Precise geoid determination over Sweden using the Stokes–Helmert method and improved topographic corrections

 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     

A new strategy for the atmospheric gravity effect in gravimetric geoid determination

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.     

On the ellipsoidal correction to the spherical Stokes solution of the gravimetric geoid

The solutions of four ellipsoidal approximations for the gravimetric geoid are reviewed: those of Molodenskii et al., Moritz, Martinec and Grafarend, and Fei and Sideris. The numerical results from synthetic tests indicate that Martinec and Grafarends solution is the most accurate, while the other three solutions contain an approximation error which is characterized by the first-degree surface spherical harmonic. Furthermore, the first 20 degrees of the geopotential harmonic series contribute approximately 90% of the ellipsoidal correction. The determination of a geoid model from the generalized Stokes scheme can accurately account for the ellipsoidal effect to overcome the first-degree surface spherical harmonic error regardless of the solution used.     

A formula for gravimetric terrain corrections using powers of topographic height

Journal of Geodesy - ?The application of Stokes' formula to create geoid undulations requires no masses outside the geoid. However, due to the existence of the topography, terrain...     

On the accuracy of modified Stokes's integration in high-frequency gravimetric geoid determination

 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     

Unification of vertical datums by GPS and gravimetric geoid models with application to Fennoscandia

The second Baltic Sea Level (BSL) GPS campaign was run for one week in June 1993. Data from 35 tide gauge sites and five fiducial stations were analysed, for three fiducial stations (Onsala, Mets?hovi and Wettzell) fixed at the ITRF93 system. On a time-scale of 5 days, precision was several parts in 10⁹ for the horizontal and vertical components. Accuracies were about 1 cm in comparison with the International GPS Geodynamical Service (IGS) coordinates in three directions. To connect the Swedish and the Finnish height systems, our numerical application utilises three approaches: a rigorous approach, a bias fit and a three-parameter fit. The results between the Swedish RH70 and the Finnish N 60 systems are estimated to −19.3 ± 6.5, −17 ± 6 and −15 ± 6 cm, respectively, by the three approaches. The results of the three indirect methods are in an agreement with those of a direct approach from levelling and gravity measurements. Received: 3 April 1996 / Accepted: 4 August 1997     

The ellipsoidal correction to the Stokes kernel for precise geoid determination

Solving the geodetic boundary-value problem (GBVP) for the precise determination of the geoid requires proper use of the fundamental equation of physical geodesy as the boundary condition given on the geoid. The Stokes formula and kernel are the result of spherical approximation of this fundamental equation, which is a violation of the proper relation between the observed quantity (gravity anomaly) and the sought function (geoid). The violation is interpreted here as the improper formulation of the boundary condition, which implies the spherical Stokes kernel to be in error compared with the proper kernel of integral transformation. To remedy this error, two correction kernels to the Stokes kernel were derived: the first in both closed and spectral forms and the second only in spectral form. Contributions from the first correction kernel to the geoid across the globe were [−0.867 m, +1.002 m] in the low-frequency domain implied by the GRIM4-S4 purely satellite-derived geopotential model. It is a few centimeters, on average, in the high-frequency domain with some exceptions of a few meters in places of high topographical relief and sizable geological features in accordance with the EGM96 combined geopotential model. The contributions from the second correction kernel to the geoid are [−0.259 m, +0.217 m] and [−0.024 m, +0.023 m] in the low- and high-frequency domains, respectively.     

Improved convergence rates for the truncation error in gravimetric geoid determination

 When Stokes's integral is used over a spherical cap to compute a gravimetric estimate of the geoid, a truncation error results due to the neglect of gravity data over the remainder of the Earth. Associated with the truncation error is an error kernel defined over these two complementary regions. An important observation is that the rate of decay of the coefficients of the series expansion for the truncation error in terms of Legendre polynomials is determined by the smoothness properties of the error kernel. Previously published deterministic modifications of Stokes's integration kernel involve either a discontinuity in the error kernel or its first derivative at the spherical cap radius. These kernels are generalised and extended by constructing error kernels whose derivatives at the spherical cap radius are continuous up to an arbitrary order. This construction is achieved by smoothly continuing the error kernel function into the spherical cap using a suitable degree polynomial. Accordingly, an improved rate of convergence of the spectral series representation of the truncation error is obtained. Received: 21 April 1998 / Accepted: 4 October 1999     

Orthometric corrections from leveling, gravity, density and elevation data: a case study in Taiwan

A new orthometric correction (OC) formula is presented and tested with various mean gravity reduction methods using leveling, gravity, elevation, and density data. For mean gravity computations, the Helmert method, a modified Helmert method with variable density and gravity anomaly gradient, and a modified Mader method were used. An improved method of terrain correction computation based on Gaussian quadrature is used in the modified Mader method. These methods produce different results and yield OCs that are greater than 10 cm between adjacent benchmarks (separated by 2 km) at elevations over 3000 m. Applying OC reduces misclosures at closed leveling circuits and improves the results of leveling network adjustments. Variable density yields variation of OC at millimeter level everywhere, while gravity anomaly gradient introduces variation of OC of greater than 10 cm at higher elevations, suggesting that these quantities must be considered in OC. The modified Mader method is recommended for computing OC.Acknowledgments.This study is supported by the Ministry of the Interior (MOI), Taiwan, under the project `Measuring gravity on first-order benchmarks'. The authors are grateful to F.S. Ning and his colleagues at BSB (Base Survey Battalion) for their precision work in collecting gravity data, and to R. Forsberg for the terrain correction program. They also thank the Institute of Agricultural and Forestry Aerial Survey for elevation data and MOI for leveling data. Dr. Will Featherstone and three anonymous reviewers are thanked for their constructive comments.     

