Journal of Geodesy (1999) 73: 10 ± 22

An analysis of vertical de¯ections derived from high-degree spherical harmonic models

C. Jekeli Department of Civil and Environmental Engineering and Geodetic Science, The Ohio State University, 2070 Neil Ave, Columbus, OH 43210, USA e-mail: [email protected]; Tel.: +1 614 292 7117; Fax: +1 614 292 2957

Received: 9 December 1997 / Accepted: 21 August 1998

Abstract. The theoretical di€erences between the Hel- from the obvious limited resolution imposed by the mert de¯ection of the vertical and that computed from a degree of truncation, vertical de¯ections from spherical truncated spherical harmonic series of the gravity ®eld, harmonic models di€er, in principle, from their geomet- aside from the limited spectral content in the latter, ric counterpart mostly because of the curvature of the include the curvature of the normal plumb line, the normal plumb line. This is a systematic e€ect that is permanent tidal e€ect, and datum origin and orientation usually small, but may be signi®cant at very high ¯ight o€sets. A numerical comparison between de¯ections altitudes: for example, if the spherical harmonic models derived from spherical harmonic model EGM96 and are used to compensate inertial navigation systems astronomic de¯ections in the conterminous United (INS) for the e€ects of gravitation. A rigorous compar- States (CONUS) shows that correcting these systematic ison is presented here that identi®es this and all other e€ects reduces the mean di€erences in some areas. di€erences, including the question of permanent tide Overall, the mean di€erence in CONUS is reduced from e€ects. Subsequently, a numerical comparison between 0:219 arcsec to 0:058 arcsec for the south±north astronomical and gravimetric de¯ections is used to de¯ection, and from ‡0:016 arcsec to ‡0:004 arcsec for quantify these e€ects for the conterminous United the west±east de¯ection. Further analysis of the root- States and to evaluate the high-degree spectrum of the mean-square di€erences indicates that the high-degree EGM96 spherical harmonic model. spectrum of the EGM96 model has signi®cantly less power than implied by the de¯ection data. 2 The geometric de¯ection of the vertical

Key words. De¯ection of the vertical Á Spherical The de¯ection of the vertical is an angle that describes harmonic model Á Normal plumb-line curvature the deviation of the true vertical, as de®ned by the direction of Earth's gravity vector, with respect to some reference direction. The reference direction may be de®ned purely geometrically or physically. The classic geometric de®nition of the reference direction identi®es it as the perpendicular to an ellipsoid (a geodetic datum) 1 Introduction that approximates the geoid, either locally or globally. Alternatively, the reference direction may be associ- High-degree spherical harmonic models of the Earth's ated with the direction of a gravity vector belonging to geopotential, whose development is motivated primarily some reference (or normal) gravity ®eld; this is the by needs for accurate global geoid modeling, may also physical de®nition. The normal gravity ®eld is de®ned to be used to compute other quantities, such as the be that ®eld generated by an ellipsoid containing all the de¯ection of the vertical. The ampli®cation of errors in Earth's mass (including atmosphere) and rotating with such computations coming from inaccurate high-degree the Earth, and such that the ellipsoid is an equipotential coecients will not be considered here; this may be surface of the ®eld thus generated. Furthermore, the studied independently in the context of ``ill-posed'' ellipsoid should be a best approximation of the geoid problems. On the other hand, such gravimetric de¯ec- and should have its center at the Earth's center of mass. tions di€er not only quantitatively, but also in de®ni- The geometric ellipsoid (geodetic datum) and the el- tion, from geometric de¯ections of the vertical. Aside lipsoid of the reference gravity ®eld may be identical. 11

However, in general, they di€er in any or all of the where U; K† are astronomic coordinates (latitude and following: size, shape, location, and orientation. In that longitude) of the point and /; k† are the corresponding case, standard formulas exist to transform de¯ections of geodetic coordinates. In Eq. (1), it is assumed that the the vertical from one to the other geometric reference north poles of the astronomic and geodetic coordinate directions (see e.g. Heiskanen and Moritz 1967, p. 207; systems coincide. Rapp 1992, p. 85). The de¯ection components are small angles, on the The de®nitions of the de¯ection of the vertical are order of several arcseconds; and the higher-order terms described in the following paragraphs (Torge 1991; see in Eq. (1) refer to terms proportional to powers of the also Fig. 1). The Helmert de®nition is perhaps the most de¯ection components. Neglecting the second- and common and corresponds to the classic geometric de®- higher-order terms, the de®nitions of Eq. (1) conform nition alluded to above. According to this de®nition, the also to the astronomically determined de¯ection, redes- vertical de¯ection is the angle at a point, P, between the ignated as gravity vector that, extended upward, de®nes the as- tronomic zenith, and a straight line that passes through nastro ˆ U /; gastro ˆ K k† cos /; the point, P, and is normal to the reference ellipsoid; this 2† Hastro ˆ nastro; gastro†T line de®nes the geodetic zenith. In terms of south±north, n, and west±east, g, components of the de¯ection, one where Hastro is a two-component vector whose magnitude has (Pick et al. 1973, p. 432) is the total de¯ection angle (to ®rst-order approxima- 1 tion). The Helmert, or astronomic, de¯ection of the n ˆ U / ‡ g2 tan / ‡ 3rd order terms 2 1† vertical is the de¯ection that, when scaled by the magnitude of gravity, is required in the inertial navigation g ˆ K k† cos / ‡ 3rd order terms equations for horizontal positions and velocities (Britting 1971). It represents the horizontal component of the gravity vector with respect to the local vertical in the navigation frame, which is aligned with the ellipsoidal normal. Another de®nition is named after Pizzetti, who pre- scribed the de¯ection of the vertical as the angle between the gravity vector at a point on the geoid and the per- pendicular to the ellipsoid. (The Helmert de¯ection at a point on the geoid is also the Pizzetti de¯ection.) Pi- zzetti's de®nition may be extended to points above the geoid according to Molodensky's de®nition of the de- ¯ection as the angle at a point, P, between the gravity vector at P and the normal gravity vector at the conju- gate point, Q. This point, Q, is the intersection of a spheropotential surface (spherop) with its normal through P, where the normal potential of this spherop equals the actual gravity potential at P. The di€erence between the Helmert and Molodensky de¯ections at a point is due to the curvature of the normal plumb line. This e€ect occurs only in the south± north component and is given to ®rst order by (Heis- kanen and Moritz 1967, p. 196)

