Non-singular expression for the gradient of the Earth gravitational potential in the local north-oriented ellipsoidal reference frame

In this contribution we continue our earlier research, concerning the ellipsoidal harmonic expansions of the Earth disturbing gravitational potential and its derivatives on an external reference ellipsoid confocal with respect to the normal ellipsoid and close to it. One of the results of the previous investigation is represented by a new expression for the derivative of the Jekeli’s Legendre function of the second kind, entering the ellipsoidal harmonics in the potential derivative. The derived expression depends on two Gauss hypergeometric functions which converge better than the hypergeometric functions of other authors. In the present paper we construct another expression for the derivative of the Jekeli’s Legendre function, depending on two alternative hypergeometric functions. While our earlier hypergeometric series in the expression for the derivative of this function converge better when the orders of the terms do not exceed a half of their degrees, the series constructed in the present paper converge more rapidly when the orders surpass a half of the degrees. We deduce an improved expression for the derivative of the Jekeli’s Legendre function by combining these results and then construct a corresponding new expression for the derivative of the disturbing potential. This expression is applied for constructing non-singular expressions for the components of the gradient of the potential in the local north-oriented ellipsoidal reference frame. The new expressions for these components have no these deficiencies and the expression for the potential gradient depends on very quickly convergent hypergeometric series.     

Improved expressions for the ellipsoidal harmonic series representing the Earth gravitational potential and its first and second derivatives on and outside the reference ellipsoid of revolution

The Legendre functions of the second kind, renormalized by Jekeli, are considered in the external space on a set of ellipsoids of revolution which are confocal with respect to the normal ellipsoid. Among these ellipsoids a reference one is chosen which bounds the Earth. New expressions for the first and second order derivatives of the Legendre functions are derived. They depend on two very quickly convergent Gauss hypergeometric series which are obtained by transforming the slowly convergent initial hypergeometric series. The derived expressions are applied for constructing the ellipsoidal harmonic series for the Earth disturbing gravitational potential and its derivatives of the first and second orders. Since outside the chosen reference ellipsoid there are no Earth masses (as compared to the normal ellipsoid) then it is more appropriate for constructing the boundary-value equation and solving it on the basis of surface gravity data reduced to this ellipsoid.     

Alternative expressions for gravity gradients in local north-oriented frame and tensor spherical harmonics

The traditional expressions for gravity gradients in local north-oriented frame and tensor spherical harmonics have complicated forms involved with first- and second-order derivatives of spherical harmonics and also singular terms. In this paper we present alternative expressions for these quantities, which are simpler and contain no singular terms. The presented formulas are useful for those disciplines of geosciences which are involved with potential theory, tensor spherical harmonics and second-order derivatives of spherical harmonic series in the local north-oriented frame. A simple numerical test on the solution of the gradiometric boundary value problems presents the correctness of these new expressions and ability of the solutions to continue the gravity gradients from satellite level down to sea level using spherical harmonics.     

Topographic and atmospheric effects on goce gradiometric data in a local north-oriented frame: A case study in Fennoscandia and Iran

Satellite gradiometry is an observation technique providing data that allow for evaluation of Stokes’ (geopotential) coefficients. This technique is capable of determining higher degrees/orders of the geopotential coefficients than can be achieved by traditional dynamic satellite geodesy. The satellite gradiometry data include topographic and atmospheric effects. By removing those effects, the satellite data becomes smoother and harmonic outside sea level and therefore more suitable for downward continuation to the Earth’s surface. For example, in this way one may determine a set of spherical harmonics of the gravity field that is harmonic in the exterior to sea level. This article deals with the above effects on the satellite gravity gradients in the local north-oriented frame. The conventional expressions of the gradients in this frame have a rather complicated form, depending on the first-and second-order derivatives of the associated Legendre functions, which contain singular factors when approaching the poles. On the contrary, we express the harmonic series of atmospheric and topographic effects as non-singular expressions. The theory is applied to the regions of Fennoscandia and Iran, where maps of such effects and their statistics are presented and discussed.     

