A new technique to determine geoid and orthometric heights from satellite positioning and geopotential numbers

Transcription

1 J Geod 6 8: 4 1 DOI 1.17/s OIGINAL ATICLE L. E. Sjöberg A new technique to determine geoid and orthometric heights from satellite positioning and geopotential numbers eceived: 19 September 5 / Accepted: 1 May 6 / ublished online: 19 August 6 Springer-Verlag 6 Abstract This paper takes advantage of space-techniquederived positions on the Earth s surface and the known normal gravity field to determine the height anomaly from geopotential numbers. A new method is also presented to downward-continue the height anomaly to the geoid height. The orthometric height is determined as the difference between the geodetic ellipsoidal height derived by space-geodetic techniques and the geoid height. It is shown that, due to the very high correlation between the geodetic height and the computed geoid height, the error of the orthometric height determined by this method is usually much smaller than that provided by standard GS/levelling. Also included is a practical formula to correct the elmert orthometric height by adding two correction terms: a topographic roughness term and a correction term for lateral topographic mass density variations. Keywords Downward continuation Geoid Geopotential number GS-levelling Orthometric height Normal height 1 Introduction Today, the figure of the Earth in terms of its topographic surface can be precisely determined by space-geodetic positioning techniques, but the mapping of the precise shape of the geoid is still a challenge. In particular, the rigorous determination of the geoid from gravimetric data requires that the density distribution of the topography is known. The alternative geometric method to reach the geoid height, GSlevelling, is apparently no real alternative in this respect, as the topographic density is also required to determine the desired orthometric height. L. E. Sjöberg Division of Geodesy, oyal Institute of Technology, 1 44 Stockholm, Sweden sjoberg@geomatics.kth.se Tel: Fax: A national height system is usually defined in terms of the orthometric height, which is determined as the ratio between the geopotential number of the computation point andthe integral mean gravity along the plumbline from the geoid to. In practice, one frequently uses elmert orthometric heights, whose accuracies are primarily limited by the imprecise topographic correction to mean gravity. Tenzer et al. 5 presented a new method to improve the determination of mean gravity, first of all by using a rigorous formulation of the topographic corrections. Unfortunately, their approach uses the gravity disturbance, which can only be precisely computed if the geodetic ellipsoidal height i.e., the height above the reference ellipsoid is known. For a major part of currently available terrestrial gravity data, this is not the case. owever, their technique could easily be converted to a solution by gravity anomalies instead of gravity disturbances to avoid this problem. Also, their method requires that topographic mass density be known. Several other historical approaches to improve orthometric heights are listed among the references given in Tenzer et al. 5. As an alternative, let us assume that the geodetic height, as well as the geopotential number, is known. In addition, of course, the reference ellipsoid and normal gravity field is precisely defined, and the normal gravity potential and all its derivatives can therefore be regarded as known in all space exterior to the reference ellipsoid. Finally, as will be shown, additional gravity data around the computation point will be needed in the most precise computations. Our goal is to use all these data for the development of a practical technique to evaluate the geoid and orthometric heights simultaneously. Ardalan and Grafarend 4 also acknowledged the potential of using geopotential numbers for geoid determination, and they downward-continued the disturbing potential from the Earth s surface to the geoid by solving oisson s integral equation. Our goal is to simplify the technique by avoiding any integral equation, and thus the downward continuation of the disturbing potential will be performed differently.

