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1.
The topographic effects by Stokes formula are typically considered for a spherical approximation of sea level. For more precise determination of the geoid, sea level is better approximated by an ellipsoid, which justifies the consideration of the ellipsoidal corrections of topographic effects for improved geoid solutions. The aim of this study is to estimate the ellipsoidal effects of the combined topographic correction (direct plus indirect topographic effects) and the downward continuation effect. It is concluded that the ellipsoidal correction to the combined topographic effect on the geoid height is far less than 1 mm. On the contrary, the ellipsoidal correction to the effect of downward continuation of gravity anomaly to sea level may be significant at the 1-cm level in mountainous regions. Nevertheless, if Stokes formula is modified and the integration of gravity anomalies is limited to a cap of a few degrees radius around the computation point, nor this effect is likely to be significant.AcknowledgementsThe author is grateful for constructive remarks by J Ågren and the three reviewers.  相似文献   

2.
A new theory for high-resolution regional geoid computation without applying Stokess formula is presented. Operationally, it uses various types of gravity functionals, namely data of type gravity potential (gravimetric leveling), vertical derivatives of the gravity potential (modulus of gravity intensity from gravimetric surveys), horizontal derivatives of the gravity potential (vertical deflections from astrogeodetic observations) or higher-order derivatives such as gravity gradients. Its algorithmic version can be described as follows: (1) Remove the effect of a very high degree/order potential reference field at the point of measurement (POM), in particular GPS positioned, either on the Earths surface or in its external space. (2) Remove the centrifugal potential and its higher-order derivatives at the POM. (3) Remove the gravitational field of topographic masses (terrain effect) in a zone of influence of radius r. A proper choice of such a radius of influence is 2r=4×104 km/n, where n is the highest degree of the harmonic expansion. (cf. Nyquist frequency). This third remove step aims at generating a harmonic gravitational field outside a reference ellipsoid, which is an equipotential surface of a reference potential field. (4) The residual gravitational functionals are downward continued to the reference ellipsoid by means of the inverse solution of the ellipsoidal Dirichlet boundary-value problem based upon the ellipsoidal Abel–Poisson kernel. As a discretized integral equation of the first kind, downward continuation is Phillips–Tikhonov regularized by an optimal choice of the regularization factor. (5) Restore the effect of a very high degree/order potential reference field at the corresponding point to the POM on the reference ellipsoid. (6) Restore the centrifugal potential and its higher-order derivatives at the ellipsoidal corresponding point to the POM. (7) Restore the gravitational field of topographic masses ( terrain effect) at the ellipsoidal corresponding point to the POM. (8) Convert the gravitational potential on the reference ellipsoid to geoidal undulations by means of the ellipsoidal Bruns formula. A large-scale application of the new concept of geoid computation is made for the Iran geoid. According to the numerical investigations based on the applied methodology, a new geoid solution for Iran with an accuracy of a few centimeters is achieved.Acknowledgments. The project of high-resolution geoid computation of Iran has been support by National Cartographic Center (NCC) of Iran. The University of Tehran, via grant number 621/3/602, supported the computation of a global geoid solution for Iran. Their support is gratefully acknowledged. A. Ardalan would like to thank Mr. Y. Hatam, and Mr. K. Ghazavi from NCC and Mr. M. Sharifi, Mr. A. Safari, and Mr. M. Motagh from the University of Tehran for their support in data gathering and computations. The authors would like to thank the comments and corrections made by the four reviewers and the editor of the paper, Professor Will Featherstone. Their comments helped us to correct the mistakes and improve the paper.  相似文献   

