Basic equations for constructing geopotential models from the gravitational potential derivatives of the first and second orders in the terrestrial reference frame

This research represents a continuation of the investigation carried out in the paper of Petrovskaya and Vershkov (J Geod 84(3):165–178, 2010) where conventional spherical harmonic series are constructed for arbitrary order derivatives of the Earth gravitational potential in the terrestrial reference frame. The problem of converting the potential derivatives of the first and second orders into geopotential models is studied. Two kinds of basic equations for solving this problem are derived. The equations of the first kind represent new non-singular non-orthogonal series for the geopotential derivatives, which are constructed by means of transforming the intermediate expressions for these derivatives from the above-mentioned paper. In contrast to the spherical harmonic expansions, these alternative series directly depend on the geopotential coefficients ${\bar{{C}}_{n,m}}$ and ${\bar{{S}}_{n,m}}$ . Each term of the series for the first-order derivatives is represented by a sum of these coefficients, which are multiplied by linear combinations of at most two spherical harmonics. For the second-order derivatives, the geopotential coefficients are multiplied by linear combinations of at most three spherical harmonics. As compared to existing non-singular expressions for the geopotential derivatives, the new expressions have a more simple structure. They depend only on the conventional spherical harmonics and do not depend on the first- and second-order derivatives of the associated Legendre functions. The basic equations of the second kind are inferred from the linear equations, constructed in the cited paper, which express the coefficients of the spherical harmonic series for the first- and second-order derivatives in terms of the geopotential coefficients. These equations are converted into recurrent relations from which the coefficients ${\bar{{C}}_{n,m}}$ and ${\bar{{S}}_{n,m}}$ are determined on the basis of the spherical harmonic coefficients of each derivative. The latter coefficients can be estimated from the values of the geopotential derivatives by the quadrature formulas or the least-squares approach. The new expressions of two kinds can be applied for spherical harmonic synthesis and analysis. In particular, they might be incorporated in geopotential modeling on the basis of the orbit data from the CHAMP, GRACE and GOCE missions, and the gradiometry data from the GOCE mission.     

Fourier-series representation and projection of spherical harmonic functions

Computations of Fourier coefficients and related integrals of the associated Legendre functions with a new method along with their application to spherical harmonics analysis and synthesis are presented. The method incorporates a stable three-step recursion equation that can be processed separately for each colatitudinal Fourier wavenumber. Recursion equations for the zonal and sectorial modes are derived in explicit single-term formulas to provide accurate initial condition. Stable computations of the Fourier coefficients as well as the integrals needed for the projection of Legendre functions are demonstrated for the ultra-high degree of 10,800 corresponding to the resolution of one arcmin. Fourier coefficients, computed in double precision, are found to be accurate to 15 significant digits, indicating that the normalized error is close to the machine round-off error. The orthonormality, evaluated with Fourier coefficients and related integrals, is shown to be accurate to O(10^?15) for degrees and orders up to 10,800. The Legendre function of degree 10,800 and order 5,000, synthesized from Fourier coefficients, is accurate to the machine round-off error. Further extension of the method to even higher degrees seems to be realizable without significant deterioration of accuracy. The Fourier series is applied to the projection of Legendre functions to the high-resolution global relief data of the National Geophysical Data Center of the National Oceanic and Atmospheric Administration, and the spherical harmonic degree variance (power spectrum) of global relief data is discussed.     

Transformation between surface spherical harmonic expansion of arbitrary high degree and order and double Fourier series on sphere

In order to accelerate the spherical harmonic synthesis and/or analysis of arbitrary function on the unit sphere, we developed a pair of procedures to transform between a truncated spherical harmonic expansion and the corresponding two-dimensional Fourier series. First, we obtained an analytic expression of the sine/cosine series coefficient of the \(4 \pi \) fully normalized associated Legendre function in terms of the rectangle values of the Wigner d function. Then, we elaborated the existing method to transform the coefficients of the surface spherical harmonic expansion to those of the double Fourier series so as to be capable with arbitrary high degree and order. Next, we created a new method to transform inversely a given double Fourier series to the corresponding surface spherical harmonic expansion. The key of the new method is a couple of new recurrence formulas to compute the inverse transformation coefficients: a decreasing-order, fixed-degree, and fixed-wavenumber three-term formula for general terms, and an increasing-degree-and-order and fixed-wavenumber two-term formula for diagonal terms. Meanwhile, the two seed values are analytically prepared. Both of the forward and inverse transformation procedures are confirmed to be sufficiently accurate and applicable to an extremely high degree/order/wavenumber as \(2^{30}\,{\approx }\,10^9\). The developed procedures will be useful not only in the synthesis and analysis of the spherical harmonic expansion of arbitrary high degree and order, but also in the evaluation of the derivatives and integrals of the spherical harmonic expansion.     

