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1.
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
2.
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
3.
Essam Ghanem LI Jiancheng 《地球空间信息科学学报》2000,3(2):53-57
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4.
Spherical harmonic expansions of the geopotential are frequently used for modelling the earth’s gravity field. Degree and
order of recently available models go up to 360, corresponding to a resolution of about50 km. Thus, the high degree potential coefficients can be verified nowadays even by locally distributed sets of terrestrial gravity
anomalies. These verifications are important when combining the short wavelength model impact, e.g. for regional geoid determinations
by means of collocation solutions. A method based on integral formulae is presented, enabling the improvement of geopotential
models with respect to non-global distributed gravity anomalies. To illustrate the foregoing, geoid computations are carried
out for the area of Iran, introducing theGPM2 geopotential model in combination with available regional gravity data. The accuracy of the geoid determination is estimated
from a comparison with Doppler and levelling data to ±1.4m. 相似文献
5.
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
6.
Due to the fact that the spectrum of a convolution is the product of the spectra of the two convolved functions, the convolution
integrals of physical geodesy can be evaluated very efficiently by the use of the fast Fourier transform (FFT) provided that gravity and/or terrain data are available on a regular grid. All Fourier transform-based methods usually treat
the gridded data as point values despite the fact that these discrete values may have been obtained by averaging and they
represent mean values over the whole area of a grid element. In the frequency domain, this fact can be taken into account
very easily, because the spectra of mean and point data are related via a two-dimensional (2D) sinc function. The paper shows explicitly this relationship using the convolution integrals for the evaluation of geoid
undulations, deflections of the vertical, and gravity and gradiometry terrain effects. Numerical tests are presented, indicating
that the differences in the two approaches may become significant when highly accurate results are wanted. The application
of the2D sinc function in the evaluation, update, and inversion of other convolution integrals is briefly discussed as well. 相似文献
7.
A new, high-resolution and high-precision geoid has been computed for the whole of Canada and part of the U.S., ranging from 35°N to about 90°N in latitude and 210°E to 320°E in longitude. The OSU91A geopotential model complete to degree and order 360 was combined with a 5 × 5 mean gravity anomaly grid and 1km × 1km topographical information to generate the geoid file. The remove-restore technique was adopted for the computation of terrain effects by Helmert's condensation reduction. The contribution of the local gravity data to the geoid was computed strictly by the 1D-FFT technique, which allows for the evaluation of the discrete spherical Stokes integral without any approximation, parallel by parallel. The indirect effects of up to second order were considered. The internal precision of the geoid, i.e. the contribution of the gravity data and the model coefficients noise, was also evaluated through error propagation by FFT. In a relative sense, these errors seem to agree quite well with the external errors and show clearly the weak areas of the geoid which are mostly due to insufficient gravity data coverage. Comparison of the gravimetric geoid with the GPS/levelling-derived geoidal heights of eight local GPS networks with a total of about 900 stations shows that the absolute agreement with respect to the GPS/levelling datum is generally better than 10 cm RMS and the relative agreement ranges, in most cases, from 4 to 1 ppm over short distances of about 20 to 100km, 1 to 0.5 ppm over distances of about 100 to 200 km, and 0.5 to 0.1 ppm for baselines of 200 to over 1000 km. Other existing geoids, such as UNB90, GEOID90 and GSD91, were also included in the comparison, showing that the new geoid achieves the best agreement with the GPS/levelling data.Presented at theIAG General Meeting, Beijing, P.R. China, Aug. 6–13, 1993 相似文献
8.
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
9.