2 A new technique to determine geoid and orthometric heights 5 The geoid height.1 General considerations Let us assume that the ellipsoidal height of the topography above the defined reference ellipsoid h and the geopotential number C are given. Then, Molodensky s normal height can be determined iteratively by the formula eiskanen and Moritz 1967, p. 17 N C = + a N + b N, 1a is normal gravity at the reference ellipsoid, and the coefficients a and b are given by a = 1 γ h mgal/km 1b and b = 1 γ 6 h γ.4 mgal/km 1c r r is the geocentric radius of the computation point. A more precise estimate of the coefficient a is given by Eqs. 4 4 of eiskanen and Moritz 1967, p. 17. Alternatively, Eq. 1a can be treated as a second-order equation with the solution N C b N = + γ + 4aC 4ab N, which should converge even faster than Eq. 1a. As Eq. 1a is the result of a series expansion of γ in powers of N, truncated at the third power, the solution could even be provided as the closed solution to a third-order equation, but that solution turns out to be rather complicated and impractical. As the geodetic height above the reference ellipsoid h and the normal potential U and its derivatives are known, we can determine the height anomaly ζ = h N as well as the disturbing potential at the computation point by Bruns s formula T = ζγ Q, 4 U γ Q = 5 h h= N is normal gravity at the normal height. Also, as the geoid height is given by Bruns s formula N = T g, 6 the remainder of the problem to determine N is to compute the correction from T to T g. The computations to determine the geoid height can thus be formulated N = ζ nt + dζdwc nt + dn I, 7a ζ nt = T V t = ζ + γ Q ζ V t 7b is the rescaled no-topography i.e., topographic masses have been removed height anomaly with scaling factor γ Q replaced by, dζdwc nt is the effect of downward-continuing ζ nt to sea level to be discussed in Sect.., and dn I is the indirect topographic effect on the geoid height to be presented in Sect... In Eq. 7b, V t is the topographic potential also to be presented in Sect.... The downward continuation of ζ nt In order to compute the downward continuation dwc effect on ζ nt, defined by the second term on the right-hand side of Eq. 7a, we first reduce T by subtracting from it the topographic potential V t, yielding the no-topography disturbing potential T nt = T V t. 8 The removal of V t implies a very significant change of the disturbing potential, and, as there is no topography related with T nt, it is much smoother than the original disturbing potential T, and its reduction to sea level is performed in free space. LetusfirstuseT nt to express the dwc effect as the difference between two Bruns Stokes s formulas: dwc = T g nt T nt dζ nt = 4π Sψ Sr,ψ g nt d, 9 Tg nt and T nt are the no-topography disturbing potentials at the geoid and topography levels, respectively, is mean Earth sphere radius, r = +, is the surface of the unit sphere, Sψ and Sr,ψ are Stokes s original and extended functions, respectively, ψ is the geocentric angle between the computation point and the running point on the sphere, and g nt denotes the downward-continued no-topography gravity anomaly, i.e. g nt = g + δ g dir, 1 δ g dir is the direct topographic effect on g Sjöberg. From the boundary condition of physical geodesy eiskanen and Moritz 1967, pp. 9 94, we get g nt T nt = h + T nt γ γ h, 11 yielding ζ nt = δgnt = gnt ζ nt γ +, 1 h γ h

3 6 L. E. Sjöberg δg nt is the no-topography gravity disturbance, but from the extended Stokes s formula we also obtain ζ nt = Sr,ψ g nt d. 1 h 4π h ence, by adding the right-hand side of Eq. 1 and subtracting the right-hand side of Eq. 1, both equations multiplied by the orthometric height, when applied to Eq. 9, we thus obtain dζdwc nt = g nt + 4π ζ nt γ γ h Sψ Sr,ψ + Sr,ψ h g nt d 14a or, in view of Eq. 1, dζdwc nt = δg nt + I, 14b I is the same as the integral term of Eq. 14a. We can assume not for the highest mountain areas that the contribution from the integral I will be small see Appendix A1, which suggests that the integral either be approximated by zero or g nt can be substituted by the more practical surface gravity anomaly g nt. For high mountain areas, these approximations may be too crude for the most accurate computations at the mm level. If so, we may instead decompose I into a low-degree component I M and a high-degree component I M,i.e. I = I M + I M, I M = M 1 n= and I M = 4π ere r 15a n+1 n + 1 n+1 Tn nt r r Sr,ψ Sψ Sr,ψ+ h 15b M g nt d. 15c M g nt = g