The Meissl scheme for the geodetic ellipsoid

We present a variant of the Meissl scheme to relate surface spherical harmonic coefficients of the disturbing potential of the Earth’s gravity field on the surface of the geodetic ellipsoid to surface spherical harmonic coefficients of its first- and second-order normal derivatives on the same or any other ellipsoid. It extends the original (spherical) Meissl scheme, which only holds for harmonic coefficients computed from geodetic data on a sphere. In our scheme, a vector of solid spherical harmonic coefficients of one quantity is transformed into spherical harmonic coefficients of another quantity by pre-multiplication with a transformation matrix. This matrix is diagonal for transformations between spheres, but block-diagonal for transformations involving the ellipsoid. The computation of the transformation matrix involves an inversion if the original coefficients are defined on the ellipsoid. This inversion can be performed accurately and efficiently (i.e., without regularisation) for transformation among different gravity field quantities on the same ellipsoid, due to diagonal dominance of the matrices. However, transformations from the ellipsoid to another surface can only be performed accurately and efficiently for coefficients up to degree and order 520 due to numerical instabilities in the inversion.     

On the computation and approximation of ultra-high-degree spherical harmonic series

Spherical harmonic series, commonly used to represent the Earth’s gravitational field, are now routinely expanded to ultra-high degree (> 2,000), where the computations of the associated Legendre functions exhibit extremely large ranges (thousands of orders) of magnitudes with varying latitude. We show that in the degree-and-order domain, (ℓ,m), of these functions (with full ortho-normalization), their rather stable oscillatory behavior is distinctly separated from a region of very strong attenuation by a simple linear relationship: , where θ is the polar angle. Derivatives and integrals of associated Legendre functions have these same characteristics. This leads to an operational approach to the computation of spherical harmonic series, including derivatives and integrals of such series, that neglects the numerically insignificant functions on the basis of the above empirical relationship and obviates any concern about their broad range of magnitudes in the recursion formulas that are used to compute them. Tests with a simulated gravitational field show that the errors in so doing can be made less than the data noise at all latitudes and up to expansion degree of at least 10,800. Neglecting numerically insignificant terms in the spherical harmonic series also offers a computational savings of at least one third.     

Harmonic analysis of the Earth's gravitational field by means of semi-continuous ephemerides of a low Earth orbiting GPS-tracked satellite. Case study: CHAMP

An algorithm for the determination of the spherical harmonic coefficients of the terrestrial gravitational field representation from the analysis of a kinematic orbit solution of a low earth orbiting GPS-tracked satellite is presented and examined. A gain in accuracy is expected since the kinematic orbit of a LEO satellite can nowadays be determined with very high precision, in the range of a few centimeters. In particular, advantage is taken of Newton's Law of Motion, which balances the acceleration vector with respect to an inertial frame of reference (IRF) and the gradient of the gravitational potential. By means of triple differences, and in particular higher-order differences (seven-point scheme, nine-point scheme), based upon Newton's interpolation formula, the local acceleration vector is estimated from relative GPS position time series. The gradient of the gravitational potential is conventionally given in a body-fixed frame of reference (BRF) where it is nearly time independent or stationary. Accordingly, the gradient of the gravitational potential has to be transformed from spherical BRF to Cartesian IRF. Such a transformation is possible by differentiating the gravitational potential, given as a spherical harmonics series expansion, with respect to Cartesian coordinates by means of the chain rule, and expressing zero- and first-order Ferrer's associated Legendre functions in terms of Cartesian coordinates. Subsequently, the BRF Cartesian coordinates are transformed into IRF Cartesian coordinates by means of the polar motion matrix, the precession–nutation matrices and the Greenwich sidereal time angle (GAST). In such a way a spherical harmonic representation of the terrestrial gravitational field intensity with respect to an IRF is achieved. Numerical tests of a resulting Gauss–Markov model document not only the quality and the high resolution of such a space gravity spectroscopy, but also the problems resulting from noise amplification in the acceleration determination process.     

