Journal of Geodesy (1998) 72: 684±697

Performance of three types of Stokes's kernel in the combined solution for the geoid

P. VanõÂ cÏ ek1, W. E. Featherstone2 1 Department of Geodesy and Geomatics Engineering, University of New Brunswick, P.O. Box 4400, Fredericton, New Brunswick, E3B 5A3, Canada e-mail: [email protected]; Tel: +1 506 453 4698; Fax: +1 506 453 4943 2 School of Spatial Sciences, Curtin University of Technology, GPO Box U1987, Perth, WA 6845, Australia e-mail: [email protected]; Tel: +61 8 9266 2734; Fax: +61 8 9266 2703

Received: 11 August 1997 / Accepted: 18 August 1998

Z Z Abstract. When regional gravity data are used to 2p p compute a gravimetric geoid in conjunction with a N ˆ j S w†Dg sin w dw da 1† geopotential model, it is sometimes implied that the 0 0 terrestrial gravity data correct any erroneous wave- where j ˆ R= 4pc†, R is the mean Earth radius, c is normal lengths present in the geopotential model. This assertion gravity on the reference ellipsoid, w; a† are the spherical is investigated. The propagation of errors from the low- distance and azimuth, respectively, of each gravity frequency terrestrial gravity ®eld into the geoid is anomaly (Dg)1 from each computation point, and S w† derived for the spherical Stokes integral, the spheroidal is the spherical Stokes kernel, which is given by an in®nite Stokes integral and the Molodensky-modi®ed spheroi- Fourier series of Legendre polynomials, Pn cos w†,as dal Stokes integral. It is shown that error-free terrestrial X1 2n ‡ 1 gravity data, if used in a spherical cap of limited extent, S w†ˆ P cos w† 2† n 1 n cannot completely correct the geopotential model. Using nˆ2 a standard norm, it is shown that the spheroidal and Molodensky-modi®ed integration kernels o€er a prefer- The orthogonality relations between Legendre polyno- able approach. This is because they can ®lter out a large mials over the sphere allow Eq. (1) to be expressed in its amount of the low-frequency errors expected to exist in spectral form as terrestrial gravity anomalies and thus rely more on the X1 2 low-frequency geopotential model, which currently of- N ˆ c Dg 3† n 1 n fers the best source of this information. nˆ2

where c ˆ R= 2c† and Dgn is the n-th degree surface Key words. Geoid determination Á Modi®ed spherical harmonic of the gravity anomalies. In practice, kernels Á Error propagation Á High-pass ®ltering these are computed using a set of fully normalised potential coecients that de®ne a global geopotential model, such that

GM a n Xn DgG ˆ n 1† dC cos mk ‡ S sin mk† n r2 r nm nm mˆ0 1 Introduction  P nm cos h† 4† Formally, Stokes's integral must be evaluated over the where GM is the product of the Newtonian gravitational whole Earth using gravity anomalies that have been constant and mass of the Earth, a is the equatorial radius downward-continued to the geoid, which is assumed to be of the reference ellipsoid, (r; h; k) are the geocentric polar spherical, to give the gravimetric geoid height (N)ata coordinates of each computation point, dCnm and Snm are single point. This is the fully normalised potential coecients of degree n and order m, which have been reduced by the even zonal harmonics of the reference ellipsoid, and P nm cos h† are the fully normalised associated Legendre functions. Using Eq. (4) in Eq. (3) for the region 2  n  M 1The derivation or exact kind of gravity anomalies are not dis- gives the low-frequency contribution of the potential cussed in this paper. coecients to the geoid as 685

XM 2 high-frequency gravity anomalies (DgM ), which are ter- N ˆ c DgG T M n 1 n restrial gravity anomalies (Dg ) that have been reduced by nˆ2 the gravity anomalies implied by a geopotential model G GM XM a nXn 5† (Dg ) of spherical harmonic degree and order (M) ˆ dC cos mk rc r nm according to nˆ2 mˆ0 XM ‡ S sin mk†P cos h† M T G nm nm Dg ˆ Dg Dgn 7† nˆ2 where M is the degree of some spherical harmonic expansion of the geopotential model. One objective of this paper is to express Eq. (6) completely The incomplete global coverage of terrestrial gravity in a spectral form, using spherical harmonics, so that the data has driven the use of a geopotential model in the propagation of errors into the geoid from the two sources determination of the geoid. This has been called the of gravity ®eld information may be investigated. In order generalised Stokes scheme by Vanõ cÏ ek and SjoÈ berg to achieve this in a straightforward manner, the spherical (1991). Such an approach reduces the e€ect of not using a Stokes kernel, Eq. (2) as used in Eq. (6) is expressed as global coverage of terrestrial gravity data and also re-  duces the impact of the spherical approximations inher- S w† for 0  w  w S w; w †ˆ o 8† ent to Stokes's formula (Heiskanen and Moritz 1967, p. o 0 for wo < w  p 97). The latter is achieved because most of the geoid's power is contained within the low frequencies; see Eq. (3). As with Eq. (2), Eq. (8) can be expressed as an in®nite However, the combination of a geopotential model Fourier series of Legendre polynomials by with terrestrial gravity data in a spherical cap of limited extent via Stokes's integral is sometimes presented in a X1 2n ‡ 1 S w; w †ˆ s w †P cos w† 9† way that implies that the terrestrial gravity data improve o 2 n o n any errors present in the geopotential model (e.g. Sideris nˆ2 and Schwarz 1987). This assertion is examined in this where sn wo† are coecients which will be derived later in paper by comparing three implementations of Stokes's this section. integration when combined with a geopotential model. Using Eq. (8) in Eq. (6) yields an identical relation- These comprise the use of the spherical Stokes kernel ship to Eq. (6), except that the integration domain is (Eq. 2), the spheroidal Stokes kernel and the Moloden- now over the whole Earth (sphere), viz. sky modi®ed Stokes kernel (Vanõ cÏ ek and Kleusberg Z 2pZ p 1987). Other modi®cations to Stokes kernel exist, such M as those of Meissl (1971), SjoÈ berg (1984, 1986, 1991), N1 ˆ NM ‡ j S w; wo†Dg sin w dw da 10† Heck and GruÈ ninger (1987), Vanõ cÏ ek and SjoÈ berg 0 0 (1991), and Featherstone et al. (1998), but they will not Equations (5) and (9) are substituted in Eq. (10), and then be discussed here for lack of space. However, they will using the orthogonality relations to replace the integral be discussed in a sequel to this paper, which is currently term yields in preparation. By transforming these Stokes's integrals to the fre- XM 2 XM N ˆ c DgG ‡ c s w †DgM quency domain, in terms of surface spherical harmonics, 1 n 1 n n o n nˆ2 nˆ2 it will be shown that when the terrestrial gravity 11† anomalies are used from a region of limited extent, they X1 M can only correct the low-frequency errors present in the ‡ c sn wo†Dgn geopotential model to a limited extent. It will also be nˆM‡1 shown that the impact on the geoid of the low-frequency The preceding mathematical development has disre- errors, known to be present in the terrestrial gravity garded the fact that all the gravity anomaly harmonics anomalies, can be reduced when the spheroidal or Mo- used are only estimates of their true values, due to the lodensky-modi®ed forms of Stokes's integral are used. existence of measurement and data reduction errors, for instance. Therefore, the gravity anomalies used in 2 The spherical Stokes kernel Eq. (11) must be replaced by their estimates from the G geopotential model (Dg^n ) and the terrestrial gravity T Many authors combine a global geopotential model ± observations (Dg^n ). As the terrestrial gravity data have typically to maximum degree and order M ˆ 360 ± with already been reduced by the geopotential model [ac- regional gravity data using the spherical Stokes kernel, cording to Eq. (7)], this gives the estimate of the geoid as Eq. (2). In this instance, the geoid is estimated via XM XM 2 G T G Z 2Zp w N^ ˆ c Dg^ ‡ c s w † Dg^ Dg^ † o 1 n 1 n n o n n N ˆ N ‡ j S w†DgM sin w dw da 6† nˆ2 nˆ2 1 M 12† 0 0 X1 T where the Stokes integration is performed over a limited ‡ c sn wo†Dg^n nˆM‡1 surface spherical cap bound by wo. This also uses only the 686

