The Relation Between Rigorous and Helmert's Definitions of Orthometric

J Geodesy DOI 10.1007/s00190-006-0086-0

ORIGINAL ARTICLE

The relation between rigorous and Helmert’s deﬁnitions of orthometric heights

M. C. Santos · P. Vanícekˇ · W. E. Featherstone · R. Kingdon · A. Ellmann · B.-A. Martin · M. Kuhn · R. Tenzer

Received: 8 November 2005 / Accepted: 28 July 2006 © Springer-Verlag 2006

Abstract Following our earlier definition of the rig- which is embedded in the vertical datums used in numer- orous orthometric height [J Geod 79(1-3):82–92 (2005)] ous countries. By way of comparison, we also consider we present the derivation and calculation of the differ- Mader and Niethammer’s refinements to the Helmert ences between this and the Helmert orthometric height, orthometric height. For a profile across the Canadian Rocky Mountains (maximum height of ∼2,800 m), the ∼ M. C. Santos (B) · P. Va n í cekˇ · R. Kingdon · A. Ellmann · rigorous correction to Helmert’s height reaches 13 cm, B.-A. Martin whereas the Mader and Niethammer corrections only Department of Geodesy and Geomatics Engineering, reach ∼3 cm. The discrepancy is due mostly to the rig- University of New Brunswick, P.O. Box 4400, orous correction’s consideration of the geoid-generated Fredericton, NB, Canada E3B 5A3 e-mail: [email protected] gravity disturbance. We also point out that several of the terms derived here are the same as those used in P. Va n í cekˇ e-mail: [email protected] regional gravimetric geoid models, thus simplifying their implementation. This will enable those who currently R. Kingdon e-mail: [email protected] use Helmert orthometric heights to upgrade them to a more rigorous height system based on the Earth’s grav- Present Address: A. Ellmann ity field and one that is more compatible with a regional Department of Civil Engineering, geoid model. Tallinn University of Technology, Ehitajate tee 5, 19086 Tallinn, Estonia Keywords Orthometric height · Geoid · Mean e-mail: [email protected] gravity · Plumbline Present Address: B.-A. Martin Critchlow Associates Ltd., 1 Introduction 61 Molesworth Street, Wellington, New Zealand

W. E. Featherstone · M. Kuhn The orthometric height is defined as the metric length Western Australian Centre for Geodesy, along the curved plumbline from the geoid to the Earth’s Curtin University of Technology, GPO Box U1987, surface. To calculate an orthometric height from spirit- Perth, WA 6845, Australia e-mail: [email protected] levelling data and/or geopotential numbers requires that the mean value of the gravity along the plumbline M. Kuhn e-mail: [email protected] between the Earth’s surface and the geoid should be known. This mean value is strictly defined in an integral R. Tenzer sense (e.g. Heiskanen and Moritz 1967, p. 166). Faculty of Aerospace Engineering, In the past, three main approximations have been ap- Physical and Space Geodesy, Kluyverweg 1, 2629 HS Delft, P.O. Box 5058, 2600 GB Delft, The Netherlands plied in practice to evaluate this integral-mean value of e-mail: [email protected] gravity. The Helmert method, as described in Heiska- M. C. Santos et al. nen and Moritz (1967, chap. 4), applies the simplified raphy-generated gravity is divided among the spherical Poincaré–Prey vertical gradient of gravity, which uses Bouguer shell, the terrain roughness residual to the normal gravity and a Bouguer shell of constant topo- Bouguer shell, and the topographical mass–density graphic mass–density, to the observed gravity at the variations. Earth’s surface in order to obtain an approximated mean The aim of this paper is to provide the theoretical value halfway down the plumbline. Niethammer (1932) background and practical methods with which to con- and Mader (1954) refined Helmert’s model by includ- vert Helmert orthometric heights [as described in, e.g. ing the effect of local variations in the terrain roughness Heiskanen and Moritz (1967, chap. 4)], which are used relative to the Bouguer shell. Mader (1954), considering as the height system embedded in the vertical datum only the linear change of the gravimetric terrain correc- adopted in numerous countries, to the more rigorous tion with respect to depth, used the simple mean of the orthometric heights presented in Tenzer et al. (2005). terrain effect at the geoid and at the Earth’s surface, With this in mind, we have presented some preliminary whereas Niethammer (1932) used the integral mean of derivations and results for various components of the terrain effects evaluated at discrete points along the correction in Vanícekˇ et al. (2001) and in Tenzer and plumbline. Dennis and Featherstone (2003) evaluated Vanícekˇ (2003). Kingdon et al. (2005) present a numer- these three approximations, showing that the accuracy ical evaluation over a part of Canada. This paper now is ordered Niethammer, Mader then Helmert, which presents and reviews the complete methodology. reflects the levels of approximation used. In addition, the mean topographical mass–density 1.1 Notation and terminology −3 ρ0 = 2,670 kg m , used in the Helmert, Niethammer and Mader approximations of the actual distribution of In the sequel, the dummy argument represents the topographical mass–density, is not sufficiently accurate geocentric spherical coordinates φ and λ of a point (in the original manuscript, Helmert (1890) refers to a [ φ ∈ −π/2, π/2 , λ ∈ 0, 2π] and r denotes its geo- mass–density value of 2,400 kg m−3). Attempts to re- centric radius. The radius of a point is a function of loca- fine the Helmert orthometric height in this regard have tion being represented by r = r(). The symbols rg() included varying topographical mass–density data (e.g. and rt() represent the geocentric radii of the geoid and Sünkel 1986; Allister and Featherstone 2001) and bore- the Earth’s surface, respectively, and will be abbreviated hole gravimetry (Strange 1982) to better approximate to rg and rt where there is no ambiguity. The orthomet- the integral-mean of gravity along the plumbline. ric height of a point is also a function of location, and is To the best of our knowledge, no attempts have been represented by HO(). made to include topographical mass–density data in the The gravity at a point is a function of both the Mader and Niethammer heights [even though Nietham- radius and the horizontal geocentric coordinates , mer (1932) already mentioned the necessity to use vary- being represented by g(r(), ) or in a simplified form ing density information for a more rigorous treatment]. used throughout the paper as g(r,). The remaining In this paper, we show that to arrive at a more rigor- gravity-related notation used throughout this paper is ous orthometric height, one must take into account not summarized in Table 1. Where relevant, overbars will only the effect of terrain roughness and normal gravity, be used to denote the integral-mean quantities between but also those additional effects coming from the masses the geoid and the Earth’s surface. contained within the geoid (herein termed the geoid- We use the term terrain roughness to represent the generated gravity) not accounted for by the Helmert ap- irregular part of topography with respect to the Bou- proach and from the mass–density variations within the guer shell, i.e.the geometric variations in the shape of topography. This is necessary because the mean value topography. There are many other terms found in the of gravity along the plumbline between the geoid and literature to indicate the same, such as topographical the Earth’s surface depends on all these quantities (cf. roughness or simply terrain, but here we choose the Tenzer et al. 2005). terminology of ‘terrain roughness’. Mean gravity along the plumbline is thus evaluated as the sum of the integral-mean values of the geoid- generated gravity and the topography-generated 2 Recapitulation of the rigorous orthometric height gravitational attraction. For practical evaluation, the geoid-generated gravity is further divided into normal The orthometric height HO() of a point on the Earth’s gravity and the geoid-generated gravity disturbance, i.e. surface (rt, ) is defined as the length of the curved the gravity disturbance in the so-called no-topography plumbline between the geoid rg() and the Earth’s sur- ∼ O (NT) space (cf. Vanícekˇ et al. 2004). Likewise, the topog- face rt() = rg() + H (), and is given by [e.g. The relation between rigorous and Helmert’s definitions of orthometric heights

