E ects of the spherical terrain on gravity and the geoid

Transcription

1 Journal of Geodesy 001) 75: 491±504 E ects of the spherical terrain on gravity and the geoid P. Nova k 1, P. Vanõ cï ek 1,. Martinec,M.Ve ronneau 3 1 Department of Geodesy and Geomatics Engineering, University of New Brunswick, PO BOX 4400, Fredericton, E3B 5A3 Canada Department of Geophysics, Charles University, V HolesÏ ovicï kaâ ch, 18000Praha 8, Czech epublic 3 Geodetic Survey Division, Natural esources Canada, 615 Booth Street, Ottawa, K1A 0E9 Canada eceived: 16 June 000 / Accepted: 7 April 001 Abstract. The determination of the gravimetric geoid is based on the magnitude of gravity observed at the topographical surface and applied in two boundary value problems of potential they: the Dirichlet problem f downward continuation of gravity anomalies from the topography to the geoid) and the Stokes problem f transfmation of gravity anomalies into the disturbing gravity potential at the geoid). Since both problems require involved functions to be harmonic everywhere outside the geoid, proper reduction of gravity must be applied. This contribution deals with far-zone e ects of the global terrain on gravity and the geoid in the Stokes±Helmert scheme. A spherical harmonic model of the global topography and a Molodenskij-type spectral approach are used f a derivation of suitable computational fmulae. Numerical results f a part of the Canadian ocky Mountains are presented to illustrate the signi cance of these e ects in precise i.e. centimetre) geoid computations. Their omission can be responsible f a long-frequency bias in the geoid, especially over mountainous areas. Due to the rough topography of the testing area, these numerical values can be used as maximum global estimates of the e ects maybe with the exception of the Himalayas). This study is a continuation of e ts to model adequately the topographical e ects on gravity and the geoid, especially of a comparing the e ects of the planar topographical plate and the spherical topographical shell on gravity and the geoid [Vanõ cï ek, Nova k, Martinec 001) J Geod 75: 10±15]. Key wds: Gravimetric Geoid ± Helmert's eduction of Gravity ± Spherical Terrain E ects Crespondence to: P. Nova k Department of Geomatics Engineering, The University of Calgary, 500 University Drive NW, Calgary, TN 1N4 Canada pnovak@ucalgary.ca Tel.: Fax: Introduction Vanõ cï ek et al. 001) discussed signi cant di erences in magnitude of the topographical e ects on gravitational potential and gravitational acceleration generated by an in nite planar plate and a global spherical shell. This manuscript was submitted separately because it focuses on something quite di erent and only incidentally fms a foundation f the present contribution. Two basic conclusions of Vanõ cï ek et al.'s investigation, relevant f this contribution, can be summarized as follows: although the choice of either the plate the shell does not signi cantly e ect the classical Bouguer reduction of gravity, it has rather imptant consequences f the harmonization of the Earth's gravity eld above the geoid which disquali es the plate model from its use in precise geoid computations. The e ects discussed in Vanõ cï ek et al. 001) represent an approximation of rst der which must be crected f all deviations of the actual topography from the spherical shell, referred to as a terrain. In the following, we aim to discuss the spherical terrain e ects on gravity and the consequences of their adoption f the geoid. When speaking about the spherical model of topography terrain), we really mean the spherical approximation of the geoid from which topographical masses are de ned. Topographical e ects in the Stokes±Helmert scheme The e ect of topography in the context of the Stokes± Helmert approach Heck 199; see also Vanõ cï ek and Martinec 1994) is evaluated in terms of three separate e ects: the direct topographical e ect DTE) on gravity, the primary indirect topographical e ect PITE) on the geoid, and the secondary indirect topographical e ect SITE) on gravity. These three topographical e ects can easily be de ned using the Helmert disturbing gravity potential T h, which is related to the disturbing gravity potential T of the Earth's gravity eld as follows [Martinec et al. 1993, Eq. 8)]:

