Evaluation of the fourth-order tesseroid formula and new combination approach to precisely determine gravitational potential

Calculating topographic gravitational potential (GP) is a time-consuming process in terms of efficiency. Prism, mass-point, mass-line, and tesseroid formulas are generally used to calculate the topographic GP effect. In this study, we reformulate the higher-order formula of the tesseroid by Taylor series expansion and then evaluate the fourth-order formula by numerical tests. Different simulation computations show that the fourth-order formula is reliable. Using the conventional approach in numerical calculations, the approximation errors in the areas of the north and south poles are extremely large. Thus, in this study we propose an approach combining the precise numerical formula and tesseroid formulas, which can satisfactorily solve the calculation problem when the computation point is located in the polar areas or areas very near the surface. Furthermore, we suggest a “best matching choice” of new combination approach to calculate the GP precisely by conducting various experiments. Given the computation point at different positions, we may use different strategies. In the low latitude, we use a precise numerical formula, the fourth-order tesseroid formula, the second-order tesseroid formula, and the zero-order formula, in the 1° range (from the computation point), 1° to 15° range, 15° to 40° range, and the range outside 40°, respectively. The accuracy can reach 2 × 10^?5 m² s^?2. For the high latitude, we use the precise numerical formula, fourth-order tesseroid, second-order tesseroid, and zero-order tesseroid formulas in the ranges of 0° to 1°, 1° to 10°, 10° to 30°, and the zones outside 30°, respectively. However, if an accuracy level of 2 × 10^?5 m² s^?2 is required, the zero-order tesseroid formulas should not be used and the second-order tesseroid formula should be used in the region outside 15° for the low latitude and in the region outside 10° for the high latitude.     

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.     

利用地球重力位模型计算重力和重力梯度

高阶高精度地球重力场模型具有广泛的用途。本文利用地球重力位模型计算重力和重力梯度在应用中很有实用阶值,同时也是计算重力场其它量的关键。利用伪局部笛卡尔坐标与球坐标的关系计算了重力与重力梯度在伪局部笛卡尔坐标系下的分量;利用张量变换的原理给出了已知重力与重力梯度在某一坐标系下的分量求它们在另一坐标系下分量的方法,并具体给出了重力与重力梯度在局部笛卡儿坐标系下的分量计算公式,同时还给出计算重力场五参量与垂线偏差的计算公式,本研究推进了地球重力场的可视化进程。     

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.     

The Rock–Water–Ice Topographic Gravity Field Model RWI_TOPO_2015 and Its Comparison to a Conventional Rock-Equivalent Version

RWI_TOPO_2015 is a new high-resolution spherical harmonic representation of the Earth’s topographic gravitational potential that is based on a refined Rock–Water–Ice (RWI) approach. This method is characterized by a three-layer decomposition of the Earth’s topography with respect to its rock, water, and ice masses. To allow a rigorous separate modeling of these masses with variable density values, gravity forward modeling is performed in the space domain using tesseroid mass bodies arranged on an ellipsoidal reference surface. While the predecessor model RWI_TOPO_2012 was based on the \(5'\times 5'\) global topographic database DTM2006.0 (Digital Topographic Model 2006.0), the new RWI model uses updated height information of the \(1'\times 1'\) Earth2014 topography suite. Moreover, in the case of RWI_TOPO_2015, the representation in spherical harmonics is extended to degree and order 2190 (formerly 1800). Beside a presentation of the used formalism, the processing for RWI_TOPO_2015 is described in detail, and the characteristics of the resulting spherical harmonic coefficients are analyzed in the space and frequency domain. Furthermore, this paper focuses on a comparison of the RWI approach to the conventionally used rock-equivalent method. For this purpose, a consistent rock-equivalent version REQ_TOPO_2015 is generated, in which the heights of water and ice masses are condensed to the constant rock density. When evaluated on the surface of the GRS80 ellipsoid (Geodetic Reference System 1980), the differences of RWI_TOPO_2015 and REQ_TOPO_2015 reach maximum amplitudes of about 1 m, 50 mGal, and 20 mE in terms of height anomaly, gravity disturbance, and the radial–radial gravity gradient, respectively. Although these differences are attenuated with increasing height above the ellipsoid, significant magnitudes can even be detected in the case of the satellite altitudes of current gravity field missions. In order to assess their performance, RWI_TOPO_2015, REQ_TOPO_2015, and RWI_TOPO_2012 are validated against independent gravity information of current global geopotential models, clearly demonstrating the attained improvements in the case of the new RWI model.     

