On the Geoid and Orthometric Height Vs. Quasigeoid and Normal Height

Home , Earth radius, Height, Orthometric height

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, quasigeoid Abstract: The geoid, but not the quasigeoid, is an equipotential surface in the Earth’s gravity field that can serve both as a geodetic datum and a reference surface in geophysics. 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

∫︁∫︁ [︁ ]︁ 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 distribution, 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 ∫︁∫︁ [︁ ]︁ R * ζ ζ dζ dNdwc = S(ψ) ∆gg (Q) − ∆g (Q) dσQ P ≈ ( P) + P , (6a) 4π훾0 σ ∫︁∫︁ where R [︁ * * ]︁ = 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 result 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. 3.5.3). between topographic potentials at the surface point P and Finally, the quasigeoid height follows from the geode- the geoid along the plumb-line through P, yielding the ex- tic height (h) determined by GNSS: pression:

N BO T T ζ = h − H (8) ∆g Vg − VP GQC ≈ + . (10) 훾¯ 훾¯ This shows that both normal height and height anomaly/quasigeoid can be determined from GNSS and Sjöberg (2010) showed that a minor term is missing in levelling (alone). this equation. See also Sjöberg (2012). Hence, the beauty of M. S. Molodensky’s introduc- The practical problem with Eq. (10) is to estimate the tion of normal height and quasigeoid is that these compo- topographic potential difference. We will return to this is- nents can be determined without any information about sue later. the Earth’s density distribution. Another approach is as follows: Proposition 1: For a constant topographic density ρ at the computation point

∆g ∑︁∞ H k+1 (︂ ∂k δg )︂ πGρ H3 4 The difference N − ζ GQC B H 1 (− P) 4 P = P − k + 훾0 훾0 (k + 1)! ∂H P 3훾0 R k=1 Taking the difference between Eqs. (1a)-(1c), (2b) and (3) (11) on one hand and Eqs. (6a), (6b) and (6d) on the other and Proof. The GQC can be found by Stokes’ original and ex- omitting some minor terms, one arrives at “the geoid-from- tended functions S (ψ) and S(rP , ψ) as quasigeoid correction” (GQC): ∫︁∫︁ Tg T R 2 (︂ )︂ GQC P S ψ S r ψ ∆g* dσ ∆gB HP ∂∆g = 훾 − 훾 = 훾 [ ( ) − ( P , )] g GQC = N − ζ ≈ HP − , (9a) 0 4π 0 σ 훾0 2훾0 ∂H P * bias(Tg) TP 훾 − 훾0 where we have introduced the simple Bouguer gravity − + , (12) 훾0 훾 훾0 anomaly by where the bias term, given in Eq. (3), accounts for the bias ∆gB = ∆g − 2πGρH (9b) in analytical continuation of the disturbing potential to the geoid. As Stokes’ function can be developed into a Taylor and omitted some minor terms. 