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1.
Recurrence relations have been derived for truncation error coefficients of the extended Stokes' function and its partial
derivatives required in the computation of the disturbing gravity vector at any elevation above the earth's surface. The corresponding
formulae, the example of values of the truncation error coefficients for H=30.1 km and ψ0=30∘ and the estimations of truncation error are given in this article.
Received: 26 January 1996 / Accepted: 11 June 1997 相似文献
2.
Z. Martinec 《Journal of Geodesy》1998,72(7-8):460-472
Green's function for the boundary-value problem of Stokes's type with ellipsoidal corrections in the boundary condition for
anomalous gravity is constructed in a closed form. The `spherical-ellipsoidal' Stokes function describing the effect of two
ellipsoidal correcting terms occurring in the boundary condition for anomalous gravity is expressed in O(e
2
0)-approximation as a finite sum of elementary functions analytically representing the behaviour of the integration kernel
at the singular point ψ=0. We show that the `spherical-ellipsoidal' Stokes function has only a logarithmic singularity in
the vicinity of its singular point. The constructed Green function enables us to avoid applying an iterative approach to solve
Stokes's boundary-value problem with ellipsoidal correction terms involved in the boundary condition for anomalous gravity.
A new Green-function approach is more convenient from the numerical point of view since the solution of the boundary-value
problem is determined in one step by computing a Stokes-type integral. The question of the convergence of an iterative scheme
recommended so far to solve this boundary-value problem is thus irrelevant.
Received: 5 June 1997 / Accepted: 20 February 1998 相似文献
3.
This paper generalizes the Stokes formula from the spherical boundary surface to the ellipsoidal boundary surface. The resulting
solution (ellipsoidal geoidal height), consisting of two parts, i.e. the spherical geoidal height N
0 evaluated from Stokes's formula and the ellipsoidal correction N
1, makes the relative geoidal height error decrease from O(e
2) to O(e
4), which can be neglected for most practical purposes. The ellipsoidal correction N
1 is expressed as a sum of an integral about the spherical geoidal height N
0 and a simple analytical function of N
0 and the first three geopotential coefficients. The kernel function in the integral has the same degree of singularity at
the origin as the original Stokes function. A brief comparison among this and other solutions shows that this solution is
more effective than the solutions of Molodensky et al. and Moritz and, when the evaluation of the ellipsoidal correction N
1 is done in an area where the spherical geoidal height N
0 has already been evaluated, it is also more effective than the solution of Martinec and Grafarend.
Received: 27 January 1999 / Accepted: 4 October 1999 相似文献
4.
关于Stokes公式的球面卷积和平面卷积的注记 总被引:2,自引:0,他引:2
晁定波 《武汉大学学报(信息科学版)》2003,28(6):651-654
讨论了Stokes公式球面卷积和平面卷积形式的近似性和严密性问题,分析了Stokes函数球面卷积形式和平面卷积形式的关系,推导了其间的差值表达式,估算了最大差值及其对计算大地水准面差距的误差影响。同时指出,将顾及Stokes函数全项的平面卷积公式称为严密公式的提法,仅仅是相对仅顾及Stokes函数首项的简单平面卷积公式而言,认为更合理的提法应该是“高精度Stokes平面近似卷积公式”。理论分析表明,球面卷积不可能严格转化为等效的平面卷积。 相似文献
5.
Improved convergence rates for the truncation error in gravimetric geoid determination 总被引:2,自引:2,他引:0
When Stokes's integral is used over a spherical cap to compute a gravimetric estimate of the geoid, a truncation error results
due to the neglect of gravity data over the remainder of the Earth. Associated with the truncation error is an error kernel
defined over these two complementary regions. An important observation is that the rate of decay of the coefficients of the
series expansion for the truncation error in terms of Legendre polynomials is determined by the smoothness properties of the
error kernel. Previously published deterministic modifications of Stokes's integration kernel involve either a discontinuity
in the error kernel or its first derivative at the spherical cap radius. These kernels are generalised and extended by constructing
error kernels whose derivatives at the spherical cap radius are continuous up to an arbitrary order. This construction is
achieved by smoothly continuing the error kernel function into the spherical cap using a suitable degree polynomial. Accordingly,
an improved rate of convergence of the spectral series representation of the truncation error is obtained.
Received: 21 April 1998 / Accepted: 4 October 1999 相似文献
6.