2 49 T h r; X ˆT r; X dv t r; X 1 A triplet of the curvilinear codinates r; u; k ˆ r; X de nes a position of the computation point in the geocentric spherical codinate system. The parameter dv t is the residual topographical potential, which can be de ned as a sum of the potential of the Bouguer shell, plus the potential of the terrain, minus the potential of the condensed Bouguer shell, minus the potential of the condensed terrain [Martinec 1998, Eqs. 3.1) and 3.6)]. Values of the function T h at the geoid approximated in the following by the reference sphere of radius ) are solved f in the Stokes±Helmert scheme. The geoid N and the PITE on the geoid P can then be derived from Eq. 1) by the spherical Bruns theem as follows: N X ˆT h ; X dv t ; X c ˆ T h ; X P X c where c is nmal gravity at the reference ellipsoid. Applying the negative radial derivative to Eq. 1), we obtain the cresponding topographical e ects on gravity dg h r; X ˆdg r; X o dv t r; X 3 where symbols dg and dg h stand f gravity disturbances de ned in the real and the Helmert gravity elds. Since gravity disturbances dg di er from gravity anomalies Dg only by a vertical change of nmal gravity along the separation between the actual equipotential surface real Helmert) and the equivalent nmal equipotential surface, we obtain f the spherical approximation of a vertical gradient of nmal gravity the following expression VanõÂ cï ek et al. 1999, Sect. 3): Dg h r; X T h r; X ˆDg r; X T r; X o dv t r; X r r 4 Equation 4) can easily be re-arranged using the DTE on gravity, D, and the SITE on gravity, S, Dg h r; X ˆDg r; X dv t r; X odv t r; X ˆ Dg r; X S r; X D r; X We will then discuss these three e ects sequentially. 3 The direct topographical e ect on gravity r 5 The spherical fm of the DTE was fmulated by Martinec and VanõÂ cï ek 1994a). The DTE can be evaluated as a sum of the e ect of the Bouguer shell A b, plus the e ect of the terrain A t, minus the e ect of the condensed Bouguer shell A cb, minus the e ect of the condensed terrain A ct D H; X ˆA b H; X A t H; X A cb H; X A ct H; X 6 All terms in Eq. 6) are related to the topography of the geocentric radius H, where H is the thometric height of topography. We will deal only with the two terrain e ects, A t and A ct, as the other two e ects were discussed in VanõÂ cï ek et al. 001). Since the e ects A b and A cb cancel each other if the mass conservation scheme is used Wichiencharoen 198), the di erence D ter ˆ A t A ct, called in the following the `direct terrain e ect' DTerE), accounts f the entire DTE on gravity. It was shown by Martinec and VanõÂ cï ek 1994a) that f the evaluation of this di erence ± they called it the roughness term ± it does not matter which of the condensation schemes is used. Di erent condensation schemes were discussed by, f instance, Wichiencharoen 198) see also VanõÂ cï ek et al. 001). Here, the mass conservation scheme only will be deployed. D ter is then given by the following fmula Martinec and VanõÂ cï ek 1994a): D ter H; X ˆ G. t o G. t o X X o L 1 r; w; n n dn dx 0 rˆ n dn o L 1 r; w; dx 0 rˆ 7 Here, G stands f the universal gravitational constant,. t o is the mean topographical density, n is the integration parameter, and X is the full spatial angle. The Newtonian integration kernel L 1 in Eq. 7) is of the fm L 1 r; w; n ˆ r n 1 r n cos w 8 where w is a spherical distance between the geocentric directions X ˆ u; k and X 0 ˆ u 0 ; k which can easily be computed by the law of cosines as follows: cos w ˆ cos u cos u 0 sin u sin u 0 cos k k 9 We will now show that, contrary to our earlier belief, it is not possible to stop integrating at a small value of the cap radius i.e. truncation of the integration domain X to a spherical cap of radius w o equal to 1,3 5 ). To show this, we can evaluate each of the two integrals in Eq. 7) in two parts: the integral over the spherical cap of radius w o, i.e. 0 < w < w o near-zone contribution), and the integral over the rest of the wld, i.e. w o w p far-zone contribution). The integral over the near zone can clearly be evaluated from the available DTM through a standard -D numerical integration