118 Ë Lars E. Sjöberg series of its extended function: the Earth’s surface is known, e.g. expressed by its laterally ∞ k k variable geocentric radius, the height anomaly can be de- ∑︁ (−H) ∂ S (R + H, ψ) S(ψ) = , (13) termined by TP /훾. As can be seen from Eqs. (6a) – (6d), k! ∂Hk k=0 this solution does not need knowledge of the topographic and also by considering density, leading to a simpler solution vs. that for the geoid. When comparing the above solutions for N and ζ one ∂T T 훾 훾 δg ∆g δg − 0 can see that both types of models are dependent on the = − ∂H , = + 훾 훾 (14) 0 (mean) topographic height variations. For the theoretician as well as Eqs. (7b) and (10), one finally arrives at the (but also for the practically minded surveyor requesting a proposition, where the last term hardly exceeds one very high resolution of the surface) it could be of interest centimetre in the highest mountains. See also Sjöberg to investigate which of the two surfaces is smoothest. In- (2015a). tuitively one would expect that the deflections of the vertical are smoother at the geoid than at the telluroid. On the Corollary 1: For a general topographic density distribution other hand, all terms of the GQC of Eq. (9a) belong to the ρ (r) along the vertical at the computation point the last geoid formula, so the answer to this question is not obvi- term in Eq. (11) should be substituted by ous. To get some further insight to the problem we consider R+HP two points (denoted with subscripts 1 and 2) at the Earth’s ∫︁ (︂ r2 )︂ πG ρ r r dr πGρH2 s −4 ( ) R − + 2 P . surface separated by a small lateral distance and heights R H1 and H2. Then the differential difference between the height anomalies can be deduced from Eq. (4) by Cf. Sjöberg (2007, Corollary 2) for this solution. ⃒ Note that the topographic bias does not include a ter- ∂T T s ∆H T δg ∆H ⃒ s T δg ∆H 훾 θ s 2 ( , ) = 1 − 1 + ∂s ⃒ = 1 − 1 − 1 1 , rain correction, because the topographic bias is not depen- ⃒1 dent on the mass distribution of the terrain. Actually, the (15) terrain correction is already accounted for in the analyt- ∆H H H δg θ ical continuation. However, for an accurate solution the where = 2 − 1, and are the gravity disturbance s bias should be corrected for a variable density distribution and deflection of the vertical in the direction . Then Eq. (4) along the vertical (see Sjöberg 2007 and 2009 a, b). yields GQC The can also be determined directly by compar- T2 T1 T2 − T1 T2 훾1 − 훾2 ∆ζ = ζ2 − ζ1 = − = + ing the quasigeoid and geoid heights determined by the 훾2 훾1 훾1 훾2 훾1 ⃒ external and internal Earth Gravitational Models at to- δg ∆H T ∂훾 ⃒ ∆g ∆H θ s 1 2 ⃒ ∆H θ s 1 = − 1 − 훾 − 훾 훾 ∂H ≈ − 1 − 훾 . pographic and sea levels, respectively. The internal dis- 1 2 1 ⃒1 1 turbing potential can either be determined by a remove- (16) compute-restore technique or by analytical continuation (Sjöberg 2015b, Foroughi and Tenzer 2017). A numerical From Eq. (9a), approximated to first order of the po- comparison can be found in the latter article. tential, one obtains for the corresponding geoid difference