Ellipsoidal vertical deflections and ellipsoidal gravity disturbance: Case studies

First, we present three different definitions of the vertical which relate to (i) astronomical longitude and astronomical latitude as spherical coordinates in gravity space, (ii) Gauss surface normal coordinates (also called geodetic coordinates) of type ellipsoidal longitude and ellipsoidal latitude and (iii) Jacobi ellipsoidal coordinates of type spheroidal longitude and spheroidal latitude in geometry space. Up to terms of second order those vertical deflections agree to each other. Vertical deflections and gravity disturbances relate to a reference gravity potential. In order to refer the horizontal and vertical components of the disturbing gravity field to a reference gravity field, which is physically meaningful, we have chosen the Somigliana-Pizzetti gravity potential as well as its gradient. Second, we give a new closed-form representation of Somigliana-Pizzetti gravity, accurate to the sub Nano Gal level. Third, we represent the gravitational disturbing potential in terms of Jacobi ellipsoidal harmonics. As soon as we take reference to a normal potential of Somigliana-Pizzetti type, the ellipsoidal harmonics of degree/order (0,0), (1,0), (1, − 1), (1,1) and (2,0) are eliminated from the gravitational disturbing potential. Fourth, we compute in all detail the gradient of the gravitational disturbing potential, in particular in orthonormal ellipsoidal vector harmonics. Proper weighting functions for orthonormality on the International Reference Ellipsoid are constructed and tabulated. In this way, we finally arrive at an ellipsoidal harmonic representation of vertical deflections and gravity disturbances. Fifth, for an ellipsoidal harmonic Gravity Earth Model (SEGEN: http://www.uni-stuttgart.de/gi/research/paper/coefficients/coefficients.zip) up to degree/order 360/360 we compute the global maps of ellipsoidal vertical deflections and ellipsoidal gravity disturbances which transfer a great amount of geophysical information in a properly chosen equiareal ellipsoidal map projection.     

Layer-Based Modelling of the Earth’s Gravitational Potential up to 10-km Scale in Spherical Harmonics in Spherical and Ellipsoidal Approximation

Global forward modelling of the Earth’s gravitational potential, a classical problem in geophysics and geodesy, is relevant for a range of applications such as gravity interpretation, isostatic hypothesis testing or combined gravity field modelling with high and ultra-high resolution. This study presents spectral forward modelling with volumetric mass layers to degree 2190 for the first time based on two different levels of approximation. In spherical approximation, the mass layers are referred to a sphere, yielding the spherical topographic potential. In ellipsoidal approximation where an ellipsoid of revolution provides the reference, the ellipsoidal topographic potential (ETP) is obtained. For both types of approximation, we derive a mass layer concept and study it with layered data from the Earth2014 topography model at 5-arc-min resolution. We show that the layer concept can be applied with either actual layer density or density contrasts w.r.t. a reference density, without discernible differences in the computed gravity functionals. To avoid aliasing and truncation errors, we carefully account for increased sampling requirements due to the exponentiation of the boundary functions and consider all numerically relevant terms of the involved binominal series expansions. The main outcome of our work is a set of new spectral models of the Earth’s topographic potential relying on mass layer modelling in spherical and in ellipsoidal approximation. We compare both levels of approximations geometrically, spectrally and numerically and quantify the benefits over the frequently used rock-equivalent topography (RET) method. We show that by using the ETP it is possible to avoid any displacement of masses and quantify also the benefit of mapping-free modelling. The layer-based forward modelling is corroborated by GOCE satellite gradiometry, by in-situ gravity observations from recently released Antarctic gravity anomaly grids and degree correlations with spectral models of the Earth’s observed geopotential. As the main conclusion of this work, the mass layer approach allows more accurate modelling of the topographic potential because it avoids 10–20-mGal approximation errors associated with RET techniques. The spherical approximation is suited for a range of geophysical applications, while the ellipsoidal approximation is preferable for applications requiring high accuracy or high resolution.     

A new ellipsoidal gravimetric, satellite altimetry and astronomic boundary value problem, a case study: The geoid of Iran

A new gravimetric, satellite altimetry, astronomical ellipsoidal boundary value problem for geoid computations has been developed and successfully tested. This boundary value problem has been constructed for gravity observables of the type (i) gravity potential, (ii) gravity intensity (i.e. modulus of gravity acceleration), (iii) astronomical longitude, (iv) astronomical latitude and (v) satellite altimetry observations. The ellipsoidal coordinates of the observation points have been considered as known quantities in the set-up of the problem in the light of availability of GPS coordinates. The developed boundary value problem is ellipsoidal by nature and as such takes advantage of high precision GPS observations in the set-up. The algorithmic steps of the solution of the boundary value problem are as follows:

- Application of the ellipsoidal harmonic expansion complete up to degree and order 360 and of the ellipsoidal centrifugal field for the removal of the effect of global gravity and the isostasy field from the gravity intensity and the astronomical observations at the surface of the Earth.

- Application of the ellipsoidal Newton integral on the multi-cylindrical equal-area map projection surface for the removal from the gravity intensity and the astronomical observations at the surface of the Earth the effect of the residual masses at the radius of up to 55 km from the computational point.