3 493 using the discretized fm of the cresponding integrals e.g. by the Gaussian quadrature rule); see, f example, Nova k 000). As this is a fairly standard procedure integration is replaced accding to the mean-value theem by discrete summation), we will not discuss it any further here. The far-zone contribution is a di erent matter all together; it must be evaluated using a di erent approach. In der to obtain the far-zone contribution to the rst integral in Eq. 7), we expand the spatial fm of the integral o L 1 r; w; n n dn rˆ 10 to its spectral fm. This is done by means of the technique introduced by Molodenskij et al. 1960), which yields see Appendix A) X nh1 pg. t o t n w; w o H n X H X X nh1 u n w; w H X o # 1 H n X H X X nh v n w; w o H n X O H 3 n 11 F the far-zone contribution to the second integral in Eq. 7), we obtain similarly see Appendix A) pg. t o 1 X nh1 X nh w n H; w; w o H n X w n H; w; w o Hn X O H n 3 # 1 Coe cients t n, u n, v n and w n are the Molodenskij truncation coe cients weights) de ned in Appendix A [see Eqs. A14)± A17)]. onal coe cients H n f the spherical harmonic representation of heights in Eqs. 11) and 1) to their maximum available degree nh1 can be computed from a global elevation model such as TUG87 Wieser 1987) H n X ˆ Xn mˆ n H n;m Y n;m X 8n ˆ 0;...; nh1 13 and the cresponding coe cients Hn heights f squared Hn Xn X ˆ mˆ n Hn;m Y n;m X ; 8 n ˆ 0;...; nh 14 Y n;m stands f the Laplace spherical harmonics Heiskanen and Mitz 1967). Maximum degrees of nh1 ˆ 180and nh ˆ 90were used in our computations. The far-zone contribution to the DTerE is given by the di erence of the series of Eqs. 11) and 1) D ter H; X X nh1 ˆ pg. t o t n w; w o H n X H X H X Xnh1 Xnh1 1 u n w; w o H n X H X w n H; w; w o H n X 1 X nh X nh v n w; w o H n X w n H; w; w o H n X O H n 3 # ; 8 w w o > 0 15 The terms of the der H n cancel eventually each other after expanding H X into a series fm. The same would apply to Eqs. 1) and 6). F our test area see Fig. 1), selected to cover the most challenging part of the Canadian ocky Mountains, and f w o ˆ 3, we obtain the numerical results shown in Fig.. Clearly, in the mountains, the e ect of the far zone is very signi cant even f a relatively large near zone, so that it certainly cannot be neglected. Our further computations not given here) have shown that the far-zone e ect is similarly signi cant f the whole territy of Canada and cannot be neglected anywhere, if an accurate geoid is to be computed. It must be noted that the far-zone contribution is of a longwavelength character. Continuing these values down to the geoid and applying Stokes' integration, cresponding e ects on the geoid can be obtained see Fig. 3). The range of both values, including basic statistical parameters mean and standard deviation), is shown in Tables 1 and. Another spherical harmonic expression f the DTE on gravity up to der Hn 3 was derived by Nahavandchi and SjoÈ berg [1998, Eq. 0)], expanding the Newtonian kernel into a series of Legendre polynomials. To account f a near-zone terrain, a new fmula, representing a compromise between the local integration and the spherical harmonic expression was derived in Nahavandchi [000, Eq. 1)]. 4 The primary indirect topographical e ect on the geoid The mathematical expression f the spherical fm of the PITE on the geoid was derived by Martinec and Vanõ cï ek 1994b). This e ect is merely a re-scaled value by nmal gravity) of the residual topographical potential evaluated at the reference sphere, see Eq. ). Following the same approach as that we applied f the DTE, this e ect can be evaluated as a sum of the e ect of the Bouguer shell P b, plus the e ect of the terrain P t, minus the e ect of the condensed Bouguer shell P cb, minus the e ect of the condensed terrain P ct P X ˆP b X P t X P cb X P ct X 16 All terms are related to the reference sphere of radius which approximates the geoid. The Bouguer shell e ects

4 494 Fig. 1. Spectral topography TUG87) of the Canadian ocky Mountains m) Fig.. Far-zone direct terrain e ect on gravity D ter mgal) were also discussed in VanõÂ cï ek et al. 001). We focus our attention on the di erence P ter ˆ P t P ct, called in the following the `primary indirect terrain e ect' PITerE). It is given by the following fmula Martinec and VanõÂ cï ek 1994b):

5 495 Fig. 3. Far-zone direct terrain e ect on the geoid m) Table 1. Far-zone terrain e ects on gravity mgal) Parameter Minimum Maximum Mean value P ter X ˆG c.t o G c.t o X X L 1 ; w; n n dn dx 0 n dnl 1 ; w; dx 0 Standard deviation A t ) D ter ) ) S ter ) ) Table. Far-zone terrain e ects on the geoid m) Parameter Minimum Maximum Mean value Standard deviation A t D ter ) ) S ter ) ) P ter ) The near-zone contribution to the PITerE can also be computed by a -D numerical integration. The far-zone contribution to the PITerE can then be obtained again by the spectral approach. In der to obtain the far-zone contribution to the rst integral in Eq. 17), the integration kernel L 1 r; w; n n dn 18 is developed into the spectral fm see Appendix B), which yields p G c.t o Xnh1 a n g; w; w o H n X H X Xnh1 c n g; w; w o H n X Xnh b n g; w; w o Hn X Xnh d n g; w; w o Hn X O H n 3 # 19 Similarly, the Molodenskij spectral approach gives f the second integral in Eq. 17) the following series see Appendix B): p G c.t o Xnh Xnh1 e n w; w o H n X e n w; w o Hn X O H n 3 # 0 The Molodenskij truncation coe cients a n, b n, c n, d n and e n are de ned in Appendix B [see Eqs. B14)± B18)]. The far-zone primary indirect terrain e ect is then the di erence of the series de ned by Eqs. 19) and 0)