∆gB2H2 − ∆gB1H1 ∆N = N2 − N1 = ζ2 − ζ1 + 훾0 ∆g2H2 − ∆g1H1 (︁ 2 2)︁ 5 Discussions = ζ2 − ζ1 + − 2πGρ H2 − H1 (17) 훾0

Formally, in view of that the topographic density distribu- Considering Eq. (16) one finally obtains tion is not well known, the problem of determining the (∆g2 − ∆g1) H2 4πGρ geoid is a gravimetric inverse problem. By assuming that ∆N ≈ −θ1s + − H∆H¯ , (18) 훾0 훾0 the topography and its density distribution are known, the problem can be altered to solving a (free) boundary value where H¯ = (H1 + H2) /2. problem. This approach is standard in one way or another As θ is the deflection of the vertical, it does not change in physical geodesy. One solution is shown in Eqs. (1a-c), much with the slope of the terrain. Hence, as the resolu- (2a) and (3). tion of the solution increases (and s decreases), the first In contrast, the problem of determining the quasi- term in Eqs. (16) and (18) go towards zero with s. However, geoid is a forward problem that does not rely on an esti- the last terms in each of the models are both related with mated topographic density distribution model. Hence, if the slope/roughness of the terrain. Obviously, one cannot On the geoid and orthometric height vs. quasigeoid and normal height Ë 119 generally state that one of the surfaces is smoother than can always be determined from precise levelling, having the other at high resolution, at least not without further its zero-level at the reference ellipsoid, and the latter by studies. GNSS-levelling, both without any information about the Foroughi and Tenzer (2017,Fig. 18) calculated the per- Earth’s density distribution, even if they are ambiguous. formances of the quasigeoid and geoid in the Himalayas, In contrast, the geoid datum is related with orthometric ∘ showing the details along a meridional profile near 88 E heights, which depend on topographic density and need a ∘ from latitude 25 N (at lower Ganges river close to sea fixed geoid model as the zero-level, while the zero-level for level), passing the extreme elevations of more than 8 km the normal height is the well-defined surface of the refer- around Mt. Everest, to 40 N (with topographic heights of ence ellipsoid. less than 1 km). They concluded that the geoid geometry is For a star-shaped earth model we could not state that modified at least 10 % less than the quasigeoid. However, one of the two surfaces (geoid or quasigeoid) is smoother their conclusion is based on the dominating very long- than the other at high resolution, as not only the quasi- wavelength variation of the (quasi)geoid surfaces over the geoid but both surfaces vary with the slope of the topogra- profile as part of the Earth’s largest geoid low causedby phy. However, generally the geoid is more correlated with the huge lower mantle density low south of India. In ad- density variations in the Earth’s interior than the quasi- dition, as these geoid and quasigeoid extremes are nega- geoid. tive, actually the geoid is more deformed than the quasi- In case of the real non-star shaped Earth the geoid geoid. Generally, in accord with Newton’s law, thegeoid is is still unique, while the quasigeoid will be ambiguous more sensitive to density structures below the crust than in some limited areas. Although this problem might be the quasigeoid (except for the limited regions with nega- rather academic and limited to a very high numerical reso- tive topographic heights). lution of the quasigeoid model, a unique model (the mean If one, on the other hand, studies the short- quasigeoid model) can still be defined by using mean to- wavelength variations of the two surfaces, which are re- pographic heights in ambiguous areas. lated with smoothness, one can actually see from their We have shown that recent formulas for estimating the Fig. 18 that the quasigeoid plot is smoother. However, GQC, using both the refined Bouguer gravity anomaly and one should bear in mind that this calculation is a single a difference between topographic potentials at the geoid numerical result that cannot be used for drawing firm and surface, do not lead to improvements, but using the conclusion. simple Bouguer anomaly (possibly corrected for variations As stated earlier, this study is based on a star-shaped of density along the vertical down to sea level) by the first Earth model, which does not fully agree with reality as term in Eq. (17), possibly refined by terms including ver- there are exceptional topographic areas with more than tical gradients of increasing order of the free-air gravity one point of intersection between the plumb-line and the anomaly, should be preferred from a practical point of surface, making the quasigeoid solution ambiguous. In view. More convenient for regional and global studies GQC this case a unique quasigeoid height can still be defined maps can be determined from the disturbing potential dif- by the mean elevation even if the block size approaches ference at the Earth’s surface and sea level evaluated by zero. The geoid solution is still unique when replacing the an Earth Gravitational Model and a topographic potential combine topographic effect of Eq. (3) with the correspond- model. ing general formula that allows for a variable topographic density as described in Sjöberg (2007). Acknowledgement: Acknowledgement. The paper was written while the author was a visiting professor at the Uni- versity of Gävle. 6 Conclusions

There is no doubt that the geoid but not the quasigeoid can References serve both as a geodetic datum and a reference surface in geophysics. The geoid is also a natural surface in the sense Ågren J., 2004, Regional geoid determination methods for the era of that it is an equipotential surface in the Earth’s gravity field satellite geodesy. PhD thesis in geodesy, Royal Institute of Tech- nology, Stockholm that agrees with the undisturbed mean sea level. Duquenne H., 2007, A data set to test geoid computation methods. One advantage of M. S. Molodensky’s genius introduc- Proc. 1st Int. Symp. of the Int. Gravity Field Services, Istanbul, tion of normal height and quasigeoid is that the former Harita Dergisi, Special Issue 18: 61-65 120 Ë Lars E. Sjöberg

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