nt M n= n 1T nt n r 16 can be computed from g nt and an Earth gravity model EGM to degree and order, say, M = 6, which can be used to approximate the corresponding downward-continued gravity anomaly of the strict integral, Eq. 15c. ence, by considering that in the last integral both the square-bracketed terms in Eqs. 15b and 15c and M g nt are residual quantities, we may usually use this approximation without significant loss of accuracy. A numerical estimate of the order of this integral is presented in Appendix A1. Also, the Laplace harmonic of the disturbing potential T n, determined from the EGM, is converted into the no-topography harmonic as used in Eqs. 15b and 16 by adding the direct topographic effect Vn t,i.e. Tn nt = T n Vn t. 17. Direct and indirect topographic effects In the case of the no-topography potential, the indirect topographic effect on the potential at point is equal to the topographic potential V t. Denoting the product of the gravitational constant and the topographic mass density assumed to be constant in the radial direction to sea level, but laterally variable by µ,thetopographicpotentialat apointis given by Martinec 1998; Sjöberg V t = f r, t = µf r, td, 18a r s r dr l = r s + r t l + r t l +r t ln r s r t + l. 18b r t + l r s is the radius of the Earth s surface, l = r + r rr t, l = r + r s r r s t, l = r + r t and t = cos ψ. Alternatively, we may expand 1/l of Eq. 18b as an external-type harmonic series, yielding V t = n+1 Vn t, 19a r rs Vn t = µ n + = 4π n + n + n + 1 +O ere υ n n = m= n n+ 1 n cos ψd n. 19b υ nmy nm ; ν = 1,, a

4 A new technique to determine geoid and orthometric heights 7 υ = 1 µ υ Y nm d ; ν = 1,, b nm 4π and Y nm is a fully normalized surface spherical harmonic. It follows that the direct topographic effect, needed to create ζ nt from ζ by Eq. 7b, is V t /,andthe indirect effect i.e., the effect of restoring the topography on the geoid becomes cf. Eq. 6 δn I = V g t. 1 By summing up the direct, dwc and indirect topographic effects, we obtain the total topographic effect on the geoid height: dn tot = dζdwc nt + V g t V t, a or dn tot = T g T. b Frequently e.g., Wichiencharon 198; Tenzer et al. 5, the topographic potential is decomposed into the contribution from a Bouguer shell and a residual terrain roughness term terrain correction. Such a representation has the major problem that the Bouguer shell contributes too much, and the residual term must therefore be integrated all over the whole globe to include all significant contributions. Alternatively, as shown in Appendix A, the potential difference of the last term of Eq. a can also be decomposed into V t g V t = µ l d + DV t, a the first term dominates the signal, and the second term DV t = µ f, t f r, t d b l is a roughness term corresponding to a terrain correction of order /, which becomes significant for rough topography. ere, l = 1 t. In a similar way, the gravitational attraction of the topography can be decomposed into the contribution from a Bouguer plate plus a small term and the roughness term dg t cf. Appendix A: g t t = V = πµ + µ d +dg t h l, 4a dg t = µ The function Jx, t =. Jr, t + + d. l f x,t x 4b is given explicitly in Sjöberg.4 Correction for lateral density variation The error in d ζ dwc nt should be small when evaluating it by the technique discussed in Sect.. and provided that the topographic reduction is performed correctly. owever, the uncertainty in the topographic mass density distribution may yield significant error contributions both to the dwc effect and the indirect effect. If we assume that the topographic mass density is constant and equal to its standard value µ, but the proper density is µ + µ, the total topographic effect Eq.a should be corrected by cf. Sjöberg 4 δn µ = µ V t + V t V t + V t g µ = µ µ V t g V t π µ, 5 we after the first equality sign have added each of the contributions from the direct, dwc and indirect topographic effects. The topographic potential V t is the external potential downward-continued to the geoid. As the geoid is located within the topographic masses, it differs from the true potential at the geoid V t g. If we assume that the topographic mass density is in error by 1%, the maximum error obtained for Mt. Everest is of the order of.9 m. For elevations of 1 and 4 km, the total geoid effect will be in error by 1.1 and 18 cm, respectively. In reality, the true density correction µ and the related geoid correction can certainly be much bigger than demonstrated by this simple example. Note that we have assumed that the radial topographic mass density is constant; an assumption that is not realistic but used with lack of widespread knowledge of the vertical density distribution..5 A practical formula for N From Eq. 7a, we obtain the following practical formula for estimating the geoid height: N = γ Q ζ + DN, 6a ζ is determined from space-geodetic positioning and levelling via Eq., and DN = dn tot + δn µ. 6b Determination of orthometric heights Once the geoid height is known, it is a simple task to also determine the orthometric height from the GS-derived geodetic height h by the formula = h N, 7