From Eq. (12), it is evident that the propagation of For convenience during the subsequent comparisons, errors into the geoid from the geopotential model and Eq. (18) is rewritten so as to distinguish the error terrestrial gravity data is controlled, in part, by the co- contribution to the geoid from the errors in the ter- ecients sn wo†. Therefore, it is important to have a restrial gravity data, errors in the geopotential coe- knowledge of their magnitudes. This can be achieved cients, and the truncation error associated with the through a rearrangement of Eq. (9) using the orthogo- limited integration domain (wo) in Eq. (6), respectively. nality relations over the sphere, such that This gives Z  p X1 XM s w †ˆ S w; w †P cos w† sin w dw 13† 2 T G n o o n N1 ˆ c Qn w †  ‡ c Qn w † 0 n 1 o n o n nˆ2 nˆ2 19† Introducing the spherical (Molodensky et al. 1962) X1 truncation coecients T c Qn wo†Dgn Z p nˆM‡1 Q w †ˆ S w†P cos w† sin w dw 14† n o n The truncation error, or remote zone contribution, w o comprises the last term in Eq. (19), and represents a bias and using these, together with the relation of Eq. (8), in that must be considered when using Eq. (6) to compute Eq. (13) yields the geoid. It results from the neglect of high-frequency Z p (n > M) gravity anomalies in the remote zones outside the integration domain (w < w  p). As this discussion is sn wo†ˆ S w†Pn cos w† sin w dw Qn wo† o 0 15† only concerned with the question of how the terrestrial 2 gravity data can attenuate any errors in the geopotential ˆ Qn w †8n  2 model, the truncation error term need not be considered n 1 o as it only a€ects the high-frequency gravity ®eld. Substituting this result in Eq. (12) produces Figure 1a shows the magnitudes of the coecients that control the propagation of the degree-errors from M  X 2 the geopotential model (G) and the terrestrial gravity N^ ˆ c Q w †Dg^G ‡ Q w † Dg^T n 1 n o n n 1 n o n anomalies (T ) through Eq. (19) in relation to the ge- nˆ2 n  16† neric 2= n 1† term [cf. Eq. (3)]. This generic term X1 2 shows how the errors propagate if the original Stokes ‡ c Q w † Dg^T n 1 n o n integral, Eq. (1), is used to compute the geoid. The nˆM‡1 spherical truncation coecients, Eq. (14), were com- Next, the degree-errors in the geopotential model- puted using the recursive routines of Paul (1973). The G values of M ˆ 360 and w ˆ 6 were chosen as they derived gravity anomalies (n ) and the terrestrial gravity o T provide a representative example to show the relative anomalies (n ) are introduced into Eq. (16) to yield magnitude of the terms in Eq. (19). XM h In Fig. 1a, the dominance of the generic 2= n 1† ^ G G N1 ˆ c Qn wo†fDgn ‡ n g term, which is asymptotic, tends to obscure the be- nˆ2  haviour of the coecients in the high degrees. There- 2 ÈÉi fore, a normalisation has been used to produce Fig. 1b, ‡ Q w † DgT ‡ T n 1 n o n n where each coecient is multiplied by n 1†=2; the  inverse of the so-called generic term. In Fig. 1b, the X1 2 ÈÉ generic term becomes unity, the spherical truncation ‡ c Q w † DgT ‡ T 17† n 1 n o n n coecients that apply to error in the geopotential nˆM‡1 G model (n ) oscillate about zero, and the coecients that T where the superscripts G and T are retained simply to apply to the terrestrial gravity data errors (n ) oscillate indicate the source of the gravity anomalies. Subtract- about unity. This form of presentation allows a clearer ing Eq. (3) from Eq. (17) 8n  2 leaves the error that comparison among the three approaches investigated occurs in the geoid (N1) when terrestrial gravity (see later). anomalies and geopotential coecients are combined to compute the geoid using the approach given in Eq. (6). This is 2.1 Special cases of Eq. (19)  XM 2 Á Next, three special cases of Eq. (19) are considered. The  ˆ c T ‡ Q w † G T N1 n 1 n n o n n ®rst is for integration of Eq. (6) over the whole Earth (i.e. nˆ2 wo ˆ p), where the spherical truncation coecients re- 1  X 2 duce to Qn p†ˆ0, 8n  2. This can be proven from ‡ c T Q w †DgT Q w †T n 1 n n o n n o n Eq. (14) using the orthogonality relations between Leg- nˆM‡1 endre polynomials over the sphere. Thus, Eq. (19) reduces 18† to 687

XM 2 X1 2  ˆ c G c DgT 21† N1 n 1 n n 1 n nˆ2 nˆM‡1 where the truncation error [second term on the right-hand side of Eq. (21)] attains its maximum value, and the error from the geopotential passes undiminished into the geoid according to the generic 2= n 1† term. This result gives an error equivalent to using only the geopotential model to compute the geoid [cf. Eqs (21) and (5)]. Also, the truncation error term in Eq. (21) is equal to the omission error of the geopotential model. When an error-free geopotential model is considered G (i.e. n ˆ 0 for 2  n  M), together with the Stokesian integration over a cap bound by wo, Eq. (19) reduces to  X1 2  ˆ c Q w † T N1 n 1 n o n nˆ2 22† X1 T c Qn wo†Dgn nˆM‡1 Equation (22) shows that the terrestrial gravity data errors propagate into the geoid 8n  2, and are controlled by the coecients † 2= n 1†Qn wo† ; see Fig. 1a, b. Therefore, even if the geopotential model is error-free, the inclusion of terrestrial gravity data via Eq. (6) allows low- frequency (2  n  M) terrestrial gravity anomaly errors to leak into the solution and thus degrade the geoid in this region.

