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.     

The computation of aliasing effects in local gravity field approximation

The aliasing effects in local gravity field computations are presented in this study. First the relation between the power spectral density of a 2-D continuous signal and its corresponding sampled version is derived. Then the power spectral density of the aliasing errors related to non band-limited signals is derived. Finally the variance of these aliasing errors is computed using gravity anomalies at different grid spacings. This computation prerequires some known gravity anomaly power spectral density model. The model used in this study corresponds to a second-order Gauss-Markov covariance function for the anomalous potential. Editor’s notice: Comments on this paper will follow in the next issue of Bulletin Géodésique.     

Determination of the potential of homogeneous polyhedral bodies using line integrals

For the determination of the potential of irregular inhomogeneous bodies they can be decomposed into (polyhedral) parts of homogeneous density. Efficient formulas for the computation of the gravitational potential (and its first and second derivatives) of homogeneous polyhedral bodies are presented. They are obtained using a transformation of the volume integral into line integrals. The most important property of the solution is that all ten quantities under consideration (potential, 3 components of the gravitation vector, 6 components of the tensor of the second derivatives) can be represented by using only two different line integrals. Furthermore, all coordinate transformations needed in the evaluation are chosen in such a way that they do not appear in the final result. The consequence, favorable for efficient programming, is that the same transcendental expressions along each edge of the polyhedron are needed for all ten quantities; even the same linear combinations of them for individual surfaces are appearing in different formulas. The expressions obtained are probably the simplest possible, which is also reflected in the fact that for the special case of a right rectangular prism they may easily be specialized to the usual well-known formulas. Received 28 Juni 1994; Accepted 13 September 1996     

The computation of gravity at elevation on a large scale

This paper briefly describes the techniques and computer programs being used to compute gravity at points located at aircraft elevations. Methods of mean anomaly prediction, the upward continuation of the gravity anomalies, and finally, the method used for normal gravity computations are outlined.     

The optimum expression for the gravitational potential of polyhedral bodies having a linearly varying density distribution

When topography is represented by a simple regular grid digital elevation model, the analytical rectangular prism approach is often used for a precise gravity field modelling at the vicinity of the computation point. However, when the topographical surface is represented more realistically, for instance by a triangular irregular network (TIN) model, the analytical integration using arbitrary polyhedral bodies (the analytical line integral approach) can be implemented directly without additional data pre-processing (gridding or interpolation). The analytical line integral approach can also facilitate 3-D density models created for complex geometrical bodies. For the forward modelling of the gravitational field generated by the geological structures with variable densities, the analytical integration can be carried out using polyhedral bodies with a varying density. The optimal expression for the gravitational attraction vector generated by an arbitrary polyhedral body having a linearly varying density is known. In this article, the corresponding optimal expression for the gravitational potential is derived by means of line integrals after applying the Gauss divergence theorem.     

Using gravity and topography-implied anomalies to assess data requirements for precise geoid computation

Many regions around the world require improved gravimetric data bases to support very accurate geoid modeling for the modernization of height systems using GPS. We present a simple yet effective method to assess gravity data requirements, particularly the necessary resolution, for a desired precision in geoid computation. The approach is based on simulating high-resolution gravimetry using a topography-correlated model that is adjusted to be consistent with an existing network of gravity data. Analysis of these adjusted, simulated data through Stokes’s integral indicates where existing gravity data must be supplemented by new surveys in order to achieve an acceptable level of omission error in the geoid undulation. The simulated model can equally be used to analyze commission error, as well as model error and data inconsistencies to a limited extent. The proposed method is applied to South Korea and shows clearly where existing gravity data are too scarce for precise geoid computation.     

Local geoid computation from gravity using the fast fourier transform technique

Model computations were performed for the study of numerical errors which are interjected into local geoid computations byFFT. The gravity field model was generated through the attractions of granitic prisms derived from actual geology. Changes in sampling interval introduced only0.3 cm variation in geoid heights. Although zero padding alone provided an improvement of more than5 cm in theFFT generated geoid, the combination of spectral windowing (tapering) and padding further reduced numerical errors. For theGPS survey of Franklin County, Ohio, the parameters selected as a result of model computations, allow large reduction in local data requirements while still retaining the centimeter accuracy when tapering and padding is applied.     

Forward calculation of gravity and its gradient using polyhedral representation of density interfaces: an application of spherical or ellipsoidal topographic gravity effect

A density interface modeling method using polyhedral representation is proposed to construct 3-D models of spherical or ellipsoidal interfaces such as the terrain surface of the Earth and applied to forward calculating gravity effect of topography and bathymetry for regional or global applications. The method utilizes triangular facets to fit undulation of the target interface. The model maintains almost equal accuracy and resolution at different locations of the globe. Meanwhile, the exterior gravitational field of the model, including its gravity and gravity gradients, is obtained simultaneously using analytic solutions. Additionally, considering the effect of distant relief, an adaptive computation process is introduced to reduce the computational burden. Then features and errors of the method are analyzed. Subsequently, the method is applied to an area for the ellipsoidal Bouguer shell correction as an example and the result is compared to existing methods, which shows our method provides high accuracy and great computational efficiency. Suggestions for further developments and conclusions are drawn at last.     

Terrain effects in the atmospheric gravity and geoid corrections

In view of the smallness of the atmospheric mass compared to the mass variations within the Earth, it is generally assumed in physical geodesy that the terrain effects are negligible. Subsequently most models assume a spherical or ellipsoidal layering of the atmosphere. The removal and restoring of the atmosphere in solving the exterior boundary value problems thus correspond to gravity and geoid corrections of the order of 0.9 mGal and -0.7 cm, respectively.We demonstrate that the gravity terrain correction for the removal of the atmosphere is of the order of 50µGal/km of elevation with a maximum close to 0.5 mGal at the top of Mount Everest. The corresponding effect on the geoid may reach several centimetres in mountainous regions. Also the total effect on geoid determination for removal and restoring the atmosphere may contribute significantly, in particular by long wavelengths. This is not the case for the quasi geoid in mountainous regions.     