5 8 L. E. Sjöberg we disregard a small correction not exceeding 1.5 mm due to the fact that in contrast to h iscurved along the plumbline. Although the accuracy of h evaluated by standard methods from GS positioning is not better than some cm, the uncertainty of, originating from the error in GS data, is usually negligible when applying our method. This result stems from the fact that h and N are highly correlated, which can be seen if we insert Eq. 6a into Eq. 7, yielding = h 1 γ Q + N γ Q DN, 8 DN was given by Eq. 6b. From Eq. 8, the orthometric height can be determined from the GS -derived geodetic height h and the levelled normal height N. If we write Eq. 8 in the form = N + h N 1 γ Q DN, 9 we notice that the orthometric height is obtained from the normal height by adding two corrections. As γ Q / 1 /r differs from unity only within 1, the error of is very insensitive to errors of h, and in practice it is therefore limited to the errors of N and DN. This result holds only for the method outlined here, and it does not apply when the geoid height is given by independent data; also see the discussion below and Sect. 4. As h N equals the height anomaly, it follows also that the order of the second term on the right-hand side of Eq. 9 is within ± cm. The error of DN is primarily dependent on the knowledge of topographic mass density. The error stemming from lateral topographic density variations was discussed in Sect..4. We conclude this section by verifying that the right-hand side of Eq. 9 indeed yields the orthometric height. For simplicity, we assume that dn µ is zero implying that correct topographic mass-density is used. We will use Eq. b, and we will also use the well-known relations eiskanen and Moritz 1967, pp h = + N = N + ζ. Then we obtain for the right-hand side of Eq. 9: M = + N γ Q ζ dn tot = + T g T T g + T =, 1 and we have verified Eq. 9. Again, Eq. disregards that the orthometric height, in contrast to the geodetic and normal heights, is slightly curved along the plumbline. 4 The correction to elmert s orthometric height Let us slightly rewrite Eq. 9 in the form = N + ζ 1 γ Q DN. This result will now be compared with the well-known elmert height, given by the formula = C ḡ, a ḡ = g + c ; c = 1 γ h πµ. b From Eqs. and a, we thus obtain the correction to elmert s height: d = = N + ζ 1 γ Q or d = ḡ γ γ DN, 4 + ζ 1 γ Q DN, 5 γ is the mean normal gravity along the ellipsoidal normal from the ellipsoid to normal height, which without loss of significance can be approximated by the truncated Taylor series the first omitted term is less than of γ 1 N N = + N γ γ N + γ h x + γ x h dx γ N h + γ 6 h, or, to the same level of approximation, p N γ h + γ h. 6a 6b In Eq. 6, we assume that the derivatives of γ are computed at the reference ellipsoid. Inserting Eqs. b and 6b into Eq. 5 with the substitution N,weobtain d g + c+1/ γ/ h + p / γ/ h γ +ζ 1 γ Q DN. 7 Equation 7 can be further simplified. Let us first approximate γ 1 by γ 1 with a relative error less than Then, by inserting Eq. 6b and considering Eqs. a and 14b, we obtain d c + 1 γ h + g t I V t g V t dn µ. 8

6 A new technique to determine geoid and orthometric heights 9 Moreover, by inserting g t and c according to Eqs. 4a and b, respectively, we obtain d µ d l + dg t γ V g t V t I dn µ, 9 or, in view of Eq. a: d dg t DVt I dn µ. 4 As both dg t and DV t are roughness terms, the approximate formula for flat terrain reduces to d I dn µ Comparison with rigorous traditional levelling Let us now consider the determination of the orthometric height by the rigorous traditional levelling in using the formula = C ḡ, 4 ḡ is mean gravity along the plumbline from the surface point to the geoid. It can be determined by the equation cf. Tenzer et al. 5 ḡ =ḡ nt +ḡ t, 4 ḡ nt and ḡ t are mean no-topography and topographic gravity, respectively. The