dnnorm:curv h†ˆd/norm:curv h†0:17 hkm sin 2/‰arcsecŠ 3†

where hkm is the normal height in units of km. This e€ect on the de¯ection is in the positive sense as a consequence of the convergence from equator to pole of the equipotential surfaces (spherops) of the normal gravity ®eld. With reference to Fig. 1, the following relationship holds:

astro Helmert Molodensky n  n ˆ n ‡ dnnorm:curv 4† Clearly, the e€ect of the normal curvature increases with Fig. 1. Relationships between di€erent de¯ections of the vertical altitude of the point. Usually it is of little concern in low- (n component) elevation terrain; however, the correction can become 12 signi®cant at high altitudes, e.g. those navigated by an vector is the gradient of the potential, one has from Eq. INS. (5) The di€erence between the Helmert (or also Mo-  1 oW lodensky) and Pizzetti de¯ections has to do with the fact nMolodensky P†ˆ that the point of de®nition is di€erent and subsequently c M ‡ h†o/ Q P  the actual gravity vector directions are di€erent (cur- oU vature of actual plumb line). Thus, these di€erences are dn M ‡ h†o/ g due to the local non-parallelism of the equipotential  Q surfaces of the actual gravity ®eld. The Pizzetti de®ni- 1 oW gMolodensky P†ˆ tion is of no further concern here. c N ‡ h† cos / ok Q P  oU dg N ‡ h† cos / ok g 3 The gravimetric de¯ection of the vertical Q 8† The de¯ection of the vertical (speci®cally, Molodensky's version) can be determined from gravimetric data, at where N and M are the principal radii of curvature of least under certain approximations. When thus deter- the ellipsoid and h is the ellipsoidal height of the point. mined, it is given the special name gravimetric de¯ection One may de®ne the gravimetric de¯ection of the of the vertical. Using the geometry of the gravity vertical as vectors, the Molodensky de¯ection components, along 1 oT south±north and west±east directions, are given in a ngrav P†ˆ c M ‡ h† o/ local north-west-up (NWU) coordinate system (again, Q P neglecting non-linear terms) by 1 oT 9† ggrav P†ˆ c N ‡ h† cos / ok 1 1 Q P nMolodensky P† g P† c Q† / / Hgrav ˆ ngrav; ggrav†T gP cQ 1 Á The gravimetric de¯ection is obtained from the hori- ˆ g/ P†c/ Q† dng cQ zontal derivatives of the disturbing potential, T,ata 5† point. By ``horizontal'' one means perpendicular to the Molodensky 1 1 g P† gk P† c Q† ellipsoid normal that passes through the point. The g c k P Q Molodensky de¯ections, Eqs. (8), would equal the 1 gravimetric de¯ections, Eqs. (9), by ignoring the small ˆ †gk P†ck Q† dgg cQ terms, dng and dgg, and by assuming that the normal gravity vectors at P and Q are parallel, i.e. by neglecting where g/ and gk are horizontal components of the the curvature of the normal plumb line between P and gravity vector at P, and c/ and ck (ck ˆ 0!) are Q.Thus horizontal components of the normal gravity vector at Q, all in the local NWU coordinate system aligned with nMolodensky P†ˆngrav P†‡dn f †dn norm:curv P g 10† the ellipsoid normal through P. The quantities dng and Molodensky grav g P†ˆg P†dgg dgg are small e€ects due to approximating the divisor gP by cQ, and are given by where fP is the height anomaly at P. Equations (10) g P† c gP Dg show the approximations made in determining the dn ˆ / Q  n P g P Molodensky de¯ections gravimetrically [according to gP cQ cQ 6† Eqs. (9)]. g P† c g Dg k Q P P The determination of the gravimetric de¯ection dgg ˆ  gP gP cQ cQ comes directly from a solution to the disturbing poten- tial. There are two classic formulations of this solution where DgP is the gravity anomaly. in terms of a boundary-value problem, where gravi- The linear approximation in Eq. (5) is practically the metric quantities, the gravity anomalies, constitute the same as for the astronomic de¯ection, where the only boundary values. These two forms are the Stokes for- di€erence is due to the normal curvature that displaces mula (Heiskanen and Moritz 1967; it is not considered the normal zenith from the geodetic zenith; this is in- further here) and the spherical harmonic series: consequential in the second-order term in Eq. (1). Consider the disturbing potential, T, which is given kM X1 Xn a n by T r; h; k†ˆ C Y h; k† 11† r r n;m n;m nˆ2 mˆn T ˆ W U 7† where r; h; k† are spherical polar coordinates, kM is the where W is the actual gravity potential and U is the product of Newton's gravitational constant and normal gravity potential. Then, noting that the gravity the Earth's total mass (including atmosphere), a is the 13 equatorial radius of the reference ellipsoid, and the The coordinates of the point of evaluation are given in  functions Yn;m are spherical harmonic functions fully terms of geocentric latitude, w. The radial dependence of normalized such that the coecients Cn;m are given by VB is given by ZZ a2 3 r2 C ˆ Dg h; k†Y h; k† dr 12† D r†ˆ kM 15† n;m n;m B B 3 4pkM n 1† r 4 rB