6 496 P ter X ˆ p G c.t o Xnh1 Xnh Xnh Xnh1 a n g; w; w o H n X H X c n g; w; w o H n X Xnh b n g; w; w o Hn X d n g; w; w o Hn Xnh1 X e n w; w o H n X e n w; w o Hn X O H n 3 # ; 8 w w o > 0 1 F our test area and the truncation radius w o ˆ 3,we obtain the numerical results shown in Fig. 4. The range of the values from Eq. 1), including basic statistical parameters such as the mean and the standard deviation, can be found in Table. It is interesting to note that, while the far-zone contribution to the DTerE on gravity is smaller than the near-zone contribution, the cresponding contributions to the PITerE are of comparable magnitude. This may iginate in the fact that the potential is an inverse function of distance while gravity is an inverse function of distance squared and tapers o me quickly. Also, it should be noted that while the near-zone contribution is always negative, the far-zone contribution changes sign so that the resulting total PITE, although still predominantly negative, reaches positive values in some places. The far-zone contribution to the PITerE on the geoid is again of a long-wavelength character. A spherical harmonic expression f the PITE on the geoid up to der Hn 3 was derived by Nahavandchi and SjoÈ berg [1998, Eq. 4)], which is also intended f the computation of the global PITE using the full spatial angle X. A new fmula, compromising between the local integration and the spherical harmonic expression, was derived in SjoÈ berg and Nahavandchi [1998, Eq. 1)]. 5 The secondary indirect topographical e ect on gravity F the Stokes±Helmert problem, the spherical fm of the SITE on gravity was fmulated in VanõÂ cï ek et al. 1999) as a re-scaled value the scale is equal to =) of the residual topographical potential evaluated at radius H; see Eq. 5). Since the di erence of the gravitational potential of the spherical topographical shell and its condensed counterpart is f the mass conservation condensation equal to zero, the so-called secondary indirect terrain e ect SITerE accounts f the entire SITE on gravity. The SITerE on gravity S ter ) can be computed by the following expression VanõÂ cï ek et al. 1999): S ter H; X ˆ G.t o G.t o X X L 1 H; w; L 1 H; w; n n dn dx 0 n dn dx 0 Fig. 4. Far-zone primary indirect terrain e ect on the geoid P ter m)

7 497 The near-zone e ect can be computed by a standard -D numerical integration discretizing the cresponding fmulae by the Gaussian quadrature rule. To compute its far-zone contribution, the integration kernel of the rst integral in Eq. ), i.e. L 1 H; w; n n dn 3 can be transfmed into the spectral fm see Appendix C), which gives 4pG. t H X X nh1 o p n w; w o H n X H X X nh1 q n w; w o H n X 1 X nh r n w; w o Hn X O H n 3 # 4 Similarly, the second integral in Eq. ) can be developed into see Appendix C) 4pG. t o 1 X nh1 X nh s n H; w; w o H n X s n H; w; w o Hn X O H n 3 # 5 The Molodenskij truncation coe cients p n, q n, r n and s n are also de ned in Appendix C [see Eqs. C9)± C1)]. Subtracting the series of Eqs. 4) and 5), the far-zone contribution to the SITerE on gravity reads S ter H; X ˆ 4pG. t H X X nh1 o p n w; w o H n X H X Xnh1 X nh1 q n w; w o H n X 1 s n H; w; w o H n X 1 X nh X nh r n w; w o H n X s n H; w; w o H n X O H n 3 # ; 8 w w o > 0 6 The far-zone SITerE on gravity S ter, computed by Eq. 6) over the test area using the truncation radius of w o ˆ 3, is shown in Fig. 5. Cresponding e ects on the geoid, obtained by continuing these values down to the geoid and applying Stokes' integration, are shown in Fig. 6. The extreme values f the SITerE on gravity can be found in Tables 1 and, including their basic statistical parameters. The near and far-zone contributions to the SITerE are also of comparable magnitude. The same argument as in the case of the PITerE could be used here. The values of the SITerE are at the accuracy level required in current geoid computations and should still be taken into the account. The far-zone contribution to the SITerE on gravity is also of a long-wavelength character. Fig. 5. Far-zone secondary indirect terrain e ect on gravity S ter mgal)

