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1.
The goal of this paper is to present the finite element scheme for solving the Earth potential problems in 3D domains above
the Earth surface. To that goal we formulate the boundary-value problem (BVP) consisting of the Laplace equation outside the
Earth accompanied by the Neumann as well as the Dirichlet boundary conditions (BC). The 3D computational domain consists of
the bottom boundary in the form of a spherical approximation or real triangulation of the Earth’s surface on which surface
gravity disturbances are given. We introduce additional upper (spherical) and side (planar and conical) boundaries where the
Dirichlet BC is given. Solution of such elliptic BVP is understood in a weak sense, it always exists and is unique and can
be efficiently found by the finite element method (FEM). We briefly present derivation of FEM for such type of problems including
main discretization ideas. This method leads to a solution of the sparse symmetric linear systems which give the Earth’s potential
solution in every discrete node of the 3D computational domain. In this point our method differs from other numerical approaches,
e.g. boundary element method (BEM) where the potential is sought on a hypersurface only. We apply and test FEM in various
situations. First, we compare the FEM solution with the known exact solution in case of homogeneous sphere. Then, we solve
the geodetic BVP in continental scale using the DNSC08 data. We compare the results with the EGM2008 geopotential model. Finally,
we study the precision of our solution by the GPS/levelling test in Slovakia where we use terrestrial gravimetric measurements
as input data. All tests show qualitative and quantitative agreement with the given solutions. 相似文献
2.
Lars Sjöberg 《Journal of Geodesy》1982,56(1):27-39
If satellite range-rate information is given continuously on an extra terrestrial sphere (S), the spherical harmonic coefficients
of the gravitational potential of the Earth can be determined by direct integration of the data. An exact solution is given
in the case when circular satellite orbits crosses S in all directions. A simpler acquisition of the data is achieved when
the observations are restricted to polar satellite orbits. However, in this case the solutions become more complex due to
attenuation. A solution to order sin Δ, where Δ is the separation angle of the two satellites, is given for the zonal harmonics.
For tesseral hamonics a zero-order solution is derived. Corrections are given for observations out of S.
Finally, the along track configuration of the satellite pair is compared with a radially designed satellite system (one above
the other). The former is found most favourable for the recovery of geopotential coefficients. 相似文献
3.
Pavel Ditmar 《Journal of Geodesy》2018,92(12):1401-1412
Time-varying Stokes coefficients estimated from GRACE satellite data are routinely converted into mass anomalies at the Earth’s surface with the expression proposed for that purpose by Wahr et al. (J Geophys Res 103(B12):30,205–30,229, 1998). However, the results obtained with it represent mass transport at the spherical surface of 6378 km radius. We show that the accuracy of such conversion may be insufficient, especially if the target area is located in a polar region and the signal-to-noise ratio is high. For instance, the peak values of mean linear trends in 2003–2015 estimated over Greenland and Amundsen Sea embayment of West Antarctica may be underestimated in this way by about 15%. As a solution, we propose an updated expression for the conversion of Stokes coefficients into mass anomalies. This expression is based on the assumptions that: (i) mass transport takes place at the reference ellipsoid and (ii) at each point of interest, the ellipsoidal surface is approximated by the sphere with a radius equal to the current radial distance from the Earth’s center (“locally spherical approximation”). The updated expression is nearly as simple as the traditionally used one but reduces the inaccuracies of the conversion procedure by an order of magnitude. In addition, we remind the reader that the conversion expressions are defined in spherical (geocentric) coordinates. We demonstrate that the difference between mass anomalies computed in spherical and ellipsoidal (geodetic) coordinates may not be negligible, so that a conversion of geodetic colatitudes into geocentric ones should not be omitted. 相似文献
4.
