). value); we shall evaluate the convenience of having a zero-height Divergences between the estimations are basically generated by level which does not coincide with the mean sea surface and it is the applied methodologies and the input models representing the not consistent with the most recent geodetic models describing Earth’s gravity eld and the sea surface. For instance, the com- geometry and physics of the Earth. This lead us to think about a the

putations carried out by Burša et al. (e.g. 1998a, 2001, 2002, necessity of a ”better estimate” for W0 and this is the main objec- 2007a) are based on sea surface models derived by themselves tive of the Working Group on Vertical Datum Standardisation. from Topex/Poseidon and Jason 1 data. Čunderlí and Mikula (2008, 2009), Sánchez (2007, 2009) and Dayoub et al. (2012) apply models 7. Working group on Vertical Datum Standardisation. already published by other specialists, e.g. GSFC00.1 (Koblinsky et al. 1999), CLS01 (Hernandes and Schaefer 2001) or KMS04 (Ander- The Working Group on Vertical Datum Standardisation is a com- sen et al. 2004). Additional computations (e.g. Burša et al. 2007a, mon initiative of the GGOS Theme 1, the IAG Commissions 1 (Ref- Sánchez 2007, Dayoub et al. 2012) concentrate on: erence Frames) and 2 (Gravity) and the IGFS. According to the IAG nomenclature (Drewes et al. 2012), it is called Joint Working Group a) Applying yearly sea surface models to establish possible JWG 0.1.1. Its main purpose is to provide a reliable W0 value to time variations of W0. be introduced as the conventional reference level for the realisa- b) Varying latitude coverage and spatial resolutions of the tion of a global vertical reference system. Although any W0 value mean sea surface models to determine possible depen- can arbitrarily be chosen, it is expected that this value is consistent

dences of W0 on these parameters. with other dening parameters of geometric and physical models of the Earth. In this way, activities developed by JWG 0.1.1 are c) Using truncated global gravity models at diﬀerent grad and based on the state-of-the-art data and methodologies, especially order (n, m) to evaluate the inuence of the spectral reso- on the newest available representations of the Earth’s surface and lution on in the W0 determination. gravity eld. Computations carried out by JWG 0.1.1 are to be doc- umented in detail in order to guarantee the repeatability and relia- The IERS Conventions explicitly include a W value since 1992 to 0 [ ] bility of the results. This documentation shall support the adoption W0 explain the value assigned to the constant LG = 2 , which c of the obtained W0 value as an oﬃcial IAG/GGOS convention. An is used for the transformation between the Terrestrial Time and additional product will be dedicated to provide guidance on the the Geocentric Coordinate Time (see Resolution B1.9 of the XXIV usage of W0 in practice, in particular for the vertical datum unica- General Assembly of the International Astronomical Union -IAU-, tion. The activities to be faced can be summarised as follows: 2000). The rst W0 value considered by the IERS was 62 636 860 2 −2 ± 30 m s (McCarthy 1992, IERS Conventions 1992, Appendix, a) To coordinate individual initiatives for a unied W0 deter- Notes for IAU Recommendation IV, item 6). This value was changed mination: At present there are at least four groups working 2 −2 to 62 636 856.85 ± 1.0 m s in the IERS Conventions 1996 (Mc- on the estimation of a global W0 value. The idea is to bring 2 −2 Carthy 1996, Table 4) and, afterwards, to 62 636 856.0 ± 0.5 m s these groups together to elaborate an inventory describ- in the IERS Conventions 2003 (McCarthy and Petit 2004, Table 1.1). ing individual methodologies, conventions, standards, and

These two last changes were introduced following the recommen- models presently applied in their W0 computations. dations given by the IAG Special Commission on Fundamen-

tal Constants, which was responsible for identifying the ”best b) To rene the W0 estimation: it is proposed that each estimates” for the required standards (see: Burša 1995, Report group makes a new W0 computation following their own of the IAG Special Commission SC3, Fundamental Con- methodologies, but applying the most recent geodetic stants, XXI IAG General Assembly and Groten 1999, Report models (e.g. GOCE/GRACE gravity models, sea surface Journal of Geodetic Science 337

2 2 Table 3. Examples of current W0 estimations (value 62 636 000 [m /s ] has to be added). 2 2 Approach W0 Value [m /s ] Comments Reference