3.
Today the combination of Stokes formula and an Earth gravity model (EGM) for geoid determination is a standard procedure. However, the method of modifying Stokes formula varies from author to author, and numerous methods of modification exist. Most methods modify Stokes kernel, but the most widely applied method, the remove compute restore technique, removes the EGM from the gravity anomaly to attain a residual gravity anomaly under Stokes integral, and at least one known method modifies both Stokes kernel and the gravity anomaly. A general model for modifying Stokes formula is presented; it includes most of the well-known techniques of modification as special cases. By assuming that the error spectra of the gravity anomalies and the EGM are known, the optimum model of modification is derived based on the least-squares principle. This solution minimizes the expected mean square error (MSE) of all possible solutions of the general geoid model. A practical formula for estimating the MSE is also presented. The power of the optimum method is demonstrated in two special cases. AcknowledgementsThis paper was partly written whilst the author was a visiting scientist at The University of New South Wales, Sydney, Australia. He is indebted to Professor W. Kearsley and his colleagues, and their hospitality is acknowledged.  相似文献   

4.
Summary A local model of the geoid in NE Italy and its section along the Venice ground track of the ERS-1 satellite of the European Space Agency is presented. The observational data consist of geoid undulations determined with a network of 25 stations of known orthometric (by spirit leveling) and ellipsoidal (by GPS differential survey) and of 13 deflections of the vertical measured at sites of the network for which, besides the ellipsoidal (WGS84) coordinates, also astronomic coordinates were known. The network covers an area of 1×1 degrees and is tied to a vertical and horizontal datum: one vertex of the network is the tide gauge of Punta Salute, in Venice, providing a tie to a mean sea level; a second vertex is the site for mobile laser systems at Monte Venda, on the Euganei Hills, for which geocentric coordinates resulted from the analysis of several LAGEOS passes.The interpolation algorithm used to map sparse and heterogeneous data to a regular grid of geoid undulations is based on least squares collocation and the autocorrelation function of the geoid undulations is modeled by a third order Markov process on flat earth. The algorithm has been applied to the observed undulations and deflections of the vertical after subtraction of the corresponding predictions made on the basis of the OSU91A global geoid model of the Ohio State University, complete to degree and order 360. The locally improved geoid results by adding back, at the nodes of a regular grid, the predictions of the global field to the least squares interpolated values. Comparison of the model values with the raw data at the observing stations indicates that the mean discrepancy is virtually zero with a root mean square dispersion of 8 cm, assuming that the ellipsoidal heights and vertical deflections data are affected by a random error of 3 cm and 0.5 respectively. The corrections resulting from the local data and added to the background 360×360 global model are described by a smooth surface with excursions from the reference surface not larger than ±30 cm.  相似文献   

5.
W. Sun 《Journal of Geodesy》2003,77(7-8):381-387
An asymptotic theory is presented for calculating co-seismic potential and geoid changes, as an approximation of the dislocation theory for a spherical Earth. This theory is given by a closed-form mathematical expression, so that it is mathematically simple and can be applied easily. Moreover, since the asymptotic theory includes sphericity and vertical structure effects, it is physically more reasonable than the flat-Earth theory. A comparison between results calculated by three dislocation theories (the flat-Earth theory, the theory for a spherical Earth and its asymptotic solution) shows that the true co-seismic geoid changes are approximated better by the asymptotic results than by those of a flat Earth. Numerical results indicate that the sphericity effect is obvious large, especially for a tensile source on a vertical fault plane. AcknowledgementsThe author is grateful to Dr S. Okubo for his helpful suggestions and discussions. Comments by anonymous reviewers are also greatly acknowledged. This research was financially supported by JSPS research grants (C13640420) and Basic design and feasibility studies for the future missions for monitoring Earths environment.  相似文献   

