The relation between rigorous and Helmert s definitions of orthometric heights

Transcription

1 J Geodesy DOI 0.007/s OIGINAL ATICLE The reation between rigorous and Hemert s definitions of orthometric heights M. C. Santos P. Vaníček W. E. Featherstone. Kingdon A. Emann B.-A. Martin M. Kuhn. Tenzer eceived: 8 November 005 / Accepted: 8 Juy 006 Springer-Verag 006 Abstract Foowing our earier definition of the rigorous orthometric height J Geod 79-3: we present the derivation and cacuation of the differences between this and the Hemert orthometric height, M. C. Santos B P. Vaníček. Kingdon A. Emann B.-A. Martin Department of Geodesy and Geomatics Engineering, University of New Brunswick, P.O. Box 4400, Fredericton, NB, Canada E3B 5A3 e-mai: msantos@unb.ca P. Vaníček e-mai: vanicek@unb.ca. Kingdon e-mai: w4etp@unb.ca Present Address: A. Emann Department of Civi Engineering, Tain University of Technoogy, Ehitajate tee 5, 9086 Tainn Estonia e-mai: Artu.Emann@ttu.ee Present Address: B.-A. Martin Critchow Associates Ltd., 6 Moesworth Street, Weington, New Zeaand W. E. Featherstone M. Kuhn Western Austraian Centre for Geodesy, Curtin University of Technoogy, GPO Box U987, Perth, WA 6845, Austraia e-mai: W.Featherstone@curtin.edu.au M. Kuhn e-mai: M.Kuhn@curtin.edu.au. Tenzer Facuty of Aerospace Engineering, Physica and Space Geodesy, Kuyverweg, 69 HS Deft, P.O. Box 5058, 600 GB Deft, The Netherands e-mai: r.tenzer@tudeft.n which is embedded in the vertica datums used in numerous countries. By way of comparison, we aso consider Mader and Niethammer s refinements to the Hemert orthometric height. For a profie across the Canadian ocky Mountains maximum height of,800 m, the rigorous correction to Hemert s height reaches 3 cm, whereas the Mader and Niethammer corrections ony reach 3 cm. The discrepancy is due mosty to the rigorous correction s consideration of the geoid-generated gravity disturbance. We aso point out that severa of the terms derived here are the same as those used in regiona gravimetric geoid modes, thus simpifying their impementation. This wi enabe those who currenty use Hemert orthometric heights to upgrade them to a more rigorous height system based on the Earth s gravity fied and one that is more compatibe with a regiona geoid mode. Keywords Orthometric height Geoid Mean gravity Pumbine Introduction The orthometric height is defined as the metric ength aong the curved pumbine from the geoid to the Earth s surface. To cacuate an orthometric height from spiriteveing data and/or geopotentia numbers requires that the mean vaue of the gravity aong the pumbine between the Earth s surface and the geoid shoud be known. This mean vaue is stricty defined in an integra sense e.g. Heiskanen and Moritz 967, p. 66. In the past, three main approximations have been appied in practice to evauate this integra-mean vaue of gravity. The Hemert method, as described in Heiska-

2 M. C. Santos et a. nen and Moritz 967, chap. 4, appies the simpified Poincaré Prey vertica gradient of gravity, which uses norma gravity and a Bouguer she of constant topographic mass density, to the observed gravity at the Earth s surface in order to obtain an approximated mean vaue hafway down the pumbine. Niethammer 93 and Mader 954 refined Hemert s mode by incuding the effect of oca variations in the terrain roughness reative to the Bouguer she. Mader 954, considering ony the inear change of the gravimetric terrain correction with respect to depth, used the simpe mean of the terrain effect at the geoid and at the Earth s surface, whereas Niethammer 93 used the integra mean of terrain effects evauated at discrete points aong the pumbine. Dennis and Featherstone 003 evauated these three approximations, showing that the accuracy is ordered Niethammer, Mader then Hemert, which refects the eves of approximation used. In addition, the mean topographica mass density ρ 0 =,670 kg m 3, used in the Hemert, Niethammer and Mader approximations of the actua distribution of topographica mass density, is not sufficienty accurate in the origina manuscript, Hemert 890 refers to a mass density vaue of,400 kg m 3. Attempts to refine the Hemert orthometric height in this regard have incuded varying topographica mass density data e.g. Sünke 986; Aister and Featherstone 00 and borehoe gravimetry Strange 98 to better approximate the integra-mean of gravity aong the pumbine. To the best of our knowedge, no attempts have been made to incude topographica mass density data in the Mader and Niethammer heights even though Niethammer 93 aready mentioned the necessity to use varying density information for a more rigorous treatment. In this paper, we show that to arrive at a more rigorous orthometric height, one must take into account not ony the effect of terrain roughness and norma gravity, but aso those additiona effects coming from the masses contained within the geoid herein termed the geoidgenerated gravity not accounted for by the Hemert approach and from the mass density variations within the topography. This is necessary because the mean vaue of gravity aong the pumbine between the geoid and the Earth s surface depends on a these quantities cf. Tenzer et a Mean gravity aong the pumbine is thus evauated as the sum of the integra-mean vaues of the geoidgenerated gravity and the topography-generated gravitationa attraction. For practica evauation, the geoid-generated gravity is further divided into norma gravity and the geoid-generated gravity disturbance, i.e. the gravity disturbance in the so-caed no-topography NT space cf. Vaníček et a Likewise, the topography-generated gravity is divided among the spherica Bouguer she, the terrain roughness residua to the Bouguer she, and the topographica mass density variations. The aim of this paper is to provide the theoretica background and practica methods with which to convert Hemert orthometric heights as described in, e.g. Heiskanen and Moritz 967, chap. 4, which are used as the height system embedded in the vertica datum adopted in numerous countries, to the more rigorous orthometric heights presented in Tenzer et a With this in mind, we have presented some preiminary derivations and resuts for various components of the correction in Vaníček et a. 00 and in Tenzer and Vaníček 003. Kingdon et a. 005 present a numerica evauation over a part of Canada. This paper now presents and reviews the compete methodoogy.. Notation and terminoogy In the seque, the dummy argument represents the geocentric spherica coordinates φ and λ of a point φ π/, π/, λ 0, π and r denotes its geocentric radius. The radius of a point is a function of ocation being represented by r = r. The symbos r g and r t represent the geocentric radii of the geoid and the Earth s surface, respectivey, and wi be abbreviated to r g and r t where there is no ambiguity. The orthometric height of a point is aso a function of ocation, and is represented by H O. The gravity at a point is a function of both the radius and the horizonta geocentric coordinates, being represented by gr, or in a simpified form used throughout the paper as gr,. The remaining gravity-reated notation used throughout this paper is summarized in Tabe. Where reevant, overbars wi be used to denote the integra-mean quantities between the geoid and the Earth s surface. We use the term terrain roughness to represent the irreguar part of topography with respect to the Bouguer she, i.e.the geometric variations in the shape of topography. There are many other terms found in the iterature to indicate the same, such as topographica roughness or simpy terrain, but here we choose the terminoogy of terrain roughness. ecapituation of the rigorous orthometric height The orthometric height H O of a point on the Earth s surface r t, is defined as the ength of the curved pumbine between the geoid r g and the Earth s surface r t = r g + H O, and is given by e.g.