Table 1 Gravity-related notation used throughout this paper g(r, ) Gravity at an arbitrary point gNT(r, ) Gravity generated by masses contained within the geoid, i.e. with the topography removed and in the NT-space gT(r, ) Gravitation generated by masses contained within the topography only, i.e. those between the geoid and Earth’s surface gδρ (r, ) Effect on gravitation due to lateral mass–density variations inside the topography with respect to the reference value of −3 ρ0 = 2,670 kg m γ (r, ) Normal gravity generated by the geocentric reference ellipsoid δgNT(r, ) Gravity disturbances generated by masses contained within the geoid T ( ) gB r, Gravitation generated by a spherical Bouguer shell T ( ) gR r, Gravitation generated by the terrain roughness, i.e. topographical undulations relative to the spherical Bouguer shell εg() Correction to Helmert’s approximation of integral-mean gravity along the plumbline ε () HO Correction to Helmert’s orthometric height to convert it to the rigorous orthometric height (Tenzer et al. 2005)

Heiskanen and Moritz 1967, Eqs. (4)–(21)] C(r , ) HO() = t (1) g() where g¯() is the integral-mean value of gravity along the plumbline between the geoid and the Earth’s surface

rt 1 g¯() = g(r, )dr (2) HO() rg and C(r , ) is the geopotential number, which is t Fig. 1 A (very simple) conceptual decomposition model of actual the difference between the Earth’s gravity potential gravity into the geoid-generated component (internal white area) W0 [= constant] at the geoid and W(rt, ) at the Earth’s and the topography-generated component (dark area); the topog- surface raphy is exaggerated for the sake of clarity ( ) = − ( ) C r, W0 W r, .(3)model, provided of course the same corrections have Concerning g¯(), since the actual value of gravity been computed from the same data. ( ) g(r, ) along the plumbline cannot be measured at all The gravity acceleration at a point g r, can be points, the integral-mean gravity g¯() generally has to decomposed into two terms; one comprising gravity generated by the masses inside geoid gNT(r, ), i.e. in be computed from the observed surface gravity g(rt, ), together with some realistic and physically meaningful the NT-space (Vanícekˇ et al. 2004), and another com- model of g(r, ) along the plumbline. This computation prising the gravitational attraction generated by topog- T( ) can be achieved in practice by reducing the observed raphy g r, , gravity according to some accepted model of terrain g(r, ) = gNT(r, ) + gT(r, ).(4) roughness and the topographical mass–density distribution between the geoid and the Earth’s surface. Figure 1 schematically shows a simplistic cross section of the Earth with the decomposition in Eq. (4), where the white internal area shows the contribution that comes 3 Decomposition of actual gravity from all masses within the geoid and the dark area shows the contribution due to the topographic masses. In order to formulate the corrections to Helmert’s The geoid-generated gravity can be further decom- orthometric height in a way that can be computed from posed into the contribution from normal gravity and the the datasets currently available (i.e. terrestrial gravity gravity disturbance caused by only the masses inside the observations, a digital elevation model (DEM) and geoid, the so-called NT gravity disturbance (cf. Vanícekˇ lateral density variations interpreted from geological et al. 2004). Likewise, the topography-generated grav- maps or databases), we use the following decomposi- itation can be further decomposed into the Bouguer tion of gravity. The primary pragmatic benefit of this shell contribution and the terrain roughness term resid- approach is that these are the same data used to com- ual to this shell. These two terms can also be adapted to pute a gravimetric geoid model, thus making the rigor- include (lateral) topographical mass–density variations −3 ous orthometric heights more compatible with the geoid from the standard value of ρ0 = 2,670 kg m (Sect. 5.5), M. C. Santos et al. this being the way in which topographic mass–density where ∂γ/∂h is the linear vertical gradient of normal data is normally derived from geological maps. gravity, evaluated at the topographical surface, h is the The geoid-generated gravity is represented by the geodetic height, G is the Newtonian gravitational con- sum of normal gravity γ (r, ) and the geoid-generated stant, and ρ0 is the (assumed-constant) topographical gravity disturbance δgNT(r, ) mass–density. It is worth mentioning that, in this paper, we fol- gNT(r, ) = γ(r, ) + δgNT(r, ).(5) low the expression for Helmert’s orthometric height The topography-generated gravitational acceleration is (Eq. (9)) as given in Heiskanen and Moritz (1967, chap. represented by the sum of that generated by Bouguer 4). This is of most interest because this is the way in T( ) shell gB r, , the terrain roughness residual to the which most (if not all) geodesists have assumed T ( ) Bouguer shell gR r, , and the lateral variations Helmert’s definition, and using a planar approximation in mass–density from the assumed average of the terrain. In his original work, however, Helmert −3 δρ (ρ0 = 2,670 kg m ) within the topography g (r, ) (1890) considered the gravitational effect of the com- δρ plete topographic masses, delineating that the varying gT(r, ) ≈ gT(r, ) + gT (r, ) + g (r, ).(6) B R density within the topographic masses and the masses Inserting Eqs. (5) and (6) into Eq. (4) gives a complete below the geoid should be considered in a rigorous treat- expression representing the total gravity as (cf. Tenzer ment. While this is described in Helmert’s (1890) text, et al. 2005) his mathematical formulation is simpler, thus probably explaining why the simplification in Eq. (9) has been g(r, ) ≈ γ(r, ) + δgNT(r, ) + gT(r, ) B adopted in practice. + T ( ) + δρ ( ) gR r, g r, .(7)Using the numerical values of ∂γ/∂h =−0.3086 The approximation sign reflects the fact that two addi- mGal/m (the linear vertical gradient of normal grav- π ρ tional effects are omitted from Eqs. (6) and (7): the gravity in free air) and 2 G 0 = + 0.1119 mGal/m (the lin- itational effects of atmospheric masses and the radial ear vertical gravity gradient from the Bouguer shell for ρ −3 variation of the topographic mass–density. The former 0 = 2,670 kg m ) in Eq. (9) gives is omitted here because it is very small (cf. Tenzer et al. g¯H() = g(r , ) + 0.0424HO(). (10) 2005, Appendix B) but will be reintroduced in Sect. 5.3; t the latter is very difficult to quantify because there is Therefore, Eqs. (9) and (10) effectively attempt to not enough reliable information on the radial distribu- reduce surface gravity to a point halfway down the tion of mass–density within the topography. As such, plumbline, using the Poincaré–Prey approximation of we shall only consider lateral topographic mass–density the vertical gravity gradient, to give an approximation variations (cf. Martinec 1993; Sjöberg 2004). This is also of the integral-mean value along the plumbline between consistent with the treatment of the geoid in the Stokes– the geoid and the Earth’s surface. Note that this approx- Helmert scheme (e.g. Vanícekˇ and Martinec 1994). imation embeds a constant topographic mass–density Finally, the integral-mean gravity along the plumbline for the Bouguer shell and completely neglects terrain g¯(), given by the integral-mean of g(r, ) according to roughness residual to the Bouguer shell. Eq. (2) when applied to Eq. (7), is Making use of the general Equation (7) at the Earth’s = () ≈ γ()+ δ NT() + T() + T () + δρ () surface (i.e. r rt), from Eq. (9) we obtain g g gB gR g . g¯H() = γ(r , ) + δgNT(r , ) + gT(r , ) + gT (r , ) (8) t t B t R t δρ 1 ∂γ + g (r , ) − + 4πGρ HO() (11) t 2 ∂h 0 4 Helmert’s and other approximations It is also worthwhile relating the rigorous orthomet- of the orthometric height ric height to the Niethammer (1932) and Mader (1954) orthometric heights. This is curiosity driven, since these By way of comparison, the expression for the approx- height systems are not in wide practical use to the best imated mean gravity along the plumbline used in the of our knowledge. Both systems attempt to take terrain Helmert orthometric height (i.e. computed using the roughness, residual to the Bouguer shell, into account simplified Poincaré-Prey reduction) is [cf. Heiskanen when determining the integral-mean value of gravity and Moritz 1967, Eqs. (4)–(25)] along the plumbline. ∂γ H 1 O Both Mader and Niethammer orthometric heights g¯ () = g(rt, ) − + 4πGρ0 H (),(9) 2 ∂h include a term in the computation of mean gravity to The relation between rigorous and Helmert’s definitions of orthometric heights