6.
Recent papers in the geodetic literature promote the reduction of gravity for geoid determination according to the Helmert condensation technique where the entire reduction is made in place before downward continuation. The alternative approach, primarily developed by Moritz, uses two evaluation points, one at the Earths surface, the other on the (co-)geoid, for the direct topographic effect. Both approaches are theoretically legitimate and the derivations in each case make use of the planar approximation and a Lipschitz condition on height. Each method is re-formulated from first principles, yielding equations for the direct effect that contain only the spherical approximation. It is shown that neither method relies on a linear relationship between gravity anomalies and height (as claimed by some). Numerical tests, however, show that the practical implementations of these two approaches yield significant differences. Computational tests were performed in three areas of the USA, using 1×1 grids of gravity data and 30×30 grids of height data to compute the gravimetric geoid undulation, and GPS/leveled heights to compute the geometric geoid undulation. Using the latter as a control, analyses of the gravimetric undulations indicate that while in areas with smooth terrain no substantial differences occur between the gravity reduction methods, the Moritz–Pellinen (MP) approach is clearly superior to the Vanicek–Martinec (VM) approach in areas of rugged terrain. In theory, downward continuation is a significant aspect of either approach. Numerically, however, based on the test data, neither approach benefited by including this effect in the areas having smooth terrain. On the other hand, in the rugged, mountainous area, the gravimetric geoid based on the VM approach was improved slightly, but with the MP approach it suffered significantly. The latter is attributed to an inability to model the downward continuation of the Bouguer anomaly accurately in rugged terrain. Applying the higher-order, more accurate gravity reduction formulas, instead of their corresponding planar and linear approximations, yielded no improvement in the accuracy of the gravimetric geoid undulation based on the available data.  相似文献   

7.
Geoid determination in Turkey (TG-91)   总被引：1，自引：0，他引：1  
It is considered that precise geoid determination is one of the main current geodetic problems in Turkey since GPS defined coordinates require geoidal heights in practice. In order to determine the geoid by least squares collocation (LSC) the area covering Turkey was divided into 114 blocks of size 1° × 1°. LSC approximation to the geoid based upon the tailored geopotential model GPM2-T1 is constructed within each block. The model GPM2-T1 complete to degree and order 200 has been developed by tailoring of the model GPM2 to mean free-air anomalies and mean heights of one degree blocks in Turkey. Terrain effect reduced point gravity data spaced 5 × 5 within each block which the sides extended 0°.5 were used in LSC. Residual terrain model (RTM) depends on point heights at 15×20 griding and 5×5 and 15×15 mean heights has been carried out in terrain effect reduction. Indirect effect of RTM on geoid is also taken into account. The geoid, called Turkish Geoid 1991 (TG-91), referenced to GRS-80 ellipsoid has been computed at 3 × 3 griding nodes within each block. The quality of the TG-91 is also evaluated by comparing computed and GPS derived geoidal height differences, and 2.1 – 2.6 ppm accuracy for average baseline lenght of 45 km is obtained.  相似文献   

8.
On Helmert’s methods of condensation   总被引：2，自引：0，他引：2  
B. Heck 《Journal of Geodesy》2003,77(3-4):155-170
Helmerts first and second method of condensation are reviewed and generalized in two respects: First, the point at which the effects of topographical and condensation masses are calculated may be situated on or outside the topographical surface; second, the depth of the condensation layer below the geoid is arbitrary. While the first extension permits the application of the generalized model to the evaluation of airborne and satellite data, the second one gives an additional degree of freedom which can be used to provide a smooth gravity field after reducing the observation data. The respective formulae are derived for the generalized condensation model in both planar and spherical approximation. A comparison of the planar and the spherical model shows some structural differences, which are primarily visible in the out-of-integral terms. Considering the respective formulae for the combined topographic–condensation reduction on the background of the density structure of the Earths lithosphere, the consequences for the residual gravity field are investigated; it is shown that the residual field after applying Helmerts second model of reduction is very rough, making this procedure unfavourable for downward continuation. Further considerations refer to the question of which sets of formulae should be used in geoid and quasigeoid determination. It is concluded that for high-precision applications the generalized spherical model, involving a depth of the condensation layer of between 20 and 30 km, should be superior to Helmerts second model of condensation, although it requires the direct calculation of the indirect effect, which is larger than in the case of Helmerts second method of condensation.  相似文献   

9.
Summary The least-squares collocation method has been used for the computation of a geoid solution in central Spain, combining a geopotential model complete to degree and order 360, gravity anomalies and topographic information. The area has been divided in two 1°× 1° blocks and predictions have been done in each block with gravity data spacing about 5 × 5 within each block, extended 1/2°. Topographic effects have been calculated from 6 × 9 heights using an RTM reduction with a reference terrain model of 30 × 30 mean heights.  相似文献   