3 The reation between rigorous and Hemert s definitions of orthometric heights Tabe Gravity-reated notation used throughout this paper gr, Gravity at an arbitrary point g NT r, Gravity generated by masses contained within the geoid, i.e. with the topography removed and in the NT-space g T r, Gravitation generated by masses contained within the topography ony, i.e. those between the geoid and Earth s surface g δρ r, Effect on gravitation due to atera mass density variations inside the topography with respect to the reference vaue of ρ 0 =,670 kg m 3 γ r, Norma gravity generated by the geocentric reference eipsoid δg NT r, Gravity disturbances generated by masses contained within the geoid gb T r, Gravitation generated by a spherica Bouguer she g T r, Gravitation generated by the terrain roughness, i.e. topographica unduations reative to the spherica Bouguer she ε g Correction to Hemert s approximation of integra-mean gravity aong the pumbine ε H O Correction to Hemert s orthometric height to convert it to the rigorous orthometric height Tenzer et a. 005 Heiskanen and Moritz 967, Eqs. 4 H O = Cr t, g where ḡ is the integra-mean vaue of gravity aong the pumbine between the geoid and the Earth s surface r t ḡ = H O gr, dr r g and Cr t, is the geopotentia number, which is the difference between the Earth s gravity potentia W 0 = constant at the geoid and Wr t, at the Earth s surface Cr, = W 0 Wr,. 3 Concerning ḡ, since the actua vaue of gravity gr, aong the pumbine cannot be measured at a points, the integra-mean gravity ḡ generay has to be computed from the observed surface gravity gr t,, together with some reaistic and physicay meaningfu mode of gr, aong the pumbine. This computation can be achieved in practice by reducing the observed gravity according to some accepted mode of terrain roughness and the topographica mass density distribution between the geoid and the Earth s surface. 3 Decomposition of actua gravity In order to formuate the corrections to Hemert s orthometric height in a way that can be computed from the datasets currenty avaiabe i.e. terrestria gravity observations, a digita eevation mode DEM and atera density variations interpreted from geoogica maps or databases, we use the foowing decomposition of gravity. The primary pragmatic benefit of this approach is that these are the same data used to compute a gravimetric geoid mode, thus making the rigorous orthometric heights more compatibe with the geoid Fig. A very simpe conceptua decomposition mode of actua gravity into the geoid-generated component interna white area and the topography-generated component dark area; the topography is exaggerated for the sake of carity mode, provided of course the same corrections have been computed from the same data. The gravity acceeration at a point gr, can be decomposed into two terms; one comprising gravity generated by the masses inside geoid g NT r,, i.e. in the NT-space Vaníček et a. 004, and another comprising the gravitationa attraction generated by topography g T r,, gr, = g NT r, + g T r,. 4 Figure schematicay shows a simpistic cross section of the Earth with the decomposition in Eq. 4, where the white interna area shows the contribution that comes from a masses within the geoid and the dark area shows the contribution due to the topographic masses. The geoid-generated gravity can be further decomposed into the contribution from norma gravity and the gravity disturbance caused by ony the masses inside the geoid, the so-caed NT gravity disturbance cf. Vaníček et a Likewise, the topography-generated gravitation can be further decomposed into the Bouguer she contribution and the terrain roughness term residua to this she. These two terms can aso be adapted to incude atera topographica mass density variations from the standard vaue of ρ 0 =,670 kg m 3 Sect. 5.5,