3 The spheroidal Stokes kernel

Fig. 1. a The generic 2= n 1† term 8n  2(solid line), the spherical The spherical Stokes kernel, Eq. (2), used in Eq. (6) is now  truncation coecients Qn 6 † for 2  n  360 (dashed line), and their replaced by its spheroidal counterpart. This is referred to  di€erence † 2= n 1†Qn 6 † for n  2(dotted line). b The  as the spheroidal Stokes kernel by Vanõ cÏ ek and Kleusberg normalised spherical truncation coecients n 1†=2†Qn 6 †† for Â Ï 2  n  360 (dashed line), and their normalised di€erence (1987) and Vanõcek and SjoÈ berg (1991). In this approach,  the low-degree Legendre polynomials (2  n  M) are n 1†=2† †2= n 1†Qn 6 † for n  2(dotted line) removed from the spherical Stokes kernel to yield

XM 2n ‡ 1 SM w†ˆS w† P cos w† n 1 n X1 2 nˆ2  ˆ c T 20† 23† N1 n 1 n X1 2n ‡ 1 nˆ2 ˆ P cos w† n 1 n Equation (20) shows that when terrestrial gravity data are nˆM‡1 used over the whole Earth in Eq. (6), the resulting error in Using this spheroidal kernel gives the estimate of the the geoid is controlled entirely by the generic 2= n 1† geoid as term, as if the original Stokes formula, Eq. (1), had been Z Z used over the whole Earth [cf. Eqs. (20) and (3)]. Note also 2p wo M M that the truncation error reduces to zero in this instance, N2 ˆ NM ‡ j S w†Dg sin wdwda 24† as would be expected. Curiously, this result is equivalent 0 0 to completely disregarding the geopotential model during Note that in Eq. (24), the degree of the spheroidal kernel the geoid computations. has been chosen to equal the degree by which the In the case of no Stokes integration [i.e. wo ˆ 0in terrestrial gravity anomalies have been reduced Eq. (6)], the spherical truncation coecients attain their [Eq. (7)]. In this discussion, the spheroidal kernel is used 2 maximum value of Qn 0†ˆn1, 8n  2. Again, this can as the intrinsic kernel for the combination of a global be proven from Eq. (14) in conjunction with the or- geopotential model of degree M with terrestrial gravity thogonality relations between Legendre polynomials data (cf. Vanõ cÏ ek and SjoÈ berg 1991). Thus, the spheroidal over the sphere. Thus, Eq. (19) becomes kernel should not be regarded as a modi®ed kernel. 688 Z However, this does not preclude the use of a di€erent p QM w †ˆ SM w†P cos w† sin w dw degree L  M† in Eq. (23), which could then be n o n w considered as a spheroidally modi®ed kernel (e.g. Wong o Z p XM 2k ‡ 1 31† and Gore 1969). ˆ Q w † P cos w† Using an approach similar to that applied to Eq. (2), n o k 1 k wo kˆ2 the spheroidal Stokes kernel, Eq. (23), is rewritten as  P cos w† sin wdw  n M M S w† for 0  w  wo The relations given by Eqs. (23), (25) and (31) are then S w; wo†ˆ 25† 0 for wo < w  p used in Eq. (30) to yield Z or as an in®nite Fourier series of Legendre polynomials by X1 2n ‡ 1 p sM w †ˆ ‰P cos w†Š2 sin w dw n o n 1 n X1 nˆM‡1 0 M 2n ‡ 1 M S w; w †ˆ s w †Pn cos w† 26† M o 2 n o Qn wo† nˆ2 or M where sn wo† are coecients associated with the spheroi-  M dal kernel, which will be determined later in this section. M Qn wo† for 2  n  M sn wo†ˆ 2 M 32† Using the de®nition in Eq. (25), the spheroidal n1 Qn wo† for M < n < 1 Stokes integral in conjunction with the geopotential model, Eq. (24), is now given by the following integra- when using the orthogonality relations. tion over the whole Earth. Inserting this result in Eq. (29), together with the de- gree-errors in the geopotential model-derived gravity Z 2pZ p anomalies (G) and terrestrial gravity anomalies (T ), gives M M n n N2 ˆ NM ‡ j S w; w †Dg sin w dw da 27† " o M  0 0 X 2 ÈÉ N^ ˆ c ‡ QM w † DgG ‡ G 2 n 1 n o n n Substituting Eqs. (5) and (26) in Eq. (27), then using the nˆ2 # orthogonality relations over the sphere, gives its spectral ÈÉ M T T 33† form as Qn wo† Dgn ‡ n  XM 2 XM X1 G M M 2 M T T N2 ˆ c Dg ‡ c s w †Dg ‡ c Q w † Dg ‡  n 1 n n o n n 1 n o n n nˆ2 nˆ2 28† nˆM‡1 X1 M M Again, the superscripts G and T are retained to show the ‡ c sn wo†Dgn nˆM‡1 origin of the gravity anomalies used. A comparison of Eq. (33) with Eq. (3) indicates the error in the geoid that which is analogous with Eq. (11). now occurs when using the spheroidal Stokes integral in The estimates of the spectral components (surface conjunction with a global geopotential model [Eq. (24)]. spherical harmonics) of the gravity anomalies implied by This is the geopotential model (Dg^G) and the terrestrial gravity  n XM 2 Á data (Dg^T ) are inserted into Eq. (28) to give  ˆ c G ‡ QM w † G T n N2 n 1 n n o n n  nˆ2  XM 2 X1 ^ M G M T 2 T M T M T N2 ˆ c s wo† Dg^ ‡ s wo†Dg^ ‡ c  Q w †Dg Q w † n 1 n n n n n 1 n n o n n o n nˆ2 29† nˆM‡1 X1 M T 34† ‡ c sn wo†Dg^n nˆM‡1 As was done for Eq. (19), Eq. (34) is divided among For the spheroidal Stokes integral, Eq. (24), the propa- the error contributions from the geopotential model, the gation of errors into the geoid from the terrestrial gravity terrestrial gravity data, and the truncated integration data and geopotential model are now governed by the domain, respectively, to give coecients sM w †. These are therefore derived by  n o XM 2 rearranging Eq. (26), in conjunction with the orthogo-  ˆ c ‡ QM w † G N2 n 1 n o n nality relations, to give nˆ2 M 1  Z p X X 2 M M T M T M c Qn wo†n ‡ c Qn wo† n sn wo†ˆ S w; wo†Pn cos w† sin w dw 30† n 1 0 nˆ2 nˆM‡1 X1 M T The spheroidal truncation coecients [cf. Eq. (14)] are c Qn wo†Dgn 35† de®ned by Vanõ cÏ ek and Kleusberg (1987) as nˆM‡1 689