地球重力场模型对低轨卫星轨道积分的影响

介绍3种不同的地球重力场模型及其(约化)动力学定轨中所涉及的动力学模型,并基于Collocation轨道积分方法对CHAMP卫星进行数值积分,然后将轨道积分结果与JPL快速精密星历相比较。实验结果表明,由CHAMP卫星SST数据反演生成的EIGEN-2模型引力位系数具有较高的精度,能够满足低轨卫星精密定轨的需要。     

Wavelet filter analysis of local atmospheric pressure effects on gravity variations

An efficient method is proposed for the analysis of atmospheric pressure effects on gravity variations. It processes gravity and pressure signals using an orthogonal filter bank derived from high-degree Daubechies wavelets. The method introduces the atmospheric pressure admittance, which is both time- and frequency-dependent, and thus provides more information about when and how the frequency components in the pressure signal influence gravity variations. We demonstrate the efficiency of the wavelet method by applying it to observations from the Wuhan (China) superconducting gravimeter station. The analysis of gravity and pressure signals in 14 sub-bands with different bandwidths covering a frequency range from 0.176 to 720 cpd (cycles per day) reveals that local atmospheric pressure fluctuations start to induce obvious effects on gravity variations in the seismic band 0.52–1.04 mHz (periods 16 to 32 min) and highly correlate with gravity variation in the long-period seismic mode band 0.26–0.52 mHz (periods 32–64 min). The harmonics of solar-heating-induced atmospheric tides play a leading role in interfering with the variation of gravity residuals in the frequency band 0.704–11.25 cpd (periods 128 min to 1.42 day). Local atmospheric pressure effects on gravity variation are very strong in the frequency band 0.176–0.704 cpd (periods 1.42–5.69 day). Accurately filtering quarter-diurnal tides into a narrow band further demonstrates the efficiency of the wavelet method. After removing secular gravity changes and long-period atmospheric pressure variations, we show that there are obvious variations of local pressure admittance on time scales of hours to days. We also reveal seasonal variability of pressure admittances in the band 0.176–0.352 cpd (periods 2.84–5.69 day) after removing the effects of solar-heating atmospheric tides.     

Evaluation of topographical effects in precise geoid computation from densely sampled heights

In this paper we investigate the behaviour of Newton's kernel in the integration for topographical effects needed for solving the boundary value problem of geodesy. We follow the standard procedure and develop the kernel into a Taylor series in height and look at the convergence of this series when the integral is evaluated numerically on a geographical grid, as is always the case in practice. We show that the Taylor series converges very rapidly for the integration over the distant zone, i.e., the zone well removed from the point of interest. We also show that the series diverges in the vicinity of the point of interest when the grid becomes too dense. Generally, when the grid step is smaller than either the height of the point of interest, or the difference between its height and those of the neighbouring points. Thus we claim that the Taylor series version of Newton's kernel cannot be used for evaluating topographical effects on too dense a topographical mesh.     

Integral formulas for geopotential coefficient determination from gravity anomalies versus gravity disturbances

Integral formulas are derived for the determination of geopotential coefficients from gravity anomalies and gravity disturbances over the surface of the Earth. First order topographic corrections to spherical formulas are presented. In addition new integral formulas are derived for the determination of the external gravity field from surface gravity. Taking advantage of modern satellite positioning techniques, it is suggested that, in general, the external gravity field as well as individual coefficients are better determined from gravity disturbances than from gravity anomalies.     

Filter design for GOCE gravity gradients

The GOCE satellite observes gravity gradients with unprecedented accuracy and resolution. The GOCE observations are reliable within a well-defined measurement bandwidth. In this study, different finite and infinite impulse response filters have been designed to obtain the demanded pass. Exhaustive time and frequency domain investigations prove that the proposed infinite impulse response filter can be a real competitor of the existing solution of the filtering problem.     

On the combination of gravity anomalies and gravity disturbances for geoid determination in Western Australia

The geoid gradient over the Darling Fault in Western Australia is extremely high, rising by as much as 38 cm over only 2 km. This poses problems for gravimetric-only geoid models of the area, whose frequency content is limited by the spatial distribution of the gravity data. The gravimetric-only version of AUSGeoid98, for instance, is only able to resolve 46% of the gradient across the fault. Hence, the ability of GPS surveys to obtain accurate orthometric heights is reduced. It is described how further gravity data were collected over the Darling Fault, augmenting the existing gravity observations at key locations so as to obtain a more representative geoid gradient. As many of the gravity observations were collected at stations with a well-known GRS80 ellipsoidal height, the opportunity arose to compute a geoid model via both the Stokes and the Hotine approaches. A scheme was devised to convert free-air anomaly data to gravity disturbances using existing geoid models, followed by a Hotine integration to geoid heights. Interestingly, these results depended very weakly upon the choice of input geoid model. The extra gravity data did indeed improve the fit of the computed geoid to local GPS/Australian Height Datum (AHD) observations by 58% over the gravimetric-only AUSGeoid98. While the conventional Stokesian approach to geoid determination proved to be slightly better than the Hotine method, the latter still improved upon the gravimetric-only AUSGeoid98 solution, supporting the viability of conducting gravity surveys with GPS control for the purposes of geoid determination. AcknowledgementsThe author would like to thank Will Featherstone, Ron Gower, Ron Hackney, Linda Morgan, Geoscience Australia, Scripps Oceanographic Institute and the three anonymous reviewers of this paper. This research was funded by the Australian Research Council.