ε () include the mean ‘terrain effect’. Niethammer (1932) easier to numerically evaluate HO in later sections performs a discrete evaluation of the integral-mean of this paper terrain effect at a series of discrete points at even inter- O −H () εg() vals along the plumbline, while Mader (1954) assumes ε O () = , (16) H ( ) + O() the terrain effect to vary linearly between the geoid and g rt, 0.0424H ε () O() ε () the surface, and so uses a simple mean of the values where HO and H are in metres, and g and of the effect evaluated for the Earth’s surface and the g(rt, ) areinmGal. geoid. In our terminology, and using our approach to evalu- 5.1 Second-order correction for normal gravity ate the terrain roughness term, Mader’s (1954) approximated mean value of gravity along the plumbline For the terms involving normal gravity, we seek a sim- method is pliﬁcation of T ( ρ ) − T ( ρ ) ∂γ M H gR rg, ; 0 gR rt, ; 0 1 O g () = g () + , (12) A = γ¯() − γ(rt, ) + H () . (17) 2 2 ∂h and according to Niethammer’s (1932) method, it is The integral-mean value of normal gravity along the plumbline γ¯() is evaluated using a second-order Tay- N() = H() − T ( ρ ) g g gR rt, ; 0 lor expansion for the analytical downward continuation + O() R H of normal gravity from the Earth’s surface γ(rt, ) to the 1 + gT (r, )dr. (13) geoid. Using a formulation in terms of geodetic coordi- O() R H nates, this is r=R h()=N()+HO() 1 ∂γ γ()¯ ≈ γ(h(), )+ O() ∂ 5 Corrections to the Helmert orthometric height H h h=h() h()=N() 2 To establish the relationships between the more rigor- 1 ∂ γ ×[n − h()]+ [n − h()]2 dn. ∂ 2 ous mean gravity given by Eq. (8) and Helmert’s approx- 2 h h=h() imate (i.e. Poincaré–Prey) formula given by Eq. (11), we (18) subtract them, grouping like terms. The resulting difference is called the correction to Helmert’s mean gravity where h() is the geodetic (ellipsoidal) height of the ( ) εg(): point rt, , N( ) is the geoid height at , and n is an element along the ellipsoidal plumbline (cf. Jekeli 2000). ε () = g() − gH() g Performing the integration, applying the integration ∂γ limits, and expressing normal gravity in terms of the ε () = γ¯() − γ( ) + 1 O() g rt, ∂ H geocentric radius of the Earth’s surface at gives 2 h + T() − T( ) + π ρ O() 1 ∂γ gB gB rt, 2 G 0H γ¯() ≈ γ(h(), ) − HO() ∂ 2 h h=h() + δ NT() − δ NT( ) g g rt, ∂2γ 2 + 1 O() 2 H . (19) + T () − T ( ) 6 ∂h = () gR gR rt, h h δρ δρ Inserting Eq. (19) in Eq. (17) yields + g () − g (rt, ) (14) ∂2γ 2 1 O After being computed, εg() can be used to apply a cor- A ≈ H () . (20) ∂ 2 ε () 6 h h=h() rection to Helmert’s orthometric height HO using (cf. Heiskanen and Moritz 1967, p. 169) Assuming the spherically approximated value of the second-order free-air gravity gradient [Heiskanen and −HO() ε () = ε () Moritz 1967, Eqs. (2)–(122)], Eq. (20) reduces to HO () g , (15) g O() 2 O() 2 ε () H H to an accuracy of <<1 mm in HO . A = γ ≈ γ , (21) rt() a Since εg() is small, the actual mean gravity in Eq. (15) can (ironically) be computed using Helmert’s where a is the major semi-axis of the reference approximation (Eq. (10)). This will make it considerably ellipsoid. Taking HO() = 8.8 km (Mount Everest), M. C. Santos et al.

