Methods of harmonic synthesis for global geopotential models and their first-, second- and third-order gradients

Four widely used algorithms for the computation of the Earth’s gravitational potential and its first-, second- and third-order gradients are examined: the traditional increasing degree recursion in associated Legendre functions and its variant based on the Clenshaw summation, plus the methods of Pines and Cunningham–Metris, which are free from the singularities that distinguish the first two methods at the geographic poles. All four methods are reorganized with the lumped coefficients approach, which in the cases of Pines and Cunningham–Metris requires a complete revision of the algorithms. The characteristics of the four methods are studied and described, and numerical tests are performed to assess and compare their precision, accuracy, and efficiency. In general the performance levels of all four codes exhibit large improvements over previously published versions. From the point of view of numerical precision, away from the geographic poles Clenshaw and Legendre offer an overall better quality. Furthermore, Pines and Cunningham–Metris are affected by an intrinsic loss of precision at the equator and suffer from additional deterioration when the gravity gradients components are rotated into the East-North-Up topocentric reference system. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

On the evaluation of the gravity effects of polyhedral bodies and a consistent treatment of related singularities

We show that the singularities which can affect the computation of the gravity effects (potential, gravity and tensor gradient fields) can be systematically addressed by invoking distribution theory and suitable formulas of differential calculus. Thus, differently from previous contributions on the subject, the use of a-posteriori corrections of the formulas derived in absence of singularities can be ruled out. The general approach presented in the paper is further specialized to the case of polyhedral bodies and detailed for a rectangular prism having a constant mass density. With reference to this last case, we derive novel expressions for the related gravitational field, as well as for its first and second derivative, at an observation point coincident with a prism vertex and show that they turn out to be more compact than the ones reported in the specialized literature.     

Satellite geodesy on curved space–time manifolds. Earth-fixed charts

The reformulation of geodetic measurement processes within the framework of general relativity is discussed. The metric tensor plays an important role in general relativity and has to be represented with respect to a set of appropriate charts. Almost every quantity of interest in geodetic or geophysical applications refers to a geocentric, Earth-fixed coordinate system (chart), therefore they are of great importance in geodesy and geophysics. The space–time metric with respect to an Earth-fixed chart is derived at first post-Newtonian order. The field equations determining the terrestrial gravitational field are derived and its explicit representation is outlined. The impact of the results on the modelling of geodetic measurement processes including space–time positioning scenarios as well as the high-precision gravitational field estimation is outlined. Received: 7 January 1998 / Accepted: 17 August 1999     

Methodology and use of tensor invariants for satellite gravity gradiometry

Although its use is widespread in several other scientific disciplines, the theory of tensor invariants is only marginally adopted in gravity field modeling. We aim to close this gap by developing and applying the invariants approach for geopotential recovery. Gravitational tensor invariants are deduced from products of second-order derivatives of the gravitational potential. The benefit of the method presented arises from its independence of the gradiometer instrument’s orientation in space. Thus, we refrain from the classical methods for satellite gravity gradiometry analysis, i.e., in terms of individual gravity gradients, in favor of the alternative invariants approach. The invariants approach requires a tailored processing strategy. Firstly, the non-linear functionals with regard to the potential series expansion in spherical harmonics necessitates the linearization and iterative solution of the resulting least-squares problem. From the computational point of view, efficient linearization by means of perturbation theory has been adopted. It only requires the computation of reference gravity gradients. Secondly, the deduced pseudo-observations are composed of all the gravitational tensor elements, all of which require a comparable level of accuracy. Additionally, implementation of the invariants method for large data sets is a challenging task. We show the fundamentals of tensor invariants theory adapted to satellite gradiometry. With regard to the GOCE (Gravity field and steady-state Ocean Circulation Explorer) satellite gradiometry mission, we demonstrate that the iterative parameter estimation process converges within only two iterations. Additionally, for the GOCE configuration, we show the invariants approach to be insensitive to the synthesis of unobserved gravity gradients.     