华南地区时变重力场建模实验和异常分析

陆面时变重力测量是监测地壳内部密度变化和物质运移的重要手段.为确定华南时变重力观测网络的场源监测能力和重力场变化特征,本文采用球面六面体单元构建重力场模型,开展重力场建模实验,对比不同建模方法与噪声条件下的局部重力场恢复效果,并对2015-2017年来5期实测流动重力观测数据进行计算和分析.结果表明,最小二乘配置方法的...     

Forward computation of the magnetic field of a 3D body with arbitrary boundary and continually varying magnetization

The forward computation of the gravitational and magnetic fields due to a 3D body with an arbitrary boundary and continually varying density or magnetization is an important problem in gravitational and magnetic prospecting. In order to solve the inverse problem for the arbitrary components of the gravitational and magnetic anomalies due to an arbitrary 3D body under complex conditions, including an uneven observation surface, the existence of background anomalies and very little or no a priori information, we used a spherical coordinate system to systematically investigate forward methods for such anomalies and developed a series of universal spherical harmonic expansions of gravitational and magnetic fields. For the case of a 3D body with an arbitrary boundary and continually varying magnetization, we have also given the surface integral expressions for the common spherical harmonic coefficients in the expansion of the magnetic field due to the body, and a very precise numerical integral algorithm to calculate them. Thus a simple and effective method of solving the forward problem for magnetic fields due to 3D bodies of this kind has been found, and in this way a foundation is laid for solving the inverse problem of these magnetic fields. In addition, by replacing the parameters and unit vectors in the spherical harmonic expansion of a magnetic field by gravitational parameters and a downward unit vector, we have also derived a forward method for the gravitational field (similar to that for the magnetic case) of a 3D body with an arbitrary boundary and continually varying density.     

Far-zone gravity field contributions corrected for the effect of topography by means of molodensky’s truncation coefficients

A spectral representation of the topographic corrections to gravity field quantities is formulated in terms of spherical height functions. When computing the far-zone contributions to the topographic corrections, various types of the truncation coefficients are applied to a spectral representation of Newton’s integral. In this study we utilise Molodensky’s truncation coefficients in deriving the expressions for the far-zone contributions to topographic corrections. The expressions for computing the far-zone gravity field contributions corrected for the effect of topography are then obtained by combining the expressions for the far-zone contributions to the gravity field quantities and to the respective topographic corrections, both expressed in terms of Molodensky’s truncation coefficients. The numerical examples of the far-zone contributions to the topographic corrections and to the topography-corrected gravity field quantities are given over the study area situated in the Canadian Rocky Mountains with adjacent planes. Coefficients of the global elevation and geopotential models are used as the input data.     

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.     

Multiscale Gravitational Field Recovery from GPS-Satellite-to-Satellite Tracking

The purpose of GPS-satellite-to-satellite tracking (GPS-SST) is to determine the gravitational potential at the earth's surface from measured ranges (geometrical distances) between a low-flying satellite and the high-flying satellites of the Global Positioning System (GPS). In this paper, GPS-satellite-to-satellite tracking is reformulated as the problem of determining the gravitational potential of the earth from given gradients at satellite altitude. The uniqueness and stability of the solution are investigated. The essential tool is to split the gradient field into a normal part (i.e. the first-order radial derivative) and a tangential part (i.e. the surface gradient). Uniqueness is proved for polar, circular orbits corresponding to both types of data (first radial derivative and/or surface gradient). In both cases gravity recovery based on satellite-to-satellite tracking turns out to be an exponentially ill-posed problem. Regularization in terms of spherical wavelets is proposed as an appropriate solution method, based on the knowledge of the singular system. Finally, the extension of this method is generalized to a nonspherical earth and a non-spherical orbital surface, based on combined terrestrial and satellite data.     

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.     

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\(^{-1/2}\).     