Figure 2 shows the numerical values of the coe- the geoid with a greater magnitude for the spheroidal cients in Eq. (35) that control the propagation of the kernel than for the spherical kernel. This provides the G degree-errors from the geopotential model (n ) and the rationale for using a low-degree geopotential model T terrestrial gravity anomalies (n ). All coecients have when computing the geoid. been normalised by the n 1†=2 term. Again, the val- 2. Conversely, the low- and medium-frequency errors in ues of M ˆ 360 and wo ˆ 6° have been chosen to show a the terrestrial gravity data are attenuated to a greater representative case of the coecients' behaviour, and to extent when using the spheroidal kernel than for the allow for a direct comparison with Figs. 1b and 4. spherical Stokes kernel. This is because An important point to observe from Fig. 2 is that the M numerical values of the Q wo† coecients become n 1 n 1 n QM w † < 1 Q w † unstable in the vicinity of M ˆ 360. This is a conse- 2 n o 2 n o quence of the Gibbs phenomenon, where the spherical for most degrees in the region 2  n < 320. As such, Stokes kernel is made discontinuous by the removal of the low-frequency errors in the terrestrial gravity data the Legendre polynomials. This e€ect is actually quite pass into the geoid with a lower magnitude for the small, but has been enhanced by the normalisation of spheroidal kernel than for the spherical kernel. n 1†=2 used purely for the sake of the presentation of the ®gures. It is argued that this combination o€ers a preferable The truncation error [the last term on the right-hand scenario because the low-frequency errors in the geo- side of Eq. (35)] must clearly also be considered when potential model are expected to be smaller in magnitude the spheroidal Stokes integral, Eq. (24), is used to than those in the terrestrial gravity anomalies. This is compute the geoid. However, the impact of this term is because the low-frequency geopotential coecients are M determined from the perturbations of arti®cial Earth reduced because jQn wo†j < jQn wo†j for most of the degrees n > 360 (cf. Figs. 1b, 2, and also see Fig. 3). satellites, which are the best source of this information Therefore, one of the bene®ts of using the spheroidal available. On the other hand, terrestrial gravity data Stokes kernel is that it reduces the magnitude of the su€er from the existence of inconsistencies between truncation error. vertical datums, systematic errors in geodetic levelling There are also other di€erences between Eqs. (35) networks used to reduce the gravity observations, and and (19), which can be seen by comparing Figs. 1b and irregular data coverage with some large areas devoid of 2, as follows. data. These and other systematic, low-frequency e€ects on the accuracy of terrestrial gravity anomalies are de- 1. The propagation of errors from the geopotential scribed in more detail by Heck (1990). G model (n ) is attenuated less than for the spherical kernel because

n 1 n 1 3.1 Special cases of Eq. (35) 1 ‡ QM w † > Q w † 2 n o 2 n o for most degrees in the region 2  n <  320. As As was done for Eq. (19), three special cases of Eq. (35) such, the errors pass from the geopotential model into are considered as follows. When the spheroidal Stokes

