共查询到20条相似文献，搜索用时 32 毫秒
1.
In the numerical evaluation of geodetic convolution integrals, whether by quadrature or discrete/fast Fourier transform (D/FFT) techniques, the integration kernel is sometimes computed at the centre of the discretised grid cells. For singular kernels—a common case in physical geodesy—this approximation produces significant errors near the computation point, where the kernel changes rapidly across the cell. Rigorously, mean kernels across each whole cell are required. We present one numerical and one analytical method capable of providing estimates of mean kernels for convolution integrals. The numerical method is based on Gauss-Legendre quadrature (GLQ) as efficient integration technique. The analytical approach is based on kernel weighting factors, computed in planar approximation close to the computation point, and used to convert non-planar kernels from point to mean representation. A numerical study exemplifies the benefits of using mean kernels in Stokes’s integral. The method is validated using closed-loop tests based on the EGM2008 global gravity model, revealing that using mean kernels instead of point kernels reduces numerical integration errors by a factor of ~5 (at a grid-resolution of 10 arc min). Analytical mean kernel solutions are then derived for 14 other commonly used geodetic convolution integrals: Hotine, Eötvös, Green-Molodensky, tidal displacement, ocean tide loading, deflection-geoid, Vening-Meinesz, inverse Vening-Meinesz, inverse Stokes, inverse Hotine, terrain correction, primary indirect effect, Molodensky’s G1 term and the Poisson integral. We recommend that mean kernels be used to accurately evaluate geodetic convolution integrals, and the two methods presented here are effective and easy to implement. 相似文献
2.
Gravity field terrain effect computations by FFT 总被引：2，自引：2，他引：2
René Forsberg 《Journal of Geodesy》1985,59(4):342-360
The widespread availability of detailed gridded topographic and bathymetric data for many areas of the earth has resulted
in a need for efficient terrain effect computation techniques, especially for applications in gravity field modelling. Compared
to conventional integration techniques, Fourier transform methods provide extremely efficient computations due to the speed
of the Fast Fourier Transform (FFT. The Fourier techniques rely on linearization and series expansions of the basically unlinear terrain effect integrals, typically
involving transformation of the heights/depths and their squares. TheFFT methods will especially be suited for terrain reduction of land gravity data and satellite altimetry geoid data.
In the paper the basic formulas will be outlined, and special emphasis will be put on the practial implementation, where a
special coarse/detailed grid pair formulation must be used in order to minimize the unavoidable edge effects ofFFT, and the special properties ofFFT are utilized to limit the actual number of data transformations needed. Actual results are presented for gravity and geoid
terrain effects in test areas of the USA, Greenland and the North Atlantic. The results are evaluated against a conventional
integration program: thus, e.g., in an area of East Greenland (with terrain corrections up to10 mgal), the accuracy ofFFT-computed terrain corrections in actual gravity stations showed anr.m.s. error of0.25 mgal, using height data from a detailed photogrammetric digital terrain model. Similarly, isostatic ocean geoid effects in the
Faeroe Islands region were found to be computed withr.m.s. errors around0.03 m 相似文献
3.
The use of the fast Fourier transform algorithm in the evaluation of the Molodensky series terms is demonstrated in this paper.
The solution by analytical continuation to point level has been reformulated to obtain convolution integrals in planar approximation
which can be efficiently evaluated in the frequency domain. Preliminary results show that the solution by Faye anomalies is
not sufficient for highly accurate deflections of the vertical and height anomalies. The Molodensky solution up to at least
the second-order term must be carried out. Part of the unrecovered deflection and height anomaly signal appears to be due
to density variations, verifying the essential role of density modelling. A remove-restore technique for the terrain effects
can improve the convergence of the series and minimize the interpolation errors.
Paper presented at theI Hotine-Marussi Symposium on Mathematical Geodesy, Rome, June 3–6, 1985. 相似文献
4.
推导了运用地球重力场模型计算单点、格网点以及格网平均的扰动重力梯度复组合分量的公式;提出了广义球谐函数及其定积分的新算法,并利用EGM96地球重力场模型试算了全球地区卫星轨道面上的重力梯度分量的格网平均观测值;通过对角线分量满足Laplace方程的精度,验证了该算法的有效性和实用性。 相似文献
5.
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
6.