Iterative Spherical Downward Continuation Applied to Magnetic and Gravitational Data from Satellite

In the last few decades, satellites have acquired various potential data sets hundreds of kilometers above the Earth’s surface. Conventionally, these global magnetic and gravitational data sets are approximated by using spherical harmonics that allow straightforward work with both fields outside the Earth’s mass. In this article, we present an alternative approach for working with potential data in mass-free space given over a regular coordinate grid on a spherical surface. The algorithm is based on an iterative scheme and the Poisson integral equation for the sphere. With help from the Fourier transform, global potential (magnetic or gravitational) data can efficiently be continued from a mean orbital sphere down to a reference surface without using the spherical harmonics. This is illustrated both with simulated magnetic field data and with real data from the satellite gradiometry mission GOCE. In the case of simulated magnetic data and the downward continuation for 450 km, we have achieved a root mean square at the level of 0.05 nT, while it was <1 E (eotvos) for real GOCE data continued for 250 km. The crucial point is to apply the algorithm twice as a large part of noise can be removed from the input data.     

Gravitational effect of distant Earth relief within the territory of former Czechoslovakia

We analyzed the gravitational effect of topography and bathymetry beyond the angular distance of approximately 1.5 degrees (referred to as the distant relief effect or DRE), and its impact on measured gravity values in the region of the former Czechoslovakia. Our work was strongly motivated by the contents of the pioneering contribution of outstanding Czech geophysicists Miloš Pick, Jan Pícha and Vincenc Vyskočil, which appeared at the turn of the 1950’s and 1960’s. Our numerical calculations were based upon the direct evaluation of the gravitational effects of compartments of a spherical layer, while the respective heights and depths were obtained from the 2 × 2 minutes digital elevation model (DEM) ETOPO2, taking into consideration also the influence of distant bathymetry. Our results are in close agreement with, but not identical to, those of the above cited authors. We also analyzed the influence of the grid cell size of the involved DEM upon the calculation results. We introduced an approximation of the analyzed effect, based on a simple linear relationship between the calculation point height, the DRE and its vertical gradient (VGDRE). Since when calculated at zero elevation the involved quantities DRE and VGDRE are smooth functions of latitude and longitude and can be easily interpolated, the approximation gives acceptable results in terms of desired accuracy of several μGal (1 μGal = 10⁻⁸ m/s²). In general, we can state that within the territories of the Czech and Slovak Republics the studied distant relief effect has negligible impact upon local gravity survey data. However, when applied to regional gravity studies, there could be a question of its possible influence in the form of a quasilinear W-E trend ranging approximately from −106.6 to −102.5 mGal within the territory of former Czechoslovakia. If we wanted to correct for this phenomenon, we should subtract this negative quantity from the standard Bouguer anomalies as they have been defined in the recent geophysical literature, thereby considerably increasing their values. But, instead of straightforward correcting the Bouguer anomalies for DRE only, we would rather recommend to wait until after the crustal and even lithospheric effects have been studied more carefully based upon the present day independent knowledge about the deep seated sources of those effects.     

RTM-based omission error corrections for global geopotential models: Case study in Central Europe

The aim of this paper is to evaluate the effects of residual terrain model (RTM) on potential and on gravity and to point out how significant can the omission error of global geopotential models (GGMs) be and how it can influence their testing. The RTM for Central Europe is computed in the spherical approximation. The topography is modelled by spherical tesseroids and the gravitational effect of the topography is obtained as a sum of their partial gravitational effects. A detailed picture of RTM in Slovakia is shown. The testing of GOCE (Gravity Field and Steady-State Ocean Circulation Explorer) global geopotential models in Central Europe published earlier is re-evaluated with the more rigorous omission error estimation. Experimental results show significantly better agreement between the gravity anomalies computed from global geopotential models with the omission-error estimation and gravity anomalies obtained from the direct measurements. On the other hand, for height anomalies such an improvement is not observed. The results are discussed in context of the other previously published studies.     

