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For over 60 years, an approximation involving spherical geopotentials has underlain the representation of gravity in global numerical models of Earth's atmosphere and oceans. This article explores how departures from sphericity can be allowed by assuming spheroidal geopotentials instead. A route to more accurate model formulation is indicated, and the theoretical basis of the classical spherical‐geopotential approximation is illuminated too. An overview—from the time of Newton to the present—is given of the development of zeroth‐ and first‐order approximations to the geopotential field in terms of two small non‐dimensional parameters. The importance of using geopotential coordinate systems in atmospheric and oceanic models is emphasized. Early suggestions for such systems using non‐spherical coordinates involved qualitatively inappropriate choices of ellipsoids derived from families of confocal ellipses. Specific examples of appropriate ellipsoid choices are considered before presentation of the recently developed Geophysically Realistic, Ellipsoidal, Analytically Tractable (GREAT) system. This is based on a suitably constructed geopotential field approximation, first order accurate, from which—without further approximation—may be analytically derived equations for geopotential surfaces and surfaces orthogonal to them. GREAT coordinates satisfy stated desiderata for geopotential coordinate systems and are applicable both above and below Earth's geoid (assumed to coincide with the WGS 84 [2004, https://gis‐lab.info/docs/nima‐tr8350.2‐wgs84fin.pdf] reference ellipsoid to an excellent approximation). GREAT‐coordinate analysis provides justification for the classical spherical‐geopotential approximation: It is revealed as a mathematical limit, but not a physically realizable one. Attention is drawn to a certain partially spherical limit that is realizable physically. Plain Language Summary Gravity is by far the dominant external force in the equations of motion for Earth's atmosphere and oceans. It is crucially important that it be adequately represented in global atmospheric and oceanic models for climate and weather prediction. In principle, one simply applies Newton's inverse square law of gravitational attraction to represent gravity in the equations of motion. In practice, this is far too complicated to do exactly—due principally to the rotating Earth being closer to spheroidal than spherical in shape, with an inhomogeneous mass distribution—thereby necessitating approximation. The dominant nature of gravity means that forecast accuracy is enhanced if one constructs an orthogonal coordinate system to integrate the equations of motion, whereby gravity only acts in the vertical and not in the horizontal. This is termed a geopotential coordinate system and, to further complicate matters, its construction is intrinsically coupled to a suitable approximation of gravity. We first review the basic concepts to represent gravity in global atmospheric and oceanic models. Next, we discuss the importance and principles of geopotential coordinates for modeling purposes. Various geopotential coordinate systems of varying accuracy are then compared. Finally, we outline some possible developments for representing gravity in future models. Key Points We review the derivation from first principles of geopotential approximations to represent gravity in global atmospheric and oceanic models Various geopotential coordinate systems of varying accuracy are compared, leading to one that satisfies all desiderata for such a system The low‐order, ubiquitous, classical, spherical‐geopotential approximation is shown to be the asymptotic limit of a more accurate one

We unify and extend classical results from function approximation theory and consider their utility in astrodynamics. Least square approximation, using the classical Chebyshev polynomials as basis functions, is reviewed for discrete samples of the to-be-approximated function. We extend the orthogonal approximation ideas to n-dimensions in a novel way, through the use of array algebra and Kronecker operations. Approximation of test functions illustrates the resulting algorithms and provides insight into the errors of approximation, as well as the associated errors arising when the approximations are differentiated or integrated. Two sets of applications are considered that are challenges in astrodynamics. The first application addresses local approximation of high degree and order geopotential models, replacing the global spherical harmonic series by a family of locally precise orthogonal polynomial approximations for efficient computation. A method is introduced which adapts the approximation degree radially, compatible with the truth that the highest degree approximations (to ensure maximum acceleration error < 10−9 m s−2, globally) are required near the Earth’s surface, whereas lower degree approximations are required as radius increases. We show that a four order of magnitude speedup is feasible, with efficiency optimized using radial adaptation.

In order to move the polar singularity of arbitrary spherical harmonic expansion to a point on the equator, we rotate the expansion around the y -axis by 90 ∘ such that the x -axis becomes a new pole. The expansion coefficients are transformed by multiplying a special value of Wigner D -matrix and a normalization factor. The transformation matrix is unchanged whether the coefficients are 4 π fully normalized or Schmidt quasi-normalized. The matrix is recursively computed by the so-called X-number formulation (Fukushima in J Geodesy 86: 271–285, 2012a ). As an example, we obtained 2190 × 2190 coefficients of the rectangular rotated spherical harmonic expansion of EGM2008. A proper combination of the original and the rotated expansions will be useful in (i) integrating the polar orbits of artificial satellites precisely and (ii) synthesizing/analyzing the gravitational/geomagnetic potentials and their derivatives accurately in the high latitude regions including the arctic and antarctic area.