Fig. 3. The normalised truncation coecients in the region  Fig. 2. The normalised spheroidal truncation coecients 360  n  720 for the spherical Stokes kernel Qn 6 † (solid line), 360  360  Án 1†=2†Qn 6 † for 2  n  360 (dashed line), the sum the spheroidal Stokes kernel Qn 6 † (dotted line)andtheL ˆ 20 360  360à  1 ‡ n Á1†=2†Qn 6 † for 2  n  360 (dotted line), and the Molodensky-modi®ed spheroidal Stokes kernel Qn 6 † (dashed 360  di€erence 1 n 1†=2†Qn 6 † for n > 360 (solid line) line) 690 integration is performed over the whole Earth [i.e. wo ˆ p for most degrees in the region 2  n < 320. This is in Eq. (24)], the spheroidal truncation coecients evident by comparing Figs. 1b and 2, and again illustrates M Qn wo†ˆ0, 8n  2, and Eq. (35) becomes that it is preferable to use a spheroidal Stokes's integral in the presence of low-frequency errors in the terrestrial XM 2 X1 2 gravity data (see earlier arguments).  ˆ c G ‡ c T 36† N2 n 1 n n 1 n nˆ2 nˆM‡1 4 The Molodensky-modi®ed spheroidal Stokes kernel This result shows that the errors in the geopotential model and the terrestrial gravity data both pass undiminished Equation (6) is now used in conjunction with the into the geoid, for the low and high frequencies, respec- Molodensky-modi®ed spheroidal Stokes kernel [SMà w†] tively. When Eq. (36) is compared with Eq. (20), this according to the generalised scheme described by Vanõ cÏ ek clearly shows the high-pass ®ltering property of the and SjoÈ berg (1991), such that spheroidal kernel, where the spheroidal kernel is insensi- Z Z 2p wo tive to the low frequencies in the terrestrial gravity ®eld. Mà M However, this ability of the kernel to completely ®lter out N3 ˆ NM ‡ j S w†Dg sin w dw da 39† the low frequencies breaks down when using a spherical 0 0 M Mà cap of limited radius wo due to the presence of the Qn wo† where S w† is the Molodensky-modi®ed spheroidal coecients in Eq. (35). Stokes kernel given by Vanõ cÏ ek and Kleusberg (1987). As argued earlier, the low-frequency errors in the This type of kernel modi®cation was designed speci®cally geopotential model are expected to be smaller than those to reduce the upper bound of the truncation error in a in the terrestrial gravity anomalies. This provides a clear least-squares sense (Vanõ cÏ ek and SjoÈ berg 1991). rationale for the use of at least a low-degree spheroidal The degree of modi®cation to the kernel (L) can be Stokes kernel, such that the undesirable e€ect of low- less or equal to the degree of spheroid used (M). frequency errors in the terrestrial gravity data on the However, it can never be greater than the degree of the geoid is suppressed as much as possible. spheroid, since this violates the boundary value prob- For the case of no integration [i.e. wo ˆ 0 in Eq. (24)], lem and thus requires the inclusion of an additional M the spheroidal truncation coecients Qn wo†ˆ0, term. This term will not be discussed in this paper. 8n  2, and Eq. (35) becomes Therefore, the degree of modi®cation (L  M) is in- troduced to give XM 2 X1 2  ˆ c G ‡ c DgT 37† XL 2k ‡ 1 N2 n 1 n n 1 n SMà w†ˆSM w† t w †P cos w† 40† nˆ2 nˆM‡1 2 k o k kˆ2 where the truncation error reaches its maximum value (i.e. where tk wo† are the coecients that are used to apply the it is equal to the omission error of the geopotential model), spheroidal Molodensky modi®cation. These are deter- and the error in the geopotential coecients again passes mined for 2  n  L using the following linear system of undiminished into the geoid. Equation (37) is identical to equations (Vanõ cÏ ek and Kleusberg 1987): Eq. (21) and is equivalent to using only the geopotential model to compute the geoid. XL 2k ‡ 1 t w †e w †ˆQL w † When an error-free geopotential model is considered 2 k o nk o n o G kˆ2 (i.e. n ˆ 0 for 2  n  M), together with the spheroidal Stokesian integration over a limited spherical cap bound XL 2k ‡ 1 ˆ Qn wo† enk wo† by wo, then Eq. (35) becomes k 1 kˆ2  XM X1 2 41†  ˆ c QM w † T c QM w † T N2 n o n n 1 n o n where the coecients nˆ2 nˆM‡1 Z X1 p M T enk w †ˆ Pn cos w†Pk cos w† sin w dw 42† c Q wo†Dg 38† o n n w nˆM‡1 o Also note that in the region 2  n  L; Eq. (41) is equal to In this result, the low-frequency errors in the terrestrial Eq. (31). It is therefore acknowledged that the expression gravity data also leak into the low-frequency geoid. for the tk wo† coecients shown by Vanõ cÏ ek and SjoÈ berg However, their e€ect is now attenuated by the coecients (1991) has a typographical error in the denominator, M Qn wo†. This represents a greater amount of attenuation where 2 is used instead of k 1†. However, the de®nition than for the spherical Stokes integral [cf. Eqs. (38) and derived herein and the original de®nition shown in (22)] because Vanõ cÏ ek and Kleusberg (1987) are correct. As for the previous two implementations of Stokes's n 1 n 1 QM w † < 1 Q w † integral, the Molodensky-modi®ed spheroidal Stokes 2 n o 2 n o kernel in Eq. (40) is rewritten as 691  Mà SMà w† for 0  w  w Using the de®nition in Eq. (23) and the orthogonality S w; w †ˆ o 43† o relations to replace the integral terms in Eq. (49), then 0 for wo < w  p separating the low- (2  n  L), medium- (L < n  M) and the corresponding Fourier series of Legendre poly- and high-frequency (M  n < 1) components, gives nomials is 8 t w †QMà w † for 2  n  L X1 > n o n o Mà 2n ‡ 1 Mà < S w; wo†ˆ sn wo†Pn cos w† 44† sMà w †ˆ QMà w † for L < n  M 2 n o > n o nˆ2 :> 2 QMà w † for M < n < 1 Mà n1 n o where sn are the coecients associated with this modi®ed kernel, and will be de®ned later in this section. 50† Using the same procedures as for the spherical and By de®nition, the Molodensky-modi®ed spheroidal spheroidal Stokes kernels, Eq. (39) is transformed to a Mà truncation coecients Qn wo† are zero in the region spectral form, then the estimates of the gravity anoma- 2  n  L [cf. Eqs. (41) and (47) herein, or Eqs. (38) and G lies as implied by the geopotential model (Dg^n )and (22) of Vanõ cÏ ek and SjoÈ berg (1991)]. Therefore, the terrestrial gravity data (Dg^T ) are inserted, to give Mà n sn wo† coecients are given by  XM h 2 8 N^ ˆ c sMà w † Dg^G > tn wo† for 2  n  L 3 n 1 n o n < nˆ2 Mà Mà i sn wo†ˆ Qn wo† for L < n  M 51† Mà T > ‡ sn wo†Dg^n 45† : 2 Mà n1 Qn wo† for M < n < 1 X1 Mà T Using this result and dividing the contribution among the ‡ c sn wo†Dg^n nˆM‡1 three frequency bands, Eq. (45) becomes

Equation (45) shows that the propagation of errors " # XL 2 ^ G T through Eq. (39) is now controlled by the coecients N3 ˆ c ‡ tn w † Dg^ tn w †Dg^ Mà n 1 o n o n sn wo†. Therefore, these coecients are derived by re- nˆ2 arranging Eq. (40) in conjunction with the orthogonality " # relations to give XM 2 ‡ c ‡ QMà w † Dg^G QMà w †Dg^T Z n 1 n o n n o n p nˆL‡1 Mà Mà sn wo†ˆ S w; wo†Pn cos w† sin w dw 46†  0 X1 2 ‡ c QMà Dg^T 52† n 1 n n The Molodensky-modi®ed spheroidal truncation coe- nˆM‡1 Mà cients [Qn wo†] are de®ned by Vanõ cÏ ek and Kleusberg (1987) as Next, the degree-errors in the geopotential model- G Z derived gravity anomalies (n ) and the terrestrial gravity p T Mà Mà anomalies (n ) are introduced into Eq. (52), the terms Qn wo†ˆ S w†Pn cos w† sin w dw 47† w expanded, and Eq. (3) subtracted. As before, the error is o then divided among the contribution from the geopo- [cf. Eqs. (14) and (31)]. Inserting Eqs. (47) and (43) in tential model, the terrestrial gravity data and the trun- Eq. (46) gives cation error. This leaves the error in the geoid that Z p occurs when using the Molodensky-modi®ed spheroidal Mà Mà Mà Stokes formula [Eq. (39)] as sn wo†ˆ S w†Pn cos w† sin w dw Qn wo† 0  48† XL 2  ˆ c ‡ t w † G N3 n 1 n o n Then, substituting Eq. (40) for the Molodensky-modi®ed nˆ2  Stokes kernel, recalling that L  M, gives XM 2 ‡ c ‡ QMà w † G Z n 1 n o n p nˆL‡1 sMà w †ˆ SM w†P cos w† sin w dw n o n XL XM 0 c t w †T c QMà w †T 53† Z n o n n o n XM p nˆ2 nˆL‡1 2k ‡ 1  tk w0†Pk cos w†Pn cos w† X1 2 2 Mà T kˆ2 0 ‡ c Q w †  n 1 n o n nˆM‡1  sin w dw X1 c QMà w †DgT Mà n o n Qn wo† 49† nˆM‡1 692