The geodetic boundary value problem is formulated which uses as boundary values the differences between the geopotential of
points at the surface of the continents and the potential of the geoid. These differences are computed by gravity measurements
and levelling data. In addition, the shape of the geoid over the oceans is assumed to be known from satellite altimetry and
the shape of the continents from satellite results together with three-dimensional triangulation. The boundary value problem
thus formulated is equivalent to Dirichlet's exterior problem except for the unknown potential of the geoid. This constant
is determined by an integral equation for the normal derivative of the gravitational potential which results from the first
derivative of Green's fundamental formula. The general solution, which exists, of the integral equation gives besides the
potential of the geoid the solution of the geodetic boundary value problem. In addition approximate solutions for a spherical
surface of the earth are derived. 相似文献
5.
Spherical harmonic modelling to ultra-high degree of Bouguer and isostatic anomalies 总被引：9，自引：4，他引：5
The availability of high-resolution global digital elevation data sets has raised a growing interest in the feasibility of obtaining their spherical harmonic representation at matching resolution, and from there in the modelling of induced gravity perturbations. We have therefore estimated spherical Bouguer and Airy isostatic anomalies whose spherical harmonic models are derived from the Earth’s topography harmonic expansion. These spherical anomalies differ from the classical planar ones and may be used in the context of new applications. We succeeded in meeting a number of challenges to build spherical harmonic models with no theoretical limitation on the resolution. A specific algorithm was developed to enable the computation of associated Legendre functions to any degree and order. It was successfully tested up to degree 32,400. All analyses and syntheses were performed, in 64 bits arithmetic and with semi-empirical control of the significant terms to prevent from calculus underflows and overflows, according to IEEE limitations, also in preserving the speed of a specific regular grid processing scheme. Finally, the continuation from the reference ellipsoid’s surface to the Earth’s surface was performed by high-order Taylor expansion with all grids of required partial derivatives being computed in parallel. The main application was the production of a 1′ × 1′ equiangular global Bouguer anomaly grid which was computed by spherical harmonic analysis of the Earth’s topography–bathymetry ETOPO1 data set up to degree and order 10,800, taking into account the precise boundaries and densities of major lakes and inner seas, with their own altitude, polar caps with bedrock information, and land areas below sea level. The harmonic coefficients for each entity were derived by analyzing the corresponding ETOPO1 part, and free surface data when required, at one arc minute resolution. The following approximations were made: the land, ocean and ice cap gravity spherical harmonic coefficients were computed up to the third degree of the altitude, and the harmonics of the other, smaller parts up to the second degree. Their sum constitutes what we call ETOPG1, the Earth’s TOPography derived Gravity model at 1′ resolution (half-wavelength). The EGM2008 gravity field model and ETOPG1 were then used to rigorously compute 1′ × 1′ point values of surface gravity anomalies and disturbances, respectively, worldwide, at the real Earth’s surface, i.e. at the lower limit of the atmosphere. The disturbance grid is the most interesting product of this study and can be used in various contexts. The surface gravity anomaly grid is an accurate product associated with EGM2008 and ETOPO1, but its gravity information contents are those of EGM2008. Our method was validated by comparison with a direct numerical integration approach applied to a test area in Morocco–South of Spain (Kuhn, private communication 2011) and the agreement was satisfactory. Finally isostatic corrections according to the Airy model, but in spherical geometry, with harmonic coefficients derived from the sets of the ETOPO1 different parts, were computed with a uniform depth of compensation of 30?km. The new world Bouguer and isostatic gravity maps and grids here produced will be made available through the Commission for the Geological Map of the World. Since gravity values are those of the EGM2008 model, geophysical interpretation from these products should not be done for spatial scales below 5 arc minutes (half-wavelength). 相似文献
6.
Lars E. Sjöberg 《Journal of Geodesy》1989,63(2):213-221
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. 相似文献
7.
M. Ivan 《Journal of Geodesy》1996,70(11):755-767
Summary A procedure is derived for the upward continuation of unevenly spaced gravity data. The topographic relief is approximated by a polyhedron with triangular faces and vertices placed at small distances around the surface of a sphere. The usual Fredholm integral equation of the second kind is modified considering the discontinuity of the normal vector. It is solved by successive approximations assuming the unknown function is linear inside each face at every step of the iteration process. An approximate formula to obtain the anomalous potential from the Bouguer anomaly is discussed. The potential of a homogeneous polyhedron is derived and used to compute relief corrections to the geoid undulations. Numerical applications are presented with respect to the Romanian territory.Partially presented at theJoint Symposium of IGC and IGC, Graz, Austria, 11–17 September 1994 相似文献
8.