4 M. C. Santos et a. this being the way in which topographic mass density data is normay derived from geoogica maps. The geoid-generated gravity is represented by the sum of norma gravity γ r, and the geoid-generated gravity disturbance δg NT r, g NT r, = γr, + δg NT r,. 5 The topography-generated gravitationa acceeration is represented by the sum of that generated by Bouguer she gb T r,, the terrain roughness residua to the Bouguer she g T r,, and the atera variations in mass density from the assumed average ρ 0 =,670 kg m 3 within the topography g δρ r, g T r, g T B r, + gt r, + gδρ r,. 6 Inserting Eqs. 5 and 6 into Eq. 4 gives a compete expression representing the tota gravity as cf. Tenzer et a. 005 gr, γr, + δg NT r, + gb T r, + g T r, + gδρ r,. 7 The approximation sign refects the fact that two additiona effects are omitted from Eqs. 6 and 7: the gravitationa effects of atmospheric masses and the radia variation of the topographic mass density. The former is omitted here because it is very sma cf. Tenzer et a. 005, Appendix B but wi be reintroduced in Sect. 5.3; the atter is very difficut to quantify because there is not enough reiabe information on the radia distribution of mass density within the topography. As such, we sha ony consider atera topographic mass density variations cf. Martinec 993; Sjöberg 004. This is aso consistent with the treatment of the geoid in the Stokes Hemert scheme e.g. Vaníček and Martinec 994. Finay, the integra-mean gravity aong the pumbine ḡ, given by the integra-mean of gr, according to Eq. when appied to Eq. 7, is g γ + δg NT + g T B + gt + gδρ. 8 4 Hemert s and other approximations of the orthometric height By way of comparison, the expression for the approximated mean gravity aong the pumbine used in the Hemert orthometric height i.e. computed using the simpified Poincaré-Prey reduction is cf. Heiskanen and Moritz 967, Eqs. 4 5 ḡ H = gr t, γ h + 4πGρ 0 H O, 9 where γ/ h is the inear vertica gradient of norma gravity, evauated at the topographica surface, h is the geodetic height, G is the Newtonian gravitationa constant, and ρ 0 is the assumed-constant topographica mass density. It is worth mentioning that, in this paper, we foow the expression for Hemert s orthometric height Eq. 9 as given in Heiskanen and Moritz 967, chap. 4. This is of most interest because this is the way in which most if not a geodesists have assumed Hemert s definition, and using a panar approximation of the terrain. In his origina work, however, Hemert 890 considered the gravitationa effect of the compete topographic masses, deineating that the varying density within the topographic masses and the masses beow the geoid shoud be considered in a rigorous treatment. Whie this is described in Hemert s 890 text, his mathematica formuation is simper, thus probaby expaining why the simpification in Eq. 9 has been adopted in practice. Using the numerica vaues of γ/ h = mga/m the inear vertica gradient of norma gravity in free air and πgρ 0 = mga/m the inear vertica gravity gradient from the Bouguer she for ρ 0 =,670 kg m 3 in Eq. 9 gives ḡ H = gr t, H O. 0 Therefore, Eqs. 9 and 0 effectivey attempt to reduce surface gravity to a point hafway down the pumbine, using the Poincaré Prey approximation of the vertica gravity gradient, to give an approximation of the integra-mean vaue aong the pumbine between the geoid and the Earth s surface. Note that this approximation embeds a constant topographic mass density for the Bouguer she and competey negects terrain roughness residua to the Bouguer she. Making use of the genera Equation 7 at the Earth s surface i.e. r = r t, from Eq. 9 we obtain ḡ H = γr t, + δg NT r t, + gb T r t, + g T r t, + g δρ r t, γ h + 4πGρ 0 H O It is aso worthwhie reating the rigorous orthometric height to the Niethammer 93 and Mader 954 orthometric heights. This is curiosity driven, since these height systems are not in wide practica use to the best of our knowedge. Both systems attempt to take terrain roughness, residua to the Bouguer she, into account when determining the integra-mean vaue of gravity aong the pumbine. Both Mader and Niethammer orthometric heights incude a term in the computation of mean gravity to

5 The reation between rigorous and Hemert s definitions of orthometric heights incude the mean terrain effect. Niethammer 93 performs a discrete evauation of the integra-mean terrain effect at a series of discrete points at even intervas aong the pumbine, whie Mader 954 assumes the terrain effect to vary ineary between the geoid and the surface, and so uses a simpe mean of the vaues of the effect evauated for the Earth s surface and the geoid. In our terminoogy, and using our approach to evauate the terrain roughness term, Mader s 954 approximated mean vaue of gravity aong the pumbine method is g M = g H + gt r g, ; ρ 0 g T r t, ; ρ 0, and according to Niethammer s 93 method, it is g N = g H g T r t, ; ρ 0 + +H O H O g T r, dr. 3 r= 5 Corrections to the Hemert orthometric height To estabish the reationships between the more rigorous mean gravity given by Eq. 8 and Hemert s approximate i.e. Poincaré Prey formua given by Eq., we subtract them, grouping ike terms. The resuting difference is caed the correction to Hemert s mean gravity ε g : ε g = g g H ε g = γ γr t, + γ h HO + g T B gt B r t, + πgρ 0 H O + δg NT δg NT r t, + g T gt r t, + g δρ g δρ r t, 4 After being computed, ε g can be used to appy a correction to Hemert s orthometric height ε H O using cf. Heiskanen and Moritz 967, p. 69 ε H O = HO ε g, 5 g to an accuracy of << mm in ε H O. Since ε g is sma, the actua mean gravity in Eq. 5 can ironicay be computed using Hemert s approximation Eq. 0. This wi make it consideraby easier to numericay evauate ε H O in ater sections of this paper H O ε g ε H O = gr t, H O, 6 where ε H O and H O are in metres, and ε g and gr t, areinmga. 5. Second-order correction for norma gravity For the terms invoving norma gravity, we seek a simpification of A = γ γr t, + γ h HO. 7 The integra-mean vaue of norma gravity aong the pumbine γ is evauated using a second-order Tayor expansion for the anaytica downward continuation of norma gravity from the Earth s surface γr t, to the geoid. Using a formuation in terms of geodetic coordinates, this is γ H O h =N +H O h =N n h + γh, + γ h h=h γ h n h dn. h=h 8 where h is the geodetic eipsoida height of the point r t,, N is the geoid height at, and n is an eement aong the eipsoida pumbine cf. Jekei 000. Performing the integration, appying the integration imits, and expressing norma gravity in terms of the geocentric radius of the Earth s surface at gives γ γh, γ h H O h=h + γ. 6 h H O 9 h=h Inserting Eq. 9 in Eq. 7 yieds A γ. 6 h H O 0 h=h Assuming the sphericay approximated vaue of the second-order free-air gravity gradient Heiskanen and Moritz 967, Eqs., Eq. 0 reduces to H O H O A = γ γ, r t a where a is the major semi-axis of the reference eipsoid. Taking H O = 8.8 km Mount Everest,