The geoid-to-quasigeoid correction has been traditionally computed approximately as a function of the planar Bouguer gravity anomaly and the topographic height. Recent numerical studies based on newly developed theoretical models, however, indicate that the computation of this correction using the approximate formula yields large errors especially in mountainous regions with computation points at high elevations. In this study we investigate these approximation errors at the study area which comprises Himalayas and Tibet where this correction reaches global maxima. Since the GPS-leveling and terrestrial gravity datasets in this part of the world are not (freely) available, global gravitational models (GGMs) are used to compute this correction utilizing the expressions for a spherical harmonic analysis of the gravity field. The computation of this correction can be done using the GGM coefficients taken from the Earth Gravitational Model 2008 (EGM08) complete to degree 2160 of spherical harmonics. The recent studies based on a regional accuracy assessment of GGMs have shown that the combined GRACE/GOCE solutions provide a substantial improvement of the Earth's gravity field at medium wavelengths of spherical harmonics compared to EGM08. We address this aspect in numerical analysis by comparing the gravity field quantities computed using the satellite-only combined GRACE/GOCE model GOCO02S against the EGM08 results. The numerical results reveal that errors in the geoid-to-quasigeoid correction computed using the approximate formula can reach as much as ~1.5 m. We also demonstrate that the expected improvement of the GOCO02S gravity field quantities at medium wavelengths (within the frequency band approximately between 100 and 250) compared to EGM08 is as much as ±60 mGal and ±0.2 m in terms of gravity anomalies and geoid/quasigeoid heights respectively.

We present an alternate mathematical technique than contemporary spherical harmonics to approximate the geopotential based on triangulated spherical spline functions, which are smooth piecewise spherical harmonic polynomials over spherical triangulations. The new method is capable of multi-spatial resolution modeling and could thus enhance spatial resolutions for regional gravity field inversion using data from space gravimetry missions such as CHAMP, GRACE or GOCE. First, we propose to use the minimal energy spherical spline interpolation to find a good approximation of the geopotential at the orbital altitude of the satellite. Then we explain how to solve Laplace’s equation on the Earth’s exterior to compute a spherical spline to approximate the geopotential at the Earth’s surface. We propose a domain decomposition technique, which can compute an approximation of the minimal energy spherical spline interpolation on the orbital altitude and a multiple star technique to compute the spherical spline approximation by the collocation method. We prove that the spherical spline constructed by means of the domain decomposition technique converges to the minimal energy spline interpolation. We also prove that the modeled spline geopotential is continuous from the satellite altitude down to the Earth’s surface. We have implemented the two computational algorithms and applied them in a numerical experiment using simulated CHAMP geopotential observations computed at satellite altitude (450 km) assuming EGM96 ( n max  = 90) is the truth model. We then validate our approach by comparing the computed geopotential values using the resulting spherical spline model down to the Earth’s surface, with the truth EGM96 values over several study regions. Our numerical evidence demonstrates that the algorithms produce a viable alternative of regional gravity field solution potentially exploiting the full accuracy of data from space gravimetry missions. The major advantage of our method is that it allows us to compute the geopotential over the regions of interest as well as enhancing the spatial resolution commensurable with the characteristics of satellite coverage, which could not be done using a global spherical harmonic representation.

The exact estimates with respect to the uniform (Chebyshev) norm of the general term Vn of the Laplace series in spherical harmonics for the gravitational potential of a planet are known to be functions of the differential properties of distribution of masses. However, it is difficult to use them in practice because the norm for large n is difficult to calculate. The mean square (Euclidean) norm, however, is used almost entirely in applied research, and so the translation of the estimates from one norm to the other can be effected only in particular cases. In the present paper, a power law estimate of the general term of the Laplace series with respect to the mean square norm is obtained, and its parameters are evaluated numerically. For the EGM2008 geopotential model (Pavlis et al., 2008), the exponent ranges between 2.7 and 3.4. Preliminary calculations are made for a model body for which the harmonic factors are known exactly. A cautious conclusion is made: the geopotential model put forward is capable of adequately describing spherical harmonics for n ≤ 1000; for 1000 ≤ n ≤ 2000 the model gives a correct description, at least qualitatively; for larger n the spherical harmonics of the model do not correspond to reality.

Use of the primitive equations in spectral models of the atmosphere raises certain questions about the representation of the horizontal components of velocity therein. A barotropic model is constructed and integrated using two types of velocity representation. In the first, (u,v) are expanded directly in spherical harmonics while the second represents (u cosθ, v cosθ) in like manner, where θ is latitude. These integrations are compared and it is concluded that the direct representation of (u,v) is an allowable procedure.