Topographic and atmospheric corrections of gravimetric geoid determination with special emphasis on the effects of harmonics of degrees zero and one

 The topographic and atmospheric effects of gravimetric geoid determination by the modified Stokes formula, which combines terrestrial gravity and a global geopotential model, are presented. Special emphasis is given to the zero- and first-degree effects. The normal potential is defined in the traditional way, such that the disturbing potential in the exterior of the masses contains no zero- and first-degree harmonics. In contrast, it is shown that, as a result of the topographic masses, the gravimetric geoid includes such harmonics of the order of several centimetres. In addition, the atmosphere contributes with a zero-degree harmonic of magnitude within 1 cm. Received: 5 November 1999 / Accepted: 22 January 2001     

Ellipsoidal geoid computation

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.     

Applications of downward-continuation in gravimetric geoid modeling: case studies in Western Canada

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.     

How to handle topography in practical geoid determination: three examples

 Three different methods of handling topography in geoid determination were investigated. The first two methods employ the residual terrain model (RTM) remove–restore technique, yielding the quasigeoid, whereas the third method uses the classical Helmert condensation method, yielding the geoid. All three methods were used with the geopotential model Earth Gravity Model (1996) (EGM96) as a reference, and the results were compared to precise global positioning system (GPS) levelling networks in Scandinavia. An investigation of the Helmert method, focusing on the different types of indirect effects and their effects on the geoid, was also carried out. The three different methods used produce almost identical results at the 5-cm level, when compared to the GPS levelling networks. However, small systematic differences existed. Received: 18 March 1999 / Accepted: 21 March 2000     

A method to validate gravimetric-geoid computation software based on Stokes's integral formula

A method is presented with which to verify that the computer software used to compute a gravimetric geoid is capable of producing the correct results, assuming accurate input data. The Stokes, gravimetric terrain correction and indirect effect formulae are integrated analytically after applying a transformation to surface spherical coordinates centred on each computation point. These analytical results can be compared with those from geoid computation software using constant gravity data in order to verify its integrity. Results of tests conducted with geoid computation software are presented which illustrate the need for integration weighting factors, especially for those compartments close to the computation point. Received: 6 February 1996 / Accepted: 19 April 1997     

Local geoid determination from airborne vector gravimetry

Methods are illustrated to compute the local geoid using the vertical and horizontal components of the gravity disturbance vector derived from an airborne GPS/inertial navigation system. The data were collected by the University of Calgary in a test area of the Canadian Rocky Mountains and consist of multiple parallel tracks and two crossing tracks of accelerometer and gyro measurements, as well as precise GPS positions. Both the boundary-value problem approach (Hotines integral) and the profiling approach (line integral) were applied to compute the disturbing potential at flight altitude. Cross-over adjustments with minimal control were investigated and utilized to remove error biases and trends in the estimated gravity disturbance components. Final estimation of the geoid from the vertical gravity disturbance included downward continuation of the disturbing potential with correction for intervening terrain masses. A comparison of geoid estimates to the Canadian Geoid 2000 (CGG2000) yielded an average standard deviation per track of 14 cm if they were derived from the vertical gravity disturbance (minimally controlled with a cross-over adjustment), and 10 cm if derived from the horizontal components (minimally controlled in part with a simulated cross-over adjustment). Downward continuation improved the estimates slightly by decreasing the average standard deviation by about 0.5 cm. The application of a wave correlation filter to both types of geoid estimates yielded significant improvement by decreasing the average standard deviation per track to 7.6 cm.     

局部地形校正的精密算法

对局部地形校正中的中央区奇异积分部分进行非奇变换,使用Simpson公式和Cotes公式推导出两种中央区的数值计算方法。试算结果表明,新方法可有效地提高地形校正的精度。     

Review and numerical assessment of the direct topographical reduction in geoid determination

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.     

Airborne gravimetry used in precise geoid computations by ring integration

Two detailed geoids have been computed in the region of North Jutland. The first computation used marine data in the offshore areas. For the second computation the marine data set was replaced by the sparser airborne gravity data resulting from the AGMASCO campaign of September 1996. The results of comparisons of the geoid heights at on-shore geometric control showed that the geoid heights computed from the airborne gravity data matched in precision those computed using the marine data, supporting the view that airborne techniques have enormous potential for mapping those unsurveyed areas between the land-based data and the off-shore marine or altimetrically derived data. Received: 7 July 1997 / Accepted: 22 April 1998     

视景地形仿真及其应用

本文主要讨论了视景地形仿真的特点、数据基础及其实现的整个过程，列举了视景地形仿真在军事上和其它领域的应用实例。实验表明，本文所介绍的视景地形仿真过程简单而容易实现，仿真的地形景观具有良好的视觉效果。