The objective of this study is to evaluate two approaches, which use different representations of the Earth’s gravity field for downward continuation (DC), for determining Helmert gravity anomalies on the geoid. The accuracy of these anomalies is validated by 1) analyzing conformity of the two approaches; and 2) converting them to geoid heights and comparing the resulting values to GPS-leveling data. The first approach (A) consists of evaluating Helmert anomalies at the topography and downward-continuing them to the geoid. The second approach (B) downward-continues refined Bouguer anomalies to the geoid and transforms them to Helmert anomalies by adding the condensed topographical effect. Approach A is sensitive to the DC because of the roughness of the Helmert gravity field. The DC effect on the geoid can reach up to 2 m in Western Canada when the Stokes kernel is used to convert gravity anomalies to geoid heights. Furthermore, Poisson’s equation for DC provides better numerical results than Moritz’s equation when the resulting geoid models are validated against the GPS-leveling. On the contrary, approach B is significantly less sensitive to the DC because of the smoothness of the refined Bouguer gravity field. In this case, the DC (Poisson’s and Moritz’s) contributes only at the decimeter level to the geoid model in Western Canada. The maximum difference between the geoid models from approaches A and B is about 5 cm in the region of interest. The differences may result from errors in the DC such as numerical instability. The standard deviations of the h−H−N for both approaches are about 8 cm at the 664 GPS-leveling validation stations in Western Canada. 相似文献
10.
A 2×2 arc-minute resolution geoid model, CARIB97, has been computed covering the Caribbean Sea. The geoid undulations refer
to the GRS-80 ellipsoid, centered at the ITRF94 (1996.0) origin. The geoid level is defined by adopting the gravity potential
on the geoid as W
0=62 636 856.88 m2/s2 and a gravity-mass constant of GM=3.986 004 418×1014 m3/s2. The geoid model was computed by applying high-frequency corrections to the Earth Gravity Model 1996 global geopotential
model in a remove-compute-restore procedure. The permanent tide system of CARIB97 is non-tidal. Comparison of CARIB97 geoid
heights to 31 GPS/tidal (ITRF94/local) benchmarks shows an average offset (h–H–N) of 51 cm, with an Root Mean Square (RMS) of 62 cm about the average. This represents an improvement over the use of a global
geoid model for the region. However, because the measured orthometric heights (H) refer to many differing tidal datums, these comparisons are biased by localized permanent ocean dynamic topography (PODT).
Therefore, we interpret the 51 cm as partially an estimate of the average PODT in the vicinity of the 31 island benchmarks.
On an island-by-island basis, CARIB97 now offers the ability to analyze local datum problems which were previously unrecognized
due to a lack of high-resolution geoid information in the area.
Received: 2 January 1998 / Accepted: 18 August 1998 相似文献
11.
The latest gravimetric geoid model for Japan, JGEOID2000, was successfully combined with the nationwide net of GPS at benchmarks,
yielding a new hybrid geoid model for Japan, GSIGEO2000. The least-squares collocation (LSC) method was applied as an interpolation
for fitting JGEOID2000 to the GPS/leveling geoid undulations. The GPS/leveling geoid undulation data were reanalyzed in advance,
in terms of three-dimensional positions from GPS and orthometric heights from leveling. The new hybrid geoid model is, therefore,
compatible with the new Japanese geodetic reference frame. GSIGEO2000 was evaluated internally and independently and the precision
was estimated at 4 cm throughout nearly the whole region.
Received: 15 October 2001 / Accepted: 27 March 2002
Acknowledgments. Messrs. Toshio Kunimi and Tadashi Saito at the Third Geodetic Division of the Geographical Survey Institute (GSI) mainly
carried out the computations of most of the updated leveled heights. With regard to the reanalysis of GPS data, the discussions
with Messrs. Yuki Hatanaka and Shoichi Matsumura of GSI were of great help in building the analysis strategy. Messrs. Kazuyuki
Tanaka and Hiromi Shigematsu collaborated in the preparatory stages of GPS data computation. The authors' thanks are extended
to these colleagues. Some plots were made by GMT software (Wessel and Smith 1991).
Correspondence to: Y. Kuroishi 相似文献
12.
