首页 | 本学科首页   官方微博 | 高级检索  
相似文献
 共查询到20条相似文献，搜索用时 46 毫秒
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.
This paper takes advantage of space-technique-derived positions on the Earth’s surface and the known normal gravity field to determine the height anomaly from geopotential numbers. A new method is also presented to downward-continue the height anomaly to the geoid height. The orthometric height is determined as the difference between the geodetic (ellipsoidal) height derived by space-geodetic techniques and the geoid height. It is shown that, due to the very high correlation between the geodetic height and the computed geoid height, the error of the orthometric height determined by this method is usually much smaller than that provided by standard GPS/levelling. Also included is a practical formula to correct the Helmert orthometric height by adding two correction terms: a topographic roughness term and a correction term for lateral topographic mass–density variations.  相似文献   

3.
The method of analytical downward continuation has been used for solving Molodensky’s problem. This method can also be used to reduce the surface free air anomaly to the ellipsoid for the determination of the coefficients of the spherical harmonic expansion of the geopotential. In the reduction of airborne or satellite gradiometry data, if the sea level is chosen as reference surface, we will encounter the problem of the analytical downward continuation of the disturbing potential into the earth, too. The goal of this paper is to find out the topographic effect of solving Stoke’sboundary value problem (determination of the geoid) by using the method of analytical downward continuation. It is shown that the disturbing potential obtained by using the analytical downward continuation is different from the true disturbing potential on the sea level mostly by a −2πGρh 2/p. This correction is important and it is very easy to compute and add to the final results. A terrain effect (effect of the topography from the Bouguer plate) is found to be much smaller than the correction of the Bouguer plate and can be neglected in most cases. It is also shown that the geoid determined by using the Helmert’s second condensation (including the indirect effect) and using the analytical downward continuation procedure (including the topographic effect) are identical. They are different procedures and may be used in different environments, e.g., the analytical downward continuation procedure is also more convenient for processing the aerial gravity gradient data. A numerical test was completed in a rough mountain area, 35°<ϕ<38°, 240°<λ<243°. A digital height model in 30″×30″ point value was used. The test indicated that the terrain effect in the test area has theRMS value ±0.2−0.3 cm for geoid. The topographic effect on the deflections of the vertical is around1 arc second.  相似文献   

4.
J. Ågren 《Journal of Geodesy》2004,78(4-5):314-332
One important application of an Earth Gravity Model (EGM) is to determine the geoid. Since an EGM is represented by an external-type series of spherical harmonics, a biased geoid model is obtained when the EGM is applied inside the masses in continental regions. In order to convert the downward-continued height anomaly to the corresponding geoid undulation, a correction has to be applied for the analytical continuation bias of the geoid height. This technique is here called the geoid bias method. A correction for the geoid bias can also be utilised when an EGM is combined with terrestrial gravity data, using the combined approach to topographic corrections. The geoid bias can be computed either by a strict integral formula, or by means of one or more terms in a binomial expansion. The accuracy of the lowest binomial terms is studied numerically. It is concluded that the first term (of power H2) can be used with high accuracy up to degree 360 everywhere on Earth. If very high mountains are disregarded, then the use of the H2 term can be extended up to maximum degrees as high as 1800. It is also shown that the geoid bias method is practically equal to the technique applied by Rapp, which utilises the quasigeoid-to-geoid separation. Another objective is to carefully consider how the combined approach to topographic corrections should be interpreted. This includes investigations of how the above-mentioned H2 term should be computed, as well as how it can be improved by a correction for the residual geoid bias. It is concluded that the computation of the combined topographic effect is efficient in the case that the residual geoid bias can be neglected, since the computation of the latter is very time consuming. It is nevertheless important to be able to compute the residual bias for individual stations. For reasonable maximum degrees, this can be used to check the quality of the H2 approximation in different situations.Acknowledgement The author would like to thank Prof. L.E. Sjöberg for several ideas and for reading two draft versions of the paper. His support and constructive remarks have improved its quality considerably. The valuable suggestions from three unknown reviewers are also appreciated.  相似文献   

