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.     

Compact non-singular expansions in the exterior space of a reference ellipsoid for the gravitational gradients in the local north-oriented ellipsoidal and spherical reference frames

Due to the complicated structure of their expressions, the ellipsoidal harmonic series for the derivatives of the Earth’s gravitational potential are commonly applied only on a reference ellipsoid. They depend on the first- and second-order derivatives of the associated Legendre functions of both kinds and contain a few singular terms. We construct ellipsoidal harmonic expansions in the exterior space for the first and second potential derivatives, which are similar to the series on the reference ellipsoid enveloping the Earth. We take a point P at an arbitrary altitude above the reference ellipsoid and construct the ellipsoid of revolution confocal to it, which passes through this point. The conventional complicated singular expressions for the first and second potential derivatives in the local north-oriented ellipsoidal reference frame, with the origin at the point P, are transformed into non-singular ellipsoidal harmonic series, which do not contain the first- and second-order derivatives of the associated Legendre functions. The resulting series have an accuracy of the squared eccentricity. These series can be applied for constructing a geopotential model, which is based, simultaneously, on the surface gravity data and the data of satellite missions, which provide measurements of the accelerations and/or the gravitational gradients. When the eccentricity of the considered external ellipsoid is equated to zero, the ellipsoid becomes an external sphere passing through the point P and the constructed ellipsoidal harmonic expansions are converted into non-singular spherical harmonic series for the first and second potential derivatives in the local north-oriented spherical reference frame.     

Spectral Combination of Spherical Gradiometric Boundary-Value Problems: A Theoretical Study

The Earth’s gravity potential can be determined from its second-order partial derivatives using the spherical gradiometric boundary-value problems which have three integral solutions. The problem of merging these solutions by spectral combination is the main subject of this paper. Integral estimators of biased- and unbiased-types are presented for recovering the disturbing gravity potential from gravity gradients. It is shown that only kernels of the biased-type integral estimators are suitable for simultaneous downward continuation and combination of gravity gradients. Numerical results show insignificant practical difference between the biased and unbiased estimators at sea level and the contribution of far-zone gravity gradients remains significant for integration. These contributions depend on the noise level of the gravity gradients at higher levels than sea. In the cases of combining the gravity gradients, contaminated with Gaussian noise, at sea and 250?km levels the errors of the estimated geoid heights are about 10 and 3 times smaller than those obtained by each integral.     

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.     

Interpretation of general geophysical patterns in Iran based on GRACE gradient component analysis

Only with satellites it is possible to cover the entire Earth densely with gravity field related measurements of uniform quality within a short period of time. However, due to the altitude of the satellite orbits, the signals of individual local masses are strongly damped. Based on the approach of Petrovskaya and Vershkov we determine the gravity gradient tensor directly from the spherical harmonic coefficients of the recent EIGEN-GL04C combined model of the GRACE satellite mission. Satellite gradiometry can be used as a complementary tool to gravity and geoid information in interpreting the general geophysical and geodynamical features of the Earth. Due to the high altitude of the satellite, the effects of the topography and the internal masses of the Earth are strongly damped. However, the gradiometer data, which are nothing else than the second order spatial derivatives of the gravity potential, efficiently counteract signal attenuation at the low and medium frequencies. In this article we review the procedure for estimating the gravity gradient components directly from spherical harmonics coefficients. Then we apply this method as a case study for the interpretation of possible geophysical or geodynamical patterns in Iran. We found strong correlations between the cross-components of the gravity gradient tensor and the components of the deflection of vertical, and we show that this result agrees with theory. Also, strong correlations of the gravity anomaly, geoid model and a digital elevation model were found with the diagonal elements of the gradient tensor.     

Determination of gravity anomaly at sea level from inversion of satellite gravity gradiometric data

Gravity gradients can be used to determine the local gravity field of the Earth. This paper investigates downward continuation of all elements of the disturbing gravitational tensor at satellite level using the second-order partial derivatives of the extended Stokes formula in the local-north oriented frame to determine the gravity anomaly at sea level. It considers the inversion of each gradient separately as well as their joint inversion. Numerical studies show that the gradients T_zz, T_xx, T_yy and T_xz have similar capability of being continued downward to sea level in the presence of white noise, while the gradient T_yz is considerably worse than the others. The bias-corrected joint inversion process shows the possibility of recovering the gravity anomaly with 1 mGal accuracy. Variance component estimation is also tested to update the observation weights in the joint inversion.     

