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1.
 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   

2.
The accumulation of good quality satellite altimetry missions allows us to have a precise geoid with fair resolution and to compute free air gravity anomalies easily by fast Fourier transform (FFT) techniques.In this study we are comparing two methods to get gravity anomalies. The first one is to establish a geoid grid and transform it into anomalies using inverse Stokes formula in the spectral domain via FFT. The second one computes deflection of the vertical grids and transforms them into anomalies.The comparison is made using different data sets: Geosat, ERS-1 and Topex-Poseidon exact repeat misions (ERMs) north of 30°S and Geosat geodetic mission (GM) south of 30°S. The second method which transforms the geoid gradients converted into deflection of the vertical values is much better and the results have been favourably evaluated by comparison with marine gravity data.  相似文献   

3.
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   

4.
The vertical component obtained from the Global Positioning System (GPS) observations is from the ellipsoid (a mathematical surface), and therefore needs to be converted to the orthometric height, which is from the geoid (represented by the mean sea level). The common practice is to use existing bench marks (around the four corners of a project area and interpolate for the rest of the area), but in many areas bench marks may not be available, in which case an existing geoid undulation is used. Present available global geoid undulation values are not generally as detailed as needed, and in many areas they are not known better than ±1 to ±5 m, because of many limitations. This article explains the difficulties encountered in obtaining precise geoid undulation with some example computations, and proposes a technique of applying corrections to the best available global geoid undulations using detailed free-air gravity anomalies (within a 2° × 2° area) to get relative centimeter accuracy. Several test computations have been performed to decide the optimal block sizes and the effective spherical distances to compute the regional and the local effects of gravity anomalies on geoid undulations by using the Stokes integral. In one test computation a 2° × 2° area was subdivided into smaller surface elements. A difference of 37.34 ± 1.6 cm in geoid undulation was obtained over the same 2° × 2° area when 1° × 1° block sizes were replaced by a combination of 5' × 5' and 1' × 1' subdivision integration elements (block sizes).  相似文献   

5.
 The use of GPS for height control in an area with existing levelling data requires the determination of a local geoid and the bias between the local levelling datum and the one implicitly defined when computing the local geoid. If only scarse gravity data are available, the heights of new data may be collected rapidly by determining the ellipsoidal height by GPS and not using orthometric heights. Hence the geoid determination has to be based on gravity disturbances contingently combined with gravity anomalies. Furthermore, existing GPS/levelling data may also be used in the geoid determination if a suitable general gravity field modelling method (such as least-squares collocation, LSC) is applied. A comparison has been made in the Aswan Dam area between geoids determined using fast Fourier transform (FFT) with gravity disturbances exclusively and LSC using only the gravity disturbances and the disturbances combined with GPS/levelling data. The EGM96 spherical harmonic model was in all cases used in a remove–restore mode. A total of 198 gravity disturbances spaced approximately 3 km apart were used, as well as 35 GPS/levelling points in the vicinity and on the Aswan Dam. No data on the Nasser Lake were available. This gave difficulties when using FFT, which requires the use of gridded data. When using exclusively the gravity disturbances, the agreement between the GPS/levelling data were 0.71 ± 0.17 m for FFT and 0.63 ± 0.15 for LSC. When combining gravity disturbances and GPS/levelling, the LSC error estimate was ±0.10 m. In the latter case two bias parameters had to be introduced to account for a possible levelling datum difference between the levelling on the dam and that on the adjacent roads. Received: 14 August 2000 / Accepted: 28 February 2001  相似文献   

6.
 A new method for computing gravitational potential and attraction induced by distant, global masses on a global scale has been developed. The method uses series expansions and the well known one-dimensional fast Fourier transform (1-D FFT) method. It has been proven to be significantly faster than quadrature while being equally accurate. Various quantities were studied to cover the two primary applications of the Stokes–Helmert scheme of modeling effects. These two applications (or paths), given the names R/r/D and R/D/r, are briefly discussed, although the primary objective of the paper is to provide computational information to either path, rather than choosing one path as preferable to the other. It is further shown that the impact of masses outside a 4-degree cap can impact the absolute computation of the geoid at more than 1 cm, and should therefore be included in all local geoid computations seeking that accuracy. Received: 13 December 2000 / Accepted: 3 September 2001  相似文献   