5.
The determination of the gravimetric geoid is based on the magnitude of gravity observed at the surface of the Earth or at airborne altitude. To apply the Stokes’s or Hotine’s formulae at the geoid, the potential outside the geoid must be harmonic and the observed gravity must be reduced to the geoid. For this reason, the topographic (and atmospheric) masses outside the geoid must be “condensed” or “shifted” inside the geoid so that the disturbing gravity potential T fulfills Laplace’s equation everywhere outside the geoid. The gravitational effects of the topographic-compensation masses can also be used to subtract these high-frequent gravity signals from the airborne observations and to simplify the downward continuation procedures. The effects of the topographic-compensation masses can be calculated by numerical integration based on a digital terrain model or by representing the topographic masses by a spherical harmonic expansion. To reduce the computation time in the former case, the integration over the Earth can be divided into two parts: a spherical cap around the computation point, called the near zone, and the rest of the world, called the far zone. The latter one can be also represented by a global spherical harmonic expansion. This can be performed by a Molodenskii-type spectral approach. This article extends the original approach derived in Novák et al. (J Geod 75(9–10):491–504, 2001), which is restricted to determine the far-zone effects for Helmert’s second method of condensation for ground gravimetry. Here formulae for the far-zone effects of the global topography on gravity and geoidal heights for Helmert’s first method of condensation as well as for the Airy-Heiskanen model are presented and some improvements given. Furthermore, this approach is generalized for determining the far-zone effects at aeroplane altitudes. Numerical results for a part of the Canadian Rocky Mountains are presented to illustrate the size and distributions of these effects.  相似文献   

6.
The analytical, or harmonic, downward continuation of the external gravity potential into the topographic masses gives rise to a bias, which is called the analytical (downward) continuation (ADC) bias (Ågren in J Geod 78:314–332, 2004a) or the topographic bias (Sjöberg in J Geod, 2006). In Sjöberg (J Geod, 2006), a proof is presented that this bias is exactly equal to a simple two-term expression, which depends only on the topographic height and density in the evaluation point P. The expression is simple and inexpensive to evaluate. In this paper, we wish to question the validity of the expression given in Sjöberg (J Geod, 2006) for realistic terrains. The topographic bias is commonly defined as the difference between the true (internal) and the analytically downward continued external geopotential, evaluated at sea level. Typically both are evaluated as external or internal spherical harmonic (SH) expansions, which may however not always converge. If they do converge, they have been well known in the literature (e.g., Ågren (J Geod 78:314–332, 2004a), Wang (J Geod 71:70–82, 1997)) to produce a bias that contains additional terms over and beyond the simple expression. Below we analyze the additional terms that arise when applying the method to realistic terrains. Also, for realistic terrains, analytical downward continuation may not even be strictly possible. In practice, for discrete data sets, it is always possible, but then, an implicit smoothing of the terrain, or terrain potential, always takes place.  相似文献   

7.
The topographic bias is defined as the error/bias committed by continuing the external gravity field inside the topographic masses by a harmonic function. We study the topographic bias given by a digital terrain model defined by a spherical template, and we show that the topographic bias is given only by the potential of an inner-zone cap, and it equals the bias of the Bouguer shell, independent of the size of the cap. Then we study the effect on the real Earth by decomposing its topography into a template, and we show also in this case that the topographic bias is that of the Bouguer shell, independent of the shape of the terrain. Finally, we show that the topographic potential of the terrain at the geoid can be determined to any precision by a Taylor expansion outside the Earth’s surface. The last statement is demonstrated by a Taylor expansion to fourth order.  相似文献   

8.
Any errors in digital elevation models (DEMs) will introduce errors directly in gravity anomalies and geoid models when used in interpolating Bouguer gravity anomalies. Errors are also propagated into the geoid model by the topographic and downward continuation (DWC) corrections in the application of Stokes’s formula. The effects of these errors are assessed by the evaluation of the absolute accuracy of nine independent DEMs for the Iran region. It is shown that the improvement in using the high-resolution Shuttle Radar Topography Mission (SRTM) data versus previously available DEMs in gridding of gravity anomalies, terrain corrections and DWC effects for the geoid model are significant. Based on the Iranian GPS/levelling network data, we estimate the absolute vertical accuracy of the SRTM in Iran to be 6.5 m, which is much better than the estimated global accuracy of the SRTM (say 16 m). Hence, this DEM has a comparable accuracy to a current photogrammetric high-resolution DEM of Iran under development. We also found very large differences between the GLOBE and SRTM models on the range of −750 to 550 m. This difference causes an error in the range of −160 to 140 mGal in interpolating surface gravity anomalies and −60 to 60 mGal in simple Bouguer anomaly correction terms. In the view of geoid heights, we found large differences between the use of GLOBE and SRTM DEMs, in the range of −1.1 to 1 m for the study area. The terrain correction of the geoid model at selected GPS/levelling points only differs by 3 cm for these two DEMs.  相似文献   

