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1.
Optimized formulas for the gravitational field of a tesseroid 总被引：7，自引：3，他引：4
Various tasks in geodesy, geophysics, and related geosciences require precise information on the impact of mass distributions on gravity field-related quantities, such as the gravitational potential and its partial derivatives. Using forward modeling based on Newton’s integral, mass distributions are generally decomposed into regular elementary bodies. In classical approaches, prisms or point mass approximations are mostly utilized. Considering the effect of the sphericity of the Earth, alternative mass modeling methods based on tesseroid bodies (spherical prisms) should be taken into account, particularly in regional and global applications. Expressions for the gravitational field of a point mass are relatively simple when formulated in Cartesian coordinates. In the case of integrating over a tesseroid volume bounded by geocentric spherical coordinates, it will be shown that it is also beneficial to represent the integral kernel in terms of Cartesian coordinates. This considerably simplifies the determination of the tesseroid’s potential derivatives in comparison with previously published methodologies that make use of integral kernels expressed in spherical coordinates. Based on this idea, optimized formulas for the gravitational potential of a homogeneous tesseroid and its derivatives up to second-order are elaborated in this paper. These new formulas do not suffer from the polar singularity of the spherical coordinate system and can, therefore, be evaluated for any position on the globe. Since integrals over tesseroid volumes cannot be solved analytically, the numerical evaluation is achieved by means of expanding the integral kernel in a Taylor series with fourth-order error in the spatial coordinates of the integration point. As the structure of the Cartesian integral kernel is substantially simplified, Taylor coefficients can be represented in a compact and computationally attractive form. Thus, the use of the optimized tesseroid formulas particularly benefits from a significant decrease in computation time by about 45 % compared to previously used algorithms. In order to show the computational efficiency and to validate the mathematical derivations, the new tesseroid formulas are applied to two realistic numerical experiments and are compared to previously published tesseroid methods and the conventional prism approach. 相似文献
2.
A new methodology for computing the gravitational effect of a spherical tesseroid has been devised and implemented. The methodology is based on the rotation from the global Earth-Centred Rotational reference frame to the local Earth-Centred P-Rotational reference frame, referred to the computation point P, and it requires knowledge of the height and the angular extension of each topographic column. After rotation, the gravitational effect of the tesseroid is computed via the effect of a sector of the spherical zonal band. In this respect, two possible procedures for handling the rotated tesseroids have been proposed and tested. The results obtained with the devised methodology are in good agreement with those derived by applying other existing methodologies. 相似文献
3.
F. Wild-Pfeiffer 《Journal of Geodesy》2008,82(10):637-653
Topographic and isostatic mass anomalies affect the external gravity field of the Earth. Therefore, these effects also exist
in the gravity gradients observed, e.g., by the satellite gravity gradiometry mission GOCE (Gravity and Steady-State Ocean Circulation Experiment). The downward continuation of the gravitational signals is rather difficult because of the high-frequency behaviour
of the combined topographic and isostatic effects. Thus, it is preferable to smooth the gravity field by some topographic-isostatic
reduction. In this paper the focus is on the modelling of masses in the space domain, which can be subdivided into different
mass elements and evaluated with analytical, semi-analytical and numerical methods. Five alternative mass elements are reviewed
and discussed: the tesseroid, the point mass, the prism, the mass layer and the mass line. The formulae for the potential,
the attraction components and the Marussi tensor of second-order potential derivatives are provided. The formulae for different
mass elements and computation methods are checked by assuming a synthetic topography of constant height over a spherical cap
and the position of the computation point on the polar axis. For this special situation an exact analytical solution for the
tesseroid exists and a comparison between the analytical solution of a spherical cap and the modelling of different mass elements
is possible. A comparison of the computation times shows that modelling by tesseroids with different methods produces the
most accurate results in an acceptable computation time. As a numerical example, the Marussi tensor of the topographic effect
is computed globally using tesseroids calculated by Gauss–Legendre cubature (3D) on the basis of a digital height model. The
order of magnitude in the radial-radial component is about ± 8 E.U.
Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users. 相似文献
4.
5.
G. L. Ramillien 《Journal of Geodesy》2017,91(8):881-895
A radial integration of spherical mass elements (i.e. tesseroids) is presented for evaluating the six components of the second-order gravity gradient (i.e. second derivatives of the Newtonian mass integral for the gravitational potential) created by an uneven spherical topography consisting of juxtaposed vertical prisms. The method uses Legendre polynomial series and takes elastic compensation of the topography by the Earth’s surface into account. The speed of computation of the polynomial series increases logically with the observing altitude from the source of anomaly. Such a forward modelling can be easily applied for reduction of observed gravity gradient anomalies by the effects of any spherical interface of density. An iterative least-squares inversion of measured gravity gradient coefficients is also proposed to estimate a regional set of juxtaposed topographic heights. Several tests of recovery have been made by considering simulated gradients created by idealistic conical and irregular Great Meteor seamount topographies, and for varying satellite altitudes and testing different levels of uncertainty. In the case of gravity gradients measured at a GOCE-type altitude of \(\sim \)300 km, the search converges down to a stable but smooth topography after 10–15 iterations, while the final root-mean-square error is \(\sim \)100 m that represents only 2 % of the seamount amplitude. This recovery error decreases with the altitude of the gravity gradient observations by revealing more topographic details in the region of survey. 相似文献
6.
The topographic effects by Stokes formula are typically considered for a spherical approximation of sea level. For more precise determination of the geoid, sea level is better approximated by an ellipsoid, which justifies the consideration of the ellipsoidal corrections of topographic effects for improved geoid solutions. The aim of this study is to estimate the ellipsoidal effects of the combined topographic correction (direct plus indirect topographic effects) and the downward continuation effect. It is concluded that the ellipsoidal correction to the combined topographic effect on the geoid height is far less than 1 mm. On the contrary, the ellipsoidal correction to the effect of downward continuation of gravity anomaly to sea level may be significant at the 1-cm level in mountainous regions. Nevertheless, if Stokes formula is modified and the integration of gravity anomalies is limited to a cap of a few degrees radius around the computation point, nor this effect is likely to be significant.AcknowledgementsThe author is grateful for constructive remarks by J Ågren and the three reviewers. 相似文献
7.
There exist three types of convolution formulae for the efficient evaluation of gravity field convolution integrals, i.e., the planar 2D convolution, the spherical 2D convolution and the spherical 1D convolution. The largest drawback of both the planar and the spherical 2D FFT methods is that, due to the approximations in the kernel function, only inexact results can be achieved. Apparently, the reason is the meridian convergence at higher latitudes. As the meridians converge, the ??,?λ blocks do not form a rectangular grid, as is assumed in 2D FFT methods. It should be pointed out that the meridian convergence not only leads to an approximation error in the kernel function, but also causes an approximation error during the implementation of 2D FFT in computer. In order to meet the increasing need for precise determination of the vertica deflections, this paper derives a more precise planar 2D FFT formula for the computation of the vertical deflections. After having made a detailed comparison between the planar and the spherical 2D FFT formulae, we find out the main source of errors causing the loss in accuracy by applying the conventional spherical 2D FFT method. And then, a modified spherical 2D FFT formula for the computation of the vertical deflections is developed in this paper. A series of numerical tests have been carried out to illustrate the improvement made upon the old spherical 2D FFT. The second part of this paper is to discuss the influences of the spherical harmonic reference field, the limited capsize, and the singular integral on the computation of the vertical deflections. The results of the vertical deflections over China by applying the spherical 1D FFT formula with different integration radii have been compared to the astro-observed vertical deflections in the South China Sea to obtain a set of optimum deflection computation parameters. 相似文献
8.
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9.