10.
Assuming that the gravity anomaly and disturbing potential are given on a reference ellipsoid, the result of Sjöberg (1988, Bull Geod 62:93–101) is applied to derive the potential coefficients on the bounding sphere of the ellipsoid to order e 2 (i.e. the square of the eccentricity of the ellipsoid). By adding the potential coefficients and continuing the potential downward to the reference ellipsoid, the spherical Stokes formula and its ellipsoidal correction are obtained. The correction is presented in terms of an integral over the unit sphere with the spherical approximation of geoidal height as the argument and only three well-known kernel functions, namely those of Stokes, Vening-Meinesz and the inverse Stokes, lending the correction to practical computations. Finally, the ellipsoidal correction is presented also in terms of spherical harmonic functions. The frequently applied and sometimes questioned approximation of the constant m, a convenient abbreviation in normal gravity field representations, by e 2/2, as introduced by Moritz, is also discussed. It is concluded that this approximation does not significantly affect the ellipsoidal corrections to potential coefficients and Stokes formula. However, whether this standard approach to correct the gravity anomaly agrees with the pure ellipsoidal solution to Stokes formula is still an open question.  相似文献   

11.
Prior to Stokes integration, the gravitational effect of atmospheric masses must be removed from the gravity anomaly g. One theory for the atmospheric gravity effect on the geoid is the well-known International Association of Geodesy approach in connection with Stokes integral formula. Another strategy is the use of a spherical harmonic representation of the topography, i.e. the use of a global topography computed from a set of spherical harmonics. The latter strategy is improved to account for local information. A new formula is derived by combining the local contribution of the atmospheric effect computed from a detailed digital terrain model and the global contribution computed from a spherical harmonic model of the topography. The new formula is tested over Iran and the results are compared with corresponding results from the old formula which only uses the global information. The results show significant differences. The differences between the two formulas reach 17 cm in a test area in Iran.  相似文献   

12.
Since the earth is closer to a revolving ellipsoid than a sphere, it is very important to study directly the original model of the Stokes' BVP on the reference ellipsoid, where denotes the reference ellipsoid, is the Somigliana normal gravity, andh is the outer normal direction of. This paper deals with: 1) simplification of the above BVP under preserving accuracy to , 2) derivation of computational formula of the elliptical harmonic series, 3) solving the BVP by the elliptical harmonic series, and 4) providing a principle for finding the elliptical harmonic model of the earth's gravity field from the spherical harmonic coefficients ofg. All results given in the paper have the same accuracy as the original BVP, that is, the accuracy of the BVP is theoretically preserved in each derivation step.  相似文献   

13.
In a modern application of Stokes formula for geoid determination, regional terrestrial gravity is combined with long-wavelength gravity information supplied by an Earth gravity model. Usually, several corrections must be added to gravity to be consistent with Stokes formula. In contrast, here all such corrections are applied directly to the approximate geoid height determined from the surface gravity anomalies. In this way, a more efficient workload is obtained. As an example, in applications of the direct and first and second indirect topographic effects significant long-wavelength contributions must be considered, all of which are time consuming to compute. By adding all three effects to produce a combined geoid effect, these long-wavelength features largely cancel. The computational scheme, including two least squares modifications of Stokes formula, is outlined, and the specific advantages of this technique, compared to traditional gravity reduction prior to Stokes integration, are summarised in the conclusions and final remarks. AcknowledgementsThis paper was written whilst the author was a visiting scientist at Curtin University of Technology, Perth, Australia. The hospitality and fruitful discussions with Professor W. Featherstone and his colleagues are gratefully acknowledged.  相似文献   

14.
A new isostatic model for the Earths gravity field is presented based on a simple hypothesis of layers approximating constant density contrasts. The spherical layer distribution used to describe the hydrostatic equilibrium of the Earths masses leads to a new set of spherical harmonic coefficients for the gravitational potential. First attempts to quantify the information content of these coefficients led to the outcome that they seem to explain the observed gravity field for a certain wavelength band, while they are insufficient for short and very long wavelengths. A synthesis of the derived coefficients over specific degree ranges provided a computation of band-limited geoid undulations on a global scale. The association of these potential quantities with known tectonic structures, such as the topography of the core–mantle boundary, strengthens the belief that the interpretation of Earth gravity models, especially those arising from global digital elevation models, should be considered in close relation with deep-Earth structure.  相似文献   