Michael G. Sideris 《Journal of Geodesy》1995,70(1-2):2-12
The fast Fourier transform (FFT) and, recently, the fast Hartley transform (FHT) have been extensively used by geodesists for efficient geoid determination. For this kind of efficiency, data must be given on a regular grid and, consequently, a pre-processing step of interpolation is required when only point measurements are available. This paper presents a way of computing a grid of geoid undulations N without explicitly gridding the data. The method is applicable to all FFT or FHT techniques of geoid or terrain effects determination, and it works with planar as well as spherical formulas. This method can be used not only for, e.g., computing a grid of undulations from irregular gravity anomalies g but it also lends itself to other applications, such as the gridding of gravity anomalies and, since the contribution of each data point is computed individually, the update of N- or g-grids as soon as new point measurements become available. In the case that there are grid cells which contain no measurements, the results of gravity interpolation or geoid estimation can be drastically improved by incorporating into the procedure a frequency-domain interpolating function. In addition to numerical results obtained using a few simple interpolating functions, the paper presents briefly the mathematical formulas for recovering missing grid values and for transforming values from one grid to another which might be rotated and/or scaled with respect to the first one. The geodetic problems where these techniques may find applications are pointed out throughout the paper.Presented at theIAG General Meeting, Beijing, P.R. China, Aug. 6–13, 1993 相似文献
7.
基于重力场水平分量-垂线偏差对地形信息敏感的特点,根据边值理论由重力与地形数据确定格网垂线偏差模型,在此基础上,首先利用三维重力矢量-格网垂线偏差与格网重力异常,联合格网高程数据求得格网点间高程异常差,然后通过GPS/水准点的控制,构成紧密的几何条件,进行严密平差,从而获得高分辨率、高精度似大地水准面的数值模型。按照本文方法,利用我国6600多个GPS/水准点、1'×1'的格网垂线偏差、格网重力异常、格网高程数据,整体平差计算了我国陆海统一的似大地水准面模型,经GPS/水准点检核,全国似大地水准面的绝对精度达到了4 cm,相对精度优于7 cm。 相似文献
8.
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
9.
L. E. Sjöberg 《Journal of Geodesy》2000,74(2):255-268
The topographic potential and the direct topographic effect on the geoid are presented as surface integrals, and the direct
gravity effect is derived as a rigorous surface integral on the unit sphere. By Taylor-expanding the integrals at sea level
with respect to topographic elevation (H) the power series of the effects is derived to arbitrary orders. This study is primarily limited to terms of order H
2. The limitations of the various effects in the frequently used planar approximations are demonstrated. In contrast, it is
shown that the spherical approximation to power H
2 leads to a combined topographic effect on the geoid (direct plus indirect effect) proportional to H˜2 (where terms of degrees 0 and 1 are missing) of the order of several metres, while the combined topographic effect on the
height anomaly vanishes, implying that current frequent efforts to determine the direct effect to this order are not needed.
The last result is in total agreement with Bjerhammar's method in physical geodesy. It is shown that the most frequently applied
remove–restore technique of topographic masses in the application of Stokes' formula suffers from significant errors both
in the terrain correction C (representing the sum of the direct topographic effect on gravity anomaly and the effect of continuing the anomaly to sea
level) and in the term t (mainly representing the indirect effect on the geoidal or quasi-geoidal height).
Received: 18 August 1998 / Accepted: 4 October 1999 相似文献
10.
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
11.
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
12.
Gottfried Gerstbach 《Journal of Geodesy》1988,62(4):541-563
The short wavelength geoid undulations, caused by topography, amount to several decimeters in mountainous areas. Up to now
these effects are computed by means of digital terrain models in a grid of 100–500m. However, for many countries these data are not yet available or their collection is too expensive.
This problem can be overcome by considering the special behaviour of the gravity potential along mountain slopes. It is shown
that 90 per cent of the topographic effects are represented by a simple summation formula, based on the average height differences
and distances between valleys and ridges along the geoid profiles,
δN=[30.H.D.+16.(H−H′).D] in mm/km, (error<10%), whereH, H′, D are estimated in a map to the nearest 0.2km. The formula is valid for asymmetric sides of valleys (H, H′) and can easily be corrected for special shapes. It can be used for topographic refinement of low resolution geoids and for
astrogeodetic projects.
The “slope method” was tested in two alpine areas (heights up to 3800m, astrogeodetic deflection points every 170km
2) and resulted in a geoid accuracy of ±3cm. In first order triangulation networks (astro points every 1000km
2) or for gravimetric deflections the accuracy is about 10cm per 30km. Since a map scale of 1∶500.000 is sufficient, the method is suitable for developing countries, too. 相似文献
13.
利用分形随机算法建立平地、丘陵和山地3种精细地形仿真场景,将DEM逐级重采样为不同网格间距,分析不同DEM网格间距对3种地形的重力近区地形改正误差影响。发现随着DEM网格间距的增加,近区地形改正误差随之增大。对于平地,使用1∶10 000的DEM,网格间距为5m仍能够满足规范要求;对于丘陵地,使用1∶5 000的DEM,网格间距为2.5m能够满足规范要求;对于山地,使用1∶1 000的DEM,网格间距1m能够满足规范要求。通过消费级无人机获取丘陵地精细地形,验证地形仿真的结论,同时说明消费级无人机能够应用于重力近区地形改正。 相似文献
14.