6 M. C. Santos et a. r t = 6,37 km and γ =9.8ms, A is about.87 mga. Using Eq. 6, this causes a maximum correction of about.5 cm to the Hemert orthometric height. 5. Second-order correction for the Bouguer she For the terms invoving the spherica Bouguer she of thickness H O, we seek a simpification of the term B = g T B gt B r t, + πgρ 0 H O. The gravitationa attraction of the spherica Bouguer she at the Earth s surface reads Martinec 993, Eq. 4.6 gb T r t, = 4πGρ 0 H O + H O + HO + H O. 3 3 From Wichiencharoen 98 cited by Martinec 998, Eq. 3.4,the gravitationa potentia inside the spherica Bouguer she is r g r r t : +H VB T r, = πgρ 0 O 3 3r r 3, 4 where is the inner radius of the she in this case, = r g and r is a dummy point inside the she. ecognizing that gb Tr, = VT B r, r, the integra mean g T B in Eq. aong the radia between the geoid r g = and approaching the Earth s surface r t + H O from within the Bouguer she gives g T B = H O r= r g +H O r=r g V T B r, r VT B rg, VB Tr t, H O. 5 As with the norma gravity term, this is a more rigorous formuation for the spherica Bouguer she, where r and H are aong the same radia i.e. H O = r t r g. As such, there is no need to worry about the deviation of the radia from the pumbine in this case cf. Tenzer et a. 005, Appendix A. Inserting the integration imits in Eq. 4, then inserting the resuts into Eq. 5, after some agebraic manipuation, gives ḡb T = πgρ 0H O H O 3 + H O. 6 dr Here we acknowedge the typographica error in the first term of Tenzer et a. 005, Eq.. Inserting Eqs. 3 and 6 in Eq. gives B = 4 3 π Gρ H O 0 + H O HO + H O. 7 Using the earier exampe of Mount Everest, a constant topographica mass density of ρ 0 =,670 kg m 3 and Eq. 6, the second-order Bouguer term Eq. 7 affects the orthometric height by as much as.6 cm. Thus, Eq. 4 now becomes ε g = δg NT δg NT r t, + g T gt r t, + g δg g δg r t, H O γ r t + 4πGρ 0 H O 3 + H O HO + H O, 8 which represents the integra-mean vaue of gravity aong the pumbine expressed in terms of corrections to Hemert s approximate mean vaue. Term A takes a negative sign, indicating the radia derivative of Eq. 7 is taken on the eipsoid. These correction terms comprise: mean and surface effects on gravity coming respectivey from masses inside the geoid, terrain roughness, ateray variabe density distribution, second-order free-air effects, and second-order Bouguer she effects. A these terms must be computed to appy a rigorous correction to Hemert s orthometric height. 5.3 The geoid-generated gravity disturbance In this subsection, we sha concentrate on the term C = δg NT δg NT r t,, 9 which deas with the corrections to the Hemert orthometric height coming from the geoid-generated gravity disturbance, comprising the mean vaue aong the pumbine δg NT and vaue on the Earth s surface δg NT r t,. The integra-mean vaue of the geoid-generated gravity disturbance aong the pumbine between the geoid

7 The reation between rigorous and Hemert s definitions of orthometric heights and the Earth surface can be represented in anaogy to Eq. by δg NT = 4π H O O +H O r= δg NT = H O = r t r g δg NT r, dr +H O H O δg NT r, dr, 30 r= where the geocentric radius of the geoid surface r g is approximated by, the mean radius of the Earth, which shoud not be confused with the subscript in the terrain roughness term. Since the geoid-generated gravity disturbance δg NT r, mutipied by r is harmonic above the geoid since the NT space contains no topographica masses above the geoid, and again negecting the atmosphere, δg NT can be evauated by averaging Poisson s equation for upward continuation e.g. Keogg 99 in an integra sense. The Poisson equation reads δg NT r, = 4π r O K r, ψ,, δg NT, d, 3 where 0 is the soid ange, is the dummy eement and ψ, represents the spherica distance or geocentric ange between the computation and integration points. The required gravity disturbance δg NT,, referred to the geoid, is a part of the sub-integra function. The spatia form of the Poisson integra kerne Kr, ψ,, is given by e.g. Keogg 99 Kr, ψ, r, = 3 r, ψ,,, 3 where the Eucidean spatia distance is given by = r + r rr cos ψ,, 33 Inserting Eq. 3 into Eq. 30, the mean gravity disturbance δg NT becomes cf. Tenzer et a. 005, Eq. 8 r K r, ψ,, dr δg NT, d. 34 Performing the radia integration of Poisson s integra kerne K r, ψ,,, mutipied by r, the foowing expression can be found for the averaged Poisson s kerne e.g. Vaníček et a. 004, Tenzer et a H O r= r K r, ψ,, dr +H O r = r 3 r, ψ,, dr r= = r, ψ,, r cos ψ, + r, ψ,, + n r sin ψ, +H O r=. 35 for ψ = 0. Substituting Eq. 35 into Eq. 34, the mean gravity disturbance δg NT aong the pumbine takes the foowing form: δg NT = 4π H O 0, ψ,, r t, ψ,, rt cos ψ, + r t, ψ,, + n r t cos ψ, +, ψ,, δg NT, d. 36 Equation 36 can be simpified as δg NT = 4π H O K + H O, ψ,, O δg NT, d 37 where K stands for the intermediary integration kerne. It can be shown, in the first approximation, that this kerne equas K + H, ψ,, = + n H, 38 where stands for,ψ,and stands for +H O, ψ,. The derivation of this kerne is given in Appendix A. Equation 37 is somewhat cumbersome because it requires the NT gravity disturbance to be known on