15.
Harmonic maps     
Harmonic maps are generated as a certain class of optimal map projections. For instance, if the distortion energy over a meridian strip of the International Reference Ellipsoid is minimized, we are led to the Laplace–Beltrami vector-valued partial differential equation. Harmonic functions x(L,B), y(L,B) given as functions of ellipsoidal surface parameters of Gauss ellipsoidal longitude L and Gauss ellipsoidal latitude B, as well as x(,q), y(,q) given as functions of relative isometric longitude =L–L0 and relative isometric latitude q=Q–Q0 gauged to a vector-valued boundary condition of special symmetry are constructed. The easting and northing {x(b,),y(b,)} of the new harmonic map is then given. Distortion energy analysis of the new harmonic map is presented, as well as case studies for (1) B[–40°,+40°], L[–31°,+49°], B0= ±30°, L0=9° and (2) B[46°,56°], L{[4.5°, 7.5°]; [7.5°, 10.5°]; [10.5°,13.5°]; [13.5°,16.5°]}, B0= 51°, L0 {6°,9°,12°,15°}.  相似文献   

16.
The regularized solution of the external sphericalStokes boundary value problem as being used for computations of geoid undulations and deflections of the vertical is based upon theGreen functions S 1(0, 0, , ) ofBox 0.1 (R = R 0) andV 1(0, 0, , ) ofBox 0.2 (R = R 0) which depend on theevaluation point {0, 0} S R0 2 and thesampling point {, } S R0 2 ofgravity anomalies (, ) with respect to a normal gravitational field of typegm/R (free air anomaly). If the evaluation point is taken as the meta-north pole of theStokes reference sphere S R0 2 , theStokes function, and theVening-Meinesz function, respectively, takes the formS() ofBox 0.1, andV 2() ofBox 0.2, respectively, as soon as we introduce {meta-longitude (azimuth), meta-colatitude (spherical distance)}, namely {A, } ofBox 0.5. In order to deriveStokes functions andVening-Meinesz functions as well as their integrals, theStokes andVening-Meinesz functionals, in aconvolutive form we map the sampling point {, } onto the tangent plane T0S R0 2 at {0, 0} by means ofoblique map projections of type(i) equidistant (Riemann polar/normal coordinates),(ii) conformal and(iii) equiareal.Box 2.1.–2.4. andBox 3.1.– 3.4. are collections of the rigorously transformedconvolutive Stokes functions andStokes integrals andconvolutive Vening-Meinesz functions andVening-Meinesz integrals. The graphs of the correspondingStokes functions S 2(),S 3(r),,S 6(r) as well as the correspondingStokes-Helmert functions H 2(),H 3(r),,H 6(r) are given byFigure 4.1–4.5. In contrast, the graphs ofFigure 4.6–4.10 illustrate the correspondingVening-Meinesz functions V 2(),V 3(r),,V 6(r) as well as the correspondingVening-Meinesz-Helmert functions Q 2(),Q 3(r),,Q 6(r). The difference between theStokes functions / Vening-Meinesz functions andtheir first term (only used in the Flat Fourier Transforms of type FAST and FASZ), namelyS 2() – (sin /2)–1,S 3(r) – (sinr/2R 0)–1,,S 6(r) – 2R 0/r andV 2() + (cos /2)/2(sin2 /2),V 3(r) + (cosr/2R 0)/2(sin2 r/2R 0),, illustrate the systematic errors in theflat Stokes function 2/ or flatVening-Meinesz function –2/2. The newly derivedStokes functions S 3(r),,S 6(r) ofBox 2.1–2.3, ofStokes integrals ofBox 2.4, as well asVening-Meinesz functionsV 3(r),,V 6(r) ofBox 3.1–3.3, ofVening-Meinesz integrals ofBox 3.4 — all of convolutive type — pave the way for the rigorousFast Fourier Transform and the rigorousWavelet Transform of theStokes integral / theVening-Meinesz integral of type equidistant, conformal and equiareal.  相似文献   