8 498 Fig. 6. Far-zone secondary indirect terrain e ect on the geoid m) A comparable spherical harmonic expression f the SITE on gravity up to der Hn 3 was again derived by Nahavandchi and SjoÈ berg [1998, Eq. 5)]. 6 The terrain crection Clearly, Eq. 11) by itself describes the terrain crection to gravity used in the evaluation of the re ned Bouguer anomaly and the Fay anomaly Heiskanen and Mitz 1967). It is thus also wth looking at it alone ± in addition to looking at the di erence of Eqs. 11) and 1). Figure 7 shows the values of the far-zone contribution to the terrain crection A t f our test area. It is very large and again of a long-wavelength character. It is of interest to compare the spherical terrain crection with the standard planar terrain crection used in practice. Figure 8 shows the di erence between the standard planar terrain crection TC), and our terrain crection computed from a spherical model taking into account the topography from all around the wld. The di erences are very signi cant, but they are predominantly of a long-wavelength nature. They thus do not a ect the routine geophysical interpretation of the Bouguer gravity anomalies, which seeks shallow density anomalies. If deeper-seeded anomalies are of interest, the spherical terrain crection must be considered. 7 Conclusions In this contribution, the far-zone spherical terrain e ects both on gravity and the geoid, which iginate in the second Helmert condensation of external topographical masses, were discussed. These e ects are usually deemed to be too small, and thus often neglected in practical geoid computations. As a consequence of this belief, the planar approximation of the terrain is usually deployed f the evaluation of the terrain e ects on gravity and the geoid. Based on the numerical values obtained in our computations, we believe that the planar approximation is not adequate and its use can be responsible f the long-wavelength errs in the geoid solutions. They can be directly transfmed into the geoidal undulations through the low-pass ltering in the Stokes integration. F comparable conclusions see, f example, SjoÈ berg and Nahavandchi 1998) and Nahavandchi 000). The far-zone spherical terrain e ects, due to their large magnitude, are very imptant f the evaluation of the centimetre geoid. The direct and the primary indirect terrain e ects have the most signi cant contribution to the geoid. The far-zone direct terrain e ect can easily reach values on the decimetre level see Table ). The far-zone primary indirect terrain e ect has the largest contribution on the metre level but its average magnitude is signi cantly smaller again see Table ). The far-zone secondary indirect terrain e ect, although the smallest of all the terrain e ects, should still be taken into the account f the evaluation of the centimetre geoid although its magnitude is the smallest of all the far-terrain e ects we introduced in this contribution. An imptant nding is represented by the values of the far-zone spherical terrain crection. F applications where the compensation is not required such as the derivation of the re ned Bouguer anomalies in geophysical applications), the di erence between the planar and the

9 499 Fig. 7. Far-zone terrain crection to gravity A t mgal) Fig. 8. Spherical vs planar terrain crection to gravity mgal) spherical model can be very signi cant see Fig. 8). In Fig. 8, the total terrain crection to gravity along the selected parallel across the Coastal Mountains in western Canada, computed using the planar and the spherical terrain model, is shown. It is the far-zone e ect clearly missing in the planar model) which distinguishes the planar and the spherical values. Therefe we should keep in mind the fact that the use of the planar and the spherical model of the topography may have serious consequences if applied in speci c applications.