The zero-degree term n ˆ 0† is missing in Eq. (11) where rB is the mean distance between the Earth and the under the assumptions that the normal ellipsoid contains body. DB is known as Doodson's coecient and re¯ects the mass, M, and that the geopotential on the geoid and that Eq. (13) is a second-degree interior harmonic the normal potential on the ellipsoid are equal. [These potential. assumptions are immaterial for the gravimetric de¯ec- The tidal potential, Eq. (13), varies in time, as viewed tions of the vertical, Eq. (9), since they a€ect only the at a point on the Earth, with di€erent periods: from constant bias of T.] The ®rst-degree n ˆ 1† terms are fortnightly (due to the moon) or semi-annually (due to omitted under the assumption that the coordinate the sun) as the coordinates aB; dB† vary, to diurnally system is geocentric. because of Earth's rotation described by tG. There is also a constant part, the average over 18 years, that is not Equation (12) for the coecients Cn;m represents an idealization, being in the ®rst place a spherical approx- zero; this is the permanent tide due to the (crudely) ap- imation, where the gravity anomaly is related in spher- proximate coplanarity of the Earth±sun±moon system. ical approximation to the disturbing potential. Modern The sun and moon are the only extraterrestrial bodies of spherical harmonic models are constructed with great consequence and one may denote care to correct for these and other approximations V r; w; k†ˆV r; w; k†‡V r; w; k† 16† (Rapp and Pavlis 1990). Therefore, one may take Eq. t sun moon (11) to be devoid of all such approximations to the ex- The components of tilt of the Earth's equipotential tent possible using the available data, and assume that surface (e.g. geoid) due to the tidal potential are given by the set of spherical harmonic coecients fCn;mg has been (Vanicek 1980) correspondingly corrected. Equation (11) thus repre- oV oV sents the true disturbing potential outside a sphere dn ˆ t ; dg ˆ t 17† enclosing all terrestrial masses, where the atmosphere t cRow t cR cos wok and all extraterrestrial masses are assumed removed. That is, the disturbing potential of Eq. (11) is given in where no distinction needs to be made between geodetic the non-tide,ortide-free system. It is to be noted that and geocentric latitude. To get an idea of the rough the astronomic (Helmert) de¯ections are given in the order of magnitude of the tidal tilt, consider that, with mean-tide system, since tidal corrections, amounting to appropriate values for the sun's and moon's mass, no more than a few hundredths of an arcsecond, are not Doodson's coecient is applied (R. Anderson, NIMA, pers. comm. 1997). D R†‡D R† sun moon ˆ 0:013 arcsec 18† cR 4 The permanent tidal e€ect on de¯ections of the vertical Only the permanent tidal e€ect is of interest in the numerical analysis presented later. Using accurate eph- A brief review of the tidal e€ects is in order. The emerides (admittedly, extreme accuracy is not warrant- gravitational tidal potential due to an external point- ed) of the sun and moon over three 18-year periods, mass body, designated generically as B, is given Cartwright and Tayler (1971) analyzed the total ensuing approximately by Torge (1991) as tidal potential and arrived at amplitudes for over 500 " spectral constituents, including the constant (perma- 2 2 nent) component. Rapp (1983) used this permanent part VB r; w; k†ˆDB r† cos w cos dB cos 2tB to infer a geoid deformation given by (Heikkinen 1978 did his own analysis and obtained the same result) ‡ sin 2w sin 2d cos t B B 0 # 0 Vt 2 1 1 dNt ˆ ˆ 0:099 0:296 sin w‰mŠ 19† ‡ 3 sin2 w sin2 d c 3 B 3 The corresponding permanent tilt in the geoid is, 13† therefore, given from Eq. (17) by

0 where tB is the hour angle of the body dn ˆ 0:0096 sin 2w [arcsec] t 20† 0 dgt ˆ 0 tB ˆ k ‡ tG aB 14† Thus, only the south±north de¯ection of the vertical and aB; dB† are the right ascension and declination of experiences a permanent tidal deformation. the body, while tG is the hour angle of the vernal These results hold only for a rigid Earth. For the equinox at the Greenwich meridian (i.e. sidereal time). more realistic elastic model, the Earth's surface itself 14 deforms under the in¯uence of the tidal forces. The de- determine the low-degree error statistics using rigorous formation occurs radially as well as tangentially. Fur- least-squares adjustments of satellite orbit perturbation thermore, this deformation redistributes terrestrial mass observations. High-degree statistics of the model usually and consequently generates an additional change in comprise a set of estimated standard deviations for the gravitational potential. The tangential displacement is coecients. ^ characterized as a ratio, `, between the actual displace- Let the computed coecients be denoted by fCn;mg; ment and that of a perfectly nonviscous ¯uid Earth. If then the model for the disturbing potential is given by the Earth were ¯uid, then the horizontal displacement, kM Xnmax Xn a n expressed in terms of the tidal potential as 1=c†oVt=ow, T^ r; h; k†ˆ C^ Y h; k† 24† would cause a change in the direction of the perpen- r r n;m n;m nˆ2 mˆn dicular to the surface (equipotential, in this case) given by 1=c†oVt=ow†=R. The Earth is not ¯uid and this is The gravimetric de¯ections of the vertical are then expressed by the ratio, `, so that the horizontal dis- obtained by di€erentiation according to Eq. (9) placement is actually `=c†oVt=ow, and the change in () ^grav 1 kM Xnmax Xn a n direction of the perpendicular is `=c†oVt=ow†=R. n r; h; k† ^ ˆ Cn;m The deformation changes the gravitational potential g^grav r; h; k† c r2 r by a fraction of the tidal potential due to the mass re- ()nˆ2 mˆn 25† o distribution, so that for the deformed Earth the total oh   Y n;m h; k†† change in potential is 1 ‡ k†Vt. The unitless numbers k o and ` are named after A.E.H. Love (Love numbers) and sin h ok have observed values (Lambeck 1988) of where a further spherical approximation is made in the k ˆ 0:29;`ˆ 0:08. The total tidal e€ect on the (south± direction of the derivative, being orthogonal to the north) de¯ection is therefore radius from the origin, rather than the normal to the ellipsoid. The e€ect of this approximation applies only oVt oVt dntd ˆ 1 ‡ k† ‡ ` 21† to the south±north component and can be evaluated cRow cRow using the equation (see Fig. 2) and the permanent tilt is modi®ed to o o o ˆcos m sin m 0 M ‡ h†o/ roh or dntd ˆ 0:012 sin 2w [arcsec] 26† 22† o o dg0 ˆ 0  m td r oh or The following relationship holds: where m is the angle between the meridian radius of curvature and the spherical radius grav grav 0 n ˆ n ‡ dntd 23† mean tide tide-free 1 m ˆ / w  f 1 2f † sin 2/ 27† That is, the de¯ection becomes more positive with a further ¯attening of the equipotential surfaces due to the f is the ¯attening of the ellipsoid. The longitudinal tidal attraction. derivative is una€ected, because