New algorithms have been derived for computing terrain connections, all components of the attraction of the topography at
the topographic surface and the gradients of these attractions. These algorithms utilize fast Fourier transforms, but, in
contrast to methods currently in use, all divergences of the integrals are removed during the analysis. Sequential methods
employing a smooth intermediate reference surface have been developed to avoid the very large transforms necessary when making
computations at high resolution over a wide area.
A new method for the numerical solution of Molodensky's problem has been developed to mitigate the convergence difficulties
that occur at short wavelengths with methods based on a Taylor series expansion. A trial field on a level surface is continued
analytically to the topographic surface, and compared with that predicted from gravity observations. The difference is used
to compute a correction to the trial field and the process iterated. Special techniques are employed to speed convergence
and prevent oscillations.
Three different spectral methods for fitting a point-mass set to a gravity field given on a regular grid at constant elevation
are described. Two of the methods differ in the way that the spectrum of the point-mass set, which extends to infinite wave
number, is matched to that of the gravity field which is band-limited. The third method is essentially a space-domain technique
in which Fourier methods are used to solve a set of simultaneous equations. 相似文献
10.
基于Helmert第二压缩法进行边值解算时需要计算地形压缩对重力的直接影响和对（似）大地水准面的间接影响。计算近区直接、间接影响的传统积分算法仍是二重积分形式。该算法以网格中心点处的积分核作为网格积分核的平均值的计算模式在一定程度上引入了近似误差。另外,直接、间接影响的传统积分算法在中央区存在奇异性,需单独计算中央网格地形影响,因而增加了计算的复杂性。为此,本文推导了近区地形直接、间接影响的棱柱模型公式,一方面提高了地形影响的计算精度;另一方面中央区不存在奇异性,从而简化了计算过程。为避免棱柱模型存在的平面近似误差,可使用顾及地球曲率的棱柱模型算法计算地形影响。最后通过试验得出结论,在（似）大地水准面精度要求较高的应用中,应尽量使用顾及地球曲率的棱柱模型算法计算地形影响。 相似文献
11.
Comparisons between high-degree models of the Earth’s topographic and gravitational potential may give insight into the quality and resolution of the source data sets, provide feedback on the modelling techniques and help to better understand the gravity field composition. Degree correlations (cross-correlation coefficients) or reduction rates (quantifying the amount of topographic signal contained in the gravitational potential) are indicators used in a number of contemporary studies. However, depending on the modelling techniques and underlying levels of approximation, the correlation at high degrees may vary significantly, as do the conclusions drawn. The present paper addresses this problem by attempting to provide a guide on global correlation measures with particular emphasis on approximation effects and variants of topographic potential modelling. We investigate and discuss the impact of different effects (e.g., truncation of series expansions of the topographic potential, mass compression, ellipsoidal versus spherical approximation, ellipsoidal harmonic coefficient versus spherical harmonic coefficient (SHC) representation) on correlation measures. Our study demonstrates that the correlation coefficients are realistic only when the model’s harmonic coefficients of a given degree are largely independent of the coefficients of other degrees, permitting degree-wise evaluations. This is the case, e.g., when both models are represented in terms of SHCs and spherical approximation (i.e. spherical arrangement of field-generating masses). Alternatively, a representation in ellipsoidal harmonics can be combined with ellipsoidal approximation. The usual ellipsoidal approximation level (i.e. ellipsoidal mass arrangement) is shown to bias correlation coefficients when SHCs are used. Importantly, gravity models from the International Centre for Global Earth Models (ICGEM) are inherently based on this approximation level. A transformation is presented that enables a transformation of ICGEM geopotential models from ellipsoidal to spherical approximation. The transformation is applied to generate a spherical transform of EGM2008 (sphEGM2008) that can meaningfully be correlated degree-wise with the topographic potential. We exploit this new technique and compare a number of models of topographic potential constituents (e.g., potential implied by land topography, ocean water masses) based on the Earth2014 global relief model and a mass-layer forward modelling technique with sphEGM2008. Different to previous findings, our results show very significant short-scale correlation between Earth’s gravitational potential and the potential generated by Earth’s land topography (correlation +0.92, and 60% of EGM2008 signals are delivered through the forward modelling). Our tests reveal that the potential generated by Earth’s oceans water masses is largely unrelated to the geopotential at short scales, suggesting that altimetry-derived gravity and/or bathymetric data sets are significantly underpowered at 5 arc-min scales. We further decompose the topographic potential into the Bouguer shell and terrain correction and show that they are responsible for about 20 and 25% of EGM2008 short-scale signals, respectively. As a general conclusion, the paper shows the importance of using compatible models in topographic/gravitational potential comparisons and recommends the use of SHCs together with spherical approximation or EHCs with ellipsoidal approximation in order to avoid biases in the correlation measures. 相似文献
12.
