共查询到20条相似文献，搜索用时 15 毫秒
1.
提出了一种用于Stokes积分和Hotine积分直接离散求和的快速算法。该算法将积分核表达为计算点纬度、流动点纬度和两点间经度差的函数,充分利用核函数的对称性,相同纬度的所有计算点只需计算一组核函数,计算次数远少于普通离散求和。基于EGM2008地球重力位模型的模拟实验表明,快速算法的计算效率远高于普通算法,有效解决了离散求和计算速度太慢的数值问题,且保留了球面积分的特性,可取代一维FFT用于计算Stokes积分和Hotine积分。 相似文献
2.
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3.
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
4.
Wavelet evaluation of the Stokes and Vening Meinesz integrals 总被引：1，自引：0，他引：1
The wavelet transform is a powerful tool in evaluating some singular geodetic integrals. Due to its localization properties in both the time (space) and frequency (scale) domains, and because the kernels of some geodetic integrals have singular points and decay smoothly and quickly away from the singularities, many wavelet transform coefficients of the kernels become zeros or negligible, and only a small number of wavelet transform coefficients are significant. It is thus possible to significantly compress the kernels of these integrals on a wavelet basis by neglecting the zero coefficients and the small coefficients below a certain threshold. Therefore, wavelets provide a convenient way of efficiently evaluating these integrals in terms of fast computation and savings of computer memory. A modified algorithm for the wavelet evaluation of Stokes' integral is presented. The same modified algorithm is applied to the evaluation of the Vening Meinesz integral, whose kernel has a stronger singularity than does Stokes' kernel. Numerical examples illustrate the efficiency and accuracy of the wavelet methods.Acknowledgments.The author express their sincere thanks to Dr. Salamonwicz for providing his PhD thesis. E-mail correspondence between the authors and Dr. Barthelmes and Dr. Benciolini contributed to the work. R. Benciolini and the other two anonymous reviewers are thanked for their constructive comments. Support for this research was provided by research grants to Dr. Sideris from the Natural Sciences and Engineering Reserch Council of Canada (NSERC) and the Geomatics for Informed Decisions (GEOIDE) Network of Centres of Excellence. The MATLAB Wavelet Toolbox package was used as the platform to develop the software in this project. 相似文献
5.
R. Lehmann 《Journal of Geodesy》1997,71(9):533-540
Geodetic surface integrals play an important role in the numerical solution of geodetic boundary-value problems. In many
cases they can be evaluated using fast methods in the frequency domain (FFT). However, this is not possible in general, because
the domain of integration may be non-trivial (as is the surface of the Earth), the kernel function may not be of convolution
type, or the data distribution may be heterogeneous. Therefore, fast evaluation strategies are also required in the space
domain. They are more difficult to design because only one property is left where a more or less fast evaluation strategy
can be built upon: the potential type of the kernel function. Consequently, the idea is not to replace well-established frequency
domain techniques, but to supplement them. Our approach to this problem goes in two directions: (1) we use advanced cubature
methods where the integration nodes automatically densify in the vicinity of the evaluation points; (2) we use powerful computer
hardware, namely MIMD computers with distributed memory. This enables us to evaluate geodetic surface integrals of any practical
complexity in reasonable time and accuracy. This is shown in a numerical example.
Received: 7 May 1996 / Accepted:17 March 1997 相似文献
6.
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
7.
李叶才 《武汉大学学报(信息科学版)》1989,(1)
本文通过对Hotine积分和Stokes积分进行比较,指出Hotine积分是一种更有利于确定高精度大地水准面的方法,同时还导出了计算Hotine积分中截断系数的递推公式以及高阶截断误差的近似估计公式。 相似文献
8.