The last term on the right-hand side of Eq. (53) n 1 n 1 QMà w † < 1 Q w † comprises the truncation error. Its impact has been 2 n o 2 n o minimised by Vanõ cÏ ek and Kleusberg (1987) according to the Molodensky et al. (1962) argument. This is be- (cf. Figs. 1b, 4). Similarly, the Molodensky-modi®ed Mà M spheroidal kernel attenuates more of the geopotential cause jQn wo†j < jQn wo†j < jQn wo†j for most of the degrees n > 400 (see Fig. 3), and is supported further model error because by the numerical results shown later in Table 1. n 1 n 1 QMà w † < 1 ‡ QM w † Figure 4 shows the behaviour of the coecients that 2 n o 2 n o control the error propagation when using the Mo- lodensky-modi®ed spheroidal Stokes kernel [Eq. (40)]. (cf. Figs. 2, 4).  When considering the propagation of medium-fre- The values of wo ˆ 6 and M ˆ 360 have been used for consistency, and the value of L ˆ 20 has been used for quency terrestrial gravity data errors (21  n  360), the the degree of Molodensky modi®cation. The rationale Molodensky-modi®ed spheroidal kernel attenuates for this degree of kernel modi®cation has already been more error than the spherical Stokes kernel. This can be given by Vanõ cÏ ek and Kleusberg (1987), and will not be seen by comparing Figs. 1b and 4. Conversely, the duplicated here. Theoretically, however, it does not Molodensky-modi®ed spheroidal kernel attenuates less matter what degree of kernel modi®cation is used to error than the spheroidal Stokes kernel, because Mà M produce Fig. 4, provided that L is never greater than M. jQn wo†j > jQn wo†j for most degrees in the region From Fig. 4, the low-frequency (2  n  20) errors in 21  n < 320 that is una€ected by the Gibbs phe- G nomenon in the spheroidal kernel (cf. Figs. 2, 4). A the geopotential model (n ) now pass into the geoid with a slightly greater magnitude than for the spherical kernel numerical comparison of these e€ects will be given later because in the paper, that supports the statements made here. Another interesting point is that the Molodensky-

n 1 n 1 modi®ed spheroidal kernel is less a€ected by the Gibbs 1 ‡ t w † > Q w † 2 n o 2 n o phenomenon about M ˆ 360. This is because the ap- plication of the modi®cation makes the kernel a more (cf. Figs. 1b and 4 for 2  n  20). Conversely, the continuous function that is very much less sensitive to amount of error propagation is less than that associated this e€ect. with the spheroidal Stokes kernel because

n 1 n 1 1 ‡ t w † < 1 Q w † 2 n o 2 n o 4.1 Special cases of Eq. (52)

(cf. Figs. 2 and 4 for some of the degrees in the region When integration of Eq. (39) is performed over the whole Mà 2  n  20). Based on the earlier arguments, this is not Earth (i.e. wo ˆ p), the coecients tn wo†ˆQn wo†ˆ0, critical because the geopotential model is currently always 8n  2. Thus, Eq. (53) becomes the best source of low-frequency gravity ®eld information. XM 2 X1 2 Also from Fig. 4, the very low-frequency terrestrial  ˆ c G ‡ c T 54† gravity data errors are also attenuated more when using N3 n 1 n n 1 n nˆ2 nˆM‡1 the Molodensky-modi®ed spheroidal kernel compared to the spherical Stokes kernel. This is because which is an identical result to Eq. (36), where the errors in the geopotential model and terrestrial gravity data pass

n 1 n 1 undiminished into the geoid. Based on the earlier tn wo† < 1 Qn wo† T G 2 2 arguments that jn j > jn j for the very low frequencies, it is thus preferable to use a low-degree geopotential (cf. Figs. 1b and 4 for 2  n  20). However, the amount model to provide the low-frequency component of the of attenuation is not as great as that for the spheroidal geoid. Stokes kernel, because jtn wo†j > jQn wo†j (cf. Figs. 2 and In the case of no integration [i.e. wo ˆ 0 in Eq. (39)], Mà 2 4 for 2  n  M). This is a direct consequence of tn wo†ˆQn wo†ˆn1, 8n  2. This yields the same minimising the truncation error using the Molodensky result as Eqs. (21) and (37), speci®cally approach, and allows for more leakage of very low- XM 2 X1 2 frequency terrestrial gravity data errors into the geoid.  ˆ c G ‡ c DgT 55† Nevertheless, the amount of leakage is still much less than N3 n 1 n n 1 n that associated with the spherical Stokes kernel. nˆ2 nˆM‡1 Next, it is interesting to see what e€ect the L ˆ 20 Again, this is equivalent to using only the geopotential Molodensky modi®cation has on the medium frequen- model to compute the geoid. Therefore, if the integration cies (2  n  360) in the geopotential model and the is not used in each of the three cases discussed, it becomes terrestrial gravity data. Compared to the spherical ker- immaterial just what form of kernel is used. The second nel, the Molodensky-modi®ed spheroidal kernel atten- term on the right-hand side of Eq. (55) is the truncation uates more of the geopotential model error. This is error, which is identical to the omission error of the because geopotential model. 693

n 1 n 1 QMà w † < 1 ‡ QM w † 2 n o 2 n o (cf. Figs. 2 and 4 for 21  n < 360). This result is a bene®cial by-product of the Molodensky modi®cation, which was designed speci®cally to minimise the trunca- Mà tion error and hence Qn wo†.