−2 rt () = 6,371 km and γ =9.81ms , A is about −1.87 mGal. Here we acknowledge the typographical error in the ﬁrst Using Eq. (16), this causes a maximum correction of term of Tenzer et al. (2005, Eq. 21). about 1.5 cm to the Helmert orthometric height. Inserting Eqs. (23) and (26) in Eq. (22) gives

5.2 Second-order correction for the Bouguer shell 4 HO()2 HO() B = π Gρ 2 − . (27) 0 + O() + O() For the terms involving the spherical Bouguer shell of 3 R H R H thickness HO(), we seek a simplification of the term = T() − T( ) + π ρ O() Using the earlier example of Mount Everest, a con- B gB gB rt, 2 G 0H . (22) −3 stant topographical mass–density of ρ0 = 2,670 kg m The gravitational attraction of the spherical Bouguer and Eq. (16), the second-order Bouguer term (Eq. 27) − shell at the Earth’s surface reads (Martinec 1993, affects the orthometric height by as much as 1.6 cm. Eq. 4.16) Thus, Eq. (14) now becomes 2 T( ) = π ρ O() R gB rt, 4 G 0H ε () = δ NT() − δ NT( ) + O() 2 g g g rt, R H O() O() 2 + gT () − gT (r , ) × + H + 1 H R R t 1 . (23) R 3 R O 2 δ δ H () + g g() − g g(r , ) − γ t r () From Wichiencharoen (1982) [cited by Martinec 1998, t Eq. (3.14)],the gravitational potential inside the spheri- 4πGρ HO()2 HO() + 0 2− , (28) cal Bouguer shell is 3 R + HO() R + HO() r r r : g t 2R3 r2 which represents the integral-mean value of gravity VT(r, ) = 2πGρ R+HO() 2− − , (24) B 0 3r 3 along the plumbline expressed in terms of corrections to Helmert’s approximate mean value. Term A takes a neg- where R is the inner radius of the shell (in this case, R= ative sign, indicating the radial derivative of Eq. (17) is rg) and r is a dummy point inside the shell. taken on the ellipsoid. These correction terms comprise: ∂VT(r,) T( ) =− B mean and surface effects on gravity coming respectively Recognizing that gB r, ∂r , the integral T() from masses inside the geoid, terrain roughness, later- mean gB in Eq. (22) along the radial between the ally variable density distribution, second-order free-air geoid rg = R and approaching the Earth’s surface rt → R + HO() from within the Bouguer shell gives effects, and second-order Bouguer shell effects. All these terms must be computed to apply a rigorous correction = + O() r rg H to Helmert’s orthometric height. 1 ∂VT(r, ) gT() =− B dr B HO() ∂r r=r g T T 5.3 The geoid-generated gravity disturbance V rg, − V (rt, ) ≈ B B . (25) O() H In this subsection, we shall concentrate on the term As with the normal gravity term, this is a more rigorous formulation for the spherical Bouguer shell, where NT O = δ () − δ NT( ) r and H are along the same radial (i.e. H () = rt() − C g g rt, , (29) rg()). As such, there is no need to worry about the deviation of the radial from the plumbline in this case (cf. Tenzer et al. 2005, Appendix A). which deals with the corrections to the Helmert ortho- Inserting the integration limits in Eq. (24), then insert- metric height coming from the geoid-generated grav- ing the results into Eq. (25), after some algebraic manip- ity disturbance, comprising the mean value along the NT ulation, gives plumbline δg () and value on the Earth’s surface NT δg (rt, ). O() T O 2H The integral-mean value of the geoid-generated grav- g¯ () = 2πGρ0H () 1 − . (26) B 3 R + HO() ity disturbance along the plumbline between the geoid The relation between rigorous and Helmert’s definitions of orthometric heights R+HO() and the Earth surface can be represented [in analogy to NT 1 R δg () =∼ Eq. (2)] by 4π HO() = r R ∈O 1 NT rt × K r, ψ , ,R dr δg R, d . NT 1 r δg () = δgNT(r, ) dr HO() (34) rg Performing the radial integration of Poisson’s integral R+HO kernel K r, ψ , ,R , multiplied by r−1, the follow- 1 =∼ δgNT(r, ) dr, (30) ing expression can be found for the averaged Poisson’s HO() kernel (e.g. Vanícekˇ et al. 2004, Tenzer et al. 2005) r=R R+HO() 1 K r, ψ(, ), R) dr where the geocentric radius of the geoid surface r is r g r=R approximated by R, the mean radius of the Earth, which R+HO() should not be confused with the subscript R in the ter- 1 r2 − R2 = R dr rain roughness term. r 3[r, ψ(, ), R] = NT r R Since the geoid-generated gravity disturbance δg 2R ( ) = − r, multiplied by r is harmonic above the geoid (since [r, ψ(, ), R] the NT space contains no topographical masses above R+HO() NT R−r cos ψ(, )+ r, ψ(, ), R the geoid, and again neglecting the atmosphere), δg + ln . (35) r sin ψ(, ) () can be evaluated by averaging Poisson’s equation for r=R upward continuation (e.g. Kellogg 1929) in an integral for ψ = 0. sense. The Poisson equation reads Substituting Eq. (35) into Eq. (34), the mean grav- NT ity disturbance δg () along the plumbline takes the following form: 1 R δ NT( ) = ψ NT g r, K r, , ,R δg () 4π r ∈O = 1 R 2R − 2R NT π O() ψ( ) () ψ( ) ×δg R, d , (31) 4 H [R, , , R] [rt , , , R] ∈ 0 R R − rt() cos ψ(, ) + rt(), ψ(, ), R + ln r () R(1 − cos ψ(, )) + [R, ψ(, ), R] where 0 is the solid angle, is the dummy element t and ψ(, ) represents the spherical distance or geo- δgNT(R, )d . (36) centric angle between the computation and integration points. δ NT( ) Equation (36) can be simplified as The required gravity disturbance g R, , referred to the geoid, is a part of the sub-integral function. The NT 1 R δg () = K[R + HO(), ψ(, ), R] spatial form of the Poisson integral kernel K[r, ψ(, 4π HO() ) ] ∈ , R is given by (e.g. Kellogg 1929) O × δgNT R, d (37)