Second-order derivatives in a local frame developed in spheroidal and spherical coordinates

Second-order derivatives of a general scalar function of position (F) with respect to the length elements along a family of local Cartesian axes are developed in the spheroidal and spherical coordinate systems. A link between the two kinds of formulations is established when the results in spherical coordinates are confirmed also indirectly, through a transformation from spheroidal coordinates. IfF becomesW (earth's potential) the six distinct second-order derivatives—which include one vertical and two horizontal gradients of gravity—relate the symmetric Marussi tensor to the curvature parameters of the field. The general formulas for the second-order derivatives ofF are specialized to yield the second-order derivatives ofU (standard potential) and ofT (disturbing potential), which allows the latter to be modeled by a suitable set of parameters. The second-order derivatives ofT in which the property ΔT=0 is explicitly incorporated are also given. According to the required precision, the spherical approximation may or may not be desirable; both kinds of results are presented. The derived formulas can be used for modeling of the second-order derivatives ofW orT at the ground level as well as at higher altitudes. They can be further applied in a rotating or a nonrotating field. The development in this paper is based on the tensor approach to theoretical geodesy, introduced by Marussi [1951] and further elaborated by Hotine [1969], which can lead to significantly shorter demonstrations when compared to conventional approaches.     

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.     

Geodesy and relativity

Relativity, or gravitational physics, has widely entered geodetic modelling and parameter determination. This concerns, first of all, the fundamental reference systems used. The Barycentric Celestial Reference System (BCRS) has to be distinguished carefully from the Geocentric Celestial Reference System (GCRS), which is the basic theoretical system for geodetic modelling with a direct link to the International Terrestrial Reference System (ITRS), simply given by a rotation matrix. The relation to the International Celestial Reference System (ICRS) is discussed, as well as various properties and relevance of these systems. Then the representation of the gravitational field is discussed when relativity comes into play. Presently, the so-called post-Newtonian approximation to GRT (general relativity theory) including relativistic effects to lowest order is sufficient for practically all geodetic applications. At the present level of accuracy, space-geodetic techniques like VLBI (Very Long Baseline Interferometry), GPS (Global Positioning System) and SLR/LLR (Satellite/Lunar Laser Ranging) have to be modelled and analysed in the context of a post-Newtonian formalism. In fact, all reference and time frames involved, satellite and planetary orbits, signal propagation and the various observables (frequencies, pulse travel times, phase and travel-time differences) are treated within relativity. This paper reviews to what extent the space-geodetic techniques are affected by such a relativistic treatment and where—vice versa—relativistic parameters can be determined by the analysis of geodetic measurements. At the end, we give a brief outlook on how new or improved measurement techniques (e.g., optical clocks, Galileo) may further push relativistic parameter determination and allow for refined geodetic measurements.     

Integral transformations of gradiometric data onto a GRACE type of observable

Integral transformations of gravitational gradients onto a Gravity Recovery And Climate Experiment (GRACE) type of observable are derived in this article. The gravitational gradients represent components of the gravitational tensor in the local north-oriented frame. The GRACE type of observable corresponds to a difference between two gravitational vectors as projected onto the line of sight between the two GRACE satellites. In total, three integral transformations relating vertical–vertical, vertical–horizontal and horizontal–horizontal gravitational gradients with the GRACE type of observable are provided. Spectral and closed forms of corresponding isotropic kernels are derived for each transformation. Special cases show that the integral transformations are general and relate gravitational gradients to many other quantities of the gravitational field, such as the gravitational vector, and its radial and tangential components. Correctness of the mathematical derivations is validated in a closed-loop simulation using synthetic data.     

The spacetime gravitational field of a deformable body

The high-resolution analysis of orbit perturbations of terrestrial artificial satellites has documented that the eigengravitation of a massive body like the Earth changes in time, namely with periodic and aperiodic constituents. For the space-time variation of the gravitational field the action of internal and external volume as well as surface forces on a deformable massive body are responsible. Free of any assumption on the symmetry of the constitution of the deformable body we review the incremental spatial (“Eulerian”) and material (“Lagrangean”) gravitational field equations, in particular the source terms (two constituents: the divergence of the displacement field as well as the projection of the displacement field onto the gradient of the reference mass density function) and the `jump conditions' at the boundary surface of the body as well as at internal interfaces both in linear approximation. A spherical harmonic expansion in terms of multipoles of the incremental Eulerian gravitational potential is presented. Three types of spherical multipoles are identified, namely the dilatation multipoles, the transport displacement multipoles and those multipoles which are generated by mass condensation onto the boundary reference surface or internal interfaces. The degree-one term has been identified as non-zero, thus as a “dipole moment” being responsible for the varying position of the deformable body's mass centre. Finally, for those deformable bodies which enjoy a spherically symmetric constitution, emphasis is on the functional relation between Green functions, namely between Fourier-/ Laplace-transformed volume versus surface Love-Shida functions (h(r),l(r) versus h ^′(r),l ^′(r)) and Love functions k(r) versus k ^′(r). The functional relation is numerically tested for an active tidal force/potential and an active loading force/potential, proving an excellent agreement with experimental results. Received: December 1995 / Accepted: 1 February 1997     