10 500 The topographical e ects are the most imptant crections to both gravity and the geoid. They represent, together with accuracy and density of observed gravity data, one of the most serious limits on higher accuracy of the gravimetric geoid today. Their crect evaluation depends on the crect knowledge of input data topographical heights and the mass density of the topography), and su ciently accurate modelling of the topographical masses. The knowledge of the global topography will greatly improve in the near future due to the latest Shuttle adar Topography Mission, which used an advanced radar technique to obtain data f production of the most precise, near-global topographical map ever nearly 80% of the Earth's landmass), and which will leave us with the incomplete knowledge of the topographical mass density distribution as the maj problem in the topographical reduction of gravity data. The fmulation of the terrain e ects based on the spherical rather than the planar approximation, which allows us to account f the low-frequency e ects of the far-zone terrain on both gravity and the geoid, represents then the maj contribution of the presented research. When combined with the cresponding highfrequency e ects computed by a -D integration over the near-zone terrain, proper values of topographical e ects can be obtained. Appendix A Truncation coe cients f DTerE In this appendix, the truncation coe cients f the direct terrain e ect on gravity are derived. The integration kernel of Eq. 10) is developed in its general fm as follows Gradshteyn and yzhik 1980): o L 1 r; w; n n dn n r ˆ n 3r cos w rn 1 6 cos w L 1 r; w; n r 3cos w 1 lnjn r cos w L r; w; n j A1 The Newtonian kernel L 1 [see Eq. 8)] can be developed into a binomial series L 1 r; w; n ˆ 1 r cos w 1 f 1 1 f 1 cos w ˆ 1 r cos w 1 1 f f 1 3 cos w O f cos w A where the following substitution is used: g ˆ n r ˆ 1 f; i.e. f ˆ n r 1 A3 The parameter r is equal to H X, and the parameter n is equal either to H X to H X. The truncation of the binomial series in Eq. A) after the quadratic term is fully justi ed f the range of values of w and f used in our computations, i.e. f w w o ˆ 3, and f jfj < 0:0015 f maximum topographical height of 9 km), which yield f f 1 cos w < 0:005 1 A4 Similarly, the inverse of the Newtonian kernel, i.e. L, can be developed as follows: L r;w;n ˆr cosw 1 1 f 1 cosw f O f cosw A5 Using the substitutions of Eqs. A) and A5), the integration kernel of Eq. A1) can be written as follows: r 3 g cos w g 1 6 cos w r 1 4 cos w 6 cos w 1 g g cos w 1 cos w 1 r 3 cos w 1 ln g cos w 1 g g cos w 1 1 cos w cos w 1 A6 The logarithmic function in Eq. A6) can further be expanded into the Tayl series which, following the above reasoning can also be truncated after the quadratic term ln g cos w 1 g g cos w 1 1 cos w cos w 1 f cos w 1 f f 1 cos w 8 1 cos w ˆ 1 cos w cosw 1 1 f cos w 1 f f cos w cos w 1 1 cos w 1 cos w 3 5 O f 3 A7 Substituting the expression in Eq. A7) into Eq. A6), the integration kernel of Eq. A1) can subsequently be developed into the fm o L 1 r;w;n n dn ˆ r r f r cosw 1 n 8 f 3 10cosw 3cos w 3cosw 1 cosw 1 h i cosw 1 1 cosw cosw 1 O f 3 A8

11 501 which, after substitutions f the actual integration limits, i.e. n ˆ H X and n ˆ H X, can be written as follows: o L 1 r; w; n n dn rˆ ˆ H X 0 1 H X ŠK 1 w H X # H X H X Š H 3 K w O H A9 The integration kernel K 1 in Eq. A9) is of the following fm: K 1 w ˆ 1 cos w cos w and the integration kernel K reads A10 K w ˆ cosw 3cos w 3cosw 1 cosw 1 h i 8 cosw 1 1 cosw cosw 1 A11 The integration kernel of the second integral in Eq. 7) can simply be written n dn o L 1 r; w; rˆ ˆ H X H X H X H X J ; w; H O H 3 A1 with the function J de ned as the radial derivative of the Newtonian kernel L 1 Heiskanen and Mitz 1967, Sect. 1.16) J ; w; H ˆ cos w 1 H L 3 ; w; H A13 The truncation coe cients, used in Eqs. 11) and 1), can nally be estimated by the 1-D integration over the spherical distance w t n w; w o ˆ p K 1 w P n cos w 1Šsin w dw; 8 n ˆ 0;...; nh1 A14 u n w; w o ˆ v n w; w o ˆ p p w n H;w;w o ˆ Appendix B K w P n cos w sin w dw; 8 n ˆ 0;...; nh1 A15 K w P n cos w 1Šsin w dw; p Truncation coe cients f PITerE 8 n ˆ 0;...; nh A16 J ;w; H P n cosw 1Šsinwdw; 8n0 A17 In this appendix, the truncation coe cients f the primary indirect terrain e ect on the geoid are derived. The integration kernel of Eq. 18) can be derived as a primitive function of the Newtonian kernel L 1 Gradshteyn and yzhik 1980) L 1 r; w; n n dn ˆ 1 n 3r cos w L r; w; n n r 3 cos w 1 lnjn r cos w L r; w; n j C B1 where C stands f the integration constant. Introducing the following two unitless parameters: g ˆ H X ˆ 1 f B g 0 ˆ H X ˆ 1 f 0 B3 and using the substitutions from the previous section, we obtain ˆ L 1 r;w;n n dn g0 3cosw 1 g 0 g 0 cosw 1 3cos w 1 ln g0 cosw 1 g 0 g 0 1 cosw g cosw 1 g gcosw 1 g 3cosw 1 g gcosw 1 B4