2 2 r − R where K stands for the intermediary integration kernel. K[r, ψ(, ), R]=R , (32) 3[r, ψ(, ), R] It can be shown, in the first approximation, that this kernel equals 1 1 ∗−H where the Euclidean spatial distance is given by + () ψ( ) = − + K R H , , , R 2R ∗ ln , (38) = r2 + r 2 − 2rr cos ψ(, ), (33) where stands for (R,ψ,R)and ∗ stands for (R+HO (), ψ,R). The derivation of this kernel is given in Appendix A. Inserting Eq. (31) into Eq. (30), the mean gravity dis- Equation (37) is somewhat cumbersome because it NT turbance δg () becomes (cf. Tenzer et al. 2005, Eq. 8) requires the NT gravity disturbance to be known on M. C. Santos et al. the geoid, which is not known. Therefore, to imple- Bouguer shell, assuming for the moment a constant ment it in practice first requires the downward con- topographical mass–density (lateral density variations δ NT( ) δ NT( ) tinuation of g rt, to g rg, . In Eq. (36), the will be considered in Sect. 5.5). NT geoid-generated gravity disturbance δg rg, is The gravitational field of the terrain roughness term obtained from the geoid-generated gravity anomaly is not harmonic inside the topography. As such, it has NT g rg, referred to the geoid in the NT-space by to be calculated from an adopted model of the shape of [cf. Heiskanen and Moritz 1967, Eq. (2)–(151e); Vanícekˇ the topography (i.e. a DEM), coupled with a constant et al. 2004] mass–density assumption. T ( ) 2 We begin with the gravitational potential V r, δgNT(R, ) = gNT(R, ) + TNT(R, ) (39) of topographical masses expressed in Eq. (41). Using R a spherical approximation of the geoid and Newtonian NT where T ( R,) represents the geoid-generated dis- integration, this reads (cf. Novák and Grafarend 2005) turbing potential in the NT space r=R+HO() TNT(R, ) = T(R, ) − VT (R, ) − VA(R, ) (40) VT (r, ) ≈ G ρ(r , ) The disturbing potential T( R,) can be taken from = a regional geoid model, computed according to Bruns r R × −1( ) 2 (1878) formula for the geoid height T = N()γ0, thus r, , r , r dr d . (44) making the geoid and the corresponding orthometric height system more compatible. The negative radial derivative of topographical grav- T The second term on the right-hand side of Eq. (40) itational attraction g (r, ) is given by VT (r , ) is the gravitational potential of the topograph- g A r =R+HO( ) ical masses, and V (rg, ) is the potential of all atmo- ∂ spheric masses. The term VT (r , ) is obtained through gT(r, ) ≈−G ρ(r , ) g ∂r the Newtonian integral = r R R+H0() − × 1(r, , r , )r 2dr d . (45) T −1 V (rg, ) = G ρ(r , ) ∈ = 0 r R From Eq. (43), we are looking for the mean value 2 × R, ψ(, ), r r dr d , (41) g¯T () between the Earth’s surface and the geoid, which R where ρ r, represents the actual mass–density of the is given – by definition – as [Tenzer et al. 2005, Eqs. (16)– topographical masses, usually computed from a density (18)] distribution model. The effect due to lateral mass–den- r=R+HO() sity variation is dealt with in Sect. 5.5. 1 g¯T () = gT(r, )dr Finally, to complement Eq. (29), the gravity distur- R HO() bance at the Earth’s surface is required. This term can r=R be evaluated directly from r=R+HO() − ∂ = 1 T( ) NT NT 2 V r, dr δg (rt, ) = g (rt, ) + T(rt, ) HO() ∂r rt r=R 2 2 − − T ( ) − A( ) 1 T O T V rt, V rt, . (42) = V [R+H (), ]−V [R, ] . (46) rt rt HO() Equation (29) can then be evaluated using Eqs. (37) and T (42). Substituting for the two values of potential V from Eq. (44), we get 5.4 The terrain/roughness-generated gravity R+H O() G In this subsection, we shall concentrate on the term g¯T() ≈ ρ(r , ) HO() r=R D = gT () − gT (r , ) (43) R R t − × 1[R, , r , ]− which gives the correction to the Helmert orthometric − height from the terrain roughness residual to the × 1[R + HO(), , r , ] r 2dr d . (47) The relation between rigorous and Helmert’s definitions of orthometric heights

r=R+Ho() Let us now express the radial integral in Eq. (47) as a ∂ −1( ) T r, ; r , sum of two integrals g (rt, )≈−Gρ R 0 ∂r r=rt() ∈ r=R+Ho() 0 R+H O( ) R+HO() × 2 r dr d (52) F(r )dr = F(r )dr R R which can also be evaluated by quadrature methods. R+H O() 5.5 The lateral variation of topographical mass–density + F(r )dr (48) R+HO() In this subsection, we consider the term δρ δρ The ﬁrst integral on the right-hand side of Eq. (48) E = g¯ () − g (rt, ) . (53) describes the contribution of the Bouguer shell of con- In most gravimetric geoid computations, the topo- stant thickness HO() (e.g. Vanícekˇ et al. 2001), which graphical mass–density is generally modelled by an aver- was dealt with in Sect. 5.2. The second integral gives the age value of ρ = 2, 670 kg m−3 . Martinec (1998) posed contribution due to the terrain residual to the Bouguer 0 the question on how much a variation in topographi- shell. cal mass–density affects geoid height computation. To The density can also be written as a sum of two terms, answer this question in the context of the orthomet- one containing a contribution due to the mean density ric height, we assume only lateral variations of density, ρ and the other containing the residual density δρ(r,) 0 leaving the radial variation still to be tackled. The devel- contribution opments below follow from those of Sect. 5.4. The contribution of lateral variation of density to ρ(r, ) = ρ0 + δρ(r, ). (49) the correction to Helmert’s orthometric height is rep- The roughness term is represented by the second term resented by third term in Eq. (21) from Tenzer et al. in Eq. (21) of Tenzer et al. (2005) (2005) r=R+H() R+H O() δρ G − ρ g () = δρ(r , ){ 1[R, , r , ] ¯T () = G 0 −1[ ] Ho() gR R, , r , ∈ = Ho() 0 r R ∈ = + O() − 0r R H − 1[R + H(), , r , ]}r 2dr d (54) − − 1[R + HO(), , r , ] r 2dr d , (50) The surface gravity generated by lateral variation of density is given by This term is nothing else but the change in terrain rough- ρ r=R+H() ness of constant density of 0, from the geoid to the sur- −1 δρ ∂ [R, ; r , ] face of the Earth, divided by the orthometric height of g (rt, )=−G δρ() ∂r r=rt() the point of interest ∈0 r =R 2 1 ×r dr d , (55) g¯T () ≈ VT (R, ) − VT (R + HO(), ) . R O() R R H which follows from a more complete expression pro- (51) vided by Martinec (1998) that takes into account the radial variation in density r

These two roughness parts of topographical potential T r =R+H( ) V can be evaluated through numerical quadrature of −1 R δρ ∂ [R, ; r , ] the Newton integral (Eq. 44). Equation (51) provides g (rt, )=−G δρ(r , ) ∂r r=rt() the mean gravity generated by the terrain roughness, ∈0 r =R expressed in terms of gravitational potential. As pointed ×r 2dr d , (56) out in Sect. 5.3, it comprises a contribution from the average topographical mass–density, plus a smaller cor- Equations (54) and (55) provide the terms required in rection due to mass–density variations. Eq. (53) The other term in Eq. (43), the terrain roughness The correction to Helmert’s orthometric height due term at the Earth’s surface, is given by the second term to the laterally varying topographical mass–density is in Martinec (1998) also given by the following approximate expression M. C. Santos et al.