Solving the geodetic boundary-value problem (GBVP) for the precise determination of the geoid requires proper use of the fundamental equation of physical geodesy as the boundary condition given on the geoid. The Stokes formula and kernel are the result of spherical approximation of this fundamental equation, which is a violation of the proper relation between the observed quantity (gravity anomaly) and the sought function (geoid). The violation is interpreted here as the improper formulation of the boundary condition, which implies the spherical Stokes kernel to be in error compared with the proper kernel of integral transformation. To remedy this error, two correction kernels to the Stokes kernel were derived: the first in both closed and spectral forms and the second only in spectral form. Contributions from the first correction kernel to the geoid across the globe were [−0.867 m, +1.002 m] in the low-frequency domain implied by the GRIM4-S4 purely satellite-derived geopotential model. It is a few centimeters, on average, in the high-frequency domain with some exceptions of a few meters in places of high topographical relief and sizable geological features in accordance with the EGM96 combined geopotential model. The contributions from the second correction kernel to the geoid are [−0.259 m, +0.217 m] and [−0.024 m, +0.023 m] in the low- and high-frequency domains, respectively. 相似文献
9.
Johannes Ihde 《Journal of Geodesy》1981,55(2):99-110
The investigations refer to the compartment method by using mean terrestrial free air anomalies only. Three main error influences of remote areas (distance from the fixed point >9°) on height anomalies and deflections of the vertical are being regarded:
- The prediction errors of mean terrestrial free air anomalies have the greatest influence and amount to about ±0″.2 in each component for deflections of the vertical and to ±3 m for height anomalies;
- The error of the compartment method, which originates from converting the integral formulas of Stokes and Vening-Meinesz into summation formulas, can be neglected if the anomalies for points and gravity profiles are compiled to 5°×5° mean values.
- The influences of the mean gravimetric correction terms of Arnold—estimated for important mountains of the Earth by means of an approximate formula—on height anomalies may amount to 1–2 m and on deflections of the vertical to 0″0.5–0″.1, and, therefore, they have to be taken into account for exact calculations.
10.
Numerical calculation of weakly singular surface integrals 总被引：1，自引：0，他引：1
Roland Klees 《Journal of Geodesy》1996,70(11):781-797
We consider semi-analytical and purely numerical integration methods for weakly singular integrals with point singularities on curved smooth surfaces. The methods can be applied to many practical computations in Geodesy, e.g. terrain corrections, Stokes' and Hotines' integral, surface potentials, and the solution of geodetic boundary value problems using integral equations. Current numerical integration techniques are reviewed. The most important semi-analytical and purely numerical techniques are described. Test calcualtions are done and the techniques are compared as regards accuracy and computational efficiency. Semi-analytical methods, which are based on some regularizing parameter transformations, are superior to purely numerical techniques. The best choice are modified polar coordinates defined in the parameter domain with the singularity as pole. Triangular coordinates show similar performance if carefully tuned. Extrapolation techniques and adaptive subdivision techniques behave poorly as regards accuracy and numerical efficiency. Standard integration techniques, which ignore the singularity, completely fail. 相似文献
11.
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
12.
NING JinshengLI JianchengCHAO DingboNING Jinsheng Academician Professor School of Geo-science Surveying Engineering WTUSM Wuhan China 《地球空间信息科学学报》1998,(1)
This paper presents a method for the computation of the Stokes for-mula using the Fast Hartley Transform（FHT）techniques.The algorithm is mostsuitable for the computation of real sequence transform,while the Fast FourierTransform（FFT）techniques are more suitable for the computaton of complex se-quence transform.A method of spherical coordinate transformation is presented inthis paper.By this method the errors,which are due to the approximate term inthe convolution of Stokes formula,can be effectively eliminated.Some numericaltests are given.By a comparison with both FFT techniques and numerical integra-tion method,the results show that the resulting values of geoidal undulations byFHT techniques are almost the same as by FFT techniques,and the computation-al speed of FHT techniques is about two times faster than that of FFT techniques. 相似文献
13.
Least-squares by observation equations is applied to the solution of geodetic boundary value problems (g.b.v.p.). The procedure is explained solving the vectorial Stokes problem in spherical and constant radius approximation. The results
are Stokes and Vening-Meinesz integrals and, in addition, the respective a posteriori variance-covariances.
Employing the same procedure the overdeterminedg.b.v.p. has been solved for observable functions potential, scalar gravity, astronomical latitude and longitude, gravity gradients
Гxz, Гyz, and Гzz and three-dimensional geocentric positions. The solutions of a large variety of uniquely and overdeterminedg.b.v.p.'s can be obtained from it by specializing weights. Interesting is that the anomalous potential can be determined—up to a
constant—from astronomical latitude and longitude in combination with either {Гxz,Гyz} or horizontal coordinate corrections Δx and Δy, or both. Dual to the formulation in terms of observation equations the overdeterminedg.b.v.p.'s can as well be solved by condition equations.