5 How do terrestrial gravity data improve the geopotential model?

Now that the mathematical descriptors have been derived for the propagation of errors from the geopotential model G T (n ) and terrestrial gravity data (n ) into the geoid, the question originally posed in this paper can be addressed. Speci®cally, the claim that the terrestrial gravity data (DgT ) attenuate errors present in the geopotential model,  Fig. 4. The normalised Molodensky-modi®cation coecients tn 6 †  and to what extent this is true for each of the three kernels, for 2  n  20 (dot±dashed line), the sum † 1 ‡ n 1†=2†tn 6 † for 2 < n  20 (solid line), the normalised Molodensky-modi®cation will be considered. An empirical estimation of this e€ect  has already been considered by Sideris and Schwarz Ácoecients tn 6 † for 21  n  360 (dotted line), the sum 20à  1 ‡ n Á1†=2†Qn 6 † for 21  n  360 (dashed line), and the (1987) for the spherical Stokes kernel, but it is important 20à o di€erence 1 n 1=2Qn 6 † for n > 360 (solid line) to know how this compares with other implementations of Stokes kernel. When using an error-free geopotential model (i.e. G ˆ 0 for 2  n  M), and integration over a limited n 5.1 An error-free terrestrial gravity ®eld spherical cap, the error in the geoid is contaminated by the terrestrial gravity data errors in both the high and First, the performance of the three kernels is now low frequencies. This is given by investigated using the special case of an error-free T XL XM terrestrial gravity ®eld (i.e. n ˆ 0 for 2  n  M). The T Mà T truncation error term will be ignored, which is justi®ed N3 ˆc tn wo†n c Qn wo†n nˆ2 nˆL‡1 because this only a€ects the high-frequency geoid and  X1 2 thus gives no answer to the question in hand. Neverthe- ‡ c QMà w † T 56† less, the in¯uence of the truncation error is reduced by n 1 n o n nˆM‡1 both the spheroidal and Molodensky-modi®ed kernels, as X1 shown in Fig. 3, and this will be discussed further in the Mà T c Qn wo†Dgn next section. nˆM‡1 Inserting the above conditions in Eq. (19) gives the error originating from the geopotential model that re- In this special case, the very low-frequency errors in the mains in the geoid when computed by means of the terrestrial gravity data are attenuated to a greater extent spherical Stokes function. This is than for the spherical Stokes formula because XM n 1 n 1  ˆ c Q w †G 57† tn wo† < 1 Qn wo† N1 n o n 2 2 nˆ2 (cf. Figs. 1b and 4 for 2  n  20). However, the amount Equation (57) clearly shows that even error-free terrestrial of error attenuation is less than that experienced with the gravity data, when used in a cap, can never completely M spheroidal Stokes integral because jtn wo†j > jQn wo†j correct the errors present in the geopotential model. This (cf. Figs. 2 and 4 for 2  n  20). As stated, this is a is due to the presence of the spherical truncation consequence of minimising the truncation error. coecients Qn wo†. If terrestrial gravity data were to In the medium frequencies, the terrestrial gravity data completely correct the geopotential model, then the errors are attenuated more by the Molodensky-modi®ed Qn wo† coecients would have to be zero in the region spheroidal kernel than for the spherical or spheroidal 2  n  M. This is clearly not the case, as shown in Fig. 1, kernels. For the spherical kernel, this is because except for the points at which the coecient curve passes through zero. The Q w † coecients will only become n 1 n 1 n o QMà w † < 1 Q w † zero 8n if and only if w ˆ p (i.e. an integration over the 2 n o 2 n o o whole Earth). Moreover, the smaller the radius of (cf. Figs. 1b and 4 for 21  n  360). For the spheroidal integration (wo), the larger the magnitude of the spherical kernel, this is because truncation coecients, and thus the less the terrestrial 694 data can begin to account for any error in the geopotential rors in a geopotential model. This applies when they are model. used within a limited spherical cap surrounding the ge- When an error-free terrestrial gravity ®eld is used in oid computation point, due to the truncated integration. Eq. (35), and the truncation error neglected, the error However, given that the low-frequency errors in the originating in the geopotential model and remaining in terrestrial gravity data are expected to be larger than the the geoid when computed by the spheroidal Stokes low-frequency errors in the geopotential model, this kernel is given by raises some question as to the appropriateness of using the spherical Stokes kernel in Eq. (6). In the presence of  XM 2 low-frequency terrestrial gravity data errors, it is there-  ˆ c ‡ QM w † G 58† fore preferable to use one of the spheroidal kernels be- N2 n 1 n o n nˆ2 cause they can attenuate a larger amount of these errors. Once again, Eq. (58) shows that the terrestrial gravity data, when used in a cap, cannot completely correct the low-frequency errors in the geopotential model. In this 5.2 Numerical comparisons of the kernel ®ltering case, the errors generated by the geopotential model properties remain, almost unaltered, in the geoid because the M Next, it is informative to quantify just how e€ective each coecients Qn wo† << 2= n 1† (see Fig. 2 for 2  n < 320). In fact, the errors are diminished more form of Stokes's kernel is at correcting the errors in the e€ectively close to M ˆ 360 courtesy of the Gibbs geopotential model. In order to answer this question phenomenon. However, this must be balanced against properly, the error characteristics of each of the two G the e€ect that the Gibbs phenomenon will have on the sources of information (i.e. n for the geopotential model T propagation of the gravity signal into the geoid. and n for the terrestrial gravity data) would have to be T Equation (58) also illustrates the (partial) high-pass known. Unfortunately, however, n are largely unknown ®ltering property of the spheroidal Stokes kernel, where for the terrestrial data. Of course, some error character- the contribution of the terrestrial gravity data to the istics could be postulated, but any such postulation would geoid is attenuated according to the coecients be speculative and subject to criticism and maybe even M controversy. Instead, a less controversial approach is (2= n 1†‡Qn wo†). If the terrestrial gravity data were to completely correct the geopotential model, then the taken whereby the M ˆ 360 geopotential model is divided M into three frequency bands, as follows. Qn wo† coecients would have to be equal to 2= n 1†, which again can only be satis®ed by a global 1. Spherical harmonic degrees 2  n  20, for which the integration. geopotential model is considered to be the best source When using an error-free terrestrial gravity ®eld in of gravity ®eld information (because of the satellite- the Molodensky-modi®ed spheroidal Stokes integral, derived contribution) and the terrestrial data are a (Eq. 40), and again neglecting the truncation error, the much poorer source of this information (for the rea- geoid error originating from the geopotential model, sons explained earlier). Eq. (53), becomes 2. Spherical harmonic degrees 20 < n  120, for which  the geopotential model gives a reasonable accuracy XL 2 almost everywhere on Earth, and where the terrestrial  ˆ c ‡ t w † G N3 n 1 n o n data may o€er an improvement in certain parts of the nˆ2  59† world, only if these data are of good quality. XM 2 3. Spherical harmonic degrees 120 < n  360, for which ‡ c ‡ QMà w † G n 1 n o n the geopotential model may not be the best source of nˆL‡1 gravity ®eld information, and an improvement from Equation (59) also shows that the terrestrial gravity data terrestrial data should thus be sought. The degrada- from within a cap cannot completely correct the low- tion of the geopotential model in this region can be frequency errors in the geopotential model. In this case, seen from the error degree variances which are usually the error in the geoid propagating from the geopotential greater than 50% of the signal strength. model is diminished more than for the spheroidal Stokes The performance of each of the three kernels discussed kernel, because jt w †j < jQM w †j (cf. Figs. 2 and 3 for n o n o here is investigated separately for each of the above 2  n  20) and jQMà w †j < jQM w †j (cf. Figs. 2 and 3 n o n o frequency bands. However, an appropriate measure of the for 21  n 320). The same arguments apply to the relative performance of each kernel must be devised. spherical kernel. Once again, this illustrates that the Firstly, the following general criteria should be satis®ed. Molodensky-modi®ed spheroidal Stokes kernel attenu- ates the low frequencies in the terrestrial gravity data. (a) The total contribution of all errors should be as small However, the high-pass ®lter is imperfect and some low- as possible, frequency errors leak into the low frequencies of the geoid. (b) Whatever the errors in the geopotential model may This is a price that must be paid for the minimisation of the be, how much of the error in the terrestrial gravity truncation error. data is going to leak into the combined solution, It is clear from these three very special cases that even (c) Conversely, whatever the errors are in the terrestrial error-free terrestrial gravity data cannot correct the er- gravity data may be, how much of the error in the 695