7.
The northern European geoid: a case study on long-wavelength geoid errors   总被引：1，自引：1，他引：1  
 The long-wavelength geoid errors on large-scale geoid solutions, and the use of modified kernels to mitigate these effects, are studied. The geoid around the Nordic area, from Greenland to the Ural mountains, is considered. The effect of including additional gravity data around the Nordic/Baltic land area, originating from both marine, satellite and ground-based measurements, is studied. It is found that additional data appear to increase the noise level in computations, indicating the presence of systematic errors. Therefore, the Wong–Gore modification to the Stokes kernel is applied. This method of removing lower-order terms in the Stokes kernel appears to improve the geoid. The best fit to the global positioning system (GPS) leveling points is obtained with a degree of modification of approximately 30. In addition to the study of modification errors, the results of different methods of combining satellite altimetry gravity and other gravimetry are presented. They all gave comparable results, at the 6-cm level, when evaluated for the Nordic GPS networks. One dimensional (1-D) and 2-D fast Fourier transform (FFT) methods are also compared. It is shown that even though methods differ by up to 6 cm, the fit to the GPS is essentially the same. A surprising conclusion is that the addition of more data does not always produce a better geoid, illustrating the danger of systematic errors in data. Received: 4 July 2001 / Accepted: 21 February 2002  相似文献   

8.
Gravity field convolutions without windowing and edge effects   总被引：5，自引：0，他引：5  
A new set of formulas has been developed for the computation of geoid undulations and terrain corrections by FFT when the input gravity anomalies and heights are mean gridded values. The effects of the analytical and the discrete spectra of kernel functions and that of zero-padding on the computation of geoid undulations and terrain corrections are studied in detail.Numerical examples show that the discrete spectrum is superior to the analytically-defined one. By using the discrete spectrum and 100% zero-padding, the RMS differences are 0.000 m for the FFT geoid undulations and 0.200 to 0.000 mGal for the FFT terrain corrections compared with results obtained by numerical integration.  相似文献   

9.
In the analyses of 2D real arrays, fast Hartley (FHT), fast T (FTT) and real-valued fast Fourier transforms are generally preferred in lieu of a complex fast Fourier transform due to the advantages of the former with respect to disk storage and computation time. Although the FHT and the FTT in one dimension are identical, they are different in two or more dimensions. Therefore, first, definitions and some properties of both transforms and the related 2D FHT and FTT algorithms are stated. After reviewing the 2D FHT and FTT solutions of Stokes' formula in planar approximation, 2D FHT and FTT methods are developed for geoid updating to incorporate additional gravity anomalies. The methods are applied for a test area which includes a 64×64 grid of 3′×3′ point gravity anomalies and geoid heights calculated from point masses. The geoids computed by 2D FHT and FTT are found to be identical. However, the RMS value of the differences between the computed and test geoid is ±15 mm. The numerical simulations indicate that the new methods of geoid updating are practical and accurate with considerable savings on storage requirements. Received: 15 February 1996; Accepted: 22 January 1997  相似文献   

10.
This paper presents a set of efficient formulas to evaluate the deflections of the vertical on the sphere using gridded data. The Vening-Meinesz formula, the topographic indirect effect on the deflections of the vertical as well as the terrain corrections are expressed as both 2D and 1D convolutions on the sphere, and consequently can be evaluated by the 2D and the 1D fast Fourier transform (FFT). When compared with the results obtained from pointwise integration, the use of the 1D FFT gives identical results, and therefore these results were used as control values in this paper. The use of the spherical 2D FFT improves significantly the computational efficiency with little sacrifice of accuracy (0.6″ rms difference from the 1D FFT results). The planar 2D FFT, which is as efficient as the spherical 2D FFT, gives worse results (1.2″ rms difference from the 1D FFT results) because of the extra approximations. Received: 27 February 1996; Accepted: 24 January 1997  相似文献   

11.
 The AUSGeoid98 gravimetric geoid model of Australia has been computed using data from the EGM96 global geopotential model, the 1996 release of the Australian gravity database, a nationwide digital elevation model, and satellite altimeter-derived marine gravity anomalies. The geoid heights are on a 2 by 2 arc-minute grid with respect to the GRS80 ellipsoid, and residual geoid heights were computed using the 1-D fast Fourier transform technique. This has been adapted to include a deterministically modified kernel over a spherical cap of limited spatial extent in the generalised Stokes scheme. Comparisons of AUSGeoid98 with GPS and Australian Height Datum (AHD) heights across the continent give an RMS agreement of ±0.364 m, although this apparently large value is attributed partly to distortions in the AHD. Received: 10 March 2000 / Accepted: 21 February 2001  相似文献   