5 Practical di€erences between gravimetric ∂ and helmert de¯ections ∂r ν A ®nite set of computed spherical harmonic coecients − ∂ h (M+h)∂φ constitutes a model for the disturbing potential in free ν space, meaning that the series of Eq. (11) is truncated at − ∂ r∂ψ some degree, nmax. In addition, a particular model has errors associated with the uncertainties of the computed coecients, which, in turn, arise from data measurement uncertainties and errors due to unaccounted approxi- mations. The errors due to measurement uncertainty M may be characterized statistically by a variance/covari- ance matrix for the coecients; other errors can be quanti®ed to some extent in terms of upper bounds. ψ φ Rarely, however, are detailed and accurate error statis- tics obtainable for the entire set of coecients, due to limitations in computer power and knowledge about the actual statistics or characteristics of the data errors. This is especially true for the higher-degree parts of the model, whereas it is computationally tractable to Fig. 2. Partial derivatives along di€erent horizons 15

o o In Eqs. (33) and (34), h; /; k† are the geodetic co- ˆ 28† N ‡ h† cos / ok r sin h ok ordinates of the point of evaluation of the de¯ection components. It is noted that orientation di€erences may Thus, we have the following error terms: be due to misalignments of astronomic as well as geo- detic reference systems. ^grav grav n r; h; k†ˆn r; h; k†‡dnm ‡ dntrunc ‡ dncoeff:err grav grav g^ r; h; k†ˆg r; h; k†‡dgtrunc ‡ dgcoeff:err 6 Recapitulation 29† where, from Eq. (26) and with the de®nition of gravity Combining Eqs. (4), (10), (23), (29), (33) and (34), we disturbance, dg, the derivative error is have the following relationship between the Helmert de¯ection and the gravimetric de¯ection as computed dgP from a spherical harmonic model of the disturbing dnm ˆ m 30† potential: cQ ^astro ^grav 0 The truncation error of Eq. (25) is given by n P†ˆntide-free P†‡dntd ‡ dnnorm:curv fP † à  ‡ dnnorm:curv HP †dng dntrans dnm dn 1 kM X1 Xn a n trunc ˆ C dn dn dn dn 2 n;m trunc coeff:err rot astro:err dgtrunc c r r astro grav nˆnmax‡1 mˆn ^ ^ () 31† g P†ˆg P†dgtrunc dgcoeff:err dgg dgtrans o  oh Y h; k†† dgrot dgastro:err o n;m sin h ok 35† and the error due to harmonic coecient error is where the spherical harmonic model is assumed to be in  the tide-free system and astronomic observation errors dn 1 kM Xnmax Xn a n coeff:err ˆ dC are also included. Missing from Eq. (35) is the periodic c r2 r n;m tidal e€ect on the Helmert de¯ection (which was not dgcoeff:err nˆ2 mˆn () 32† computed for the data to be analyzed in this paper), as o  oh Y h; k†† well as the time-varying e€ects due to atmospheric and o n;m other mass redistributions. The constant atmospheric sin h ok attraction (important for terrestrial gravity anomalies, Note that if the coordinates of the point of evaluation with a maximum value of 0.87 mgal) in the horizontal are given as geodetic coordinates h; /; k† then these direction is due to the lateral mass inhomogeneities, the must be transformed to geocentric coordinates r; h; k† largest of which results from the air being displaced by before evaluating Eq. (25). Also, the normal gravity c in local terrain. This is about 0.05% (air±crustal density Eq. (25) should be evaluated at the point of computa- ratio) of the terrain correction for de¯ections and thus tion; replacing it with the approximation kM=r2 intro- less than 1 milliarcsecond and negligible. duces a potentially signi®cant error of up to 1 part in 500 Table 1 gives an otherwise full account of the di€er- of the de¯ection. ences between the two types of de¯ections, Helmert and Other di€erences between gravimetric and astro- gravimetric, Eq. (25), in terms of root-mean-square nomic de¯ections of the vertical include the o€set of the (rms) values of each of the correction terms. Some of center of mass from the ellipsoid center and the non- these di€erences are a consequence of di€erences in the parallelism of the coordinate axes. In particular, if there de®nitions, others result from approximations to the is a small translation Dx; Dy; Dz† at the geocenter be- de®nition. The values were obtained with H à ˆ 2km, tween datums, then / ˆ 45,rms Dg†ˆ42 mgal, rms dg†ˆ46 mgal, rms f†ˆ30 m and rms n†ˆ5 arcsec. Dx Dy Clearly, the truncation error followed by the pre- dn ˆ sin / cos k ‡ sin / sin k trans M ‡ h M ‡ h dicted inaccuracy in the spherical harmonic model Dz contribute the most to the di€erence between the Hel- cos / 33† mert de¯ection and the spherical harmonic model de- M ‡ h ¯ection. This is not an unexpected result, since the model Dx Dy dg ˆ sin k cos k is limited in resolution to about 50 km, while there is trans N ‡ h N ‡ h signi®cant power in the de¯ection signal at higher res- olution. Also, the astronomic observational error asso- Orientation di€erences, as represented by small rotation ciated with the Helmert de¯ection represents a angles xx; xy; xz† with the pivot point being the substantial contribution. However, regionally all of geocenter, cause de¯ection changes given by these errors take on a random character, while the e€ects of next signi®cance (numbers 1, 4, 5, 8 and 9 in Table 1) dn ˆ sin kx cos kx rot x y 34† are quite systematic. It is noted that the normal curva- dgrot ˆsin / cos kxx sin / sin kxy ‡ cos /xz ture correction is de®nitely the most prominent of these, 16