Upward continuation of Dome-C airborne gravity and comparison with GOCE gradients at orbit altitude in east Antarctica

An airborne gravity campaign was carried out at the Dome-C survey area in East Antarctica between the 17th and 22nd of January 2013, in order to provide data for an experiment to validate GOCE satellite gravity gradients. After typical filtering for airborne gravity data, the cross-over error statistics for the few crossing points are 11.3 mGal root mean square (rms) error, corresponding to an rms line error of 8.0 mGal. This number is relatively large due to the rough flight conditions, short lines and field handling procedures used. Comparison of the airborne gravity data with GOCE RL4 spherical harmonic models confirmed the quality of the airborne data and that they contain more high-frequency signal than the global models. First, the airborne gravity data were upward continued to GOCE altitude to predict gravity gradients in the local North-East-Up reference frame. In this step, the least squares collocation using the ITGGRACE2010S field to degree and order 90 as reference field, which is subtracted from both the airborne gravity and GOCE gravity gradients, was applied. Then, the predicted gradients were rotated to the gradiometer reference frame using level 1 attitude quaternion data. The validation with the airborne gravity data was limited to the accurate gradient anomalies (T_XX, T_YY, T_ZZ and T_XZ) where the long-wavelength information of the GOCE gradients has been replaced with GOCO03s signal to avoid contamination with GOCE gradient errors at these wavelengths. The comparison shows standard deviations between the predicted and GOCE gradient anomalies T_XX, T_YY, T_ZZ and T_XZ of 9.9, 11.5, 11.6 and 10.4 mE, respectively. A more precise airborne gravity survey of the southern polar gap which is not observed by GOCE would thus provide gradient predictions at a better accuracy, complementing the GOCE coverage in this region.     

Adaptation of the torus and Rosborough approach to radial base functions

The most common approach for the processing of data of gravity field satellite missions is the so-called time-wise approach. In this approach satellite data are considered as a time series and processed by a standard least-squares approach. This approach has a very strong flexibility but it is computationally very demanding. To improve the computational efficiency and numerical stability, the so-called torus and Rosborough approaches have been developed. So far, these approaches have been applied only for global gravity field determinations, based on spherical harmonics as basis functions. For regional applications basis functions with a local support are superior to spherical harmonics, because they provide the same approximation quality with much less parameters. So far, torus and Rosborough approach have been developed for spherical harmonics only. Therefore, the paper aims at the development and testing of the torus and Rosborough approach for regional gravity field improvements, based on radial basis functions as basis functions. The developed regional Rosborough approach is tested against a changing gravity field produced by simulated ice-mass changes over Greenland. With only 350 parameters a recovery of the simulated mass changes with a relative accuracy of 5% is possible.     

An efficient and adaptive approach for modeling gravity effects in spherical coordinates

The need to obtain more reliable Earth structures has been the impetus for conducting joint inversions of disparate geophysical datasets. For seismic arrival time tomography, joint inversion of arrival time and gravity data has become an important way to investigate velocity structure of the crust and upper mantle. However, the absence of an efficient approach for modeling gravity effects in spherical coordinates limits the joint tomographic analysis to only local scales. In order to extend the joint tomographic inversion into spherical coordinates, and enable it to be feasible for regional studies, we develop an efficient and adaptive approach for modeling gravity effects in spherical coordinates based on the longitudinal/latitudinal grid spacing. The complete gravity effects of spherical prisms, including gravitational potential, gravity vector and tensor gradients, are calculated by numerical integration of the Gauss–Legendre quadrature (GLQ). To ensure the efficiency of the gravity modeling, spherical prisms are recursively subdivided into smaller units according to their distances to the observation point. This approach is compatible with the parameterization of regional arrival time tomography for large areas, in which both the near- and far-field effects of the Earth's curvature cannot be ignored. Therefore, this approach can be implemented into the joint tomographic inversion of arrival time and gravity data conveniently. As practical applications, the complete gravity effects of a single anomalous density body have been calculated, and the gravity anomalies of two tomographic models in the Taiwan region have also been obtained using empirical relationships between P-wave velocity and density.     

Analytic Expressions for the Gravity Gradient Tensor of 3D Prisms with Depth-Dependent Density

Variable-density sources have been paid more attention in gravity modeling. We conduct the computation of gravity gradient tensor of given mass sources with variable density in this paper. 3D rectangular prisms, as simple building blocks, can be used to approximate well 3D irregular-shaped sources. A polynomial function of depth can represent flexibly the complicated density variations in each prism. Hence, we derive the analytic expressions in closed form for computing all components of the gravity gradient tensor due to a 3D right rectangular prism with an arbitrary-order polynomial density function of depth. The singularity of the expressions is analyzed. The singular points distribute at the corners of the prism or on some of the lines through the edges of the prism in the lower semi-space containing the prism. The expressions are validated, and their numerical stability is also evaluated through numerical tests. The numerical examples with variable-density prism and basin models show that the expressions within their range of numerical stability are superior in computational accuracy and efficiency to the common solution that sums up the effects of a collection of uniform subprisms, and provide an effective method for computing gravity gradient tensor of 3D irregular-shaped sources with complicated density variation. In addition, the tensor computed with variable density is different in magnitude from that with constant density. It demonstrates the importance of the gravity gradient tensor modeling with variable density.     