(Vanícekˇ et al. 1995) if one considers only the radial 30 gradient of the gravitational attraction generated by the 25 2500 spherical Bouguer shell of the anomalous topographical 20 2000 density δρ(), 15 1500 δρ 2 δρ() ε () ≈ 2 π G HO() . (57) 10 1000 HO g() 5 500 Correction (cm) 0 0 5.6 Summary Height -5 (m)

The correction to the Helmert orthometric height to give -10 the rigorous orthometric height defined by Tenzer et al. -15 ε () 235.0 235.5 236.0 236.5 237.0 237.5 238.0 238.5 239.0 (2005) HO is given by Eq. (15). It follows directly from the evaluation of the correction to Helmert’s mean Longitude (degrees) ε () gravity g , written below in a simplified manner as Fig. 2 Profiles of the five components of the correction to Helm- ε () = + + + + ert’s orthometric height (cm), as well as the Helmert orthometric g A B C D E (58) height (m) along a profile at 50◦N across the Canadian Ropcy Mountains. The continuous thick line represents the topographic The terms A and B can be computed from Eq. (28) as δ height; continuous thin line corresponds to ε g () (geoid-gen- HO O 2 erated gravity disturbance); dashed line corresponds to εT () H () HO A + B =−γ (terrain-roughness-generated gravity)] dotted line corresponds to r () δρ t ε () (lateral variation of topographical mass–density). The HO 4 HO()2 HO() other two components are too small to be plotted + πGρ0 2 − 3 R + HO() R + HO() (59) to show their relative contributions to the correction. The terms C, D and E can be computed from Eqs. (37) These terms are superimposed on the topographic height and (42), (51) and (42) and (54) and (55), respectively. variations (shown with the thicker line in Fig. 2) scaled Note that several of these terms would have already down by 100 m, to show that there is not always a one- been computed for a regional gravimetric geoid model to-one correspondence of the correction terms with based on the Stokes–Helmert approach (Vanícekˇ and height. All integral terms were computed over a spher- Martinec 1994). This simplifies the task, where the ical cap radius of 3◦, beyond which the far-zone contri- gridded quantities can be interpolated to the points of butions become negligible (<1 mm) for this test area. interest and applied as part of the corrections to the Inspecting Fig. 2, we see that the correction term Helmert orthometric height, provided that the horizon- from the geoid-generated gravity disturbance (C) gives tal locations of the benchmarks are known. It also makes the largest correction values, and is generally positively the rigorous orthometric heights more compatible with correlated with topography, though not perfectly. The the regional geoid model. Finally, the total correction correction due to terrain-roughness-generated gravity to the Helmert orthometric height ε O () is H (D) is the second most important contribution. How- −HO() ever, it works against the former correction, and there ε O () = (A + B + C + D + E) (60) H g() is a less strong, negative correlation with topography. The third largest term in magnitude is the correction due to lateral variation of topographical mass–density 6 Numerical tests (E), varying around zero and with maximum magnitude not greater than 5 cm. The final two terms, due Using Canadian gravity, terrain and lateral topographic to second-order correction for normal gravity (A) and mass–density data, we have computed rigorous correc- second-order correction for the Bouguer shell (B), are tions to Helmert’s orthometric heights along a profile both very small, not showing up in Fig. 2. Table 2 sum- across the Canadian Rocky Mountains. This profile marizes the statistics of these five correction terms. spans the longitudes from 235 to 239◦E along the 50◦N Figure 3 shows a comparison between the corrections parallel. Figure 2 shows each one of the terms in Eq. (60) to Helmert orthometric heights using the method de- (i.e. A: second-order free-air, B: second-order Bouguer scribed in this paper (termed rigorous), and the Mader shell, C: NT gravity disturbance, D: terrain roughness, and Niethammer approaches, for the same profile as in and E: lateral density variations) computed separately Fig. 2. The Mader and Niethammer corrections were The relation between rigorous and Helmert’s definitions of orthometric heights

Table 2 Descriptive statistics of corrections to Helmert’s Correction due Correction due Correction due to Correction due Correction due orthometric height from the to gravity to terrain- lateral variation for 2nd-order for 2nd-order proﬁle shown in Fig. 2 disturbance roughness of density normal gravity Bouguer shell Mean 6.0 −1.4 0.2 −0.0 −0.0 STD 3.3 2.5 1.1 0.0 0.0 Values in centimetres, Minimum 0.0 −11.5 −1.9 0.0 0.0 rounded to the nearest Maximum 15.4 2.5 5.2 −0.1 −0.0 millimetres

30 Table 3 Descriptive statistics of the total corrections to Helmert’s 25 2500 orthometric height from the proﬁle shown in Fig. 3

20 2000 Mader Niethammer Rigorous 15 1500 Mean −1.4 −1.7 4.4 10 1000 Standard deviation 2.5 3.1 3.1 − − − 5 500 Minimum value 11.5 15.2 4.9 Maximum value 2.5 2.9 13.0 0 0 Correction (cm) Height Values in centimetres, rounded to the nearest millimetres -5 (m)