12.
Truncated geoid and gravity inversion for one point-mass anomaly   总被引：1，自引：0，他引：1  
The truncated geoid, defined by the truncated Stokes' integral transform, an integral convolution of gravity anomalies with the Stokes' function on a spherical cap, is often used as a mathematical tool in geoid computations via Stokes' integral to overcome computational difficulties, particularly the need to integrate over the entire boundary spheroid. The objective of this paper is to demonstrate that the truncated geoid does, besides having mathematical applications, have physical interpretation, and thus may be used in gravity inversion. A very simple model of one point-mass anomaly is chosen and a method for inverting its synthetic gravity field with the use of the truncated geoid is presented. The method of inverting the synthetic field generated by one point-mass anomaly has become fundamental for the authors' inversion studies for sets of point-mass anomalies, which are published in a separate paper. More general applications are currently under investigation. Since an inversion technique for physically meaningful mass distributions based on the truncated geoid has not yet been developed, this work is not related to any of the existing gravity inversion techniques. The inversion for one point mass is based on the onset of the so-called dimple event, which occurs in the sequence of surfaces (or profiles) of the first derivative of the truncated geoid with respect to the truncation parameter (radius of the integration cap), its only free parameter. Computing the truncated geoid at various values of the truncation parameter may be understood as spatial filtering of surface gravity data, a type of weighted spherical windowing method. Studying the change of the truncated geoid represented by its first derivative may be understood as a data enhancement method. The instant of the dimple onset is practically independent of the mass of the point anomaly and linearly dependent on its depth. Received: 26 September 1996 /Accepted: 28 September 1998  相似文献   

13.
Using the spherical harmonic representations of the earth's disturbing potential and its functionals, we derive the inverse Vening Meinesz formula, which converts deflection of the vertical to gravity anomaly using the gradient of the H function. The deflection-geoid formula is also derived that converts deflection to geoidal undulation using the gradient of the C function. The two formulae are implemented by the 1D FFT and the 2D FFT methods. The innermost zone effect is derived. The inverse Vening Meinesz formula is employed to compute gravity anomalies and geoidal undulations over the South China Sea using deflections from Seasat, Geosat, ERS-1 and TOPEX//POSEIDON satellite altimetry. The 1D FFT yields the best result of 9.9-mgal rms difference with the shipborne gravity anomalies. Using the simulated deflections from EGM96, the deflection-geoid formula yields a 4-cm rms difference with the EGM96-generated geoid. The predicted gravity anomalies and geoidal undulations can be used to study the tectonic structure and the ocean circulations of the South China Sea. Received: 7 April 1997 / Accepted: 7 January 1998  相似文献   

14.
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   

15.
1　ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｅｍｉｄ＿1 980ｓ,ｔｈｅｆａｓｔＦｏｕｒｉｅｒｔｒａｎｓｆｏｒｍａｔｉｏｎ(ＦＦＴ)ｂｅｇａｎｔｏｆｉｎｄｗｉｄｅｓｐｒｅａｄｕｓｅｉｎｇｅｏｉｄｄｅｔｅｒ ｍｉｎａｔｉｏｎｂｅｃａｕｓｅｏｆｉｔｓｅｆｆｉｃｉｅｎｔｅｖａｌｕａｔｉｏｎｏｆｃｏｎ ｖｏｌｕｔｉｏｎｉｎｔｅｇｒａｌｓ,ｗｈｅｎｃｏｍｐａｒｅｄｔｏｃｌａｓｓｉｃａｌｎｕ ｍｅｒｉｃａｌｉｎｔｅｇｒａｔｉｏｎ .Ｆｏｒｍａｎｙｙｅａｒｓ,ｔｈｅｐｌａｎａｒ,2＿ＤＦＦＴｈａｄｂｅｅｎｕｓｅｄ (Ｓｃｈｗａｒｚ ,1 …  相似文献   