Stanley W. Shepperd 《Journal of Geodesy》1982,56(2):95-105
A recursive method is derived for computing the Molodenskii truncation error coefficients at altitude for the altitude-generalized
Stokes integral. Furthermore, the Cook truncation error coefficients at altitude corresponding to the generalized Vening-Meinesz
integral are derived in terms of the Molodenskii coefficients. Also, the gravity disturbance truncation error coefficients
at altitude are related to the Molodenskii coefficients. By combining these results, it is shown how the truncation error
for the complete gravity disturbance vector at altitude may be computed recursively. 相似文献
7.
Recursive algorithms for the computation of the potential harmonic coefficients of a constant density polyhedron 总被引:1,自引:0,他引:1
Dimitrios Tsoulis Olivier Jamet Jérôme Verdun Nicolas Gonindard 《Journal of Geodesy》2009,83(10):925-942
The gravitational potential of a constant density general polyhedron can be expressed both in terms of a closed analytical expression and as a series expansion involving the corresponding spherical harmonic coefficients. The latter can be obtained from two independent algorithms, which differ not only in their algorithmic architecture but in their efficiency and overall performance, especially when computing the coefficients of higher degree and order. In the present paper a comparative study of all these three approaches is carried out focusing on the numerical implementation of the recursive relations appearing in the two algorithms for the computation of the polyhedral potential harmonic coefficients. The performed numerical investigations show that the linear algorithm proposed by Jamet and Thomas (Proceedings of the second international GOCE user workshop, ‘GOCE, The Geoid and Oceanography’, ESA-ESRIN, Frascati, Italy, 8–10 March 2004, ESA SP-569, 2004), but so far not implemented, achieves a reasonable accuracy at a computational expense that opens to practical applications, for instance in the field of satellite gravimetry/gradiometry interpretation. The convergence behavior of the linear recursion algorithm is studied thoroughly and a computational procedure is proposed that enables the stable computation of potential harmonic coefficients up to degree 60 when referring to an arbitrarily shaped polyhedral body. 相似文献
8.
The derivatives of the Earth gravitational potential are considered in the global Cartesian Earth-fixed reference frame. Spherical
harmonic series are constructed for the potential derivatives of the first and second orders on the basis of a general expression
of Cunningham (Celest Mech 2:207–216, 1970) for arbitrary order derivatives of a spherical harmonic. A common structure of
the series for the potential and its first- and second-order derivatives allows to develop a general procedure for constructing
similar series for the potential derivatives of arbitrary orders. The coefficients of the derivatives are defined by means
of recurrence relations in which a coefficient of a certain order derivative is a linear function of two coefficients of a
preceding order derivative. The coefficients of the second-order derivatives are also presented as explicit functions of three
coefficients of the potential. On the basis of the geopotential model EGM2008, the spherical harmonic coefficients are calculated
for the first-, second-, and some third-order derivatives of the disturbing potential T, representing the full potential V, after eliminating from it the zero- and first-degree harmonics. The coefficients of two lowest degrees in the series for
the derivatives of T are presented. The corresponding degree variances are estimated. The obtained results can be applied for solving various
problems of satellite geodesy and celestial mechanics. 相似文献
9.
This work is an investigation of three methods for regional geoid computation: Stokes’s formula, least-squares collocation (LSC), and spherical radial base functions (RBFs) using the spline kernel (SK). It is a first attempt to compare the three methods theoretically and numerically in a unified framework. While Stokes integration and LSC may be regarded as classic methods for regional geoid computation, RBFs may still be regarded as a modern approach. All methods are theoretically equal when applied globally, and we therefore expect them to give comparable results in regional applications. However, it has been shown by de Min (Bull Géod 69:223–232, 1995. doi: 10.1007/BF00806734) that the equivalence of Stokes’s formula and LSC does not hold in regional applications without modifying the cross-covariance function. In order to make all methods comparable in regional applications, the corresponding modification has been introduced also in the SK. Ultimately, we present numerical examples comparing Stokes’s formula, LSC, and SKs in a closed-loop environment using synthetic noise-free data, to verify their equivalence. All agree on the millimeter level. 相似文献
10.
Arne Bjerhammar 《Journal of Geodesy》1962,36(3):215-220
Summary The principal formulae of the geophysical geodesy are based on the famous explicit expression of Stokes (1849). Up to now,
there has been no method for a computation of the corresponding explicit expression of a (non-spherical) surface with masses
outside the geoid. In this paper there is a solution of this problem. Another paper on this subject was presented to “Nordiska
geodetm?tet” in Copenhagen May 1959 (in Swedish). 相似文献
11.