On computing ellipsoidal harmonics using Jekeli’s renormalization

Gravity data observed on or reduced to the ellipsoid are preferably represented using ellipsoidal harmonics instead of spherical harmonics. Ellipsoidal harmonics, however, are difficult to use in practice because the computation of the associated Legendre functions of the second kind that occur in the ellipsoidal harmonic expansions is not straightforward. Jekeli’s renormalization simplifies the computation of the associated Legendre functions. We extended the direct computation of these functions—as well as that of their ratio—up to the second derivatives and minimized the number of required recurrences by a suitable hypergeometric transformation. Compared with the original Jekeli’s renormalization the associated Legendre differential equation is fulfilled up to much higher degrees and orders for our optimized recurrences. The derived functions were tested by comparing functionals of the gravitational potential computed with both ellipsoidal and spherical harmonic syntheses. As an input, the high resolution global gravity field model EGM2008 was used. The relative agreement we found between the results of ellipsoidal and spherical syntheses is 10^?14, 10^?12 and 10^?8 for the potential and its first and second derivatives, respectively. Using the original renormalization, this agreement is 10^?12, 10^?8 and 10^?5, respectively. In addition, our optimized recurrences require less computation time as the number of required terms for the hypergeometric functions is less.     

Construction of spherical harmonic series for the potential derivatives of arbitrary orders in the geocentric Earth-fixed reference frame

The derivatives of the Earth gravitational potential are considered in the global Cartesian Earth-fixed reference frame. Spherical harmonic series are constructed for the potential derivatives of the first and second orders on the basis of a general expression of Cunningham (Celest Mech 2:207–216, 1970) for arbitrary order derivatives of a spherical harmonic. A common structure of the series for the potential and its first- and second-order derivatives allows to develop a general procedure for constructing similar series for the potential derivatives of arbitrary orders. The coefficients of the derivatives are defined by means of recurrence relations in which a coefficient of a certain order derivative is a linear function of two coefficients of a preceding order derivative. The coefficients of the second-order derivatives are also presented as explicit functions of three coefficients of the potential. On the basis of the geopotential model EGM2008, the spherical harmonic coefficients are calculated for the first-, second-, and some third-order derivatives of the disturbing potential T, representing the full potential V, after eliminating from it the zero- and first-degree harmonics. The coefficients of two lowest degrees in the series for the derivatives of T are presented. The corresponding degree variances are estimated. The obtained results can be applied for solving various problems of satellite geodesy and celestial mechanics.     

A new strategy for the atmospheric gravity effect in gravimetric geoid determination

Prior to Stokes integration, the gravitational effect of atmospheric masses must be removed from the gravity anomaly g. One theory for the atmospheric gravity effect on the geoid is the well-known International Association of Geodesy approach in connection with Stokes integral formula. Another strategy is the use of a spherical harmonic representation of the topography, i.e. the use of a global topography computed from a set of spherical harmonics. The latter strategy is improved to account for local information. A new formula is derived by combining the local contribution of the atmospheric effect computed from a detailed digital terrain model and the global contribution computed from a spherical harmonic model of the topography. The new formula is tested over Iran and the results are compared with corresponding results from the old formula which only uses the global information. The results show significant differences. The differences between the two formulas reach 17 cm in a test area in Iran.     