Analysis of some systematic errors affecting altimeter-derived sea surface gradient with application to geoid determination over Taiwan 总被引：3，自引：1，他引：3
C. Hwang 《Journal of Geodesy》1997,71(2):113-130
This paper analyzes several systematic errors affecting sea surface gradients derived from Seasat, Geosat/ERM, Geosat/GM,
ERS-1/35d, ERS-1/GM and TOPEX/POSEIDON altimetry. Considering the data noises, the conclusion is: (1) only Seasat needs to
correct for the non-geocentricity induced error, (2) only Seasat and Geosat/GM need to correct for the one cycle per revolution
error, (3) only Seasat, ERS-1/GM and Geosat/GM need to correct for the tide model error; over shallow waters it is suggested
to use a local tide model not solely from altimetry. The effects of the sea surface topography on gravity and geoid computations
from altimetry are significant over areas with major oceanographic phenomena. In conclusion, sea surface gradient is a better
data type than sea surface height. Sea surface gradients from altimetry, land gravity anomalies, ship gravity anomalies and
elevation data were then used to calculate the geoid over Taiwan by least-squares collocation. The inclusion of sea surface
gradients improves the geoid prediction by 27% when comparing the GPS-derived and the predicted geoidal heights, and by 30%
when comparing the observed and the geoid-derived deflections of the vertical. The predicted geoid along coastal areas is
accurate to 2 cm and can help GPS to do the third-order leveling.
Received 22 January 1996; Accepted 13 September 1996 相似文献
13.
I. N. Tziavos 《Journal of Geodesy》1996,70(6):357-373
The Stokes formula is efficiently evaluated by the one-and two- dimensional (1D, 2D) fast Fourier transform (FFT) technique in the plane and on the sphere in order to obtain precise geoid determinatiover a large area such as Europe. Using a high-pass filtered spherical harmonic reference model (OSU91A truncated to different degrees), gridded gravity anomalies and geoid heights were produced and the anomalies were used as input in the FFT software. Various tests were performed with respect to the different kernel functions used, to the spherical computations in bands, as well as to windowing, edge effects and extent of the area. It is thus demonstrated that, in geoid computations over large regions, the 1D spherical FFT and the 2D multiband spherical FFT in combination with discrete spectra for the kernel functions and 100% zero-padding give better results than those obtained by the other transform techniques. Additionally, numerical tests were carried out at the same test area using the planar fast Hartley transform (FHT) instead of the FFT and the results obtained by the two attractive alternatives were compared regarding the requirements in both computer time and computer memory needed in geoid height computations.A slightly modified version of the paper has been presented at the XX EGS General Assembly, Hamburg, 3–7 April, 1995 相似文献
14.
William E. Strange 《Journal of Geodesy》1982,56(4):300-311
Errors are introduced in orthometric height computations by the use of standard formulas to estimate mean gravity along the
plumb line. Direct measurements of gravity between the Earth’s surface and sea level from bore hole gravimetry were used to
determine the magnitude of these errors. For the seven cases studied, errors in orthometric height, due to the use of the
Helmert method for computing mean gravity along the plumb line, were generally small (<2 cm). However, in one instance the error was substantial, being9.6 cm. The results verified the general validity of the Poincaré-Prey approach to estimation of gravity along the plumb line and
demonstrated that the suggestion byVanicek (1980) that the air gradient is more appropriate is incorrect. With sufficient topographic information to compute terrain
corrections, and density estimates from surface gravity, errors in mean gravity along the plumb line should contribute no
more than 3cm to orthometric height computation. 相似文献
15.
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
16.
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
17.
Y. M. Wang 《Journal of Geodesy》1989,63(4):359-370
The formulas for the determination of the coefficients of the spherical harmonic expansion of the disturbing potential of
the earth are defined for data given on a sphere. In order to determine the spherical harmonic coefficients, the gravity anomalies
have to be analytically downward continued from the earth's surface to a sphere—at least to the ellipsoid. The goal of this
paper is to continue the gravity anomalies from the earth's surface downward to the ellipsoid using recent elevation models.
The basic method for the downward continuation is the gradient solution (theg
1 term). The terrain correction has also been computed because of the role it can play as a correction term when calculating
harmonic coefficients from surface gravity data.