8 M. C. Santos et a. the geoid, which is not known. Therefore, to impement it in practice first requires the downward continuation of δg NT r t, to δg NT r g,. In Eq. 36, the geoid-generated gravity disturbance δg NT r g, is obtained from the geoid-generated gravity anomay g NT r g, referred to the geoid in the NT-space by cf. Heiskanen and Moritz 967, Eq. 5e; Vaníček et a. 004 δg NT, = g NT, + TNT, 39 where T NT, represents the geoid-generated disturbing potentia in the NT space T NT, = T, V T, V A, 40 The disturbing potentia T, can be taken from a regiona geoid mode, computed according to Bruns 878 formua for the geoid height T = N γ 0, thus making the geoid and the corresponding orthometric height system more compatibe. The second term on the right-hand side of Eq. 40 V T r g, is the gravitationa potentia of the topographica masses, and V A r g, is the potentia of a atmospheric masses. The term V T r g, is obtained through the Newtonian integra V T r g, = G 0 +H 0 r = ρr,, ψ,, r r dr d, 4 where ρ r, represents the actua mass density of the topographica masses, usuay computed from a density distribution mode. The effect due to atera mass density variation is deat with in Sect Finay, to compement Eq. 9, the gravity disturbance at the Earth s surface is required. This term can be evauated directy from δg NT r t, = g NT r t, + r t Tr t, r t V T r t, r t V A r t,. 4 Equation 9 can then be evauated using Eqs. 37 and The terrain/roughness-generated gravity In this subsection, we sha concentrate on the term D = g T gt r t, 43 which gives the correction to the Hemert orthometric height from the terrain roughness residua to the Bouguer she, assuming for the moment a constant topographica mass density atera density variations wi be considered in Sect The gravitationa fied of the terrain roughness term is not harmonic inside the topography. As such, it has to be cacuated from an adopted mode of the shape of the topography i.e. a DEM, couped with a constant mass density assumption. We begin with the gravitationa potentia V T r, of topographica masses expressed in Eq. 4. Using a spherica approximation of the geoid and Newtonian integration, this reads cf. Novák and Grafarend 005 V T r, G r =+H O r = ρr, r,, r, r dr d. 44 The negative radia derivative of topographica gravitationa attraction g T r, is given by g T r, G r =+H O ρr, r r = r,, r, r dr d. 45 From Eq. 43, we are ooking for the mean vaue ḡ T between the Earth s surface and the geoid, which is given by definition as Tenzer et a. 005, Eqs. 6 8 r=+h O ḡ T = H O g T r, dr r= = r=+h O H O r VT r, dr r= = { } H O V T +H O, V T,. 46 Substituting for the two vaues of potentia V T from Eq. 44, we get ḡ T G +H O H O ρr, r =,, r, + H O,, r, r dr d. 47

9 The reation between rigorous and Hemert s definitions of orthometric heights Let us now express the radia integra in Eq. 47 as a sum of two integras +H O Fr dr = +H O + +H O +H O Fr dr Fr dr 48 The first integra on the right-hand side of Eq. 48 describes the contribution of the Bouguer she of constant thickness H O e.g. Vaníček et a. 00, which was deat with in Sect. 5.. The second integra gives the contribution due to the terrain residua to the Bouguer she. The density can aso be written as a sum of two terms, one containing a contribution due to the mean density ρ 0 and the other containing the residua density δρr, contribution ρr, = ρ 0 + δρr,. 49 The roughness term is represented by the second term in Eq. of Tenzer et a. 005 ḡ T = Gρ 0 H o +H O 0 r =+H O {,, r, } + H O,, r, r dr d, 50 This term is nothing ese but the change in terrain roughness of constant density of ρ 0, from the geoid to the surface of the Earth, divided by the orthometric height of the point of interest ḡ T H O V T, VT + HO,. 5 These two roughness parts of topographica potentia V T can be evauated through numerica quadrature of the Newton integra Eq. 44. Equation 5 provides the mean gravity generated by the terrain roughness, expressed in terms of gravitationa potentia. As pointed out in Sect. 5.3, it comprises a contribution from the average topographica mass density, pus a smaer correction due to mass density variations. The other term in Eq. 43, the terrain roughness term at the Earth s surface, is given by the second term in Martinec 998 g T r t, Gρ 0 r =+H o 0 r =+H o r dr d r, ; r, r r=rt which can aso be evauated by quadrature methods. 5.5 The atera variation of topographica mass density In this subsection, we consider the term 5 E = ḡ δρ g δρ r t,. 53 In most gravimetric geoid computations, the topographica mass density is generay modeed by an average vaue of ρ 0 =, 670 kg m 3. Martinec 998 posed the question on how much a variation in topographica mass density affects geoid height computation. To answer this question in the context of the orthometric height, we assume ony atera variations of density, eaving the radia variation sti to be tacked. The deveopments beow foow from those of Sect The contribution of atera variation of density to the correction to Hemert s orthometric height is represented by third term in Eq. from Tenzer et a. 005 g δρ = G H o 0 r =+H r = δρr, {,, r, + H,, r, }r dr d 54 The surface gravity generated by atera variation of density is given by g δρ r t, = G δρ 0 r =+H r =, ; r, r r=r t r dr d, 55 which foows from a more compete expression provided by Martinec 998 that takes into account the radia variation in density r g δρ r t, = G 0 δρr,, ; r, r r =+H r = r=r t r dr d, 56 Equations 54 and 55 provide the terms required in Eq. 53 The correction to Hemert s orthometric height due to the ateray varying topographica mass density is aso given by the foowing approximate expression