Non-Singular Expressions for the Gravity Gradients in the Local North-Oriented and Orbital Reference Frames

The conventional expansions of the gravity gradients in the local north-oriented reference frame have a complicated form, depending on the first- and second-order derivatives of the associated Legendre functions of the colatitude and containing factors which tend to infinity when approaching the poles. In the present paper, the general term of each of these series is transformed to a product of a geopotential coefficient and a sum of several adjacent Legendre functions of the colatitude multiplied by a function of the longitude. These transformations are performed on the basis of relations between the Legendre functions and their derivatives published by Ilk (1983). The second-order geopotential derivatives corresponding to the local orbital reference frame are presented as linear functions of the north-oriented gravity gradients. The new expansions for the latter are substituted into these functions. As a result, the orbital derivatives are also presented as series depending on the geopotential coefficients multiplied by sums of the Legendre functions whose coefficients depend on the longitude and the satellite track azimuth at an observation point. The derived expansions of the observables can be applied for constructing a geopotential model from the GOCE mission data by the time-wise and space-wise approaches. The numerical experiments demonstrate the correctness of the analytical formulas.An erratum to this article can be found at     

The temporal variation of the spherical and Cartesian multipoles of the gravity field: the generalized MacCullagh representation

 The Cartesian moments of the mass density of a gravitating body and the spherical harmonic coefficients of its gravitational field are related in a peculiar way. In particular, the products of inertia can be expressed by the spherical harmonic coefficients of the gravitational potential as was derived by MacCullagh for a rigid body. Here the MacCullagh formulae are extended to a deformable body which is restricted to radial symmetry in order to apply the Love–Shida hypothesis. The mass conservation law allows a representation of the incremental mass density by the respective excitation function. A representation of an arbitrary Cartesian monome is always possible by sums of solid spherical harmonics multiplied by powers of the radius. Introducing these representations into the definition of the Cartesian moments, an extension of the MacCullagh formulae is obtained. In particular, for excitation functions with a vanishing harmonic coefficient of degree zero, the (diagonal) incremental moments of inertia also can be represented by the excitation coefficients. Four types of excitation functions are considered, namely: (1) tidal excitation; (2) loading potential; (3) centrifugal potential; and (4) transverse surface stress. One application of the results could be model computation of the length-of-day variations and polar motion, which depend on the moments of inertia. Received: 27 July 1999 / Accepted: 24 May 2000     

Global spherical harmonic computation by two-dimensional Fourier methods

A method is presented for performing global spherical harmonic computation by two-dimensional Fourier transformations. The method goes back to old literature (Schuster 1902) and tackles the problem of non-orthogonality of Legendre-functions, when discretized on an equi-angular grid. Both analysis and synthesis relations are presented, which link the spherical harmonic spectrum to a two-dimensional Fourier spectrum. As an alternative, certain functions of co-latitude are introduced, which are orthogonal to discretized Legendre functions. Several independent Fourier approaches for spherical harmonic computation fit into our general scheme.     

全张量重力梯度数据的谱表示方法

在文献「１」的基础上,进一步研究全张量重力梯度数据的全局和局部分量的广义球谐谱表示和轨道根数据表示,并给出了广义球谐函数与球谐函数这间的关系,从理论上得到了全张量重力梯度数据的描述方法和由全张量重力梯度网格数据恢复全球重力位谱系数的基本公式,本文对全张量重力梯度数据的谱表示和谱分析所做的工作,对由重力梯度张量的部分分量恢复全球重力位普系数有一定的参考价值。     

广义球谐函数定积分计算方法的改进

运用球谐函数定积分的基本递推公式，推导了在重力场球谐综合与球谐分析中出现的广义球谐函数定积分的计算公式；给出了其适用于超高阶次的改良型递推公式。数值试验表明，该改良公式具有较高的计算精度和计算速度，解决了超高阶次广义球谐函数定积分计算的溢出问题，拓展了这类定积分的计算公式。他们的数值实现为利用位模型计算高分辨率扰动重力场元格网平均值、重力场球谐综合分析等奠定了基础。     

Ellipsoidal corrections to order e 2 of geopotential coefficients and Stokes’ formula