The solution of the linear Molodensky problem by analytical continuation to point level is numerically the most convenient
of all the theoretically equivalent solutions. It is obtained by successively applying the same integral operator and it does
not depend explicitly on the terrain inclination. However, its dependence on the computation point restricts somehow the computational
efficiency. The use of the Fourier transform for the evaluation of the integral operator in planar approximation lessens significantly
the burden of computations. Using this spectral approach, the problem has been reformulated and solved in the frequency domain.
Moreover, it is shown that the solution can be easily split into two steps: (a) “downward” continuation to sea level, which
is independent of the computation point, and (b) “upward” continuation from sea to point level, using the values computed
at sea level. Such a treatment not only simplifies the formulas and increases the numerical efficiency but also clarifies
the physical interpretation and the theoretical equivalence of the continuation solution with respect to the other solution
types. Numerical tests have been performed to investigate which terms in the Molodensky series are of significance for geoid
and deflection computations. The practical difficulty of differences in the grid spacings of gravity and height data has been
overcome by frequency domain interpolation.
Presented at theXIX IUGG General Assembly, Vancouver, B.C., August 9–22, 1987. 相似文献
12.
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
13.
L. de Witte 《Journal of Geodesy》1967,41(1):41-53
When the values of gravity anomalies are given at the geoid, Ag can be calculated at altitude by application of Poisson’s
integral theorem. The process requires integration of Δg multiplied by the Poisson kernel function over the entire globe.
It is common practice to add to the kernel function terms that will ensure removal of any zeroth and first order components
of Δg that may be present. The effects of trancating the integration at the boundary of a spherical cap of earth central half
angle ψo have been analyzed using an adaptation of Molodenskii’s procedure. The extension process without removal terms retains the
correct effects of inaccuracies in the constant term of the gravity reference model used in the definition of Δg. Furthermore,
the effects of ignoring remote zones or unmapped areas in the integration process are very much smaller for the extension
without removal terms than for the commonly used formula with removal terms. For these reasons the Poisson vertical extension
process without removal terms is to be preferred over the extension with the zeroth order term removal. Truncation of this
process at the point recommended for the Stokes integration, namely, the first zero crossing of the Stokes kernel function,
leaves negligible truncation errors. 相似文献
14.
The first Australian gravimetric quasigeoid model with location-specific uncertainty estimates 总被引:1,自引:1,他引:0
W. E. Featherstone J. C. McCubbine N. J. Brown S. J. Claessens M. S. Filmer J. F. Kirby 《Journal of Geodesy》2018,92(2):149-168
We describe the computation of the first Australian quasigeoid model to include error estimates as a function of location that have been propagated from uncertainties in the EGM2008 global model, land and altimeter-derived gravity anomalies and terrain corrections. The model has been extended to include Australia’s offshore territories and maritime boundaries using newer datasets comprising an additional \({\sim }\)280,000 land gravity observations, a newer altimeter-derived marine gravity anomaly grid, and terrain corrections at \(1^{\prime \prime }\times 1^{\prime \prime }\) resolution. The error propagation uses a remove–restore approach, where the EGM2008 quasigeoid and gravity anomaly error grids are augmented by errors propagated through a modified Stokes integral from the errors in the altimeter gravity anomalies, land gravity observations and terrain corrections. The gravimetric quasigeoid errors (one sigma) are 50–60 mm across most of the Australian landmass, increasing to \({\sim }100\) mm in regions of steep horizontal gravity gradients or the mountains, and are commensurate with external estimates. 相似文献
15.
Richard H. Rapp 《Journal of Geodesy》1980,54(2):149-163
A gravimetric geoid computed using different techniques has been compared to a geoid derived from Geos-3 altimeter data in
two 30°×30° areas: one in the Tonga Trench area and one in the Indian Ocean. The specific techniques used were the usual Stokes
integration (using 1°×1° mean anomalies) with the Molodenskii truncation procedure; a modified Stokes integration with a modified
truncation method; and computations using three sets of potential coefficients including one complete to degree 180. In the
Tonga Trench area the standard deviation of the difference between the modified Stokes’ procedure and the altimeter geoid
was ±1.1 m while in the Indian Ocean area the difference was ±0.6 m. Similar results were found from the 180×180 potential
coefficient field. However, the differences in using the usual Stokes integration procedure were about a factor of two greater
as was predicted from an error analysis.
We conclude that there is good agreement at the ±1 m level between the two types of geoids. In addition, systematic differences
are at the half-meter level. The modified Stokes procedure clearly is superior to the usual Stokes method although the 180×180
solution is of comparable accuracy with the computational effort six times less than the integration procedures. 相似文献
16.