9.
Gravimetric geoid determination by Stokes formula requires that the effects of topographic masses be removed prior to Stokes integration. This step includes the direct topographic and the downward continuation (DWC) effects on gravity anomaly, and the computations yield the co-geoid height. By adding the effect of restoration of the topography, the indirect effect on the geoid, the geoid height is obtained. Unfortunately, the computations of all these topographic effects are hampered by the uncertainty of the density distribution of the topography. Usually the computations are limited to a constant topographic density, but recently the effects of lateral density variations have been studied for their direct and indirect effects on the geoid. It is emphasised that the DWC effect might also be significantly affected by a lateral density variation. However, instead of computing separate effects of lateral density variation for direct, DWC and indirect effects, it is shown in two independent ways that the total geoid effect due to the lateral density anomaly can be represented as a simple correction proportional to the lateral density anomaly and the elevation squared of the computation point. This simple formula stems from the fact that the significant long-wavelength contributions to the various topographic effects cancel in their sum. Assuming that the lateral density anomaly is within 20% of the standard topographic density, the derived formula implies that the total effect on the geoid is significant at the centimetre level for topographic elevations above 0.66 km. For elevations of 1000, 2000 and 5000 m the effect is within ± 2.2, ± 8.8 and ± 56.8 cm, respectively. For the elevation of Mt. Everest the effect is within ± 1.78 m.  相似文献   

10.
The application of Stokes’s formula to determine the geoid height requires that topographic and atmospheric masses be mathematically removed prior to Stokes integration. This corresponds to the applications of the direct topographic and atmospheric effects. For a proper geoid determination, the external masses must then be restored, yielding the indirect effects. Assuming an ellipsoidal layering of the atmosphere with 15% increase in its density towards the poles, the direct atmospheric effect on the geoid height is estimated to be −5.51 m plus a second-degree zonal harmonic term with an amplitude of 1.1 cm. The indirect effect is +5.50 m and the total geoid correction thus varies between −1.2 cm at the equator to 1.9 cm at the poles. Finally, the correction needed to the atmospheric effect if Stokes’s formula is used in a spherical approximation, rather than an ellipsoidal approximation, of the Earth varies between 0.3 cm and 4.0 cm at the equator and pole, respectively.  相似文献   

11.
 The analytical continuation of the surface gravity anomaly to sea level is a necessary correction in the application of Stokes' formula for geoid estimation. This process is frequently performed by the inversion of Poisson's integral formula for a sphere. Unfortunately, this integral equation corresponds to an improperly posed problem, and the solution is both numerically unstable, unless it is well smoothed, and tedious to compute. A solution that avoids the intermediate step of downward continuation of the gravity anomaly is presented. Instead the effect on the geoid as provided by Stokes' formula is studied directly. The practical solution is partly presented in terms of a truncated Taylor series and partly as a truncated series of spherical harmonics. Some simple numerical estimates show that the solution mostly meets the requests of a 1-cm geoid model, but the truncation error of the far zone must be studied more precisely for high altitudes of the computation point. In addition, it should be emphasized that the derived solution is more computer efficient than the detour by Poisson's integral. Received: 6 February 2002 / Accepted: 18 November 2002 Acknowledgements. Jonas ?gren carried out the numerical calculations and gave some critical and constructive remarks on a draft version of the paper. This support is cordially acknowledged. Also, the thorough work performed by one unknown reviewer is very much appreciated.  相似文献   

12.
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   

13.
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   

14.
 The downward continuation of the harmonic disturbing gravity potential, derived at flight level from discrete observations of airborne gravity by the spherical Hotine integral, to the geoid is discussed. The initial-boundary-value approach, based on both the direct and inverse solution to Dirichlet's problem of potential theory, is used. Evaluation of the discretized Fredholm integral equation of the first kind and its inverse is numerically tested using synthetic airborne gravity data. Characteristics of the synthetic gravity data correspond to typical airborne data used for geoid determination today and in the foreseeable future: discrete gravity observations at a mean flight height of 2 to 6 km above mean sea level with minimum spatial resolution of 2.5 arcmin and a noise level of 1.5 mGal. Numerical results for both approaches are presented and discussed. The direct approach can successfully be used for the downward continuation of airborne potential without any numerical instabilities associated with the inverse approach. In addition to these two-step approaches, a one-step procedure is also discussed. This procedure is based on a direct relationship between gravity disturbances at flight level and the disturbing gravity potential at sea level. This procedure provided the best results in terms of accuracy, stability and numerical efficiency. As a general result, numerically stable downward continuation of airborne gravity data can be seen as another advantage of airborne gravimetry in the field of geoid determination. Received: 6 June 2001 / Accepted: 3 January 2002  相似文献   