- Application of the ellipsoidal harmonic expansion complete up to degree and order 360 and ellipsoidal centrifugal field for the removal from the geoidal undulations derived from satellite altimetry the effect of the global gravity and isostasy on the geoidal undulations.

- Application of the ellipsoidal Newton integral on the multi-cylindrical equal-area map projection surface for the removal from the geoidal undulations derived from satellite altimetry the effect of the water masses outside the reference ellipsoid within a radius of 55 km around the computational point.

- Least squares solution of the observation equations of the incremental quantities derived from aforementioned steps in order to obtain the incremental gravity potential at the surface of the reference ellipsoid.

- The removed effects at the application points are restored on the surface of reference ellipsoid.

- Application of the ellipsoidal Bruns’ formula for converting the potential values on the surface of the reference ellipsoid into the geoidal heights with respect to the reference ellipsoid.

- Computation of the geoid of Iran has successfully tested this new methodology.

Spheroidal Integral Equations for Geodetic Inversion of Geopotential Gradients

The static Earth’s gravitational field has traditionally been described in geodesy and geophysics by the gravitational potential (geopotential for short), a scalar function of 3-D position. Although not directly observable, geopotential functionals such as its first- and second-order gradients are routinely measured by ground, airborne and/or satellite sensors. In geodesy, these observables are often used for recovery of the static geopotential at some simple reference surface approximating the actual Earth’s surface. A generalized mathematical model is represented by a surface integral equation which originates in solving Dirichlet’s boundary-value problem of the potential theory defined for the harmonic geopotential, spheroidal boundary and globally distributed gradient data. The mathematical model can be used for combining various geopotential gradients without necessity of their re-sampling or prior continuation in space. The model extends the apparatus of integral equations which results from solving boundary-value problems of the potential theory to all geopotential gradients observed by current ground, airborne and satellite sensors. Differences between spherical and spheroidal formulations of integral kernel functions of Green’s kind are investigated. Estimated differences reach relative values at the level of 3% which demonstrates the significance of spheroidal approximation for flattened bodies such as the Earth. The observation model can be used for combined inversion of currently available geopotential gradients while exploring their spectral and stochastic characteristics. The model would be even more relevant to gravitational field modelling of other bodies in space with more pronounced spheroidal geometry than that of the Earth.     

Some space gravity formulas and the dimensions and the mass of the Earth

Summary Adopting thePizzetti-Somigliana method and using elliptic integrals we have obtained closed formulas for the space gravity field in which one of the equipotential surfaces is a triaxial ellipsoid. The same formulas are also obtained in first approximation of the equatorial flattening avoiding the use of the elliptic integrals. Using data from satellites and Earth gravity data the gravitational and geometric bulge of the Earth's equator are computed. On the basis of these results and on the basis of recent gravity data taken around the equator between the longitudes 50° to 100° E, 155° to 180° E, and 145° to 180° W, we question the advantage of using a triaxial gravity formula and a triaxial ellipsoid in geodesy. Closed formulas for the space field in which a biaxial ellipsoid is an equipotential surface are also derived in polar coordinates and its parameters are specialized to give the international gravity formula values on the international ellipsoid. The possibility to compute the Earth's dimensions from the present Earth gravity data is the discussed and the value ofMG=(3.98603×10²⁰ cm³ sec^–2) (M mass of the Earth,G gravitational constant) is computed. The agreement of this value with others computed from the mean distance Earth-Moon is discussed. The Legendre polinomials series expansion of the gravitational potential is also added. In this series the coefficients of the polinomials are closed formulas in terms of the flattening andMG.Publication Number 327, and Istituto di Geodesia e Geofisica of Università di Trieste.     