On High-Frequency Topography-Implied Gravity Signals for a Height System Unification Using GOCE-Based Global Geopotential Models

National height reference systems have conventionally been linked to the local mean sea level, observed at individual tide gauges. Due to variations in the sea surface topography, the reference levels of these systems are inconsistent, causing height datum offsets of up to ±1–2 m. For the unification of height systems, a satellite-based method is presented that utilizes global geopotential models (GGMs) derived from ESA’s satellite mission Gravity field and steady-state Ocean Circulation Explorer (GOCE). In this context, height datum offsets are estimated within a least squares adjustment by comparing the GGM information with measured GNSS/leveling data. While the GNSS/leveling data comprises the full spectral information, GOCE GGMs are restricted to long wavelengths according to the maximum degree of their spherical harmonic representation. To provide accurate height datum offsets, it is indispensable to account for the remaining signal above this maximum degree, known as the omission error of the GGM. Therefore, a combination of the GOCE information with the high-resolution Earth Gravitational Model 2008 (EGM2008) is performed. The main contribution of this paper is to analyze the benefit, when high-frequency topography-implied gravity signals are additionally used to reduce the remaining omission error of EGM2008. In terms of a spectral extension, a new method is proposed that does not rely on an assumed spectral consistency of topographic heights and implied gravity as is the case for the residual terrain modeling (RTM) technique. In the first step of this new approach, gravity forward modeling based on tesseroid mass bodies is performed according to the Rock–Water–Ice (RWI) approach. In a second step, the resulting full spectral RWI-based topographic potential values are reduced by the effect of the topographic gravity field model RWI_TOPO_2015, thus, removing the long to medium wavelengths. By using the latest GOCE GGMs, the impact of topography-implied gravity signals on the estimation of height datum offsets is analyzed in detail for representative GNSS/leveling data sets in Germany, Austria, and Brazil. Besides considerable changes in the estimated offset of up to 3 cm, the conducted analyses show that significant improvements of 30–40% can be achieved in terms of a reduced standard deviation and range of the least squares adjusted residuals.     

中国东部大兴安岭重力梯级带域地球物理场特征及其成因

综合利用7条地学断面（GGT）资料研究了大兴安岭重力梯级带附近的壳幔地球物理特征模式.分析了形成上述地球物理特征的3种因素:东亚大陆边缘周边三大板块运动、地幔流运动和地幔热柱。对大兴安岭重力梯级带的重力异常的正演拟合结果表明，壳幔物质密度不均匀和莫霍界面超伏造成该带的重力异常，地应力场的综合作用产生了该重力梯级系列地球物理特征。最后，探讨了大兴安岭重力梯级带成因机制，提出了以“挤”、“涌”为动力的“三结点模型”。     

The Effects of the Earth’s Curvature on Gravity and Geoid Calculations

While it is obvious that large-scale gravity studies should account for the sphericity of the Earth, each case should be examined. If a geometry model is very large for the 3D-gravity calculation, it cannot be correctly defined in Cartesian coordinates. Because of the Earth’s curvature it is necessary to use spherical coordinates, the importance of which is shown in this paper. The calculation of the gravity for a cylinder reveals, 1 m above the center of the cylinder, a relative difference of 13% between the models with Cartesian and spherical coordinates.     

海洋潮汐对重力潮汐观测的影响

本文论述了海潮对重力固体潮观测的影响,把整个褶积计算分为近洋区及远洋区两部分,其中近洋区采用格林函数进行数值积分的方法,远洋区则利用全球海潮潮高的球函数展开式。着重讨论了远洋区的实施方案,导出了必要的工作公式,并就加速远洋区级数收敛问题作了探讨。利用导得的公式,计算了M₂海洋潮波对中国境内重力观测的影响。     

Noise filtering of full-gravity gradient tensor data

In oil and mineral exploration, gravity gradient tensor data include higher-frequency signals than gravity data, which can be used to delineate small-scale anomalies. However, full-tensor gradiometry (FTG) data are contaminated by high-frequency random noise. The separation of noise from high-frequency signals is one of the most challenging tasks in processing of gravity gradient tensor data. We first derive the Cartesian equations of gravity gradient tensors under the constraint of the Laplace equation and the expression for the gravitational potential, and then we use the Cartesian equations to fit the measured gradient tensor data by using optimal linear inversion and remove the noise from the measured data. Based on model tests, we confirm that not only this method removes the high-frequency random noise but also enhances the weak anomaly signals masked by the noise. Compared with traditional low-pass filtering methods, this method avoids removing noise by sacrificing resolution. Finally, we apply our method to real gravity gradient tensor data acquired by Bell Geospace for the Vinton Dome at the Texas-Louisiana border.