first component in Eq. 4 can be evaluated by either ḡ nt = γ + δḡ nt, 44a or g nt = γ + ḡ nt, 44b γ and γ are the mean normal gravities along the normal to the ellipsoid from sea-level to geodetic height and from the ellipsoid to normal height, respectively, and δḡ nt and g nt are the mean no-topography gravity disturbance and gravity anomaly, respectively, between sea-level and topographic surface along the plumbline. Both δḡ nt and ḡ nt can be evaluated by employing oisson s integral equation and formula i.e., the inverse and forward application of the formula. This requires that the gravity disturbance/anomaly is known in the area around the computation point, and all gravity disturbances/anomalies must be reduced to no-topography quantities by the direct topographic correction. The practical application of oisson s integral is an inverse problem not necessarily solvable in the strict sense, but any approximate solution is smoothed in the mean value computation of mean gravity anomaly. The correction to elmert s orthometric height can thus be written d = C ḡ C g = dḡ ḡ, 45a the mean gravity correction dḡ can be decomposed into dḡ =ḡ ḡ nt ḡ t =ḡ δḡ nt γ ḡ t. 45b ere the accuracy of δḡ nt depends not only on the quality of the gravity data in the area and on the strictness of the application of oisson s formula backward and forward, but also on how rigorously the direct topographic effect has been applied in the determination of g nt.δḡ t and ḡ nt are, of course, also dependent on the uncertainty of the topographic mass density and distribution. ence, there is a coupling between these two error components of Eq. 45b, which are both related with the uncertainty in the topographic mass distribution. Once the orthometric height is known, the geoid height can be estimated by GS-levelling neglecting a small correction for the curvature of the plumbline in the orthometric height: N = h, 46 and, as h and are determined by independent methods, all the errors in each component of Eq. 46 are propagated into the geoid height. In particular, systematic errors, stemming from datum problems among h, and N, frequently dominate the resulting error spectrum. 6 Conclusions We have shown that the known geodetic ellipsoidal height, in combination with the geopotential number and some gravity data around the computation point, can be efficiently used to determine the geoid height. The new method mainly suffers from the error originating with the uncertainty of topographic mass density, whose error becomes significant at the 1 cm level for topographic elevations exceeding 1 km. In particular, the influence of the mostly unknown vertical mass density distribution is not considered here. We have shown that the errors related with topographic mass density are practically the same as in traditional geodetic levelling. Once the geoid height is determined, the orthometric height is obtained by GS or any other space-based positioning technique and geodetic levelling. ere, we have shown that the GS or corresponding space tool error is practically not propagated to the orthometric height, but is eliminated by the high degree of correlation between the geodetic and geoidal heights. This also implies that the new method to determine orthometric height from GS is not sensitive to systematic errors related to different reference systems used for GS, geoid model and orthometric height systems. This result suggests that the so-called corrective surface, frequently used in GS-levelling, should hardly be significant in this approach. As a by-product, a new computationally efficient method to reduce the height anomaly to the geoid height has been outlined in Sect...

7 1 L. E. Sjöberg Appendix A1: the approximation error of substituting g nt for g nt in Eq. 15c By Taylor-expanding Sψaround the surface point, we obtain k k Sr,ψ Sψ = Sr,ψ+ k! r k, 47 k=1 r=r and the bracket under the integral of Eq. 15c becomes k k Sr,ψ = k! r k. 48 k= r=r Considering only the first term in Eq. 48 and inserting it for the bracket under the integral of Eq. 15c, the first omitted term of Eq. 15c becomes for G = M g nt ζdwc nt = Sr,ψ 8π r G d. 49 r=r Since G is the high-degree residual of the gravity anomaly, and that also the radial derivative of Stokes s function loses much of its power when moving away from the computation point, we may approximate the global integral of Eq. 49 by a plane integral to some radial limit s.thenwe have Sr,ψ and d sdsdα, 5 s + s and α are polar coordinates distance and azimuth, and the integral Eq. 49 takes the form ζ nt dwc 4π s π G s + sdsdα. 