Gravitational Gradients at Satellite Altitudes in Global Geophysical Studies

Due to the ESA’s satellite mission GOCE launched in March 2009, gravitational gradients sampled along the orbital trajectory approximately 250 km above the Earth’s surface have become available. Since 2010, gravitational gradients have routinely been applied in geodesy for the derivation of global Earth’s gravitational models provided in terms of fully normalized coefficients in a spherical harmonic series representation of the Earth’s gravitational potential. However, in geophysics, gravitational gradients observed by spaceborne instruments have still been applied relatively seldom. This contribution describes their possible geophysical applications in structural studies where gravitational gradients observed at satellite altitudes are compared with those derived by a spectral forward modeling technique using available global models of selected Earth’s mass components as input data. In particular, GOCE gravitational gradients are interpreted in terms of a superposition principle of gravitation as combined gravitational effects generated by a homogeneous reference ellipsoid of revolution, mean topographic and ice mass density distributions, depth-dependent mass density contrasts within bathymetry and lateral mass density anomalies with sediments and crustal layers. Respective gravitational effects are one by one removed from gravitational gradients observed at approximately 250 km elevation above ground. Removing respective gravitational gradients from observed gravitational gradients gradually reveals problematic geographic areas with model deficiencies. For the full interpretation of observed gravitational gradients, deficiencies of CRUST2.0 must be corrected and effects of deeper laying mass anomalies not included in the study considered. These findings are confirmed by parameters describing spectral properties of the gravitational gradients. The methodology can be applied for validating Earth’s gravitational models and for constraining crustal models in the development phase.     

High Resolution Constituents of the Earth’s Gravitational Field

During the General Assembly of the European Geosciences Union in April 2008, the new Earth Gravitational Model 2008 (EGM08) was released with fully-normalized coefficients in the spherical harmonic expansion of the Earth’s gravitational potential complete to degree and order 2159 (for selected degrees up to 2190). EGM08 was derived through combination of a satellite-based geopotential model and 5 arcmin mean ground gravity data. Spherical harmonic coefficients of the global height function, that describes the surface of the solid Earth with the same angular resolution as EGM08, became available at the same time. This global topographical model can be used for estimation of selected constituents of EGM08, namely the gravitational potentials of the Earth’s atmosphere, ocean water (fluid masses below the geoid) and topographical masses (solid masses above the geoid), which can be evaluated numerically through spherical harmonic expansions. The spectral properties of the respective potential coefficients are studied in terms of power spectra and their relation to the EGM08 potential coefficients is analyzed by using correlation coefficients. The power spectra of the topographical and sea water potentials exceed the power of the EGM08 potential over substantial parts of the considered spectrum indicating large effects of global isostasy. The correlation analysis reveals significant correlations of all three potentials with the EGM08 potential. The potential constituents (namely their functionals such as directional derivatives) can be used for a step known in geodesy and geophysics as the gravity field reduction or stripping. Removing from EGM08 known constituents will help to analyze the internal structure of the Earth (geophysics) as well as to derive the Earth’s gravitational field harmonic outside the geoid (geodesy).     

On a more precise stokes' series

Summary A relation is established between coefficients of an expansion of the gravitational potential into a series of Legendre's function of the second kind and coefficients of an expansion of gravity anomalies on the surface of the reference ellipsoid into a series of the same functions. This connection can be useful in geodetic computations which take into account the Earth's flattening.     

超高阶扰动场元的计算方法

空间重力测量技术的飞速发展为超高阶全球重力场模型的建立提供了条件和更迫切的应用需求.本文从完全规格化缔合勒让德函数及其一、二阶导数的标准前向列递推公式出发,研究了超高阶扰动场元球谐展开式中完全规格化缔合勒让德函数及其一、二阶导数的数值特征,改进了上述标准前向列递推关系式,提出了在微机上计算超高阶扰动场元的实用方法,避免了数字溢出.     

Gravity disturbances in regions of negative heights: A reference quasi-ellipsoid approach

Compilation of the bathymetrically and topographically corrected gravity disturbance, the so called BT disturbance, for the purpose of gravity interpretation/inversion, is investigated from the numerical point of view, with special emphasis on regions of negative heights. In regions of negative ellipsoidal (geodetic) heights, such as the Dead Sea region onshore or offshore areas of negative geoidal heights, two issues complicate the compilation and subsequently the inversion of the BT disturbance. The first is associated with the evaluation of normal gravity below the surface of the reference ellipsoid (RE). The latter is tied to the legitimacy of the harmonic continuation of the BT disturbance in these regions. These two issues are proposed to be resolved by the so called reference quasi-ellipsoid (RQE) approach. New bathymetric and topographic corrections are derived based on the RQE and the inverse problem is formulated based on the RQE. The RQE approach enables the computation of normal gravity by means of the international gravity formula, and makes the harmonic continuation in the regions of negative heights of gravity stations legitimate. The gravimetric inversion is then transformed from the RQE approach back to the RE approach, following the now legitimate harmonic upward continuation of the gravity data to stations on or above the RE. Stripping, the removal of an effect of a known density contrast, is considered in the context of the RQE approach. A numerical case study is presented for the RQE approach in a region of NW Canada.     