geopotential model is going to remain in the combined The norm adopted in this study is the Euclidean solution. norm, which is given by the following expression: v Given this, the question ``to what extent can terres- u uXj trial gravity data correct errors in a high-degree geo- kfkˆt f 2 63† potential model?'' should be posed in terms that can n nˆi more readily be translated into mathematical expres- sions. Therefore, the question is rephrased as ``which where the summation is taken over the j i spherical G kernel gives the least N and the best balance between n harmonic degrees within each space. Speci®cally, for T and n for the 2  n  360 contribution to N ?''. This F1; i ˆ 2 and j ˆ 20, for F2; i ˆ 21 and j ˆ 120, and for formulation automatically eliminates the truncation er- F3; i ˆ 121 and j ˆ 360. Other choices of norm are, of ror from the following deliberations. It also obviates the course, available. T investigation of the e€ect of n for n > 360. Neverthe- In order to be able to compare the adopted norms less, the truncation error will be brie¯y discussed at the among the three spaces (F1, F2 and F3), Eq. (63) should end of this section. be standardised by j i (the dimension of each space). A plausible way of measuring the contribution of the This gives T terrestrial data errors (n ) and geopotential model errors v G u ( ) is now devised, given that these errors are not ac- j n 1 uX tually known. The error (à ) in the ®rst M ˆ 360 terms kfkà ˆ t f 2 64† N j i n of the combined geoid can be written as nˆi XM XM An alternative way of interpreting the standardised norm à ˆ c f GG ‡ c f T T N n n n n 60† in Eq. (64) is that it gives the average value of the upper nˆ2 nˆ2 bound of the degree contribution between i and j when G G T T T G ˆ chf ; i‡chf ; i multiplied by a degree-error (i.e. n or n for i  n  j). Next, the expressions for the individual ®lter values where f is the vector of ®lter values fn which multiplies the are recalled for the three kernels. For the spherical vector of the (unknown) error values n in either the Stokes kernel, Eq. (57) yields terrestrial data (T ) or geopotential model (G), and h; i denotes a scalar product in the space F of real functions 8n ˆ 2; ...; M : f G ˆ Q w †; n n o  de®ned for integer arguments in the region 2  n  M (i.e. 65† G T T 2 F 2f 2; ...; M†!

6 Conclusions Acknowledgements. The ®rst author is funded by an NSERC op- erating grant, and the second author is funded by Australian Re- From the theoretical developments and numerical inves- search Council grant AA49331318 and a sabbatical leave allowance from Curtin University of Technology; all are gratefully ac- tigations shown in the present paper, it is clear that the knowledged. The authors would also like to thank ZdeneÏ k Mar- question as to whether terrestrial gravity data can correct tinec for providing an elegant program for computing the a global geopotential model has been answered with a truncation coecients. Finally, thanks are extended to the re- ``no''. This applies to the spherical, spheroidal and viewers (Erik de Min and Lars SjoÈ berg) and the editor (Martin Molodensky-modi®ed spheroidal kernels to a varying Vermeer) for their time taken to consider this manuscript. degree. Of note, the spherical kernels allows low-frequen- cy terrestrial gravity data errors to pass, almost undimin- ished, into the geoid. The spheroidal and spheroidal References Molodensky kernels attenuate these errors to a greater extent, with the spheroidal kernel being the most e€ective Featherstone WE, Evans JD, Olliver JG (1998) A Meissl-modi®ed Vanõ cÏ ek and Kleusberg kernel to reduce the truncation error in attenuator. The Molodensky-modi®ed spheroidal kernel geoid computations. J Geod 72(3): 154±160 is slightly less e€ective than the spheroidal kernel because Heck B (1990) An evaluation of some of the systematic error this is the price that must be paid for minimising the sources a€ecting terrestrial gravity anomalies. Bull Geod 64: 88± truncation error. However, this must be balanced against 108 the artefact introduced by the Gibbs phenomenon that Heck B, GruÈ ninger W (1987) Modi®cation of Stokes's formula by a€ects the spheroidal kernel. combining two classical approaches. Proc XIX IUGG general assembly, Vancouver, pp 299±337 It should be pointed out that these results only hold Heiskanen WH, Moritz H (1967) Physical geodesy. Freeman, San for the case when terrestrial gravity data are used in a Francisco spherical cap about the computation point, as is indeed Meissl P (1971) Preparations for the numerical evaluation of sec- customary in modern regional geoid computation. If the ond-order Molodensky-type formulas. Report 163, Dept of integration is performed over the whole Earth with any Geodetic Science and Surveying, The Ohio State University, Stokes-type kernel and error-free terrestrial gravity data, Columbus, Ohio Molodensky MS, Eremeev VF, Yurkina MI (1962) Methods for then the geopotential model is always completely cor- study of the external gravitational ®eld and ®gure of the earth. rected. This holds because each approximation reduces Israeli Programme for the Translation of Scienti®c Publications, to the original Stokes integral. Jerusalem An important result is that the spheroidal and mod- Paul MK (1973) A method of evaluating the truncation error co- i®ed Stokes kernels o€er a preferable approach to the ecients for geoidal height. Bull Geod 47: 413±425 spherical Stokes kernel in the presence of low-frequency Sideris MG, Schwarz K-P (1987) Improvement of medium and short wavelength features of geopotential solutions by local terrestrial gravity data errors. These are strongly sus- gravity data. Boll Geod Sci A€ XLVI(3): 207±221 pected to exist because terrestrial gravity data su€er SjoÈ berg LE (1984) Least squares modi®cation of Stokes's and from the existence of vertical datum inconsistencies be- Vening Meinesz's formulas by accounting for the truncation tween countries, systematic errors in geodetic levelling and potential coecient errors. Manuscr Geod 9: 209±229 networks used to reduce the gravity observations, and SjoÈ berg LE (1986) Comparison of some methods of modifying irregular data coverage with some large areas devoid of Stokes's formula. Boll Geod Sci A€ XLV(3): 229±248 SjoÈ berg LE (1991) Re®ned least squares modi®cation of Stokes's data. On the other hand, the low-frequency potential formula. Manuscr Geod 16: 367±375 coecients that de®ne the geopotential model are de- Vanõ cÏ ek P, Kleusberg A (1987) The Canadian geoid ± Stokesian termined from the perturbations of arti®cial Earth sat- approach. Manuscr Geod 12: 86-98 ellite orbits, which are the best available source of this Vanõ cÏ ek P, SjoÈ berg LE (1991) Reformulation of Stokes's theory for information. Therefore, because these low-frequency higher than second-degree reference ®eld and modi®cation of errors are not completely removed during the compu- integration kernels. J Geophys Res 96(B4): 6529±6539 Wong L, Gore R (1969) Accuracy of geoid heights from modi®ed tation of residual gravity anomalies, it is sensible to use a Stokes kernels. Geophys J R Astr Soc 18(1): 81±91 low degree of modi®ed spheroidal kernel in order to attenuate the detrimental e€ects of these low-frequency errors in terrestrial gravity data.