Table 1. Di€erences between the Helmert de¯ection compo- E€ect Equation rms value, arcsec nent, n, and the de¯ection as reference (where appropriate) given by a truncated spherical 1. Di€erence between Helmert and Molodensky de¯ection (3) 0.34 (hkm =2) harmonic series (nmax = 360) [some of these also hold for g; 2. Actual gravity approximated by normal gravity (6) 0.0002 see Eq. (35)] 3. Di€erence between Molodensky and gravimetric de¯ection (10) 0.005 4. Di€erence between mean-tide and tide-free systems (23) 0.012 5. Horizontal derivative approximation (30) 0.032 6. Harmonic series truncation at degree 360 (T/Ra model) (31) 3.7 7. Harmonic coecient error (EGM96b predicted) (32) 1.27 8. Coordinate origin di€erences (ITRF90c)WGS84) (33) <0.018 9. Coordinate axes misalignmentd (ITRF90)WGS84) (34) <0.020 10. Astronomic observation error (CONUSe) ± 0.31 a Tscherning/Rapp model, Tscherning and Rapp (1974). b Joint NASA/NIMA Earth Gravity Model, Lemoine et al. (1996). c International Terrestrial Reference Frame 1990, McCarthy (1992). d Large z-axis rotation of the astronomic system (FK4 catalogue) will cause greater di€erences. e Conterminous United States, provided by NIMA (National Imagery and Mapping Agency).

not only especially in mountainous areas, but particu- ders up to the maximum degree nmax ˆ 360†. Each co- larly at higher ¯ight altitudes. Certain e€ects can be ecient is also associated with a given standard safely neglected, for example, item numbers 2 and 3 in deviation. For the evaluation of the spherical harmonic Table 1. Also, with the possible exception of the normal model, Eq. (25), the orthometric height for each astro- curvature correction, all other systematic e€ects may be nomic de¯ection point was added to the geoid undula- neglected in the traditional reduction of geodetic quan- tion obtained from the EGM96 model in order to obtain tities to the ellipsoid. the ellipsoidal height, h; then the geodetic coordinates h; /; k† were converted to spherical polar coordinates r; h; k†. The normal gravity appearing in Eq. (25) was 7 Numerical results evaluated at the point of computation using ®rst-order upward continuation of the normal gravity formula. The object of this section is to compare astronomic Due to the vastness and variability in terrain within de¯ections of the vertical with de¯ections computed CONUS, this total area was divided into six regions from high-degree spherical harmonic models. The approximately evenly separating north from south and astronomic de¯ection data for the comparisons were central from eastern and western parts (Fig. 4). These provided by the National Imagery and Mapping Agency regions approximately delineate rough (west), moderate (NIMA, pers. commun.). The spherical harmonic mod- (east) and smooth (central) terrain, where further north± els considered are the OSU91A model (Rapp et al. 1991) south distinctions may also be anticipated. In each of and the recently constructed EGM96 model (Lemoine these areas the astronomic de¯ections and those implied et al. 1996). Both models are complete to degree and by the spherical harmonic model may be characterized order nmax ˆ 360 and are functions of geocentric spher- brie¯y by their maximum, minimum, mean and rms ical polar coordinates r; h; k†. The comparison is values, as in Table 2. The di€erences between the two restricted to the conterminous United States (CONUS). types of de¯ections, both with respect to OSU91A and The total number of astronomic de¯ections is 3678 EGM96, are similarly characterized. and they are distributed unevenly across CONUS. Each Note that H jHj is the magnitude of the de¯ection astronomic de¯ection data record consists of two com- vector, and that DH jDHj represents the magnitude of ponents, nastro; gastro, the orthometric height, and the the di€erence in de¯ection vectors, not the di€erence in geodetic latitude and longitude /; k† in the WGS84 magnitudes. Table 3 gives the corresponding statistics of reference system. Also included is a standard deviation the di€erences for each individual region. The rms val- for each de¯ection component. Of the 3678 observed ues in these tables are computed by de¯ections, 117 have a combined standard deviation, v q u 2 2 u 1 XN rn ‡ rg, that is greater than 1 arcsec. (Oddly, most of t 2 rms zj†ˆ z ; z ˆ n; g; H; Dn; Dg; DH 36† j N j these are situated along the 35th parallel and along the jˆ1 eastern Florida coast.) These observed de¯ections were deleted from the analysis, leaving a total of 3561 de- where N is the number of de¯ections in a region. ¯ections (Fig. 3). In this reduced set, the rms of the Considering just the rms values, it is seen in Table 2 standard deviations for nastro is 0.293 arcsec, and for that the rms astronomic de¯ection is signi®cantly larger gastro it is 0.394 arcsec. than the rms de¯ection implied by either spherical har- The spherical harmonic models are given in terms of monic model, where the discrepancy is largest in the ^ spherical harmonic coecients fCn;mg, with respect to mountainous regions. Also, the EGM96 model captures the GRS80 normal gravity ®eld, for all degrees and or- slightly more de¯ection information than does the 17

OSU91A model. This is veri®ed by the di€erences be- improvement, though slight, in the de¯ections computed tween the astronomic de¯ections and the spherical har- by EGM96 versus OSU91A. Of particular interest are monic de¯ections. Table 3 shows a systematic the mean di€erences, which could imply systematic

Fig. 3. Locations of 3561 astronomic de¯ections of the vertical

Fig. 4. Delineation of six area of CONUS, roughly by type of terrain 18

Table 2. Essential statistics of the astronomic and model-implied de¯ections of the vertical

Region

Northwest (489 pts) North-central (468 pts) Northeast (405 pts) Southwest (1081 pts) South-central (618 pts) Southeast (500 pts)