L. E. Sjöberg 《Journal of Geodesy》2003,77(3-4):139-147
Assuming that the gravity anomaly and disturbing potential are given on a reference ellipsoid, the result of Sjöberg (1988, Bull Geod 62:93–101) is applied to derive the potential coefficients on the bounding sphere of the ellipsoid to order e
2 (i.e. the square of the eccentricity of the ellipsoid). By adding the potential coefficients and continuing the potential downward to the reference ellipsoid, the spherical Stokes formula and its ellipsoidal correction are obtained. The correction is presented in terms of an integral over the unit sphere with the spherical approximation of geoidal height as the argument and only three well-known kernel functions, namely those of Stokes, Vening-Meinesz and the inverse Stokes, lending the correction to practical computations. Finally, the ellipsoidal correction is presented also in terms of spherical harmonic functions. The frequently applied and sometimes questioned approximation of the constant m, a convenient abbreviation in normal gravity field representations, by e
2/2, as introduced by Moritz, is also discussed. It is concluded that this approximation does not significantly affect the ellipsoidal corrections to potential coefficients and Stokes formula. However, whether this standard approach to correct the gravity anomaly agrees with the pure ellipsoidal solution to Stokes formula is still an open question. 相似文献
9.
The second-order derivatives of the Earth’s potential in the local north-oriented reference frame are expanded in series of
modified spherical harmonics. Linear relations are derived between the spectral coefficients of these series and the spectrum
of the geopotential. On the basis of these relations, recurrence procedures are developed for evaluating the geopotential
coefficients from the spectrum of each derivative and, inversely, for simulating the latter from a known geopotential model.
Very simple structure of the derived expressions for the derivatives is convenient for estimating the geopotential coefficients
by the least-squares procedure, at a certain step of processing satellite gradiometry data. Due to the orthogonality of the
new series, the quadrature formula approach can be also applied, which allows avoidance of aliasing errors caused by the series
truncation. The spectral coefficients of the derivatives are evaluated on the basis of the derived relations from the geopotential
models EGM96 and EIGEN-CG01C at a mean orbital sphere of the GOCE satellite. Various characteristics of the spectra are studied
corresponding to the EGM96 model.
Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users. 相似文献
10.
A comparison of the tesseroid,prism and point-mass approaches for mass reductions in gravity field modelling 总被引：6，自引：3，他引：6
The calculation of topographic (and iso- static) reductions is one of the most time-consuming operations in gravity field
modelling. For this calculation, the topographic surface of the Earth is often divided with respect to geographical or map-grid
lines, and the topographic heights are averaged over the respective grid elements. The bodies bounded by surfaces of constant
(ellipsoidal) heights and geographical grid lines are denoted as tesseroids. Usually these ellipsoidal (or spherical) tesseroids
are replaced by “equivalent” vertical rectangular prisms of the same mass. This approximation is motivated by the fact that
the volume integrals for the calculation of the potential and its derivatives can be exactly solved for rectangular prisms,
but not for the tesseroids. In this paper, an approximate solution of the spherical tesseroid integrals is provided based
on series expansions including third-order terms. By choosing the geometrical centre of the tesseroid as the Taylor expansion
point, the number of non-vanishing series terms can be greatly reduced. The zero-order term is equivalent to the point-mass
formula. Test computations show the high numerical efficiency of the tesseroid method versus the prism approach, both regarding
computation time and accuracy. Since the approximation errors due to the truncation of the Taylor series decrease very quickly
with increasing distance of the tesseroid from the computation point, only the elements in the direct vicinity of the computation
point have to be separately evaluated, e.g. by the prism formulas. The results are also compared with the point-mass formula.
Further potential refinements of the tesseroid approach, such as considering ellipsoidal tesseroids, are indicated. 相似文献
11.