16.
The effects of the deviations of sea surface topography from the geoid are estimated for terrestrial geoid computations as obtained from Stokes' formula. The results are based on an equal-area expansion of Lisitzin's sea surface topography data in a spherical harmonic series. It is realized that those data affect mainly the harmonics of degree n≤10. Consequently, in geoids obtained from combination solutions (where low harmonics are dominated by harmonics as obtained from differential orbit improvement) the sea surface topography effects are relatively small.  相似文献   

17.
Least-squares collocation and Stokes integral formula, as implemented using the Fast Fourier Technique, handle the harmonic downward continuation problem quite differently. FFT furthermore requires gridded data, amplifying the difference of methods.We have in this paper studied numerically the effects of downward continuation and gridding in a mountainous area in central Norway. Topographically smoothed data were used in order to reduce these effects. Despite the smoothing, it was found that the vertical gravity gradient had values up to -11 mgal/km. The corresponding differences between geoid heights and the height anomalies at altitude reached 12 cm.The differences between geoid heights obtained using collocation or FFT with gravity data at terrain level or sea level showed differences between the values of up to 10 cm r.m.s. A part of this difference was a consequence of different data areas used in the FFT and collocation solution, though.Major discrepancies between the solutions were found in areas where the topographic smoothing could not be applied (deep fjords with no depth information in the used DTM) or where there seemed to be gross errors in the data.We conclude that proper handling of harmonic continuation is important, even when we as here have used a 1 km resolution DTM for the calculation of topographic effects. The effect of data gridding, required for the FFT method, seems not to be as serious as the need to limit the data distribution area, required when least squares collocation is used with randomly distributed data.  相似文献   

18.
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   

19.
1　ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｆａｓｔＦｏｕｒｉｅｒｔｒａｎｓｆｏｒｍ (ＦＦＴ)ｔｅｃｈｎｉｑｕｅｉｓａｖｅｒｙｐｏｗｅｒｆｕｌｔｏｏｌｆｏｒｔｈｅｅｆｆｉｃｉｅｎｔｅｖａｌｕａｔｉｏｎｏｆｇｒａｖｉｔｙｆｉｅｌｄｃｏｎｖｏｌｕｔｉｏｎｉｎｔｅｇｒａｌｓ.Ｔｈａｎｋｓｔｏｔｈｅｇｏｏｄｃｏｍｐｕｔａｔｉｏｎｅｆｆｉｃｉｅｎｃｙ ,ｔｈｅＦＦＴｔｅｃｈｎｉｑｕｅ ,ｉｎｔｈｅｍｉｄ＿1 980ｓ ,ｂｅｇａｎｔｏｆｉｎｄｗｉｄｅｓｐｒｅａｄｕｓｅｉｎｇｅｏｉｄｄｅｔｅｒｍｉｎａｔｉｏｎ ,ｗｈｅｎｃｏｍｐａｒ…  相似文献   

20.
Minimization and estimation of geoid undulation errors   总被引：2，自引：1，他引：1  
The objective of this paper is to minimize the geoid undulation errors by focusing on the contribution of the global geopotential model and regional gravity anomalies, and to estimate the accuracy of the predicted gravimetric geoid.The geopotential model's contribution is improved by (a) tailoring it using the regional gravity anomalies and (b) introducing a weighting function to the geopotential coefficients. The tailoring and the weighting function reduced the difference (1) between the geopotential model and the GPS/levelling-derived geoid undulations in British Columbia by about 55% and more than 10%, respectively.Geoid undulations computed in an area of 40° by 120° by Stokes' integral with different kernel functions are analyzed. The use of the approximated kernels results in about 25 cm () and 190 cm (maximum) geoid errors. As compared with the geoid derived by GPS/levelling, the gravimetric geoid gives relative differences of about 0.3 to 1.4 ppm in flat areas, and 1 to 2.5 ppm in mountainous areas for distances of 30 to 200 km, while the absolute difference (1) is about 5 cm and 20 cm, respectively.A optimal Wiener filter is introduced for filtering of the gravity anomaly noise, and the performance is investigated by numerical examples. The internal accuracy of the gravimetric geoid is studied by propagating the errors of the gravity anomalies and the geopotential coefficients into the geoid undulations. Numerical computations indicate that the propagated geoid errors can reasonably reflect the differences between the gravimetric and GPS/levelling-derived geoid undulations in flat areas, such as Alberta, and is over optimistic in the Rocky Mountains of British Columbia.Paper presented at the IAG General Meeting, Beijing, China, August 8–13, 1993.  相似文献   

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