Accurate computation of gravitational field of a tesseroid

We developed an accurate method to compute the gravitational field of a tesseroid. The method numerically integrates a surface integral representation of the gravitational potential of the tesseroid by conditionally splitting its line integration intervals and by using the double exponential quadrature rule. Then, it evaluates the gravitational acceleration vector and the gravity gradient tensor by numerically differentiating the numerically integrated potential. The numerical differentiation is conducted by appropriately switching the central and the single-sided second-order difference formulas with a suitable choice of the test argument displacement. If necessary, the new method is extended to the case of a general tesseroid with the variable density profile, the variable surface height functions, and/or the variable intervals in longitude or in latitude. The new method is capable of computing the gravitational field of the tesseroid independently on the location of the evaluation point, namely whether outside, near the surface of, on the surface of, or inside the tesseroid. The achievable precision is 14–15 digits for the potential, 9–11 digits for the acceleration vector, and 6–8 digits for the gradient tensor in the double precision environment. The correct digits are roughly doubled if employing the quadruple precision computation. The new method provides a reliable procedure to compute the topographic gravitational field, especially that near, on, and below the surface. Also, it could potentially serve as a sure reference to complement and elaborate the existing approaches using the Gauss–Legendre quadrature or other standard methods of numerical integration.     

Wavelet Modeling of Regional and Temporal Variations of the Earth’s Gravitational Potential Observed by GRACE

This work is dedicated to the wavelet modeling of regional and temporal variations of the Earth’s gravitational potential observed by the GRACE (gravity recovery and climate experiment) satellite mission. In the first part, all required mathematical tools and methods involving spherical wavelets are provided. Then, we apply our method to monthly GRACE gravity fields. A strong seasonal signal can be identified which is restricted to areas where large-scale redistributions of continental water mass are expected. This assumption is analyzed and verified by comparing the time-series of regionally obtained wavelet coefficients of the gravitational signal originating from hydrology models and the gravitational potential observed by GRACE. The results are in good agreement with previous studies and illustrate that wavelets are an appropriate tool to investigate regional effects in the Earth’s gravitational field. Electronic Supplementary Material Supplementary material is available for this article at     

The gravitational potential and its derivatives for the prism

 As a simple building block, the right rectangular parallelepiped (prism) has an important role mostly in local gravity field modelling studies when the so called flat-Earth approximation is sufficient. Its primary (methodological) advantage follows from the simplicity of the rigorous and consistent analytical forms describing the different gravitation-related quantities. The analytical forms provide numerical values for these quantities which satisfy the functional connections existing between these quantities at the level of numerical precision applied. Closed expressions for the gravitational potential of the prism and its derivatives (up to the third order) are listed for easy reference. Received: 18 August 1999 / Accepted: 15 June 2000     

GOCE gravitational gradients along the orbit

GOCE is ESA’s gravity field mission and the first satellite ever that measures gravitational gradients in space, that is, the second spatial derivatives of the Earth’s gravitational potential. The goal is to determine the Earth’s mean gravitational field with unprecedented accuracy at spatial resolutions down to 100 km. GOCE carries a gravity gradiometer that allows deriving the gravitational gradients with very high precision to achieve this goal. There are two types of GOCE Level 2 gravitational gradients (GGs) along the orbit: the gravitational gradients in the gradiometer reference frame (GRF) and the gravitational gradients in the local north oriented frame (LNOF) derived from the GGs in the GRF by point-wise rotation. Because the V _XX , V _YY , V _ZZ and V _XZ are much more accurate than V _XY and V _YZ , and because the error of the accurate GGs increases for low frequencies, the rotation requires that part of the measured GG signal is replaced by model signal. However, the actual quality of the gradients in GRF and LNOF needs to be assessed. We analysed the outliers in the GGs, validated the GGs in the GRF using independent gravity field information and compared their assessed error with the requirements. In addition, we compared the GGs in the LNOF with state-of-the-art global gravity field models and determined the model contribution to the rotated GGs. We found that the percentage of detected outliers is below 0.1% for all GGs, and external gravity data confirm that the GG scale factors do not differ from one down to the 10⁻³ level. Furthermore, we found that the error of V _XX and V _YY is approximately at the level of the requirement on the gravitational gradient trace, whereas the V _ZZ error is a factor of 2–3 above the requirement for higher frequencies. We show that the model contribution in the rotated GGs is 2–35% dependent on the gravitational gradient. Finally, we found that GOCE gravitational gradients and gradients derived from EIGEN-5C and EGM2008 are consistent over the oceans, but that over the continents the consistency may be less, especially in areas with poor terrestrial gravity data. All in all, our analyses show that the quality of the GOCE gravitational gradients is good and that with this type of data valuable new gravity field information is obtained.     