H00 n00 g00 H00 n00 g00 H00 n00 g00 H00 n00 g00 H00 n00 g00 H00 n00 g00

Maximum astro 31.62 18.04 31.04 24.73 10.87 24.73 15.52 8.43 12.11 32.29 20.19 26.36 19.82 19.78 10.72 16.41 11.31 11.37 OSU91A 20.03 14.09 18.00 17.60 6.65 17.59 9.53 6.25 7.21 21.04 7.58 10.57 10.69 10.27 10.67 9.91 7.74 8.89 EGM96 20.70 15.26 19.32 19.06 6.25 19.03 11.31 6.77 8.05 21.65 8.87 10.80 11.92 11.44 11.34 10.89 8.48 9.77 Minimum astro 0.11 )13.42 )20.79 0.27 )15.81 )9.44 0.10 )10.58 )15.52 0.13 )25.83 )27.10 0.11 )10.31 )9.14 0.08 )12.77 )10.09 OSU91A 0.40 )15.34 )12.88 0.15 )9.65 )4.49 0.25 )7.65 )9.45 0.25 )14.82 )16.47 0.15 )7.38 )7.72 0.15 )4.94 )6.16 EGM96 0.21 )14.50 )14.12 0.04 )10.92 )5.55 0.29 )9.21 )11.11 0.27 )16.59 )17.41 0.08 )7.91 )8.42 0.19 )5.94 )6.23 Mean astro )0.63 )0.54 )0.36 3.24 +0.00 )0.45 )2.92 )2.91 0.44 0.74 0.20 2.26 OSU91A )1.02 )0.32 )0.39 3.32 )0.18 )0.52 )3.51 )3.46 0.46 0.91 0.33 2.61 EGM96 )0.97 )0.26 )0.38 3.29 )0.13 )0.41 )3.48 )3.33 0.48 0.99 0.28 2.53 rms astro 7.74 4.56 6.26 6.14 2.93 5.39 5.05 3.07 4.00 8.62 5.64 6.52 4.89 3.54 3.36 5.01 3.09 3.76 OSU91A 5.80 3.67 4.49 5.35 2.37 4.80 3.89 2.26 3.16 6.78 4.52 5.05 4.23 3.00 2.99 4.39 2.37 3.68 EGM96 6.25 3.88 4.90 5.66 2.55 5.05 4.28 2.65 3.37 6.98 4.70 5.15 4.55 3.19 3.24 4.54 2.46 3.77

Table 3. Essential statistics of the di€erences, model minus astronomic de¯ections

Region

Northwest (489 pts) North-central (468 pts) Northeast (405 pts) Southwest (1081 pts) South-central (618 pts) Southeast (500 pts)

DQ00 Dn00 Dg00 DQ00 Dn00 Dg00 DQ00 Dn00 Dg00 DQ00 Dn00 Dg00 DQ00 Dn00 Dg00 DQ00 Dn00 Dg00

Maximum OSU91A)astro 16.73 10.89 12.30 14.35 8.16 9.97 11.09 10.63 10.93 26.55 20.85 17.42 19.93 6.46 5.42 12.70 10.99 8.99 EGM96)astro 16.24 11.29 14.00 13.08 8.29 9.57 11.30 10.67 9.92 25.76 20.63 16.88 19.62 6.42 6.84 10.22 9.53 8.24 Minimum OSU91A)astro 0.08 )15.41 )16.59 0.11 )6.36 )14.21 0.06 )4.99 )8.41 0.19 )15.02 )21.27 0.10 )19.91 )3.98 0.10 )6.04 )7.23 EGM96)astro 0.02 )14.25 )16.23 0.06 )6.20 )12.87 0.07 )5.01 )7.72 0.04 )14.37 )19.13 0.03 )19.57 )4.57 0.04 )5.65 )5.65 Mean OSU91A)astro )0.40 0.22 )0.03 0.09 )0.18 )0.07 )0.59 )0.56 0.02 0.17 0.13 0.35 EGM96)astro )0.35 0.28 )0.01 0.15 )0.14 0.04 )0.56 )0.42 0.04 0.25 0.08 0.28 rms OSU91A)astro 5.28 3.33 4.10 2.67 1.73 2.04 2.50 1.68 1.85 6.34 4.02 4.90 2.25 1.78 1.37 2.33 1.72 1.57 EGM96)zastro 5.20 3.30 4.01 2.63 1.71 2.00 2.42 1.62 1.80 6.14 4.00 4.67 2.19 1.69 1.38 2.18 1.59 1.49 19 errors in one or the other (or both) de¯ection types. These mean di€erences, however, do vary from one region to the next, being somewhat larger in the more (mean)

mountainous areas. Table 4 gives the rms and mean ¢¢ 0.274 0.0043 g ) de¯ections and di€erences between model and astro- D nomic de¯ections for the entire CONUS as one region. The theoretical di€erences between astronomic de-

¯ections and spherical harmonic model de¯ections are (mean) ¢¢ given by Eq. (35). Each of the calculable systematic n 0.015 0.0006 0.018 0.141 correction terms in Eq. (35) is applied to the spherical D harmonic model EGM96. The resulting corrected model de¯ections are then compared, again, to the astronomic (mean)

de¯ections. Only the mean di€erences are computed ¢¢ 0.237 0.0038 0.0068 g ) ) since the rms, maximum and minimum di€erences will D not di€er substantively from the corresponding uncor- rected di€erences. The term dnm† is computed from

Eq. (30) using the gravity disturbance implied by (mean) ¢¢

EGM96. The combined term ‡dn † is computed n 0.0760.011 0.016 0.011 0.0065 0.0035 norm:curv D from Eq. (3) using the orthometric height plus geoid undulation implied by EGM96. The terms dntrans† and dg † are computed using Eq. (33) for the trans (mean)

WGS84 to ITRF90 coordinate translations (McCarthy ¢¢ 0.0083 0.015 0.0097 0.018 0.4240.441 0.036 0.163 0.248 0.078 0.277 g ) ) ) ) 1992) D

Dx ˆ xITRF90 xWGS84 ˆ0:06 m

Dy ˆ y y ˆ‡0:52 m 37† (mean) ITRF90 WGS84 ¢¢ 0.014 0.564 0.357 n ) ) Dz ˆ zITRF90 zWGS84 ˆ‡0:22 m D