-10 -15 normal gravity, second-order correction for the Bou- -20 235.0 235.5 236.0 236.5 237.0 237.5 238.0 238.5 239.0 guer shell, the geoid-generated gravity disturbance, the Longitude (degrees) terrain-roughness-generated gravity, and the lateral variation of topographical mass–density. These individual Fig. 3 Comparison among the rigorous, Mader and Nietham- corrections have been evaluated numerically along a mer corrections to Helmert orthometric heights along the same profile as in Fig. 2. Units in centimetres. The continuous thicker profile across the Canadian Rocky Mountains, and plot- line represents the topographic profile; continuous thin line repre- ted against the topographical height variation. sents the rigorous correction; dashed line represents Niethammer This comparison shows that the geoid-generated grav- correction; dotted line represents Mader correction ity disturbance, the terrain-roughness-generated gravity and the lateral variation of topographical mass–density are, respectively, the most important contributors to- wards obtaining a more rigorous orthometric height. computed from Eqs. (12) and (13) using the same topo- It also shows that the geoid-generated gravity distur- graphical corrections used to evaluate the rigorous cor- bance and the terrain-roughness-generated gravity work rections. approximately against each other, though not From Fig. 3, the Mader and Niethammer corrections completely, as each is not perfectly correlated with the are very similar to one another, whereas the rigorous topography. The second-order correction for normal correction is larger, which is attributed to the two addi- gravity and the Bouguer shell are negligibly small for tional terms not accounted for in Mader nor Nietham- this test, but become larger for very high elevations. mer’s approaches: geoid-generated gravity disturbance Comparisons with other refinements of Helmert or- and lateral variation of topographical mass–density. The thometric heights, namely Mader (1954) and Nietham- larger contribution comes mostly from the geoid-gener- mer (1932), have also been performed. The Mader and ated gravity disturbance (cf. Fig. 2). Table 3 summarizes Niethammer orthometric heights are very similar to one the statistics of the corrections along this profile. another, but the respective corrections are smaller than the rigorous corrections. They differ from the rigorous approach due to inclusion of the terms pertaining to the 7 Summary, discussion and conclusion geoid-generated gravity anomaly and lateral variation of topographical mass–density. We have derived expressions to transform Helmert’s Finally, it is important to point out that several of the approximation of the orthometric height into a more correction terms used here are the same as would have rigorous one (cf. Tenzer et al. 2005), taking into account been computed for a regional gravimetric geoid model effects coming from the second-order correction for based on the Stokes–Helmert approach (e.g. Vanícekˇ M. C. Santos et al. and Martinec 1994). As such, they are relatively easy Substituting this result into Eq. (A3) gives to apply to existing Helmert orthometric heights. More- over, this makes the resulting heights more compatible K R + HO(), ψ, R with a regional gravimetric geoid model based upon the Stokes–Helmert approach. = 1 − 1 2R ∗ O() 2 ∗ 1 − 1 + H 1 − 1 + R 2 R R 8 Appendix A: derivation of Eq. (38) +ln (A8) + HO() − − 1 2 + 1 R 1 1 2 R R We wish to simplify the expression for the averaging Poisson’s kernel (Eq. (35)), which reads After a few algebraic operations, we get ( + O() ψ ) K R H , , R K R + HO(), ψ, R 2R 2R =− + 1 1 (R + HO(), ψ, R) (R, ψ, R) = − 2R ∗ O O R−(R+H ()) cos ψ + (R+H (), ψ, R) +ln − + HO() − 1 2 + ∗ ( + O()) sin ψ 1 1 1 R H R 2 R R +ln R − R cos ψ + (R, ψ, R) + HO() 1 2 + −ln .(A1) 1 R 2 R R R sin ψ 1 1 ∗−HO() ψ ≈ − + For integration within a very small radius 0 of, say, 3 2R ∗ ln .(A9) arc-degrees, we can assume , ∗, HO() << R,(A2)This is the final simplified form, valid for a small (<3◦) radius ψ of integration, which will next be studied. where we have denoted ( R,ψ,R)by and 0 It should be noted that the first term in Eq. (A9) is (R+HO(), ψ, R)by *. This is permitted because of the leading term, while the second is a corrective term. the rapid decay of the Poisson kernel with spherical The leading term converges very rapidly since it holds distance, as supported by our empirical evidence (cf. most of its power in the nearest vicinity of the computa- Sect. 6). tion point. For instance, the cumulative sum of this term Now, we can rewrite Eq. (A1) as ◦ across a profile gains ∼99% power at ψ =0.1 . The mag- 1 1 nitude of the corrective (logarithmic) term comprises K(R + HO(), ψ, R) = 2R − ∗ <1% of the magnitude of the leading term. O 1 − (1 + H ()/R) cos ψ + ∗ /R One may see that the first terms on the right-hand + ln .(A3) (1 + HO()/R)(1 − cos ψ + /R) side of Eqs. (A3) and (A7) are exactly the same. Thus, the difference between the exact expression and the first The last term in Eq. (A3) should be approximation stems only from the much smaller loga- − ( + HO() ) ψ + ∗ rithmic term. As such, our subsequent numerical inves- 1 1 cos ln R R .(A4)tigations study the relationship between the term + HO() − ψ + 1 + O() 1 cos R R H − ( + HO() ) ψ + ∗ / However, due to the precision required, the approxima- 1 1 cos ln R R (A10a) O() tion in Eq. (A3) is enough since 1 + H 1 − cos ψ + R R HO() HO() 1 + ≈ 1 + (A5) R + HO() R and its approximation Realizing that ∗−HO() ψ ln = 2R sin ,(A6) . (A10b) 2 we can express cosψ in Eq. (A4) as The variables in both terms are the orthometric height ψ 2 HO() and the angular distance ψ. The behaviour of cos ψ = 1 − 2 sin2 = 1 − 1 .(A7) ◦ ◦ 2 2 R these terms within the interval 0.0001 (10 m) <ψ3 will The relation between rigorous and Helmert’s definitions of orthometric heights O -6 1 1 ∗−H () -10 2R − ∗ +ln is sufficient as the first approximation of the complicated integration term. Next, next let us have a look at the first term in -10-4 Eq. (A3). Realizing that lim ∗= 2 + HO()2,it ψ→0 can be written as -2 -10 1 1 1 1 2R − ≈ 2R − ∗ 2 + HO()2 ⎡ ⎤ − 1 -100 2R HO()2 2 = ⎣1 − 1 + ⎦ (A11) 2 0 0.5 1 1.5 2 2.5 3 Spherical distance [°] For HO() < , (the case of HO() > can be treated O() ∗ 1−(1+ H ) cos ψ+ R R in a similar way) this can be represented by a convergent Fig. A1 The behaviour of the ln O() 1+ H 1−cos ψ+ R R binomial series term (lower batch of curves) and the discrepancies ⎡ ⎤ O() ∗ − 1 1−(1+ H ) cos ψ+ ∗− O() O 2 2 R R − H 1 1 1 1 H () ln O() ln (upper batch of curves) ⎣ ⎦ 1+ H 1−cos ψ+ 2R − = 2R − 1 + R R ∗ 2 O() O() a H = 200 m dotted line; b H = 1,000 m dashed line; c HO() = 2,000 m solid thin line; d HO() = 5,000 m solid bold ∞ 2k 1 1 − 1 HO() line. Also note the negative logarithmic scale = 2R − 1+ 2 . k k=1 (A12) be numerically investigated. In these tests, the orthomet- Carrying out the required algebraic operations, we ric height takes the following constant values HO() = arrive at 200 m, HO() =1km,HO() =3km,HO() =5km.In ∞ 2k other words, the topography of the test area is assumed 1 1 2R − 1 HO() O() 2R − = 2 . (A13) to be a plateau with a constant height H . ∗ k The logarithmic term is always negative, since the k=1 ∗− O() argument H takes values between 0 and 1, The series in Eq. (A10) is alternating, and thus a con- ∗−HO() vergent series even for H and going simultaneously to i.e. 0 < < 1. Figure A1 shows the behaviour zero, thus of the logarithmic term and the discrepancies between the exact expression (Eq. (A3)) and its approximation 1 1 ◦ − (Eq. (A7)) across the 3 integration area. The lower lim 2R ∗ → 0 batch of curves in Fig. A1 indicates the magnitude of HO() → 0 the logarithmic term for each case, whereas the upper ∞ 2k batch denotes the corresponding discrepancies. 1 − 1 HO() = 2R lim 2 From Fig. A1, most of the power in the logarithmic →0 = k term is in the nearest vicinity of the computation point HO()→0 k 1 ψ ∞ and it decreases with increasing . Note that at the com- 1 − 1 putation point, the discrepancies are almost zero. The = 2R lim 2 (A14) →0 = k magnitude of the relative discrepancies increases line- HO()→0 k 1 arly with the distance (recall that a logarithmic scale is used in Fig. A1. Note, however, that in any tested As the summation is a real number, the whole expres- case and for ψ <1◦ the discrepancies are at least of two sion grows above all limits and we get orders of magnitude less than the logarithmic term itself. O ◦ ∗−H () 1 − 1 = δ( ) At ψ =3 , the error of the approximation ln lim 2R , , (A15) H→0 ∗ consists of ∼ 3% only from the exact expression. Also recall that the logarithmic term is <1% of the where δ is the Kronecker symbol for the function that whole Poisson kernel. From the above results, it is obvi- grows beyond all limits when = and equals zero ous that the expression K[R + HO(), ψ(, ), R]= for all other values of . We can thus see that M. C. Santos et al.