Theg
1 term and the terrain correction were expanded into the spherical harmonics up to180
th
order. The corrections (theg
1 term and the terrain correction) have the order of about 2% of theRMS value of degree variance of the disturbing potential per degree. The influences of theg
1 term and the terrain correction on the geoid take the order of 1 meter (RMS value of corrections of the geoid undulation) and on the deflections of the vertical is of the order 0.1″ (RMS value of correction of the deflections of the vertical). 相似文献
18.
The problem of “global height datum unification” is solved in the gravity potential space based on: (1) high-resolution local
gravity field modeling, (2) geocentric coordinates of the reference benchmark, and (3) a known value of the geoid’s potential.
The high-resolution local gravity field model is derived based on a solution of the fixed-free two-boundary-value problem
of the Earth’s gravity field using (a) potential difference values (from precise leveling), (b) modulus of the gravity vector
(from gravimetry), (c) astronomical longitude and latitude (from geodetic astronomy and/or combination of (GNSS) Global Navigation
Satellite System observations with total station measurements), (d) and satellite altimetry. Knowing the height of the reference
benchmark in the national height system and its geocentric GNSS coordinates, and using the derived high-resolution local gravity
field model, the gravity potential value of the zero point of the height system is computed. The difference between the derived
gravity potential value of the zero point of the height system and the geoid’s potential value is computed. This potential
difference gives the offset of the zero point of the height system from geoid in the “potential space”, which is transferred
into “geometry space” using the transformation formula derived in this paper. The method was applied to the computation of
the offset of the zero point of the Iranian height datum from the geoid’s potential value W
0=62636855.8 m2/s2. According to the geometry space computations, the height datum of Iran is 0.09 m below the geoid. 相似文献
19.
A synthetic Earth Gravity Model Designed Specifically for Testing Regional Gravimetric Geoid Determination Algorithms 总被引：1，自引：0，他引：1
I. Baran M. Kuhn S. J. Claessens W. E. Featherstone S. A. Holmes P. Vaníček 《Journal of Geodesy》2006,80(1):1-16
A synthetic [simulated] Earth gravity model (SEGM) of the geoid, gravity and topography has been constructed over Australia specifically for validating regional gravimetric geoid determination theories, techniques and computer software. This regional high-resolution (1-arc-min by 1-arc-min) Australian SEGM (AusSEGM) is a combined source and effect model. The long-wavelength effect part (up to and including spherical harmonic degree and order 360) is taken from an assumed errorless EGM96 global geopotential model. Using forward modelling via numerical Newtonian integration, the short-wavelength source part is computed from a high-resolution (3-arc-sec by 3-arc-sec) synthetic digital elevation model (SDEM), which is a fractal surface based on the GLOBE v1 DEM. All topographic masses are modelled with a constant mass-density of 2,670 kg/m3. Based on these input data, gravity values on the synthetic topography (on a grid and at arbitrarily distributed discrete points) and consistent geoidal heights at regular 1-arc-min geographical grid nodes have been computed. The precision of the synthetic gravity and geoid data (after a first iteration) is estimated to be better than 30 μ Gal and 3 mm, respectively, which reduces to 1 μ Gal and 1 mm after a second iteration. The second iteration accounts for the changes in the geoid due to the superposed synthetic topographic mass distribution. The first iteration of AusSEGM is compared with Australian gravity and GPS-levelling data to verify that it gives a realistic representation of the Earth’s gravity field. As a by-product of this comparison, AusSEGM gives further evidence of the north–south-trending error in the Australian Height Datum. The freely available AusSEGM-derived gravity and SDEM data, included as Electronic Supplementary Material (ESM) with this paper, can be used to compute a geoid model that, if correct, will agree to in 3 mm with the AusSEGM geoidal heights, thus offering independent verification of theories and numerical techniques used for regional geoid modelling.Electronic Supplementary Material Supplementary material is available in the online version of this article at http://dx.doi.org/10.1007/s00190-005-0002-z 相似文献