10 M. C. Santos et a. Vaníček et a. 995 if one considers ony the radia gradient of the gravitationa attraction generated by the spherica Bouguer she of the anomaous topographica density δρ, ε δρ δρ π G H O H O g Summary The correction to the Hemert orthometric height to give the rigorous orthometric height defined by Tenzer et a. 005 ε H O is given by Eq. 5. It foows directy from the evauation of the correction to Hemert s mean gravity ε g, written beow in a simpified manner as ε g = A + B + C + D + E 58 The terms A and B can be computed from Eq. 8 as H O A + B = γ r t πgρ H O 0 + H O HO + H O 59 The terms C, D and E can be computed from Eqs. 37 and 4, 5 and 4 and 54 and 55, respectivey. Note that severa of these terms woud have aready been computed for a regiona gravimetric geoid mode based on the Stokes Hemert approach Vaníček and Martinec 994. This simpifies the task, where the gridded quantities can be interpoated to the points of interest and appied as part of the corrections to the Hemert orthometric height, provided that the horizonta ocations of the benchmarks are known. It aso makes the rigorous orthometric heights more compatibe with the regiona geoid mode. Finay, the tota correction to the Hemert orthometric height ε H O is ε H O = HO A + B + C + D + E 60 g 6 Numerica tests Using Canadian gravity, terrain and atera topographic mass density data, we have computed rigorous corrections to Hemert s orthometric heights aong a profie across the Canadian ocky Mountains. This profie spans the ongitudes from 35 to 39 E aong the 50 N parae. Figure shows each one of the terms in Eq. 60 i.e. A: second-order free-air, B: second-order Bouguer she, C: NT gravity disturbance, D: terrain roughness, and E: atera density variations computed separatey Correction cm Longitude degrees Height m Fig. Profies of the five components of the correction to Hemert s orthometric height cm, as we as the Hemert orthometric height m aong a profie at 50 N across the Canadian opcy Mountains. The continuous thick ine represents the topographic height; continuous thin ine corresponds to ε δg geoid-generated gravity disturbance; dashed ine corresponds to ε T H O H O terrain-roughness-generated gravity dotted ine corresponds to ε δρ atera variation of topographica mass density. The H other O two components are too sma to be potted to show their reative contributions to the correction. These terms are superimposed on the topographic height variations shown with the thicker ine in Fig. scaed down by 00 m, to show that there is not aways a oneto-one correspondence of the correction terms with height. A integra terms were computed over a spherica cap radius of 3, beyond which the far-zone contributions become negigibe < mm for this test area. Inspecting Fig., we see that the correction term from the geoid-generated gravity disturbance C gives the argest correction vaues, and is generay positivey correated with topography, though not perfecty. The correction due to terrain-roughness-generated gravity D is the second most important contribution. However, it works against the former correction, and there is a ess strong, negative correation with topography. The third argest term in magnitude is the correction due to atera variation of topographica mass density E, varying around zero and with maximum magnitude not greater than 5 cm. The fina two terms, due to second-order correction for norma gravity A and second-order correction for the Bouguer she B, are both very sma, not showing up in Fig.. Tabe summarizes the statistics of these five correction terms. Figure 3 shows a comparison between the corrections to Hemert orthometric heights using the method described in this paper termed rigorous, and the Mader and Niethammer approaches, for the same profie as in Fig.. The Mader and Niethammer corrections were

11 The reation between rigorous and Hemert s definitions of orthometric heights Tabe Descriptive statistics of corrections to Hemert s orthometric height from the profie shown in Fig. Vaues in centimetres, rounded to the nearest miimetres Correction due Correction due Correction due to Correction due Correction due to gravity to terrain- atera variation for nd-order for nd-order disturbance roughness of density norma gravity Bouguer she Mean STD Minimum Maximum Tabe 3 Descriptive statistics of the tota corrections to Hemert s orthometric height from the profie shown in Fig Mader Niethammer igorous Correction cm Height m Mean Standard deviation Minimum vaue Maximum vaue Vaues in centimetres, rounded to the nearest miimetres Longitude degrees Fig. 3 Comparison among the rigorous, Mader and Niethammer corrections to Hemert orthometric heights aong the same profie as in Fig.. Units in centimetres. The continuous thicker ine represents the topographic profie; continuous thin ine represents the rigorous correction; dashed ine represents Niethammer correction; dotted ine represents Mader correction computed from Eqs. and 3 using the same topographica corrections used to evauate the rigorous corrections. From Fig. 3, the Mader and Niethammer corrections are very simiar to one another, whereas the rigorous correction is arger, which is attributed to the two additiona terms not accounted for in Mader nor Niethammer s approaches: geoid-generated gravity disturbance and atera variation of topographica mass density. The arger contribution comes mosty from the geoid-generated gravity disturbance cf. Fig.. Tabe 3 summarizes the statistics of the corrections aong this profie. 7 Summary, discussion and concusion We have derived expressions to transform Hemert s approximation of the orthometric height into a more rigorous one cf. Tenzer et a. 005, taking into account effects coming from the second-order correction for norma gravity, second-order correction for the Bouguer she, the geoid-generated gravity disturbance, the terrain-roughness-generated gravity, and the atera variation of topographica mass density. These individua corrections have been evauated numericay aong a profie across the Canadian ocky Mountains, and potted against the topographica height variation. This comparison shows that the geoid-generated gravity disturbance, the terrain-roughness-generated gravity and the atera variation of topographica mass density are, respectivey, the most important contributors towards obtaining a more rigorous orthometric height. It aso shows that the geoid-generated gravity disturbance and the terrain-roughness-generated gravity work approximatey against each other, though not competey, as each is not perfecty correated with the topography. The second-order correction for norma gravity and the Bouguer she are negigiby sma for this test, but become arger for very high eevations. Comparisons with other refinements of Hemert orthometric heights, namey Mader 954 and Niethammer 93, have aso been performed. The Mader and Niethammer orthometric heights are very simiar to one another, but the respective corrections are smaer than the rigorous corrections. They differ from the rigorous approach due to incusion of the terms pertaining to the geoid-generated gravity anomay and atera variation of topographica mass density. Finay, it is important to point out that severa of the correction terms used here are the same as woud have been computed for a regiona gravimetric geoid mode based on the Stokes Hemert approach e.g. Vaníček