Assuming that the gravity anomaly and disturbing potential are given on a reference ellipsoid, the result of Sjöberg (1988, Bull Geod 62:93–101) is applied to derive the potential coefficients on the bounding sphere of the ellipsoid to order e ² (i.e. the square of the eccentricity of the ellipsoid). By adding the potential coefficients and continuing the potential downward to the reference ellipsoid, the spherical Stokes formula and its ellipsoidal correction are obtained. The correction is presented in terms of an integral over the unit sphere with the spherical approximation of geoidal height as the argument and only three well-known kernel functions, namely those of Stokes, Vening-Meinesz and the inverse Stokes, lending the correction to practical computations. Finally, the ellipsoidal correction is presented also in terms of spherical harmonic functions. The frequently applied and sometimes questioned approximation of the constant m, a convenient abbreviation in normal gravity field representations, by e ²/2, as introduced by Moritz, is also discussed. It is concluded that this approximation does not significantly affect the ellipsoidal corrections to potential coefficients and Stokes formula. However, whether this standard approach to correct the gravity anomaly agrees with the pure ellipsoidal solution to Stokes formula is still an open question.     

椭球谐和球谐系数之间一个简单的转换关系

本文推导的椭球谐系数和球谐系数相互之间转换关系的核心思想是在ε~2量级下利用Legendre函数的正交性,从球谐系数求解的积分表示出发,将积分中的椭球坐标变量与球坐标变量相互转换,从而得出椭球谐系数与球谐系数之间的转换关系。本文导出的转换关系有以下优点:①对于第二类Legendre函数的计算采用Laurent级数表示,使计算第二类Legendre函数更为简单;②保留了ε~2量级下,导出的转换关系相比文献[2]的形式更简单,满足物理大地测量边值问题线性化的要求;③顾及了余纬和归化余纬的区别。     

Ellipsoidal spectral properties of the Earth’s gravitational potential and its first and second derivatives

The spectral analysis of the Earth’s gravitational potential, its first and second derivatives is performed in spherical/ellipsoidal harmonics relative to the International Reference Sphere/International Reference Ellipsoid. The highlights of the diagrammatic approach are: (1) Up- and downward continuation of incremental gravity gradients, (2) Downward continuation of incremental gravity gradients (four tensor-valued harmonic functions) to the incremental gravity potential on the International Reference Figure, (3) Direct conversion of external incremental gravity gradients to geoidal undulation by means of the spherical/ellipsoidal Bruns Formula. The International Reference Ellipsoid was chosen as an equipotential surface in the Somigliana-Pizzetti reference potential field.     

New relations among associated Legendre functions and spherical harmonics

Several new relations among associated Legendre functions (ALFs) are derived, most of which relate a product of an ALF with trigonometric functions to a weighted summation over ALFs, where the weights only depend on the degree and order of the ALF. These relations are, for example, useful in applications such as the computation of geopotential coefficients and computation of ellipsoidal corrections in geoid modelling. The main relations are presented in both their unnormalised and fully normalised (4π-normalised) form. Several approaches to compute the weights involved are discussed, and it is shown that the relations can also be applied in the case of first- and second-order derivatives of ALFs, which may be of use in analysis of satellite gradiometry data. Finally, the derived relations are combined to provide new identities among ALFs, which contain no dependency on the colatitudinal coordinate other than that in the ALFs themselves.     

Short guide to direct gravitational field modelling with Hotine’s equations

This paper presents a unified approach to the least squares spherical harmonic analysis of the acceleration vector and Eötvös tensor (gravitational gradients) in an arbitrary orientation. The Jacobian matrices are based on Hotine’s equations that hold in the Earth-fixed Cartesian frame and do not need any derivatives of the associated Legendre functions. The implementation was confirmed through closed-loop tests in which the simulated input is inverted in the least square sense using the rotated Hotine’s equations. The precision achieved is at the level of rounding error with RMS about $10^{-12}{-}10^{-14}$  m in terms of the height anomaly. The second validation of the linear model is done with help from the standard ellipsoidal correction for the gravity disturbance that can be computed with an analytic expression as well as with the rotated equations. Although the analytic expression for this correction is only of a limited accuracy at the submillimeter level, it was used for an independent validation. Finally, the equivalent of the ellipsoidal correction, called the effect of the normal, has been numerically obtained also for other gravitational functionals and some of their combinations. Most of the numerical investigations are provided up to spherical harmonic degree 70, with degree 80 for the computation time comparison using real GRACE data. The relevant Matlab source codes for the design matrices are provided.