A formula for computing the gravity disturbance from the second radial derivative of the disturbing potential 总被引：6，自引：0，他引：6
J. Li 《Journal of Geodesy》2002,76(4):226-231
A formula for computing the gravity disturbance and gravity anomaly from the second radial derivative of the disturbing potential
is derived in detail using the basic differential equation with spherical approximation in physical geodesy and the modified
Poisson integral formula. The derived integral in the space domain, expressed by a spherical geometric quantity, is then converted
to a convolution form in the local planar rectangular coordinate system tangent to the geoid at the computing point, and the
corresponding spectral formulae of 1-D FFT and 2-D FFT are presented for numerical computation.
Received: 27 December 2000 / Accepted: 3 September 2001 相似文献
13.
In gravimetric geoid determination, there are three integrals to be evaluated: Stokes, terrain correction and potential.
Using geographical grid data, the straightforward evaluation gives an `exact' summation but is very time-consuming. This paper
proposes a new method which is based on the modification of the integrated 2D or 3D function into a 1D (spherical distance
angle) function applied to an optimal quadrature. The advantages are: a) It is exact (without approximation, especially the
singularities have been removed) and can be used for all the three integrals; b) It involves gridded data and is easy to handle;
c) It greatly speeds up the computation. A great mountain area, the southern Alps, has been chosen to test the new method.
Numerical tests show that: compared with the straightforward evaluation, the new technique consumes on average only 1.23%
of CPU time for the three integrals without adversely affecting accuracy.
Received 27 March 1995; Accepted 13 September 1996 相似文献
14.
Lars E. Sjöberg 《Journal of Geodesy》2006,79(12):675-681
The application of Stokes’s formula to determine the geoid height requires that topographic and atmospheric masses be mathematically removed prior to Stokes integration. This corresponds to the applications of the direct topographic and atmospheric effects. For a proper geoid determination, the external masses must then be restored, yielding the indirect effects. Assuming an ellipsoidal layering of the atmosphere with 15% increase in its density towards the poles, the direct atmospheric effect on the geoid height is estimated to be −5.51 m plus a second-degree zonal harmonic term with an amplitude of 1.1 cm. The indirect effect is +5.50 m and the total geoid correction thus varies between −1.2 cm at the equator to 1.9 cm at the poles. Finally, the correction needed to the atmospheric effect if Stokes’s formula is used in a spherical approximation, rather than an ellipsoidal approximation, of the Earth varies between 0.3 cm and 4.0 cm at the equator and pole, respectively. 相似文献
15.
在重力归算中,局部地形改正在重力勘探、地壳结构分析和大地水准面计算等领域有着重要意义,但严格棱柱体积分公式计算效率低,而快速计算公式则会降低计算精度。本文利用CUDA并行编程平台,提出一种地形格网重新编码和严格棱柱体积分八分量拆解方法,实现了基于CPU+GPU异构并行技术的严格棱柱体积分计算地形改正快速并行算法,克服了GPU各个线程计算任务分配和线程计算超载问题,解决了局部地形改正的高分辨率、高精度严密公式的快速计算难题。通过试验,在显卡型号为Tesla V100的计算机上进行4°×6°范围,积分半径40'和分辨率1'的局部地形改正计算仅需1.5 s;分辨率10″的局部地形改正计算仅需14.6 min;进行分辨率3″的地形改正计算耗时45.7 h,而传统串行算法则难以完成计算。在保证微伽级以上计算精度的条件下,计算加速比最高达到850倍以上,有效缩短了计算耗时,提高了计算效率。本文还依据上述并行算法对全国范围地形改正量进行计算。结果表明,我国地形改正量普遍低于80 mGal（1 Gal=10-2 m/s2）,平均值1.83 mGal,最大值达到196 mGal。 相似文献
16.