12 50 The rst and third expressions on the right-hand side of Eq. B4) can be developed using g 0 3 cos w 1 g 0 g 0 cos w 1 1 f 0 3 cos w cos w 1! 1 f0 f 0 1 cos w O f 03 B5 8 1 cos w g 3 cos w 1 g g cos w 1 1 f 3 cos w cos w 1 1 f f 1 cos w O f 3 B6 8 1 cos w and the logarithmic function can be expanded as follows [cf. Eq. A7)] ln g0 cos w 1 g 0 g 0 cos w 1 g cos w 1 g g cos w 1 f 0 f cos w 1 f 0 f f0 f 1 cos w 8 1 cos w ˆ g cos w 1 g g cos w f 0 f cos w 1 f 0 f f0 f 1 cos w 8 1 cos w 4 5 g cos w 1 g g cos w 1 O f 3 B7 The expressions in Eqs. B5)± B7) can be further simpli ed using elementary algebraic operations. Their substitution into Eq. B4) yields the integration kernel of Eq. B1) in the following fm: L 1 r; w; n n dn ˆ 3 4 f0 f cos w 1 1 cos w 8 f0 f 3 cos w 1 1 cos w h i cos w 1 g cos w 1 g g cos w 1 8 f0 f 3 cos w 1 h 3 cos w cosw 1 g cos w 1 g g cos w 1 i 4 f0 f 3 cos w 1 cos w 1 g cos w 1 g g cos w 1 8 f0 f 5 3 cos w O f 3 B8 cos w 1 Substituting f the actual integration limits, Eq. B8) can be developed into the following fm which can be used f practical computations: L 1 r; w; n n dn ˆ H X H X ŠH 1 g; w H X H X H g; w H X H X Š H 3 g; w O H 3 where the integration kernels read H 1 g; w ˆ3 4 cos w 1 1 cos w h i 3 cos w 1 cos w 1 h i 4 g cos w 1 g g cos w 1 H g; w ˆ 5 3 cos w 8 cos w 1 B9 B10 3cos w 1 1 cos w h i 8 cos w 1 g cos w 1 g g cos w 1 B11 H 3 g; w h i 3 cos w 1 3 cos w cos w 1 ˆ h i B1 8 g cos w 1 g g cos w 1 The integration kernel of the second integral in Eq. 17) is simply developed into another series truncated after the quadratic term L 1 ; w; n dn ˆ H X H X H X H X L 1 ; w; O H 3 B13 Finally, the truncation coe cients in Eqs. 19) and 0) can again be evaluated by the 1-D integration over the spherical distance w a n g; w; w o ˆ p H 1 g; w P n cos w 1Šsin w dw; 8 n ˆ 0;...; nh1 B14

13 b n g; w; w o ˆ c n g; w; w o ˆ d n g; w; w o ˆ e n w; w o ˆ p p p p H g; w P n cos w 1Šsin w dw; 8 n ˆ 0;...; nh B15 H 3 g; w P n cos w sin w dw; 8 n ˆ 0;...; nh1 B16 H 3 g; w P n cos w 1Šsin w dw; 8 n ˆ 0;...; nh B17 L 1 ; w; P n cos w 1Šsin w dw; ln g cos w 1 g g cos w 1 1 cos w cos w 1 r 1 3cosw cos w 1 C3 Deploying the power series expansion of the logarithmic function, cf. Eq. A7), and perfming all algebraic operations, we obtain ˆ r L 1 r; w; n n dn 1 f cosw 1 cos w cos w 1 3r 4 f cos w cos w 1 cos w cosw 1 h i cos w 1 1 cos w cos w 1 O f 3 C4 The parameter g is equal to 8 n 0 B18 g ˆ 1 H X B19 In practice, the coe cients of Eqs. B14)± B18) are evaluated in the fm of tables f the selected reference heights. During computations, the heights of computation points are used to nd the appropriate kernel values using some of available interpolation techniques e.g. Lagrange's interpolation). Appendix C Truncation coe cients f SITerE Let the parameter g now be de ned as follows: g ˆ H X H X ˆ H 0 H ˆ r0 r and the parameter f as C1 f ˆ r0 r ˆ H 0 H r H ˆ g 1 C The integration kernel of Eq. 3) can be fmulated f the speci ed integration limits as L 1 r; w; n n dn ˆ r g 3 cos w 1 g g cos w 1 r 3 cos w 1 After substitutions f the actual integration limits, Eq. C4) can be developed into the following fm: L 1 H; w; n n dn ˆ H X H X Š H X ŠM 1 w dx 0 H X H X Š M w dx 0 O H 3 H where the cresponding integration kernels are C5 cosw 1 M 1 w ˆ h i C6 1 cos w cosw 1 M w ˆ9 1cosw 3cos w 6 1 cosw cosw 1 h i 4 cosw 1 1 cosw cosw 1 C7 The integration kernel in the second integral of Eq. ) is simply developed into the fm L 1 ; w; H n dn ˆ H X H X H X H X L 1 H; w; O H 3 C8 Finally, the truncation coe cients in Eqs. 4) and 5) can be estimated by the 1-D integration over the spherical distance w