15.
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   

16.
This study deals with the external type of topographic–isostatic potential and gravity anomaly and its vertical derivatives, derived from the Airy/Heiskanen model for isostatic compensation. From the first and the second radial derivatives of the gravity anomaly the effect on the geoid is estimated for the downward continuation of gravity to sea level in the application of Stokes' formula. The major and regional effect is shown to be of order H 3 of the topography, and it is estimated to be negligible at sea level and modest for most mountains, but of the order of several metres for the highest and most extended mountain belts. Another, global, effect is of order H but much less significant Received: 3 October 1997 / Accepted: 30 June 1998  相似文献   

17.
针对局部重力异常向上延拓计算复杂、耗时长的问题,该文基于泊松积分离散化的基本原理,提出一种快速的局部格网重力异常向上延拓的实用算法;并结合中国东北和青藏高原地区大地水准面的重力异常格网数据,采用该延拓方法分别计算了空中10、50、100km处的重力异常,将其与等高度的EIGEN-6C4模型结果对比分析。实验结果表明:在顾及边界效应影响的情况下,相对于EIGEN-6C4模型,中国东北和青藏高原地区重力异常向上延拓的最大均方根误差分别优于1.5和3.5mGal;在保证精度可用的前提下,计算效率可以有大幅度提高,证明了该方法解算局部重力异常向上延拓的适用性。  相似文献   

18.
Geoid and quasigeoid modelling from gravity anomalies by the method of least squares modification of Stokes’s formula with additive corrections is adapted for the usage with gravity disturbances and Hotine’s formula. The biased, unbiased and optimum versions of least squares modification are considered. Equations are presented for the four additive corrections that account for the combined (direct plus indirect) effect of downward continuation (DWC), topographic, atmospheric and ellipsoidal corrections in geoid or quasigeoid modelling. The geoid or quasigeoid modelling scheme by the least squares modified Hotine formula is numerically verified, analysed and compared to the Stokes counterpart in a heterogeneous study area. The resulting geoid models and the additive corrections computed both for use with Stokes’s or Hotine’s formula differ most in high topography areas. Over the study area (reaching almost 2 km in altitude), the approximate geoid models (before the additive corrections) differ by 7 mm on average with a 3 mm standard deviation (SD) and a maximum of 1.3 cm. The additive corrections, out of which only the DWC correction has a numerically significant difference, improve the agreement between respective geoid or quasigeoid models to an average difference of 5 mm with a 1 mm SD and a maximum of 8 mm.  相似文献   

19.
 Four different implementations of Stokes' formula are employed for the estimation of geoid heights over Sweden: the Vincent and Marsh (1974) model with the high-degree reference gravity field but no kernel modifications; modified Wong and Gore (1969) and Molodenskii et al. (1962) models, which use a high-degree reference gravity field and modification of Stokes' kernel; and a least-squares (LS) spectral weighting proposed by Sj?berg (1991). Classical topographic correction formulae are improved to consider long-wavelength contributions. The effect of a Bouguer shell is also included in the formulae, which is neglected in classical formulae due to planar approximation. The gravimetric geoid is compared with global positioning system (GPS)-levelling-derived geoid heights at 23 Swedish Permanent GPS Network SWEPOS stations distributed over Sweden. The LS method is in best agreement, with a 10.1-cm mean and ±5.5-cm standard deviation in the differences between gravimetric and GPS geoid heights. The gravimetric geoid was also fitted to the GPS-levelling-derived geoid using a four-parameter transformation model. The results after fitting also show the best consistency for the LS method, with the standard deviation of differences reduced to ±1.1 cm. For comparison, the NKG96 geoid yields a 17-cm mean and ±8-cm standard deviation of agreement with the same SWEPOS stations. After four-parameter fitting to the GPS stations, the standard deviation reduces to ±6.1 cm for the NKG96 geoid. It is concluded that the new corrections in this study improve the accuracy of the geoid. The final geoid heights range from 17.22 to 43.62 m with a mean value of 29.01 m. The standard errors of the computed geoid heights, through a simple error propagation of standard errors of mean anomalies, are also computed. They range from ±7.02 to ±13.05 cm. The global root-mean-square error of the LS model is the other estimation of the accuracy of the final geoid, and is computed to be ±28.6 cm. Received: 15 September 1999 / Accepted: 6 November 2000  相似文献   

20.
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   

设为首页 | 免责声明 | 关于勤云 | 加入收藏

Copyright©北京勤云科技发展有限公司  京ICP备09084417号