J. Geod. Sci. 2018; 8:115–120 Research Article Open Access Lars E. Sjöberg* On the geoid and orthometric height vs. quasigeoid and normal height https://doi.org/10.1515/jogs-2018-0011 Keywords: ambiguous quasigeoid, geoid, geoid- Received August 6, 2018; accepted November 6, 2018 quasigeoid difference, resolution, vertical datum, quasi- geoid Abstract: The geoid, but not the quasigeoid, is an equipo- tential surface in the Earth’s gravity field that can serve both as a geodetic datum and a reference surface in geo- physics. It is also a natural zero-level surface, as it agrees 1 Introduction with the undisturbed mean sea level. Orthometric heights are physical heights above the geoid, while normal heights The geoid is an important equipotential surface and ver- are geometric heights (of the telluroid) above the reference tical reference surface in geodesy and geophysics. The ellipsoid. Normal heights and the quasigeoid can be deter- quasigeoid, introduced by M.S. Moldensky (Molodensky mined without any information on the Earth’s topographic et al. 1962) is not an equipotential surface, and it has density distribution, which is not the case for orthometric no special meaning in geophysics. The geoid serves as heights and geoid. the ideal reference surface for height systems in all coun- We show from various derivations that the difference be- tries that adopt orthometric heigths, while the rest of the tween the geoid and the quasigeoid heights, being of the world uses the quasigeoid with normal height systems (or order of 5 m, can be expressed by the simple Bouguer grav- normal-orthometric heights with more or less unknown ity anomaly as the only term that includes the topographic zero-levels). The great advantage of the quasigeoid to the density distribution. This implies that recent formulas, in- geoid is that it can be determined without knowledge cluding the refined Bouguer anomaly and a difference be- of the topographic density distribution. Foroughi et al. tween topographic gravity potentials, do not necessarily (2017), in a comparison of geoid and quasigeoid heights improve the result. vs. GPS-levelling geoid/and quasigeoid heights used the Intuitively one may assume that the quasigeoid, closely re- test area/data in Auvergne, France (Duquenne 2007), to lated with the Earth’s surface, is rougher than the geoid. demonstrate that the uncertainty in topographic density For numerical studies the topography is usually divided is practically harmless in geoid estimation. However, as into blocks of mean elevations, excluding the problem shown by Sjöberg (2018) GPS/levelling geoid heights can- with a non-star shaped Earth. In this case the smoothness not be used to validate the density model used in a gravi- of both types of geoid models are affected by the slope of metric geoid model. the terrain, which shows that even at high resolutions with As the quasigeoid is closely related with the Earth’s ultra-small blocks the geoid model is likely as rough as the geometric surface, Vanicek et al. (2012) raised the ques- quasigeoid model. tion whether it can be practically determined according to In case of the real Earth there are areas where the quasi- Molodensky’s proposed method by successive approxima- geoid, but not the geoid, is ambiguous, and this prob- tions. Sjöberg (2013) agreed that this computational tech- lem increases with the numerical resolution of the re- nique will hardly be successful in a detailed modelling of quested solution. These ambiguities affect also normal the quasigeoid, but he instead suggested employing the and orthometric heights. However, this problem can be extended Stokes’ formula, which is closely related with the solved by using the mean quasigeoid model defined by original Stokes’ formula, the basis in most geoid determi- using average topographic heights at any requested reso- nations (e.g., Ellmann and Vanicek 2007). Nevertheless, lution. An exact solution of the ambiguity for the normal as the quasigeoid is related with the Earth’s irregular sur- height/quasigeoid can be provided by GNSS-levelling. *Corresponding Author: Lars E. Sjöberg: Royal Institute of Tech- nology (KTH) Stockholm, Sweden Open Access. © 2018 Lars E. Sjöberg, published by De Gruyter. This work is licensed under the Creative Commons Attribution- NonCommercial-NoDerivs 4.0 License. 