12 M. C. Santos et a. and Martinec 994. As such, they are reativey easy to appy to existing Hemert orthometric heights. Moreover, this makes the resuting heights more compatibe with a regiona gravimetric geoid mode based upon the Stokes Hemert approach. 8 Appendix A: derivation of Eq. 38 We wish to simpify the expression for the averaging Poisson s kerne Eq. 35, which reads K + H O, ψ, = + H O, ψ, +, ψ, +n +H O cos ψ ++H O, ψ, +H O sin ψ n cos ψ +, ψ, sin ψ. A For integration within a very sma radius ψ 0 of, say, 3 arc-degrees, we can assume,, H O <<, A where we have denoted,ψ, by and +H O, ψ, by*. This is permitted because of the rapid decay of the Poisson kerne with spherica distance, as supported by our empirica evidence cf. Sect. 6. Now, we can rewrite Eq. A as K + H O, ψ, = + n + H O / cos ψ + / + H O / cos ψ + / The ast term in Eq. A3 shoud be + n HO cos ψ + + HO +H O cos ψ +.. A3 A4 However, due to the precision required, the approximation in Eq. A3 is enough since + HO + H O + HO A5 eaizing that = sin ψ, A6 we can express cosψ in Eq. A4 as cos ψ = sin ψ =. A7 Substituting this resut into Eq. A3 gives K + H O, ψ, = + HO +n + HO { + + After a few agebraic operations, we get K + H O, ψ, = + HO +n { + HO } + + n H O. + } A8 A9 This is the fina simpified form, vaid for a sma <3 radius ψ 0 of integration, which wi next be studied. It shoud be noted that the first term in Eq. A9 is the eading term, whie the second is a corrective term. The eading term converges very rapidy since it hods most of its power in the nearest vicinity of the computation point. For instance, the cumuative sum of this term across a profie gains 99% power at ψ =0.. The magnitude of the corrective ogarithmic term comprises <% of the magnitude of the eading term. One may see that the first terms on the right-hand side of Eqs. A3 and A7 are exacty the same. Thus, the difference between the exact expression and the first approximation stems ony from the much smaer ogarithmic term. As such, our subsequent numerica investigations study the reationship between the term + n HO cos ψ + / + HO cos ψ + and its approximation n H O. The variabes in both terms are the orthometric height H O and the anguar distance ψ. The behaviour of these terms within the interva m <ψ3 wi

13 The reation between rigorous and Hemert s definitions of orthometric heights Spherica distance + Fig. A The behaviour of the n HO cos ψ+ + HO cos ψ+ term ower batch of curves and the discrepancies + n HO cos ψ+ + HO cos ψ+ n H O upper batch of curves a H O = 00 m dotted ine; b H O =,000 m dashed ine; c H O =,000 m soid thin ine; d H O = 5,000 m soid bod ine. Aso note the negative ogarithmic scae be numericay investigated. In these tests, the orthometric height takes the foowing constant vaues H O = 00 m, H O =km,h O =3km,H O =5km.In other words, the topography of the test area is assumed to be a pateau with a constant height H O. The ogarithmic term is aways negative, since the argument HO takes vaues between 0 and, i.e. 0 < HO <. Figure A shows the behaviour of the ogarithmic term and the discrepancies between the exact expression Eq. A3 and its approximation Eq. A7 across the 3 integration area. The ower batch of curves in Fig. A indicates the magnitude of the ogarithmic term for each case, whereas the upper batch denotes the corresponding discrepancies. From Fig. A, most of the power in the ogarithmic term is in the nearest vicinity of the computation point and it decreases with increasing ψ. Note that at the computation point, the discrepancies are amost zero. The magnitude of the reative discrepancies increases ineary with the distance reca that a ogarithmic scae is used in Fig. A. Note, however, that in any tested case and for ψ < the discrepancies are at east of two orders of magnitude ess than the ogarithmic term itsef. At ψ =3, the error of the approximation n HO consists of 3% ony from the exact expression. Aso reca that the ogarithmic term is <% of the whoe Poisson kerne. From the above resuts, it is obvious that the expression K + H O, ψ,, = +n HO is sufficient as the first approximation of the compicated integration term. Next, next et us have a ook at the first term in Eq. A3. eaizing that im = + H O,it ψ 0 can be written as + H O = + HO A For H O <, the case of H O > can be treated in a simiar way this can be represented by a convergent binomia series = + HO { = + H O k }. k k= A Carrying out the required agebraic operations, we arrive at = k= k H O k. A3 The series in Eq. A0 is aternating, and thus a convergent series even for H and going simutaneousy to zero, thus im 0 H O 0 = im 0 H O 0 = im 0 H O 0 k k= k k= H O k A4 As the summation is a rea number, the whoe expression grows above a imits and we get im H 0 = δ,, A5 where δ is the Kronecker symbo for the function that grows beyond a imits when = and equas zero for a other vaues of. We can thus see that