On the non-linear geodetic boundary value problem for a fixed boundary surface

Summary The fixed gravimetric boundary value problem of Physical Geodesy is a nonlinear, oblique derivative problem. Expanding the non-linear boundary condition into a Taylor series—based upon some reference potential field approximating the geopotential—it is shown that the numerical convergence of this series is very rapid; only the quadratic term may have some practical impact on the solution. The secondorder solution term can be described by a spherical integral formula involving the deflections of the vertical with respect to the reference field. The influence of nonlinear terms on the figure of the level surfaces (e.g. the geoid) is roughly estimated to have an order of magnitude of some few centimetres, based upon a Somigliana-Pizzetti reference field; if on the other hand some high-degree geopotential model is used as reference then the effects by non-linearity are negligible from a practical point of view.     

W0: an estimate in the Finnish Height Datum N60, epoch 1993.4, from twenty-five GPS points of the Baltic Sea Level Project

Based upon a data set of 25 points of the Baltic Sea Level Project, second campaign 1993.4, which are close to mareographic stations, described by (1) GPS derived Cartesian coordinates in the World Geodetic Reference System 1984 and (2) orthometric heights in the Finnish Height Datum N60, epoch 1993.4, we have computed the primary geodetic parameter W ₀(1993.4) for the epoch 1993.4 according to the following model. The Cartesian coordinates of the GPS stations have been converted into spheroidal coordinates. The gravity potential as the additive decomposition of the gravitational potential and the centrifugal potential has been computed for any GPS station in spheroidal coordinates, namely for a global spheroidal model of the gravitational potential field. For a global set of spheroidal harmonic coefficients a transformation of spherical harmonic coefficients into spheroidal harmonic coefficients has been implemented and applied to the global spherical model OSU 91A up to degree/order 360/360. The gravity potential with respect to a global spheroidal model of degree/order 360/360 has been finally transformed by means of the orthometric heights of the GPS stations with respect to the Finnish Height Datum N60, epoch 1993.4, in terms of the spheroidal “free-air” potential reduction in order to produce the spheroidal W ₀(1993.4) value. As a mean of those 25 W ₀(1993.4) data as well as a root mean square error estimation we computed W ₀(1993.4)=(6 263 685.58 ± 0.36) kgal × m. Finally a comparison of different W ₀ data with respect to a spherical harmonic global model and spheroidal harmonic global model of Somigliana-Pizetti type (level ellipsoid as a reference, degree/order 2/0) according to The Geodesist's Handbook 1992 has been made. Received: 7 November 1996 / Accepted: 27 March 1997     

Relation between geoidal undulation, deflection of the vertical and vertical gravity gradient revisited

The vertical gradients of gravity anomaly and gravity disturbance can be related to horizontal first derivatives of deflection of the vertical or second derivatives of geoidal undulations. These are simplified relations of which different variations have found application in satellite altimetry with the implicit assumption that the neglected terms—using remove-restore—are sufficiently small. In this paper, the different simplified relations are rigorously connected and the neglected terms are made explicit. The main neglected terms are a curvilinear term that accounts for the difference between second derivatives in a Cartesian system and on a spherical surface, and a small circle term that stems from the difference between second derivatives on a great and small circle. The neglected terms were compared with the dynamic ocean topography (DOT) and the requirements on the GOCE gravity gradients. In addition, the signal root-mean-square (RMS) of the neglected terms and vertical gravity gradient were compared, and the effect of a remove-restore procedure was studied. These analyses show that both neglected terms have the same order of magnitude as the DOT gradient signal and may be above the GOCE requirements, and should be accounted for when combining altimetry derived and GOCE measured gradients. The signal RMS of both neglected terms is in general small when compared with the signal RMS of the vertical gravity gradient, but they may introduce gradient errors above the spherical approximation error. Remove-restore with gravity field models reduces the errors in the vertical gravity gradient, but it appears that errors above the spherical approximation error cannot be avoided at individual locations. When computing the vertical gradient of gravity anomaly from satellite altimeter data using deflections of the vertical, the small circle term is readily available and can be included. The direct computation of the vertical gradient of gravity disturbance from satellite altimeter data is more difficult than the computation of the vertical gradient of gravity anomaly because in the former case the curvilinear term is needed, which is not readily available.     