Atmospheric effects on satellite gravity gradiometry data

Atmospheric masses play an important role in precise downward continuation and validation of satellite gravity gradiometry data. In this paper we present two alternative ways to formulate the atmospheric potential. Two density models for the atmosphere are proposed and used to formulate the external and internal atmospheric potentials in spherical harmonics. Based on the derived harmonic coefficients, the direct atmospheric effects on the satellite gravity gradiometry data are investigated and presented in the orbital frame over Fennoscandia. The formulas of the indirect atmospheric effects on gravity anomaly and geoid (downward continued quantities) are also derived using the proposed density models. The numerical results show that the atmospheric effect can only be significant for precise validation or inversion of the GOCE gradiometric data at the mE level.     

On the estimation of a multi-resolution representation of the gravity field based on spherical harmonics and wavelets

The gravitational potential of the Earth is usually modeled by means of a series expansion in terms of spherical harmonics. However, the computation of the series coefficients requires preferably homogeneous distributed global data sets. Since one of the most important features of wavelet functions is the ability to localize both in the spatial and in the frequency domain, regional and local structures may be modeled by means of a spherical wavelet expansion. In general, applying wavelet theory a given input data set is decomposed into a certain number of frequency-dependent detail signals, which can be interpreted as the building blocks of a multi-resolution representation. On the other hand, there is no doubt that the low-frequency part of the geopotential can be modeled appropriately by means of spherical harmonics. Hence, the main idea of this paper is to derive a combined model consisting of an expansion in spherical harmonics for the low-frequency part and an expansion in spherical wavelets for the remaining medium and high-frequency parts of the gravity field. Furthermore, an appropriate parameter estimation procedure is outlined to solve for the unknown model coefficients.     

A review of basic computations with spherical harmonics in geomagnetism

Summary A number of problems arising in geomagnetism may be successfully solved by using various recurrence relations for spherical harmonic functions. This paper combines these recurrence relations into one simple computational algorithm, and illustrates the flexibility of the algorithm by applying it to the prototype problems of evaluating spherical harmonics and their derivatives, and transforming them under changes of reference frame.     

曲线坐标系下的块体运动与应变模型研究

本文建立了顾及地球扁率和局部切标架随点变化特性的椭球坐标系下的刚体运动模型和块体运动与应变模型,以及球坐标系下顾及局部切标架随点变化特性的严密的块体运动与应变模型,分析了球坐标系下块体运动与应变模型及椭球坐标系下的块体运动与应变模型间的差异;通过计算具体讨论了地球扁率和曲线坐标系的局部切标架随点变化特性对欧拉矢量与应变张量的影响.结果表明：地球扁率对刚体欧拉矢量和应变参数的影响甚小,具体计算时可以不予考虑,但曲线坐标系的局部切标架随点变化特性对两者的影响较大,在建模过程中需要顾及,常用的Savage模型需要修正.     

Evaluation of Optimal Formulas for Gravitational Tensors up to Gravitational Curvatures of a Tesseroid

The forward modeling of the topographic effects of the gravitational parameters in the gravity field is a fundamental topic in geodesy and geophysics. Since the gravitational effects, including for instance the gravitational potential (GP), the gravity vector (GV) and the gravity gradient tensor (GGT), of the topographic (or isostatic) mass reduction have been expanded by adding the gravitational curvatures (GC) in geoscience, it is crucial to find efficient numerical approaches to evaluate these effects. In this paper, the GC formulas of a tesseroid in Cartesian integral kernels are derived in 3D/2D forms. Three generally used numerical approaches for computing the topographic effects (e.g., GP, GV, GGT, GC) of a tesseroid are studied, including the Taylor Series Expansion (TSE), Gauss–Legendre Quadrature (GLQ) and Newton–Cotes Quadrature (NCQ) approaches. Numerical investigations show that the GC formulas in Cartesian integral kernels are more efficient if compared to the previously given GC formulas in spherical integral kernels: by exploiting the 3D TSE second-order formulas, the computational burden associated with the former is 46%, as an average, of that associated with the latter. The GLQ behaves better than the 3D/2D TSE and NCQ in terms of accuracy and computational time. In addition, the effects of a spherical shell’s thickness and large-scale geocentric distance on the GP, GV, GGT and GC functionals have been studied with the 3D TSE second-order formulas as well. The relative approximation errors of the GC functionals are larger with the thicker spherical shell, which are the same as those of the GP, GV and GGT. Finally, the very-near-area problem and polar singularity problem have been considered by the numerical methods of the 3D TSE, GLQ and NCQ. The relative approximation errors of the GC components are larger than those of the GP, GV and GGT, especially at the very near area. Compared to the GC formulas in spherical integral kernels, these new GC formulas can avoid the polar singularity problem.     