Constant radius approximation can be overcome in an iterative approach. For the Stokes problem this results in the solution
of the “simple” Molodenskii problem. Finally defining an error covariance model with a Krarup-type kernel first results were
obtained for a posteriori variance-covariance and reliability analysis. 相似文献
14.
Various formulations of the geodetic fixed and free boundary value problem are presented, depending upon the type of boundary data. For the free problem, boundary data of type astronomical latitude, astronomical longitude and a pair of the triplet potential, zero and first-order vertical gradient of gravity are presupposed. For the fixed problem, either the potential or gravity or the vertical gradient of gravity is assumed to be given on the boundary. The potential and its derivatives on the boundary surface are linearized with respect to a reference potential and a reference surface by Taylor expansion. The Eulerian and Lagrangean concepts of a perturbation theory of the nonlinear geodetic boundary value problem are reviewed. Finally the boundary value problems are solved by Hilbert space techniques leading to new generalized Stokes and Hotine functions. Reduced Stokes and Hotine functions are recommended for numerical reasons. For the case of a boundary surface representing the topography a base representation of the solution is achieved by solving an infinite dimensional system of equations. This system of equations is obtained by means of the product-sum-formula for scalar surface spherical harmonics with Wigner 3j-coefficients. 相似文献
15.
Various formulations of the geodetic fixed and free boundary value problem are presented, depending upon the type of boundary
data. For the free problem, boundary data of type astronomical latitude, astronomical longitude and a pair of the triplet
potential, zero and first-order vertical gradient of gravity are presupposed. For the fixed problem, either the potential
or gravity or the vertical gradient of gravity is assumed to be given on the boundary.
The potential and its derivatives on the boundary surface are linearized with respect to a reference potential and a reference
surface by Taylor expansion. The Eulerian and Lagrangean concepts of a perturbation theory of the nonlinear geodetic boundary
value problem are reviewed. Finally the boundary value problems are solved by Hilbert space techniques leading to new generalized
Stokes and Hotine functions. Reduced Stokes and Hotine functions are recommended for numerical reasons. For the case of a
boundary surface representing the topography a base representation of the solution is achieved by solving an infinite dimensional
system of equations. This system of equations is obtained by means of the product-sum-formula for scalar surface spherical
harmonics with Wigner 3j-coefficients. 相似文献
16.
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
17.
The aim of this investigation is to study some FFT problems related to the application of FFT to gravity field convolution integrals. And the others, such as the effect of spectral leakage, edge effects, cyclic convolution and effect of padding, are also discussed. A numerical test for these problems is made. A large area of Western China selected for the test is located between 30°N～36°N and 96°E～102°E and includes 1 858 gravity observations on land. The results show that the removal of the bias in the residual gravity anomalies is important to avoid spectral leakage. One hundred percent zero padding is highly recommended for further research of the geoid to remove cyclic convolution errors and edge effects. 1-D FFT is recommended for precise local geoid determination because it does not use kernel approximation. 相似文献
18.
Traditionally, the evaluation of geoidal height by Stokes formula and the vertical deflection by Vening-Meinesz one, and the estimation of the influence of neglecting the distant zone on computing the geoidal height and the vertical deflection were done by taking the inner zone as a spherical cap. It is not very convenient from the point of view of modern numerical methods such as fast Fourier and Hartley transforms where the inner zone is not a spherical cap, but a spherical trapezoid. So, we generalized the known formulas for evaluating the geoidal height and the vertical deflection for an integration area of arbitrary shape. The corresponding formulas for computing the effects of neglecting the distant zone have been derived. Some issues on computation techniques have been investigated. As an example, the case where the inner zone is modeled as a spherical trapezoid was given special attention, and practical computations were performed. 相似文献
19.
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
20.
Essam Ghanem LI Jiancheng 《地球空间信息科学学报》2000,3(2):53-57
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