Gravity gradient modeling using gravity and DEM

A model of the gravity gradient tensor at aircraft altitude is developed from the combination of ground gravity anomaly data and a digital elevation model. The gravity data are processed according to various operational solutions to the boundary-value problem (numerical integration of Stokes’ integral, radial-basis splines, and least-squares collocation). The terrain elevation data are used to reduce free-air anomalies to the geoid and to compute a corresponding indirect effect on the gradients at altitude. We compare the various modeled gradients to airborne gradiometric data and find differences of the order of 10–20 E (SD) for all gradient tensor elements. Our analysis of these differences leads to a conclusion that their source may be primarily measurement error in these particular gradient data. We have thus demonstrated the procedures and the utility of combining ground gravity and elevation data to validate airborne gradiometer systems.     

Analytical methods for error propagation in planar space–time prisms

The space–time prism demarcates all locations in space–time that a mobile object or person can occupy during an episode of potential or unobserved movement. The prism is central to time geography as a measure of potential mobility and to mobile object databases as a measure of location possibilities given sampling error. This paper develops an analytical approach to assessing error propagation in space–time prisms and prism–prism intersections. We analyze the geometry of the prisms to derive a core set of geometric problems involving the intersection of circles and ellipses. Analytical error propagation techniques such as the Taylor linearization method based on the first-order partial derivatives are not available since explicit functions describing the intersections and their derivatives are unwieldy. However, since we have implicit functions describing prism geometry, we modify this approach using an implicit function theorem that provides the required first-order partials without the explicit expressions. We describe the general method as well as details for the two spatial dimensions case and provide example calculations.     

SGG重力场球谐系数正则解的误差估计方法

重力梯度为重力位的二阶导数,可以通过星载梯度仪进行观测。重力场球谐函数系数可以通过正则化方法由重力梯度算出。本文在对正则化方法分析的基础上提出了估计球谐函数系数正则解误差的方法,为我国今后发射重力梯度卫星提供技术准备。     

Effects of topographic–isostatic masses on gravitational functionals at the Earth’s surface and at airborne and satellite altitudes

Topographic–isostatic masses represent an important source of gravity field information, especially in the high-frequency band, even if the detailed mass-density distribution inside the topographic masses is unknown. If this information is used within a remove-restore procedure, then the instability problems in downward continuation of gravity observations from aircraft or satellite altitudes can be reduced. In this article, integral formulae are derived for determination of gravitational effects of topographic–isostatic masses on the first- and second-order derivatives of the gravitational potential for three topographic–isostatic models. The application of these formulas is useful for airborne gravimetry/gradiometry and satellite gravity gradiometry. The formulas are presented in spherical approximation by separating the 3D integration in an analytical integration in the radial direction and 2D integration over the mean sphere. Therefore, spherical volume elements can be considered as being approximated by mass-lines located at the centre of the discretization compartments (the mass of the tesseroid is condensed mathematically along its vertical axis). The errors of this approximation are investigated for the second-order derivatives of the topographic–isostatic gravitational potential in the vicinity of the Earth’s surface. The formulas are then applied to various scenarios of airborne gravimetry/gradiometry and satellite gradiometry. The components of the gravitational vector at aircraft altitudes of 4 and 10 km have been determined, as well as the gravitational tensor components at a satellite altitude of 250 km envisaged for the forthcoming GOCE (gravity field and steady-state ocean-circulation explorer) mission. The numerical computations are based on digital elevation models with a 5-arc-minute resolution for satellite gravity gradiometry and 1-arc-minute resolution for airborne gravity/gradiometry.