W0 = U0 860.850 GRS80 Moritz (2000) 858.546 Best fitting ellipsoid for the T/P mean sea surface Rapp et al. (1991) 856.88 Best fitting ellipsoid for the T/P mean sea surface Rapp (1995b) 854.18 Best fitting ellipsoid for the DNSC08 mean sea Dayoub et al. (2012) surface ∫ W −W DOT 2ds = min; DOT = 0 j 856.5 ± 3 Mean sea surface: GEOSAT, Gravity model: Burša et al. (1992) S γj GEM-T2 857.5 ± 1 Mean sea surface: GEOSAT, Gravity model: Nesvorný and Šíma (1994) JGM-2 855.8 ± 0.50 Mean sea surface: ERS1 + T/P (10.1992 - Burša et al. (1997) 12.1993) minus MDT: POCM 4B, Gravity model: EGM96 855.72 ± 0.50 Mean sea surface: T/P (1994 - 1996) minus MDT: Burša et al. (1998a) POCM 4B, Gravity model: EGM96 855.611 ± 0.008 Mean sea surface: T/P (1993-1996), Gravity Burša et al. (1998b) model: EGM96 856.161 ± 0.002 Mean sea surface: T/P (1993-2001), Gravity Burša et al. (2002) model: EGM96 853.4* Mean sea surface: T/P (2000), Gravity model: Sánchez (2007) EIGEN-GC03C Reference epoch 2000.0 853.35* Mean sea surface: CLS01 (ϕ = 60◦N/S), Gravity model: EIGEN-GC03C Reference epoch 2000.0 854.61* Mean sea surface: CLS01 (ϕ = 80◦N/S), Gravity model: EIGEN-GC03C Reference epoch 2000.0 853.24* Mean sea surface: KMS04 (ϕ = 60◦N/S), Gravity model: EIGEN-GC03C Reference epoch 2000.0 854.46* Mean sea surface: KMS04 (ϕ = 80◦N/S), Gravity model: EIGEN-GC03C Reference epoch 2000.0 855,619 ± 0.004 Mean sea surface: T/P-J1 (1993 - 2003), Gravity Burša et al. (2007a) model: EGM96 855,832 ± 0.005 Mean sea surface: T/P-J1 (1993 - 2003), Gravity model: GGM02C 855,883 ± 0.005 Mean sea surface: T/P-J1 (1993 - 2003), Gravity model: EIGEN-CG01 854.6 ± 0.004 Mean sea surface: T/P (1993 - 2003), Gravity Burša et al. (2007b) model: EGM96 853.19* Mean sea surface: CLS01 (ϕ = 60◦N/S), Gravity Dayoub et al. (2012) model: EGM2008, Reference epoch 2005.0 854.22* Mean sea surface: CLS01 (ϕ = 80◦N/S), Gravity model: EGM2008, Reference epoch 2005.0 853.43* Mean sea surface: DNSC08 (ϕ = 60◦N/S), Grav- ity model: EGM2008, Reference epoch 2005.0 854.43* Mean sea surface: DNSC08 (ϕ = 80◦N/S), Grav- ity model: EGM2008, Reference epoch 2005.0 853.31* Mean sea surface: T/P-J1 (1992 - 2009), Gravity model: EGM2008, Reference epoch 2005.0 854.2 ± 0.2 Mean sea surface: CLS01/DNSC08 minus MDT: ECCO2, Gravity model: EGM2008 ◦ W0 = U0 + ∆W0 854.38 ± 0.03 Mean sea surface: CLS01 (ϕ = 80 N/S), Gravity Sánchez 2009 model: EIGEN-GL04S 853.11 ± 0.03 Mean sea surface: CLS01 (ϕ = 60◦N/S), Gravity model: EIGEN-GL04S 857.95* Mean sea surface: KMS04, Gravity model: Čunderlík and Mikula (2009) EGM96 *No standard deviation is provided. 338 Journal of Geodetic Science

Table 4. On-going activities of the working group on Vertical Datum Standardisation. On-going activities

W0 computation based on analytical L. Sánchez (Germany) solutions of the fixed boundary value problem. W0 computation based on the solution R. Čunderlík (Slovakia) of the fixed boundary value problem Z. Faskova (Slovakia) K. after the Boundary Element Method Mikula (Slovakia) (BEM) Figure 8. Interaction of the working group on Vertical Datum Stan- W0 computation based on averaging N. Dayoub (Syria) P. dardisation with other GGOS/IAG components. potential values from a global gravity Moore (United Kingdom) model at points describing the sea surface. W0 computation based on a reference ellipsoid (U0) models derived from calibrated and combined satellite al- W0 computation based on averaging Z. Šima (Czech Republic) timetry observations, etc.). For this, JWG 0.1.1 will take into potential values from a global grav- V. Vatrt (Czech Repub- account recommendations given by the different IAG com- ity model at points describing the sea lic) M. Vojtiskova (Czech ponents in the respective eld of speciality (Fig. 8). This surface. Republic) analysis shall also include an investigation about the time- Strategies for the regional realisation J. Huang (Canada) D. of a global W Roman (USA) Y. Wang dependence of W . 0 0 (USA) J. Agren (Sweden) c) To make a proposal for a formal IAG/GGOS convention

about W0: It is expected that results obtained after applying the diﬀerent methodologies considered in the previous dtu.dk/English/Research/Scientific_data_ item are very similar. In this way, after a rigorous reliability and_models/Global_Mean_sea_surface.aspx evaluation, the JWG 0.1.1 members shall formulate a uni-

ed agreement proposing the best estimate of W0 as an Ardalan A. and Safari A., 2005, Global height datum uni- IAG/GGOS oﬃcial convention. This will be supported by a cation: a new approach in the gravity potential space. J Geod document containing the detailed computation of the rec- 79, 512-523. Springer. ommended value.

d) To provide a standard about the usage of W0 in the verti- Balasubramania N., 1994, Denition and realization of a cal datum unication: Based on the interchange of expe- global vertical datum. Ohio State University, Department of riences within the working group, it is expected to gener- Geodetic Science and Surveying. OSU Report No. 427. 112 pp. ate a document describing the most appropriate strategy to connect (unify, transform) any local height system with Barthelmes F., 2009, Denition of functionals of the geopo-

the global W0 reference level. tential and their calculation from spherical harmonic models. GFZ Scientifc Technical Report STR09/02. GFZ, Potsdam, 36 The on-going activities are summarized in Table 4. This article pro- p.www.gfz-potsdam.de - News - GFZ Publications. vides an inventory about the existing W0 estimations and it shall be understood as a very rst result of this working group. A re- Becker J. and Sandwel D., 2003, Accuracy and resolution port about the following activities and the corresponding achieve- of Shuttle Radar Topography Mission Data. Geosphys Res Lett ments will be included in a forthcoming paper. 30, 9.

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