51 Using the mean value theorem from integral calculus with Ḡ being some mean value for G, the integral can be written ζdwc nt Ḡ 1 s + s + / = Ḡ s 1 + /s /. 5 By assuming that s is 1 km and Ḡ is 1 mgal, the result of ζdwc nt for h being 1, 4 and 8.85 km is.5,.8 and 4 mm, respectively. ence, the contribution from this integral is very small, and G can thus be safely substituted by G. If we repeat the above exercise with Eq. 14a instead of Eq. 15c, the difference is that we must consider the entire gravity anomaly g nt, being, say, 1 times larger than the residual gravity anomaly used in Eq. 5. The resulting estimate for ζdwc nt will thus reach 4 8 cm for the highest mountains. Appendix A: the topographic potential and attraction Assuming a constant topographic mass-density µ,the topographic potential is given by the Newton integral V t = µ r s r dr l d. 5 Exterior to the Brillouin sphere with r > r max =maximum radius of the Earth the gravitational potential of the topography can be expressed as an external-type harmonic series: V t = n+1 Vn t, 54a V t n = r = 4πµ n + 1 and ν n = n + 1 4π 1 + / n+ 1 n td n + n + n + n + O ν n td. This implies that to order / it holds that V t 1 n + 1 Vn t 4πµ 1 n + 1 n + n + n n + 1 n 54b 54c. 55 Below we will assume that the series in Eq. 54a and its radial derivative are valid all the way down to the Earth s surface. At and below the geoid, the topographic potential can be developed into an internal-type series e.g. Sjöberg 1996 r V t = µ r n r n 1 dr n td ; r, 56 and, particularly at the geoid with r =, Vg t = 4πµ 1 n n 1 n + O. n From Eqs. 57 and 55, we thus obtain the potential difference Vg t V t = πµ n n O = µ d + O. 58 l

8 A new technique to determine geoid and orthometric heights 11 In these derivations, we have used n n + 1 = d with 4π l l = 1 t = n t. 59 Let us now estimate the magnitude of the above potential difference by using the plane approximations d sdsdα, l s, s,αare polar coordinates. Then we obtain Vg t V t µ d l µ π dα s ds = πµ s, 6 s is the limiting radius and is the mean height of the region of interest. Using the numerical values = 6, 71 km, πµ =.1119 mgal/m, = =8 km Mt. Everest, s = 1, km and =981 Gal, we obtain Vg t V t.69 m. 61 The precise determination of the potential difference can thus be written Vg t V t = µ l d + DV t, including the residual potential difference DV t = µ f, t f r, t d. l 6a 6b The exterior gravitational attraction of the topography follows from Eq. 54a by the following derivation: V g t t = = 1 n+ n + 1 Vn t r r=r r, 6 or, directly from Eqs. 54a and 54b g t = 4πµ n + 1 n + 1 n + O = πµ + µ We thus obtain g t = πµ + µ l d + O l d + dg t, a, in view of Eqs. 18a and 18b, the residual topographic gravity becomes dg t = µ Jr, t + + d 65b l with f x, t Jr, t =. 65c x x=r Considering Eqs. 54a and 54b, Eq. 65b can be written dg t = πµ n + 1n + n n +O or dg t = πµ +O n + 1, 65d n + n n + 1 n, 66 which, in view of the relation eiskanen and Moritz 1967, Eq nn k = k k d. 67 π becomes dg t = πµ µ l l d 4l +O, 68 the first term on the right-hand side is very small reaching about 1 mgal for the height of Mt. Everest. ence we obtain the approximation dg t µ d, 69 which is the traditional terrain correction to gravity. l Acknowledgements I am grateful for the critical remarks by W. Featherstone,. Tenzer and an unknown reviewer on a previous version of this manuscript. eferences Ardalan AA, Grafarend EW 4 igh-resolution regional geoid computation without applying Stokes s formula: a case study of the Iranian geoid. J Geod 78: eiskanen WA, Moritz 1967 hysical geodesy. Freemann, San Francisco

9 1 L. E. Sjöberg Martinec Z 1998 Boundary value problems for gravimetric determination of a precise geoid. Lecture notes in Earth Sciences, vol 7. Springer, Berlin eidelberg New York Sjöberg LE 1996 The terrain effect in geoid computation from satellite derived geopotential models. Boll Geod Sci Aff 554:85 9 Sjöberg LE Topographic effects by the Stokes elmert method of geoid and quasi-geoid determinations. J Geod 74:55 68 Sjöberg LE 4 The effect on the geoid of lateral topographic density variations. J Geod 78:4 9 Tenzer, Vaníček, Santos M, Featherstone WE, Kuhn M 5 The rigorous determination of orthometric heights. J Geod 79:8 9 Wichiencharoen C 198 The indirect effects on the computation of geoid undulations. eport 6, Deptartment of Geodetic Science, Ohio State University, Columbus