Solution to the Stokes Boundary-Value Problem on an Ellipsoid of Revolution

We have constructed Green's function to Stokes's boundary-value problem with the gravity data distributed over an ellipsoid of revolution. We show that the problem has a unique solution provided that the first eccentricity e₀ of the ellipsoid of revolution is less than 0·65041. The ellipsoidal Stokes function describing the effect of ellipticity of the boundary is expressed in the E-approximation as a finite sum of elementary functions which describe analytically the behaviour of the ellipsoidal Stokes function at the singular point = 0. We prove that the degree of singularity of the ellipsoidal Stokes function in the vicinity of its singular point is the same as that of the spherical Stokes function.     

Study of the anomalous gravity field in the arctic based on modern geopotential models

The results of modeling the anomalous gravity field of the Earth in the high-latitude regions, which are of interest in the context of Arctic exploration, are analyzed. The structure and accuracy characteristics of quasi-geoid heights (QGH) and gravity anomalies (GAs) in the Arctic are studied. The analysis addresses fifteen modern Russian and foreign spherical harmonic models of the gravitational potential of the Earth, including the EGM-2008 model, which provides high-order spherical harmonic representations, and the first models based on the data of satellite gradiometry.     

Gravity potential of the normal body in the outer and inner space

Summary The level rotational ellipsoid, best fitting the actual Earth, rotating with the same angular velocity around a common axis of rotation, is assumed to be a mathematical model of the real Earth. The gravity potential of this body and its derivatives in the outer space are derived by means of the generalized Pizzetti method[1]. For some analyses of the structure of the Earth we need to know the gravity anomaly and thus the gravity potential and its derivatives inside the mathematical model. These values are not defined in the classical conception. In this paper, the normal potential and its derivatives in the inner space are derived up to a certain depth, which is still of significance for gravimetric research.Dedicated to RNDr Jan Pícha, CSc., on his 60th Birthday     

Spherical Harmonic Analysis of Gravitational Curvatures and Its Implications for Future Satellite Missions

In this study we assume that a gravitational curvature tensor, i.e. a tensor of third-order directional derivatives of the Earth’s gravitational potential, is observable at satellite altitudes. Such a tensor is composed of ten different components, i.e. gravitational curvatures, which may be combined into vertical–vertical–vertical, vertical–vertical–horizontal, vertical–horizontal–horizontal and horizontal–horizontal-horizontal gravitational curvatures. Firstly, we study spectral properties of the gravitational curvatures. Secondly, we derive new quadrature formulas for the spherical harmonic analysis of the four gravitational curvatures and provide their corresponding analytical error models. Thirdly, requirements for an instrument that would eventually observe gravitational curvatures by differential accelerometry are investigated. The results reveal that measuring third-order directional derivatives of the gravitational potential imposes very high requirements on the accuracy of deployed accelerometers which are beyond the limits of currently available sensors. For example, for orbital parameters and performance similar to those of the GOCE mission, observing third-order directional derivatives requires accelerometers with the noise level of \({\sim}10^{-17}\,\hbox {m}\,\hbox {s}^{-2}\) Hz\(^{-1/2}\).     

Estimation of the gravitational potential energy of the earth based on different density models

The estimation of the Earth’s gravitational potential energy E was obtained for different density distributions and rests on the expression E = − (W_min + ΔW) derived from the conventional relationship for E. The first component W_min expresses minimum amount of the work W and the second component ΔW represents a deviation from W_min interpreted in terms of Dirichlet’s integral applied on the internal potential. Relationships between the internal potential and E were developed for continuous and piecewise continuous density distributions. The global 3D density model inside an ellipsoid of revolution was chosen as a combined solution of the 3D continuous distribution and the reference PREM radial piecewise continuous profile. All the estimates of E were obtained for the spherical Earth since the estimated (from error propagation rule) accuracy σ_E of the energy E is at least two orders greater than the ellipsoidal reduction and the contribution of lateral density inhomogeneities of the 3D global density model. The energy E contained in the 2nd degree Stokes coefficients was determined. A good agreement between E = E_Gauss derived from Gaussian distribution and other E, in particular for E = E_PREM based on the PREM piecewise continuous density model and E-estimates derived from simplest Legendre-Laplace, Roche, Bullard and Gauss models separated into core and mantle only, suggests the Gaussian distribution as a basic radial model when information about density jumps is absent or incomplete.     

Computation of the potential of a simple inhomogeneous equipotential layer with a given density on a given surface

Summary The potential of a simple equipotential layer, defined by the equation of a surface, on which the layer is spread, and by the density function in the form of harmonic expansions, is expressed with the aid of the parameters of the surface and of the density function. The integral of the product of three Legendre associated functions is treated with the aid of Gaunt's integral [3, 4].