The terms dnrot† and dgrot† are computed using

Eq. (34) for the WGS84 to ITRF90 coordinate rotations (mean) ¢¢ 0.0018 0.0023 0.017 (McCarthy 1992) g ) D xx ˆ‡0:0183 arcsec x ˆ0:0003 arcsec 38†

y (mean) ¢¢ 0.0390.012 0.157 0.011 0.1350.044 0.041 0.041 xz ˆ‡0:0070 arcsec n ) ) D It is assumed that the di€erences between ITRF90 and ITRF94 (to which EGM96 refers) are negligible. Finally, the term ‡dn0 † is computed according to (mean) td ¢¢ 0.0039 0.016 0.0064 0.018 g 0.141 ) ) Eq. (22) and terms dng† and dgg† are neglected (in D accordance with the discussion of Table 1). Table 5 gives the mean e€ects of each of these corrections per region,

as well as the total mean di€erences of the corrected and (mean) ¢¢ 0.110 0.012 0.018 0.0052 0.0060 0.0076 0.014 0.152 astronomic de¯ection components. The values that n ) +0.148 reduce these di€erences are highlighted in bold type, as D are the corrected di€erences that show improvement. Corresponding rms values of these di€erences change (mean)

insigni®cantly from the uncorrected di€erences. ¢¢ 0.00870.010 0.016 g ) ) D Table 4. Essential statistics of the astronomic and model-implied vertical de¯ections and their di€erences (mean)

CONUS (3561 pts) Mean value rms value ¢¢ 0.217 0.012 0.015 0.017 0.0011 0.345 0.276 0.082 0.257 n ) ) ) Region Northwest (489 pts)D North-central (468 pts) Northeast (405 pts) Southwest (1081 pts) South-central (618 pts) Southeast (500 pts) n00 g00 Q00 n00 g00 astro) ) astro )0.91 )0.14 6.79 4.27 5.28 trans rot

OSU91A )1.15 )0.19 5.46 3.42 4.26 dg dg

EGM96 )1.13 )0.12 5.73 3.61 4.45 Mean e€ects of corrections terms and mean di€erences between corrected model and astronomic de¯ections

00 00 curv 00 00 00 : Dn Dg DQ Dn Dg ; OSU91)aastro )0.24 )0.06 4.39 2.85 3.35 ; m norm 0 td trans rot no corrections all corrections dn dn EGM96)astro )0.22 0.02 4.27 2.81 3.22 dn dn dn ‡ ) ‡ ) mean (EGM96 Table 5. Correction 20

0.40

0.30 2 0.20 [arcsec ]

Tscherning/Rapp model 0.10 R/D-1 EGM96 R/D-2

OSU91A

0.00 100 200 300 400 500 Degree, n

Fig. 5. Degree variances for the vertical de¯ection from spherical harmonic models EGM96 and OSU91A, as well as from analytical approximations

For the entire region, CONUS, the mean di€erence the entry in the last row of Table 4, a reasonable for the south±north de¯ection component was reduced, estimate of the truncation error variance for CONUS is as a consequence of applying the systematic corrections, from 0:219 arcsec to ±0.058 arcsec; and for the west± r^2 ˆ 4:27†2 1:80†2 0:49†2 east component, it was reduced from +0.016 arcsec to H; trunc 42† +0.004 arcsec. ˆ 3:84 arcsec†2 From Table 1, the largest di€erences between the model and astronomic de¯ections are caused by the With a view toward evaluating this empirical value of truncation of the spherical harmonic series. Neglecting 2 rH; trunc, Fig. 5 shows the degree variances of the de- all systematic errors in Eq. (35), one has ¯ection vector, in the sense of Eq. (40), given by grav astro ^ ^ n ntidefree P†n P†dntrunc ‡ dncoeff:err ‡ dnastro:err X r2 ˆ n n ‡ 1† C2 43† g^grav P†g^astro P†dg ‡ dg ‡ dg 39† H; n n; m trunc coeff:err astro:err mˆn

In terms of variances, one may assume that the errors on for the OSU91A and EGM96 models, where the units of the right side are uncorrelated and obtain an approxi- radians-squared in Eq. (43) are converted to arcseconds- mate expression of the error variance for the de¯ection squared in the ®gure. Also included in this ®gure are vector. In order to work with a single number, let putative analytic approximations and continuations for r2  tr M HHT† the high degrees. They are based on the Tscherning/ H Rapp degree variance model (included for comparison; 2 2 ˆ rn ‡ rg 40† Tscherning and Rapp 1974) where M Á† represents a global average or statistical r2 T =R† expectation, as the case may be; and zero mean values H; n are assumed. Then, from Eq. (39) 1 425:28 n n ‡ 1† ˆ 0:999617†n‡2 ‰rad2Š c2 n 1† n 2† n ‡ 24† r2  r2 ‡ r2 ‡ r2 41† H; tot H; trunc H; coeff:err H; astro:err 44† 2 2 For EGM96, rH; coeff:err ˆ 1:80 arcsec† , and for the 3561 astronomic de¯ections, the rms of the error where c ˆ 9:8  105, and a linear combination of 2 2 variances is rH; astro:err ˆ 0:49 arcsec† . Therefore, using spherical reciprocal distance (R/D) models 21