14 504 p n w; w o ˆ q n w; w o ˆ r n w; w o ˆ p p p s n H; w; w o ˆ M 1 w P n cos w 1Šsin w dw; 8 n ˆ 0;...; nh1 C9 M w P n cos w sin w dw; 8 n ˆ 0;...; nh1 C10 M w P n cos w 1Šsin w dw; p 8 n ˆ 0;...; nh C11 L 1 H; w; P n cos w 1Š sin w dw; 8 n 0 C1 Acknowledgements. The research described in this paper was prepared to meet the objectives of the research contract `Theetical and Practical e nements of Precise Geoid Determination Methods' between Natural esources Canada and the University of New Brunswick. We thank Natural esources Canada f their nancial suppt and f providing us with all necessary gravity and elevation data. Additional suppt was provided by the NSEC of Canada through an operating grant, by a NATO linkage grant, and by the GEOIDE Netwk of Excellence project no. 10). A part of the presented research was presented at the CGU Annual Meeting in Ban, May It also represents a part of the Ph.D. dissertation of the seni auth. The auths would also like to acknowledge the thoughtful comments of Prof. P. Holota, Prof. C. Jekeli, Prof. B. Heck and other anonymous reviewers. eferences Gradshteyn IS, yzhik IM 1980) Tables of integrals, series, and products. Academic Press, New Yk Heck B 199) A revision of Helmert's second method of condensation in the geoid and quasi-geoid determination. 7th IAG Int Symp Geodesy and Physics of the Earth, IAG-Symposium, No.11. Potsdam, 5±10October Heiskanen WA, Mitz H 1967) Physical geodesy. WH Freeman, San Francisco Martinec, Vanõ cï ek P 1994a) Direct topographical e ect of Helmert's condensation f a spherical approximation of the geoid. Manuscr Geod 19: 57±68 Martinec, Vanõ cï ek P 1994b) Indirect e ect of topography in the Stokes±Helmert technique f a spherical approximation of the geoid. Manuscr Geod 19: 13±19 Martinec 1998) Boundary-value problems f gravimetric determination of a precise geoid. Lecture notes in Earth Sciences 73. Springer, Berlin Heidelberg New Yk Martinec, Matyska C, Grafarend EW, Vanõ cï ek P 1993) On Helmert's nd condensation method. Manuscr Geod 18: 417± 41 Molodenskij MS, Eremeev VF, Yurkina MI 1960) Methods f study of the external gravitational eld and gure of the Earth [translated from ussian by the Israel Program f Scienti c Translations, O ce of Technical Services, Department of Commerce, Washington, DC, 196] Nahavandchi H 000) The direct topographical crection in gravimetric geoid determination by the Stokes±Helmert method. J Geod 74: 488±496 Nahavandchi H, SjoÈ berg LE 1998) Terrain crections to power H 3 in gravimetric geoid determination. J Geod 7: 14±135 Nova k P 000) Evaluation of gravity data f the Stokes±Helmert solution to the geodetic boundary-value problem. Tech rep 07, Department of Geodesy and Geomatics Engineering, University of New Brunswick, Fredericton SjoÈ berg LE, Nahavandchi H 1998) On the indirect e ect in the Stokes±Helmert method of geoid determination. J Geod 73: 87± 93 Vanõ cï ek P, Martinec 1994) The Stokes±Helmert scheme f the evaluation of a precise geoid. Manuscr Geod 19: 119±18 Vanõ cï ek P, Huang J, Nova k P, Pagiatakis S, Ve ronneau M, Martinec, Featherstone W 1999) Determination of boundary values f the Stokes±Helmert problem. J Geod 73: 180±19 Vanõ cï ek P, Nova k P, Martinec 001) Geoid, topography, and the Bouguer plate shell. J Geod 75: 10±15 Wichiencharoen C 198) The indirect e ects on the computation of geoid undulations. ep 336, Department of Geodetic Science and Surveying, The Ohio State University, Columbus Wieser M 1987) The global digital terrain model TUG'87. Internal rept on set-up, igin, and characteristics. Institute of Mathematical Geodesy, Technical University, Graz