卫星重力梯度测量与地球引力场的精度研究

本文根据地球引力位的球谐函数展开式,利用重力梯度张量各分量导出了位系数模型的精度估计公式.从三方面进行了研究:假定卫星重力梯度仪测量精度,探讨用重力梯度数据确定地球重力场模型的精度;求出位系数模型和大气阻力引起的重力梯度卫星的轨道误差;最后,反求轨道误差和位系数误差对重力梯度测量值的影响.数值计算表明,与地面技术和常规卫星方法相比,卫星梯度测量可使重力场模型的精度至少提高3-5倍;利用重力梯度张量全分量求得的重力值精度比单用径向分量V_rr的结果提高40％以上;若仅顾及位系数模型和大气阻力误差,则轨道误差对梯度测量值的影响△V_i3（i=3,2,1）至少可分别在1/4和1/3弧圈内达到△V_i3≤σ（仪器精度）.     

A refined method of recovering potential coefficients from surface gravity data

Summary A new method for computing the potential coefficients of the Earth's external gravity field is presented. The gravimetric boundary-value problem with a free boundary is reduced to the problem with a fixed known telluroid. The main idea of the derivation consists in a continuation of the quantities from the physical surface to the telluroid by means of Taylor's series expansion in such a way that the terms whose magnitudes are comparable with the accuracy of today's gravity measurements are retained. Thus not only linear, but also non-linear terms are taken into account. Explicitly, the terms up to the order of the third power of the Earth's flattening are retained. The non-linear boundary-value problem on the telluroid is solved by an iteration procedure with successive approximations. In each iteration step the solution of the non-linear problem is estimated by the solutions of two linear problems utilizing the fact that the non-linear boundary condition may be split into two parts; the linear spherical approximation of the gravity anomaly whose magnitude is significantly greater than the others and the non-linear ellipsoidal corrections. Finally, in order to solve the problem in terms of spherical harmonics, the transform method composed of the fast Fourier transform and Gauss Legendre quadrature is theoretically outlined. Immediate data processing of gravity data measured on the physical Earth's surface without any continuation of gravity measurements to a reference level surface belongs to the main advantage of the presented method. This implies that no preliminary data handling is needed and that the error data propagation is, consequently, maximally suppressed.     

Local Ionospheric Modeling Using the Localized Global Ionospheric Map and Terrestrial GPS

global ionosphere maps are generated on a daily basis at CODE using data from about 200 GPS/GLONASS sites of the IGS and other institutions. The vertical total electron content is modeled in a solargeomagnetic reference frame using a spherical harmonics expansion up to degree and order 15. The spherical Slepian basis is a set of bandlimited functions which have the majority of their energy concentrated by optimization inside an arbitrarily defined region, yet remain orthogonal within the spatial region of interest. Hence, they are suitable for decomposing the spherical harmonic models into the portions that have significant strength only in the selected areas. In this study, the converted spherical harmonics to the Slepian bases were updated by the terrestrial GPS observations by use of the least-squares estimation with weighted parameters for local ionospheric modeling. Validations show that the approach adopted in this study is highly capable of yielding reliable results.     

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.     

Application of Gravity Gradients in the Process of GOCE Orbit Determination

The possibility of improving the Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) mission satellite orbit using gravity gradient observations was investigated. The orbit improvement is performed by a dedicated software package, called the Orbital Computation System (OCS), which is based on the classical least squares method. The corrections to the initial satellite state vector components are estimated in an iterative process, using dynamic models describing gravitational perturbations. An important component implemented in the OCS package is the 8th order Cowell numerical integration procedure, which directly generates the satellite orbit. Taking into account the real and simulated GOCE gravity gradients, different variants of the solution of the orbit improvement process were obtained. The improved orbits were compared to the GOCE reference orbits (Precise Science Orbits for the GOCE satellite provided by the European Space Agency) using the root mean squares (RMS) of the differences between the satellite positions in these orbits. The comparison between the improved orbits and the reference orbits was performed with respect to the inertial reference frame (IRF) at J2000.0 epoch. The RMS values for the solutions based on the real gravity gradient measurements are at a level of hundreds of kilometers and more. This means that orbit improvement using the real gravity gradients is ineffective. However, all solutions using simulated gravity gradients have RMS values below the threshold determined by the RMS values for the computed orbits (without the improvement). The most promising results were achieved when short orbital arcs with lengths up to tens of minutes were improved. For these short arcs, the RMS values reach the level of centimeters, which is close to the accuracy of the Precise Science Orbit for the GOCE satellite. Additional research has provided requirements for efficient orbit improvement in terms of the accuracy and spectral content of the measured gravity gradients.