Donald H. Eckhardt 《Journal of Geodesy》1983,57(1-4):394-409
Meissl has derived weighting functions for converting point gravity anomaly degree variances into mean anomaly variances over a circular cap on a sphere. If the cap is sufficiently small so that the cap on a sphere degenerates into a circle on a plane, the problem may be considered that of the gain of a circular filter for a surface wave whose wave number depends on the spherical harmonic degree. The Meissl weights then become replaced by diffraction integrals of optical physics. The expected gain for a square filter for waves coming from random directions is derived and shown to be close to the gain of a circular filter with the same area. The expected gains and cross-gains for rectangular filters are also derived. When weighted by an anomaly degree variance model, these gains and cross-gains can be used to determine rectangular anomaly variances and covariances for arbitrary bandwidths. Using the Tscherning-Rapp model, analytic gravity anomaly variances and covariances are calculated for 1°×1° blocks. 相似文献
12.
Cédric H David David J Gochis David R Maidment Wei Yu David N Yates Zong-Liang Yang 《Transactions in GIS》2009,13(4):363-377
The National Elevation, Hydrography and Land Cover datasets of the United States have been synthesized into a geospatial dataset called NHDPlus which is referenced to a spheroidal Earth, provides geospatial data layers for topography on 30 m rasters, and has vector coverages for catchments and river reaches. In this article, we examine the integration of NHDPlus with the Noah-distributed model. In order to retain compatibility with atmospheric models, Noah-distributed utilizes surface domain fields referenced to a spherical rather than spheroidal Earth in its computation of vertical land surface/atmosphere water and energy budgets (at coarse resolution) as well as horizontal cell-to-cell water routing across the land surface and through the shallow subsurface (at fine resolution). Two data-centric issues affecting the linkage between Noah-distributed and NHDPlus are examined: (1) the shape of the Earth; and (2) the linking of gridded landscape with a vector representation of the stream and river network. At mid-latitudes the errors due to projections between spherical and spheroidal representations of the Earth are significant. A catchment-based "pour point" technique is developed to link the raster and vector data to provide lateral inflow from the landscape to a one-dimensional river model. We conclude that, when Noah-distributed is run uncoupled to an atmospheric model, it is advantageous to implement Noah-distributed at the native spatial scale of the digital elevation data and the spheroidal Earth of the NHDPlus dataset rather than transforming the NHDPlus dataset to fit the coarser resolution and spherical Earth shape of the Noah-distributed model. 相似文献
13.
利用卫星重力数据确定地球外部重力场的一种方法及模拟实验检验 总被引：5，自引：1，他引：5
假定给定了海量的卫星重力观测数据，基于球谐展开法并应用最小二乘原理可以确定地球重力场模型EGM，由此确定的重力场模型在地球表面附近的空间区域未必有效。设想有一个包含了地球的大球Ks，假定EGM在大球的外部成立，则可根据虚拟压缩恢复法求出一种新的重力场模型NEGM，它是对原有重力场模型的进一步精化，适合于整个地球外部空间，从理论上可以解决重力场的“向下延拓”问题。初步的模拟实验检验支持虚拟压缩恢复法以及由此而引中出的“向下延拓”法。 相似文献
14.
地面激光扫描球形标靶的球心误差研究 总被引：1，自引：0，他引：1
分析了球形标靶的特点,通过激光雷达方程描述了球心误差的产生过程和性质,研究统计了球心误差随激光入射方向的分布,采取多项式拟合和模拟球面的方法确定了球心位置和误差大小。实测数据验证了球心误差的系统性特点,并提出削弱球心误差的方法。 相似文献
15.
The boundary value problem of physical geodesy has been solved with the use of a harmonic reduction down to an internal sphere using a discrete procedure. (For gravity cf. Bjerhammar 1964 and for the potential cf. Bjerhammar 1968). This was a finite-dimensional approach mostly with one-to-one correspondence between observations and unknowns on the sphere. Earlier studies were made with the use of surface elements (on the sphere) with constantgravity. Integration over the surface elements was replaced by a discrete approach with the use of the distance to a point in the centre of the surface element. See Bjerhammar (1968) and (1969). This approach was later presented as a “reflexive prediction” technique for a weakly stationary stochastic process. Bjerhammar (1974, 1976). Krarup (1969) minimized the L2-norm of the potential on the internal sphere. It will here be proved that the two solutions are identical for a proper choice of the radii of the internal spheres. The proof is given for a spherical earth with selected choice of “carrier points”. The convergence problem is discussed. The L2-norm solution is found convergent for the fully harmonic case. Uniform convergence is obtained in the non-harmonic case with the use of the original procedure applied in accordance with the theorems of Keldych-Lavrentieff and Yamabe. 相似文献
16.