In precise geoid determination by Stokes formula, direct and primary and secondary indirect terrain effects are applied for
removing and restoring the terrain masses. We use Helmert's second condensation method to derive the sum of these effects,
together called the total terrain effect for geoid. We develop the total terrain effect to third power of elevation H in the original Stokes formula, Earth gravity model and modified Stokes formula. It is shown that the original Stokes formula,
Earth gravity model and modified Stokes formula all theoretically experience different total terrain effects. Numerical results
indicate that the total terrain effect is very significant for moderate topographies and mountainous regions. Absolute global
mean values of 5–10 cm can be reached for harmonic expansions of the terrain to degree and order 360. In another experiment,
we conclude that the most important part of the total terrain effect is the contribution from the second power of H, while the contribution from the third power term is within 9 cm.
Received: 2 September 1996 / Accepted: 4 August 1997 相似文献
15.
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
16.
A function having some properties of a wavelet and being harmonic around a given point in R
3 is defined, and three models showing the local relationships between the disturbing density, the disturbing potential and
the disturbing gravity are established by using the function as the kernel function of the integrals in the models. The local
relationship has two meanings. One is that we can evaluate with a high accuracy the integrals in the models by using mainly
high-accuracy and high-resolution data in a local area. The other is that we can obtain a stable solution with high resolution
when we invert the integrals in the models because of the rapid decrease of the kernel function of the integrals. As a result,
with these models we evaluate one quantity with high resolution, in a band limited by the maximum degree of a set of geopotential
coefficients or by the resolution (spacing) of the local data, from another quantity (or quantities) in a local area, and
the resulting solution is stable.
Received: 6 April 1998 / Accepted: 16 June 1999 相似文献
17.
Erik de Min 《Journal of Geodesy》1995,69(4):223-232
Summary Basically two different evaluation methods are available to compute geoid heights from residual gravity anomalies in the inner zone: numerical integration and least squares collocation.If collocation is not applied to a global gravity data set, as is usually the case in practice, its result will not be equal to the numerical integration result. However, the cross covariance function between geoid heights and gravity anomalies can be adapted such that the geoid contribution is computed only from a small gravity area up to a certain distance
o from the computation point. Using this modification, identical results are obtained as from numerical integration.Applying this modification makes the results less dependent on the covariance function used. The difference between numerical integration and collocation is mainly caused by the implicitly extrapolated residual gravity anomaly values, outside the original data area. This extrapolated signal depends very much on the covariance function used, while the interpolated values within the original data area depend much less on it.As a sort of by-product, this modified collocation formula also leads to a new combination technique of numerical integration and collocation, in which the optimizing practical properties of both methods are fully exploited.Numerical examples are added as illustration. 相似文献
18.
I. N. Tziavos 《Journal of Geodesy》1987,61(2):177-197
Mean gravity anomalies, deflections of the vertical, and a geopotential model complete to degree and order180 are combined in order to determine geoidal heights in the area bounded by [34°≦ϕ≤42°, 18°≦λ≦28°]. Moreover, employing point
gravity anomalies simultaneously with the above data, an attempt is made to predict deflections of the vertical in the same
area. The method used in the computations is least squares collocation. Using empirical covariance functions for the data,
the suitable errors for the different sources of observations, and the optimum cap radius around each point of evaluation,
an accuracy better than±0.60m for geoidal heights and±1″.5 for deflections of the vertical is obtained taking into account existing systematic effects. This accuracy refers to the
comparison between observed and predicted values. 相似文献
19.
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
20.
The determination of potential difference by the joint application of measured and synthetical gravity data: a case study in Hungary 总被引：1，自引：1，他引：0
In an elementary approach every geometrical height difference between the staff points of a levelling line should have a corresponding
average g value for the determination of potential difference in the Earth’s gravity field. In practice this condition requires as
many gravity data as the number of staff points if linear variation of g is assumed between them. Because of the expensive fieldwork, the necessary data should be supplied from different sources.
This study proposes an alternative solution, which is proved at a test bed located in the Mecsek Mountains, Southwest Hungary,
where a detailed gravity survey, as dense as the staff point density (~1 point/34 m), is available along a 4.3-km-long levelling
line. In the first part of the paper the effect of point density of gravity data on the accuracy of potential difference is
investigated. The average g value is simply derived from two neighbouring g measurements along the levelling line, which are incrementally decimated in the consecutive turns of processing. The results
show that the error of the potential difference between the endpoints of the line exceeds 0.1 mm in terms of length unit if
the sampling distance is greater than 2 km. Thereafter, a suitable method for the densification of the decimated g measurements is provided. It is based on forward gravity modelling utilising a high-resolution digital terrain model, the
normal gravity and the complete Bouguer anomalies. The test shows that the error is only in the order of 10−3mm even if the sampling distance of g measurements is 4 km. As a component of the error sources of levelling, the ambiguity of the levelled height difference which
is the Euclidean distance between the inclined equipotential surfaces is also investigated. Although its effect accumulated
along the test line is almost zero, it reaches 0.15 mm in a 1-km-long intermediate section of the line. 相似文献