17.
Since the advent of CHAMP, the first in a series of low-altitude satellites being almost continuously and precisely tracked by GPS, a new generation of long-wavelength gravitational geopotential models can be derived. The accuracy evaluation of these models depends to a large extent on the comparison with external data of comparable quality. Here, two CHAMP-derived models, EIGEN-1S and EIGEN-2, are tested with independent long-term-averaged single satellite crossover (SSC) sea heights from three altimetric satellites (ERS-1, ERS-2 and Geosat). The analyses show that long-term averages of crossover residuals still are powerful data to test CHAMP gravity field models. The new models are tested in the spatial domain with the aid of ERS-1/-2 and Geosat SSCs, and in the spectral domain with latitude-lumped coefficient (LLC) corrections derived from the SSCs. The LLC corrections allow a representation of the satellite-orbit-specific error spectra per order of the models spherical harmonic coefficients. These observed LLC corrections are compared to the LLC projections from the models variance–covariance matrix. The excessively large LLC errors at order 2 found in the case of EIGEN-2 with the ERS data are discussed. The degree-dependent scaling factors for the variance-covariance matrices of EIGEN-1S and –2, applied to obtain more realistic error estimates of the solved-for coefficients, are compatible with the results found here.  相似文献   

18.
Ellipsoidal geoid computation   总被引：1，自引：1，他引：0  
Modern geoid computation uses a global gravity model, such as EGM96, as a third component in a remove–restore process. The classical approach uses only two: the reference ellipsoid and a geometrical model representing the topography. The rationale for all three components is reviewed, drawing attention to the much smaller precision now needed when transforming residual gravity anomalies. It is shown that all ellipsoidal effects needed for geoid computation with millimetric accuracy are automatically included provided that the free air anomaly and geoid are calculated correctly from the global model. Both must be consistent with an ellipsoidal Earth and with the treatment of observed gravity data. Further ellipsoidal corrections are then negligible. Precise formulae are developed for the geoid height and the free air anomaly using a global gravity model, given as spherical harmonic coefficients. Although only linear in the anomalous potential, these formulae are otherwise exact for an ellipsoidal reference Earth—they involve closed analytical functions of the eccentricity (and the Earths spin rate), rather than a truncated power series in e2. They are evaluated using EGM96 and give ellipsoidal corrections to the conventional free air anomaly ranging from –0.84 to +1.14 mGal, both extremes occurring in Tibet. The geoid error corresponding to these differences is dominated by longer wavelengths, so extrema occur elsewhere, rising to +766 mm south of India and falling to –594 mm over New Guinea. At short wavelengths, the difference between ellipsoidal corrections based only on EGM96 and those derived from detailed local gravity data for the North Sea geoid GEONZ97 has a standard deviation of only 3.3 mm. However, the long-wavelength components missed by the local computation reach 300 mm and have a significant slope. In Australia, for example, such a slope would amount to a 600-mm rise from Perth to Cairns.  相似文献   

19.
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20.
This paper provides numerical examples for the prediction of height anomalies by the solution of Molodensky's boundary value problem. Computations are done within two areas in the Canadian Rockies. The data used are on a grid with various grid spacings from 100 m to 5 arc-minutes. Numerical results indicate that the Bouguer or the topographicisostatic gravity anomalies should be used in gravity interpolation. It is feasible to predict height anomalies in mountainous areas with an accuracy of 10 cm (1) if sufficiently dense data grids are used. After removing the systematic bias, the differences between the geoid undulations converted from height anomalies and those derived from GPS/levelling on 50 benchmarks is 12 cm (1) when the grid spacing is 1km, and 50 cm (1) when the grid spacing is 5. It is not necessary, in most cases, to require a grid spacing finer than 1 km, because the height anomaly changes only by 3 cm (1) when the grid spacing is increased from 100 m to 1000 m. Numerical results also indicate that, only the first two terms of the Molodensky series have to be evaluated in all but the extreme cases, since the contributions of the higher order terms are negligible compared to the objective accuracy.  相似文献   

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