NT 1 R lim δg () = lim Heiskanen WA, Moritz H (1967) Physical geodesy. Freeman, San HO()→0 HO()→0 4π HO() Francisco Helmert FR (1890) Die Schwerkraft im Hochgebirge, Insbeson- dere in den Tyroler Alpen, Veröff. Königl. Preuss Geod Inst O × K[R + H (), ψ(, ), R] No.1,Berlin ∈ Jekeli C (2000) Heights, the geopotential, and vertical datums, Rep O 459, Dept. Geod. Sci., Ohio State University, Columbus × δ NT Kellogg OD (1929) Foundations of potential theory. Springer, Ber- g R, d lin Heidelberg New York Kingdon R, Vanícekˇ P, Santos M, Ellmann A, Tenzer R (2005) NT = δg (R, ) (A16) Toward an improved orthometric height system for Canada. Geomatica 59(3):241–250 (Errata: Figure 4 on Geomatica, Vol as one would expect. 60 (1):101) We note that the averaging Poisson kernel also has a Mader K (1954) Die orthometrische Schwerekorrektion des Präz- singularity for the case when H is not equal to zero. For isions-Nivellements in den Hohen Tauern, Österreichische Zeitschrift f’ur´ Vermessungswesen, Sonderheft 15, Vienna HO() > 0, we get Martinec Z (1993) Effect of lateral density variations of topo- 1 1 graphical masses in view of improving geoid model accuracy lim 2R − over Canada. Final report of contract DSS No. 23244-2-4356, → ∗ 0 Geodetic Survey of Canada, Ottawa ⎡ ⎤ Martinec Z (1998) Boundary value problems for gravimetric deter- − 1 2 ⎣1 1 2 ⎦ mination of a precise geoid. Lecture notes in Earth Sciences, = lim 2R − 1 + 73, Springer, Berlin Heidelberg New York → O()2 0 H H Niethammer T (1932) Nivellement und Schwere als Mittel zur Berechnung wahrer Meereshöhen. Schweizerische Geodäti- ∞ 1 1 − 1 2k sche Kommission, Berne = lim 2R − 1+ 2 Novák P, Grafarend EW (2005) Ellipsoidal representation of the →0 HO() k HO() topographical potential and its vertical gradient, J Geod 78(1– k=1 3):691–706. DOI 10.1007/s00190-005-0435-4 1 1 1 Sjöberg LE (2004) The effect on the geoid of lateral topographic = lim 2R − = lim (A17) density variations. J Geod 78(1-2):34–39 → O() → 0 H 0 Strange WE (1982) An evaluation of orthometric height accuracy which also grows above all limits. Thus, the averaging using borehole gravimetry. Bull Géod 56:300–311 Sünkel H (1986) Digital height and density model and its use Poisson kernel has a removable singularity of a linear for the orthometric height and gravity field determination for type (1/0) at the point of interest , whether the height Austria. In: Proceedings of international symposium on the is equal to 0 or not. The second, logarithmic term is definition of the geoid, Florence, May, pp 599–604 always negative: it equals 0 for going to 0, and it also Tenzer R, Vanícekˇ P (2003) Correction to Helmert orthometric < height due to actual lateral variation of topographical density. goes to 0 for growing . Note that the argument 0 Braz J Cart 55(2):44–47 ∗−HO() < 1. Tenzer R, Vanícekˇ P, Santos M, Featherstone WE, Kuhn M (2005) Rigorous determination of the orthometric height. J Geod 79(1–3):82–92. DOI 10.1007/s00190-005-0445-2 Acknowledgements The Canadian investigators are supported Vanícekˇ P, Kleusberg A, Martinec Z, Sun W, Ong P, Najafi M, Va- by the “GEOIDE Network of Centres of Excellence”. The Austra- jda P, Harrie L, Tomášek P, ter Horst B (1995) Compilation of lian investigators were supported by Australian Research Council a precise regional geoid. DSS Contract# 23244-1-4405/01-SS grant DP0211827. The topographical data were kindly supplied by Report for Geodetic Survey Division, Ottawa, pp 45 Geodetic Survey Division of Natural Resources Canada. We also Vanícekˇ P,Martinec Z (1994) Stokes–Helmert scheme for the eval- thank the reviewers and editors for their time and constructive uation of a precise geoid. Manuscr Geodaet 19(3):119–128 criticism. 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