申文斌 《地球空间信息科学学报》2009,12(1):1-6
Given a continuous boundary value on the boundary of a "simply closed surface"S that encloses the whole Earth, a regular harmonic fictitious field V*(P) in the domain outside an inner sphere K i that lies inside the Earth could be determined, and it is proved that V*(P) coincides with the Earth’s real field V(P) in the whole domain outside the Earth. Since in the domain outside the inner sphere Ki and the fictitious regular harmonic function V*(P) could be expressed as a uniformly convergent spherical harm... 相似文献
17.
基于$\frac{{{{\bar{P}}}_{nm}}\left( \cos \theta \right)}{\sin \theta }\left( m>0 \right)$的非奇异递推公式,给出了基于球坐标的引力矢量和垂线偏差非奇异计算公式;针对极点λ可任意取值引起的地方指北坐标系方向的不确定性问题,证明了引力矢量在转换到地心直角坐标系后不随λ的变化而变化,即与λ的取值无关。最终的数值计算结果表明,直角坐标系下的非奇异计算公式与本文提出的球坐标下的非奇异计算公式所得计算结果绝对值差异小于10-16m/s2,证明了该非奇异公式的正确性。最后总结了所有引力位球函数一阶导、二阶导非奇异性计算的一般思路。 相似文献
18.
In space-borne gravitational field determination, two challenges are inherent. First, the continuation of the data down to
the surface of the Earth is an ill-posed problem, requiring therefore regularization techniques. Second huge data sets result
requiring efficient numerical methods. In this paper, we show how locally supported wavelets on the sphere can be developed
by means of a spherical version of the so-called up function. By construction, the corresponding scaling functions and wavelets
are infinitely smooth, so that they can be used for regularization purposes. In particular, we show how the ill-posed pseudo-differential
equations coming from satellite missions can be regularized by efficient numerical schemes using locally supported wavelets.
These methods seem in particular to be interesting for regional gravity field modelling. 相似文献
19.
This paper is devoted to the spherical and spheroidal harmonic expansion of the gravitational potential of the topographic masses in the most rigorous way. Such an expansion can be used to compute gravimetric topographic effects for geodetic and geophysical applications. It can also be used to augment a global gravity model to a much higher resolution of the gravitational potential of the topography. A formulation for a spherical harmonic expansion is developed without the spherical approximation. Then, formulas for the spheroidal harmonic expansion are derived. For the latter, Legendre’s functions of the first and second kinds with imaginary variable are expanded in Laurent series. They are then scaled into two real power series of the second eccentricity of the reference ellipsoid. Using these series, formulas for computing the spheroidal harmonic coefficients are reduced to surface harmonic analysis. Two numerical examples are presented. The first is a spherical harmonic expansion to degree and order 2700 by taking advantage of existing software. It demonstrates that rigorous spherical harmonic expansion is possible, but the computed potential on the geoid shows noticeable error pattern at Polar Regions due to the downward continuation from the bounding sphere to the geoid. The second numerical example is the spheroidal expansion to degree and order 180 for the exterior space. The power series of the second eccentricity of the reference ellipsoid is truncated at the eighth order leading to omission errors of 25 nm (RMS) for land areas, with extreme values around 0.5 mm to geoid height. The results show that the ellipsoidal correction is 1.65 m (RMS) over land areas, with maximum value of 13.19 m in the Andes. It shows also that the correction resembles the topography closely, implying that the ellipsoidal correction is rich in all frequencies of the gravity field and not only long wavelength as it is commonly assumed. 相似文献