Performance of Three Types of Stokes's Kernel in the Combined Solution for the Geoid

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 j S wDg sin w dw da 1 geopotential model, it is sometimes implied that the 0 0 terrestrial gravity data correct any erroneous wave- where j 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 P cos w 2 n 1 n cannot completely correct the geopotential model. Using n2 a standard norm, it is shown that the spheroidal and Molodensky-modi®ed integration kernels oer 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 c Dg 3 n 1 n fers the best source of this information. n2 where c 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 coecients that de®ne a global geopotential model, such that GM a n Xn DgG n 1 dC cos mk S sin mk n r2 r nm nm m0 1 Introduction Â 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 coecients 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. coecients to the geoid as 685 XM 2 high-frequency gravity anomalies (DgM ), which are ter- N c DgG T M n 1 n restrial gravity anomalies (Dg ) that have been reduced by n2 the gravity anomalies implied by a geopotential model G GM XM a nXn 5 (Dg ) of spherical harmonic degree and order (M) dC cos mk rc r nm according to n2 m0 XM S sin mkP cos h M T G nm nm Dg Dg Dgn 7 n2 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 VanõÂ cÏ ek and SjoÈ berg to achieve this in a straightforward manner, the spherical (1991). Such an approach reduces the eect of not using a Stokes kernel, Eq. (2) as used in Eq. (6) is expressed as global coverage of terrestrial gravity data and also reduces the impact of the spherical approximations inher- S w for 0 w w S w; w 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 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 n2 and Schwarz 1987). This assertion is examined in this where sn wo are coecients 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 (VanõÂ 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 NM j S w; woDg sin w dw da 10 Heck and GruÈ ninger (1987), VanõÂ 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 c DgG c s w DgM quency domain, in terms of surface spherical harmonics, 1 n 1 n n o n n2 n2 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 woDgn geopotential model to a limited extent. It will also be nM1 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 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^ c Dg^ c s w Dg^ Dg^ o 1 n 1 n n o n n N N j S wDgM sin w dw da 6 n2 n2 1 M 12 0 0 X1 T where the Stokes integration is performed over a limited c sn woDg^n nM1 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- ecients 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 S w; w P cos w sin w dw 13 2 T G n o o n N1 c Qn w c Qn w 0 n 1 o n o n n2 n2 19 Introducing the spherical (Molodensky et al.

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