A detailed gravimetric geoid in the North Atlantic Ocean, named DGGNA-77, has been computed, based on a satellite and gravimetry
derived earth potential model (consisting in spherical harmonic coefficients up to degree and order 30) and mean free air
surface gravity anomalies (35180 1°×1° mean values and 245000 4′×4′ mean values). The long wavelength undulations were computed
from the spherical harmonics of the reference potential model and the details were obtained by integrating the residual gravity
anomalies through the Stokes formula: from 0 to 5° with the 4′×4′ data, and from 5° to 20° with the 1°×1° data. For computer
time reasons the final grid was computed with half a degree spacing only. This grid extends from the Gulf of Mexico to the
European and African coasts.
Comparisons have been made with Geos 3 altimetry derived geoid heights and with the 5′×5′ gravimetric geoid derived byMarsh andChang [8] in the northwestern part of the Atlantic Ocean, which show a good agreement in most places apart from some tilts which
porbably come from the satellite orbit recovery. 相似文献
17.
J. Y. Chen 《Journal of Geodesy》1982,56(1):9-26
Summary The application of combined data (satellite and terrestrial data) to the practical computation of height anomalies or the
deflections of the vertical was originally suggested by (Molodensky et al. 1962). This idea usually leads to the modification
of Stokes' or Vening-Meinesz' functions in the integration procedure. In the recent decade there were various suggestions
in this regard especially for the computation of height anomalies. For example, a considerable mathematical insight into the
modification of Stokes' function and the truncation of its integral has been provided by (Meissl 1971, Houtze et al. 1979,
Rapp 1980, Jekeli 1980).
Five different methods for computing deflections of the vertical by modifying Vening-Meinesz' function are studied and compared
with each other. The corresponding formulae, the values of the coefficients in each method and the estimations of their corresponding
potential coefficient error and truncation error are given in this article.
This paper was written at the Institut f. Angewandte Geod?sie, Technische Universit?t Graz, Austria. 相似文献
18.
L. P. Pellinen 《Journal of Geodesy》1962,36(1):57-65
A calculation of quasigeoidal heights and plumb-line deflections according to Molodensky formulae was carried out under elimination
of the effect of topography from gravity anomalies. After the masses of topography had been removed a smoothed-out surface
passing through astronomical and gravity stations was considered as representing the physical surface of the Earth. Thus it
has been practically rendered possible to use the first-approximation formulae of Molodensky, and, in many cases, also the
“zero-approximation” formulae analogous to the formulae of Stokes and Vening-Meinesz. The effect of the restored masses of
topography was then added to the quantities found; the said effect was expressed as the effect of topography condensed on
the normal equipotential surface passing through the point under investigation, plus a correction for condensation. Following
some transformations, the resulting formulae (13) and (18) were obtained which formulae differ in their “zero-approximation”
(15) and (20) from traditional formulas in that they contain terrait reductions added to free-air anomalies. Moreover, in
the calculation of plumb-line deflections directly in mountain regions a correction for differing effects of topography before
and after its condensation is to be introduced.
A tentative expansion of terrain reduction in terms of spherical harmonics up to the third order is given; it can be seen
therefrom that the Stokes series in its usual form is subject to a mean arror about 15–20%. It is also shown that the expansion
of free-air anomalies in terms of spherical functions contains a first-order harmonic with a mean values about ±0.3 mgl. The
said harmonic practically disappears in the expansion of the sum of free-air anomalies and terrain reductions. 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 Meissl scheme for the geodetic ellipsoid 总被引:2,自引:1,他引:1
We present a variant of the Meissl scheme to relate surface spherical harmonic coefficients of the disturbing potential of
the Earth’s gravity field on the surface of the geodetic ellipsoid to surface spherical harmonic coefficients of its first- and second-order normal derivatives on the same or any other ellipsoid.
It extends the original (spherical) Meissl scheme, which only holds for harmonic coefficients computed from geodetic data
on a sphere. In our scheme, a vector of solid spherical harmonic coefficients of one quantity is transformed into spherical
harmonic coefficients of another quantity by pre-multiplication with a transformation matrix. This matrix is diagonal for
transformations between spheres, but block-diagonal for transformations involving the ellipsoid. The computation of the transformation
matrix involves an inversion if the original coefficients are defined on the ellipsoid. This inversion can be performed accurately
and efficiently (i.e., without regularisation) for transformation among different gravity field quantities on the same ellipsoid,
due to diagonal dominance of the matrices. However, transformations from the ellipsoid to another surface can only be performed
accurately and efficiently for coefficients up to degree and order 520 due to numerical instabilities in the inversion. 相似文献