17.
Solutions of the linearized geodetic boundary value problem for an ellipsoidal boundary to order e
3
The geodetic boundary value problem (GBVP) was originally formulated for the topographic surface of the Earth. It degenerates to an ellipsoidal problem, for example when topographic and downward continuation reductions have been applied. Although these ellipsoidal GBVPs possess a simpler structure than the original ones, they cannot be solved analytically, since the boundary condition still contains disturbing terms due to anisotropy, ellipticity and centrifugal components in the reference potential. Solutions of the so-called scalar-free version of the GBVP, upon which most recent practical calculations of geoidal and quasigeoidal heights are based, are considered. Starting at the linearized boundary condition and presupposing a normal field of Somigliana–Pizzetti type, the boundary condition described in spherical coordinates is expanded into a series with respect to the flattening f of the Earth. This series is truncated after the linear terms in f, and first-order solutions of the corresponding GBVP are developed in closed form on the basis of spherical integral formulae, modified by suitable reduction terms. Three alternative representations of the solution are discussed, implying corrections by adding a first-order non-spherical term to the solution, by reducing the boundary data, or by modifying the integration kernel. A numerically efficient procedure for the evaluation of ellipsoidal effects, in the case of the linearized scalar-free version of the GBVP, involving first-order ellipsoidal terms in the boundary condition, is derived, utilizing geopotential models such as EGM96. 相似文献
18.
Herman van Gysen 《Journal of Geodesy》1994,68(3):173-179
Thin-plate splines — well known for their flexibility and fidelity in representing experimental data — are especially suited for the numerical evaluation of geodetic integrals in the area where these are most sensitive to the data, i.e. in the immediate vicinity of the computation point. Quadrature rules that are exact for thin-plate splines interpolating randomly spaced data are derived for the inner zone contribution (to a planar approximation) to Stokes's formula, to the formulae of Vening Meinesz and to theL
1 gradient operator in the analytical continuation solution of Molodensky's problem.The quadrature method is demonstrated by calculating the inner zone contribution to height anomalies in a mountainous area of Lesotho and carrying out a comparison with GPS-derived heights. Height anomalies are recovered with an accuracy of 6 cm. 相似文献
19.
Two numerical techniques are used in recent regional high-frequency geoid computations in Canada: discrete numerical integration
and fast Fourier transform. These two techniques have been tested for their numerical accuracy using a synthetic gravity field.
The synthetic field was generated by artificially extending the EGM96 spherical harmonic coefficients to degree 2160, which
is commensurate with the regular 5′ geographical grid used in Canada. This field was used to generate self-consistent sets of synthetic gravity anomalies and
synthetic geoid heights with different degree variance spectra, which were used as control on the numerical geoid computation
techniques. Both the discrete integration and the fast Fourier transform were applied within a 6∘ spherical cap centered at each computation point. The effect of the gravity data outside the spherical cap was computed using
the spheroidal Molodenskij approach. Comparisons of these geoid solutions with the synthetic geoid heights over western Canada
indicate that the high-frequency geoid can be computed with an accuracy of approximately 1 cm using the modified Stokes technique,
with discrete numerical integration giving a slightly, though not significantly, better result than fast Fourier transform.
Received: 2 November 1999 / Accepted: 11 July 2000 相似文献
20.
The optimum expression for the gravitational potential of polyhedral bodies having a linearly varying density distribution 总被引：2，自引：0，他引：2
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. 相似文献