116 Ë Lars E. Sjöberg ZZ h i face, one may still believe that it is much more irregular R * + S(ψ) ∆g (rP , Q) − ∆g (Q) dσQ , 4π훾0 than the geoid. This problem will be analysed by compar- σ ing formulas for geoid and quasigeoid determination. Here (2a) the KTH-method, also named Least Squares Modification of Stokes formula with Additive corrections (LSMSA; e.g. where superscript * denotes harmonic analytical continu- ∆g* Sjöberg 2003a, b; Sjöberg and Bagherbandi 2017, Chap. 6) ation (along the vertical) to geoid level in g and to com- r ∆g* r Q will be used, as it in contrast to other methods (e.g., the putation point level of radius P in ( P , ). After a few UNB technique; Ellmann and Vanicek 2007) explicitly pro- manipulations, including a Taylor expansion to first order vides the corrections needed for topographic height and in the last term, this formula can be approximated to 2 (︂ )︂ density. To simplify the discussion, the density is assumed HP ∆gP HP HP ∂∆g dNdwc ≈ + 3ζP − to be constant, and the geoid and quasigeoid are deter- 훾0 R 2훾0 ∂H P mined only from surface gravity data by Stokes-types for- R (︂ ∂∆g )︂ + (HP − HQ) dσQ . (2b) mulas. The minor effects of the Earth’s atmosphere and el- 4π훾0 ∂H Q lipsoidal shape (being almost the same for both types of H geoid modellings) are disregarded. The LSMSA technique where is the orthometric height. is based on analytical continuation of the surface gravity The combined topographic effect is the same as the anomaly to sea-level and surface level in geoid and quasi- negative of the topographic bias (Sjöberg 2007 and 2009a, geoid determinations, respectively (see the references in b; Sjöberg and Bagherbandi 2017, Sects. 5.2.3-5.2.4): (︂ 3 )︂ * Sect. 2). T 2πGρ 2 2HP bias(Tg) dNcomb = − HP + = − (3) 훾0 3R 훾0 for the constant topographic density ρ and gravitational 2 Determination of the geoid constant G. (For an arbitrary topographic density distribu- tion, see Sjöberg 2007 and 2009b.) Using the KTH-method the geoid height is given by (cf. Sjöberg and Bagherbandi 2017, Sect. 6.2.2) N = (N) + dN, (1a) 3 Determination of the quasigeoid where height T* + T* R (N) = 0 1 + ∆gdσ (1b) 3.1 Gravimetric approach 훾0 4π훾0 and The quasigeoid height (or height anomaly) at surface point P is given by (Ågren 2004, Sect. 9.5.1; Sjöberg and dN dN dNT = dwc + comb (1c) Bagherbandi 2017, Sect. 6.3.2) are the additive corrections for the downward continua- TP (T0 + T1)P rP * ζP = = + ∆g (rP , Q)dσQ tion (dwc) of the surface gravity anomaly (∆g) and the 훾 훾 4π훾 R * HP R * combined direct and indirect topographic effects. More- = ∆g (rP , Q)dσQ + ζP ≈ ∆g (rP , Q)dσQ 4π훾 rP 4π훾 over, R is the Mean Earth Radius, 훾0 is normal gravity at the 0 H reference ellipsoid, S (ψ) is Stokes’ kernel function with ar- P ζ + 3 r P , (4) gument ψ being the geocentric distance between computa- P * 훾 훾 tion and integration points and σ is the unit sphere. T0 and where is normal gravity at the telluroid, related to 0 * T1 are the analytically continued zeroth- and first-degree by disturbing potential harmonics. 훾 ≈ 훾 (1 − 2H/R). (5) The dwc effect becomes (Sjöberg 2003a, b and Sjöberg 0 and Bagherbandi 2017, Sect.5.3.1) The last step in Eq. (4) can also be decomposed into ZZ h i R * ζ ζ dζ dNdwc = S(ψ) ∆gg (Q) − ∆g (Q) dσQ P ≈ ( P) + P , (6a) 4π훾0 σ ZZ where R h * * i = S(ψ) ∆gg (Q) − ∆g (rP , Q) dσQ (T0 + T1)P R 4π훾0 (ζP) = + ∆gdσQ , (6b) σ 훾 4π훾 On the geoid and orthometric height vs. quasigeoid and normal height Ë 117 and Equation (9a) agrees with Sjöberg (1995), where the re- sult was derived in two different ways, namely by exter- R (︁ * )︁ HP dζP = ∆g (rP , Q) − ∆g(Q) dσQ + 3 ζP , (6c) 4π훾0 R nal and internal harmonic expansions of the topographic potential and by using Stokes’ original and extended for- * or, after using a Taylor expansion of ∆g to first order: mulas with Helmert gravity anomalies for the geoid and (︂ )︂ quasigeoid heights, respectively. R ∂∆g ζP dζP ≈ (HP − HQ) dσQ + 3 HP . (6d) The first term in Eq. (9a) is the traditional representa- 4π훾0 ∂H Q R tion for the GQC (Heiskanen and Moritz 1967, pp. 327-328), and the complete formula also agrees with the major terms 3.2 GNSS-levelling approach of the refined formula of Sjöberg and Bagherbandi (2017, Eq. 7.31b), derived by employing analytical continuation of The normal height can be defined by the formula (Heiska- the external disturbing potential. nen and Moritz 1967, Sect. 4-5 ) N C H = (7) 4.1 Other solutions to the GQC 훾¯ where C is the geopotential number (determined by pre- Flury and Rummel (2009) suggested replacing the simple (︁ BO)︁ cise levelling), and 훾¯ is the mean value of normal gravity Bouguer anomaly by the refined one ∆g as the basic (훾) between the reference ellipsoid and the telluroid. As 훾 contribution to the GQC, and they also refined the solution N (︁ T T)︁ decreases smoothly with height, H can easily be iterated by the difference Vg − VP /훾¯, 훾¯ being the mean value from an approximate value (e.g., Sjöberg and Bagherbandi of normal gravity at the reference ellipsoid and telluroid, 2017, Sect.