14 M. C. Santos et a. im δg NT = H O 0 im H O 0 4π H O O K + H O, ψ,, δg NT, d = δg NT, A6 as one woud expect. We note that the averaging Poisson kerne aso has a singuarity for the case when H is not equa to zero. For H O > 0, we get im 0 = im 0 = im 0 H { + H O H O + k= k } k H O = im 0 H O = im A7 0 which aso grows above a imits. Thus, the averaging Poisson kerne has a removabe singuarity of a inear type /0 at the point of interest, whether the height is equa to 0 or not. The second, ogarithmic term is aways negative: it equas 0 for going to 0, and it aso goes to 0 for growing. Note that the argument 0 < <. HO Acknowedgements The Canadian investigators are supported by the GEOIDE Network of Centres of Exceence. The Austraian investigators were supported by Austraian esearch Counci grant DP087. The topographica data were kindy suppied by Geodetic Survey Division of Natura esources Canada. We aso thank the reviewers and editors for their time and constructive criticism. eferences Aister NA, Featherstone WE 00 Estimation of Hemert orthometric heights using digita barcode eveing, observed gravity and topographic mass density data over part of Daring Scarp, Western Austraia. Geom es Aust 75:5 5 Bruns H 878 Die Figur der Erde, Pub0. Preuss Geod Inst, Berin Dennis ML, Featherstone WE 003 Evauation of orthometric and reated height systems using a simuated mountain gravity fied. In: Tziavos IN ed Gravity and geoid 00, Dept of Surv and Geodesy, Aristote University of Thessaoniki, pp Heiskanen WA, Moritz H 967 Physica geodesy. Freeman, San Francisco Hemert F 890 Die Schwerkraft im Hochgebirge, Insbesondere in den Tyroer Apen, Veröff. König. Preuss Geod Inst No.,Berin Jekei C 000 Heights, the geopotentia, and vertica datums, ep 459, Dept. Geod. Sci., Ohio State University, Coumbus Keogg OD 99 Foundations of potentia theory. Springer, Berin Heideberg New York Kingdon, Vaníček P, Santos M, Emann A, Tenzer 005 Toward an improved orthometric height system for Canada. Geomatica 593:4 50 Errata: Figure 4 on Geomatica, Vo 60 :0 Mader K 954 Die orthometrische Schwerekorrektion des Präzisions-Niveements in den Hohen Tauern, Österreichische Zeitschrift f ur Vermessungswesen, Sonderheft 5, Vienna Martinec Z 993 Effect of atera density variations of topographica masses in view of improving geoid mode accuracy over Canada. Fina report of contract DSS No , Geodetic Survey of Canada, Ottawa Martinec Z 998 Boundary vaue probems for gravimetric determination of a precise geoid. Lecture notes in Earth Sciences, 73, Springer, Berin Heideberg New York Niethammer T 93 Niveement und Schwere as Mitte zur Berechnung wahrer Meereshöhen. Schweizerische Geodätische Kommission, Berne Novák P, Grafarend EW 005 Eipsoida representation of the topographica potentia and its vertica gradient, J Geod 78 3: DOI 0.007/s Sjöberg LE 004 The effect on the geoid of atera topographic density variations. J Geod 78-:34 39 Strange WE 98 An evauation of orthometric height accuracy using borehoe gravimetry. Bu Géod 56:300 3 Sünke H 986 Digita height and density mode and its use for the orthometric height and gravity fied determination for Austria. In: Proceedings of internationa symposium on the definition of the geoid, Forence, May, pp Tenzer, Vaníček P 003 Correction to Hemert orthometric height due to actua atera variation of topographica density. Braz J Cart 55:44 47 Tenzer, Vaníček P, Santos M, Featherstone WE, Kuhn M 005 igorous determination of the orthometric height. J Geod 79 3:8 9. DOI 0.007/s Vaníček P, Keusberg A, Martinec Z, Sun W, Ong P, Najafi M, Vajda P, Harrie L, Tomášek P, ter Horst B 995 Compiation of a precise regiona geoid. DSS Contract# /0-SS eport for Geodetic Survey Division, Ottawa, pp 45 Vaníček P, Martinec Z 994 Stokes Hemert scheme for the evauation of a precise geoid. Manuscr Geodaet 93:9 8 Vaníček P, Janák J, Huang JL 00 Mean vertica gradient of gravity. In: Sideris MG ed Gravity, geoid and geodynamics 000, Springer, Berin Heideberg New York, pp 59 6 Vaníček P, Novák P, Martinec Z 00 Geoid, topography, and the Bouguer pate or she. J Geod 754:0 5. DOI 0.007/s Vaníček P, Keusberg A, Martinec Z, Sun W, Ong P, Najafi M, Vajda P, Harrie L, Tomášek P, ter Horst B 995 Compiation of a precise regiona geoid. Fina report on research done for the Geodetic Survey Division, Fredericton Vaníček P, Tenzer, Sjöberg LE, Martinec Z, Featherstone WE 004 New views of the spherica Bouguer gravity anomay. Geophys J Int 59: DOI 0./j X x Wichiencharoen C 98 The indirect effects on the computation of geoid unduations. eport 336, Dept Geod Sci, Ohio State University, Coumbus