n n ‡ 1† Xp 8 Summary r2 R=D†ˆ s2 1 q † qn ‰rad2Š 45† H; n c2R2 j j j jˆ1 A careful theoretical development of the di€erences 2 2 2 between astronomic vertical de¯ections and de¯ections where R ˆ 6 371 000 and sj is given in units of ‰m =s Š. The parameter values given in Table 6 pertain to the implied by 360-degree spherical harmonic models shows R/D models shown in Fig. 5. that several systematic di€erences are signi®cant at the These parameter values are purely empirical based on level of a few hundredths of an arcsecond, or more. By the following criteria. Both R/D model degree variances far the largest di€erence is the e€ect due to the curvature should approximate the EGM96 degree variances from of the normal plumb line. Accounting for this and other degree 100 to degree 200. R/D-1 model degree variances calculable systematic di€erences due to the permanent should yield the estimated truncation standard error tide and datum translation and rotation reduces the given in Eq. (42). R/D-2 model degree variances should mean di€erences between these two types of de¯ections approximate EGM96 degree variances from degree 100 in the conterminous US (CONUS) by about 75% , from 0:219 arcsec to 0:058 arsec and from +0.016 arcsec to degree nmax ˆ 360. Table 7 compares the truncation errors implied by to +0.004 arsec, in their respective components. It is the degree variance models to the empirical truncation seen from Table 5 that the normal plumb-line curvature error for CONUS derived from a comparison of as- correction contributes most to the reduction of the mean tronomic and EGM96 de¯ections. It is clear that this di€erences in mountainous areas, as expected. This also latter truncation error, given by Eq. (42), is consistent reinforces the fact that large curvature corrections may only with an analytic degree variance model that can- need to be applied to gravimetric de¯ections computed not ®t the observed degree variances at degrees greater at ¯ight altitudes, if these are to be used in inertial than 200. This means that the high-degree spectrum navigation systems. (greater than degree 200) of the EGM96 model has less Since de¯ections of the vertical are rich in high-fre- power than the de¯ection data would suggest. A note quency gravitational information, observed values of the of caution must accompany this conclusion, since it is astronomic (or Helmert) de¯ections are well suited to based on only a relatively small part of the globe. For assess the high-degree spectrum of a geopotential model example, the rms de¯ection from the EGM96 degree like EGM96. Figure 5 clearly shows the improvement of variances is 6.57 arcsec, whereas for CONUS, from EGM96 over OSU91A in the power of the signal at Table 4, it is 5.73 arcsec (but this would imply an rms degrees greater than 100. However, when comparing truncation error even smaller than 1.33 arcsec in Table EGM96 de¯ections with astronomic de¯ections in 7). Moreover, the distribution of astronomic de¯ec- CONUS, the EGM96 de¯ections appear to be too tions on which the analysis is based is not uniform smooth at degrees greater than 200. This conclusion is even in this small part (namely, CONUS). Neverthe- obtained by trying to ®t, without success, models of the less, this analysis represents an evaluation of power remaining in the observed de¯ections after the EGM96 model against data that have strong EGM96 is removed to that suggested by EGM96. It high-frequency informational content and that are follows that the EGM96 model may be under-powered representative of a variety of signal strengths of the in the high-degree spectrum (say, beyond degree 200). anomalous gravity ®eld. Acknowledgments. This work was supported in part by Phillips Laboratory, US Air Force, contract F19628-94-K-0005, and Na- tional Imagery and Mapping Agency. The author is grateful to NIMA for providing the astronomic de¯ection data and to Mr. Table 6. Parameter values for the R/D models Jian Yi, graduate student at Ohio State University, for performing some of the computations. The author also thanks the referees for 2 2 2 their reviews. Model sj [m /s ] qj R/D-1, p ˆ 4 0.4 0.99875 5 0.9962 70 0.989 References 780 0.969 R/D-2, p ˆ 3 60 0.988 Britting KR (1971) Inertial navigation systems analysis. John Wi- 80 0.978 ley, New York 650 0.970 Cartwright DE, Tayler RJ (1971) New computations of the tide- generating potential. Geophys J R Astr Soc 23: 45±74 Heikkinen M (1978) On the tide-generating forces. Pub Finnish Geodetic Inst No 85, Helsinki Heiskanen WA, Moritz H (1967) Physical geodesy. Freeman, San Table 7. Standard errors of truncation for di€erent models and Francisco CONUS [units = arcsec] Lambeck K (1988) Geophysical geodesy, the slow deformations of the Earth. Clarendon Press, Oxford Model Empirical T/R model R/D-1 R/D-2 Lemoine FG, Smith DE, Kunz L, Smith R, Pavlis EC, Pavlis NK, (CONUS) Klosko SM, Chinn DS, Torrence MH, Williamson RG, Cox r 3.84 5.32 3.85 1.33 CM, Rachlin KE, Wang YM, Kenyon SC, Salman R, Trimmer H;trunc R, Rapp RH, Nerem RS (1996) The development of the NASA 22

GSFC and NIMA Joint Geopotential Model. Proc Int Symp Rapp RH, Wang YM, Pavlis NK (1991) The Ohio State 1991 Gravity, geoid, and marine geodesy (GRAGEOMAR 1996), geopotential and sea surface topography harmonic coecient University of Tokyo, 30 Sept±5 Oct models. Rep No 410, Dep Geodetic Science, The Ohio State McCarthy DD (ed) (1992) IERS standards (1992). IERS Tech Note University, Columbus 13, Obsde Paris, Paris Torge W (1991) Geodesy, 2nd edn. Walter de Gruyter, Berlin Pick M, Picha J, Vyskocil V (1973) Theory of the Earth's gravity Tscherning CC, Rapp RH (1974) Closed covariance expressions for ®eld. Elsevier, Amsterdam gravity anomalies, geoid undulations and de¯ections of the Rapp RH (1983) Tidal gravity computations based on recom- vertical implied by anomaly degree variance models. Rep No. mendations of the Standard Earth Tide Committee. Bull d'Info, 208, Dep Geodetic Science, The Ohio State University, Co- Commiss Permanente des Marees Terrestres, Brussels No 89, pp lumbus 5814±5819 Vanicek P (1980) Tidal corrections to geodetic quantities. NOAA Rapp RH (1992) Geometric geodesy, part II. Dep Geodetic Science Tech Rep NOS 83 NGS 14, National Oceanic and Atmospheric and Surveying, The Ohio State University, Columbus Administration, National Ocean Survey, Washington, DC Rapp RH, Pavlis NK (1990) The development and analysis of geopotential coecient models to spherical harmonic degree 360 J Geophys Res 95 (B13): 21 885±21 911