高精度海岸带重力似大地水准面的若干问题讨论

本文针对海岸带多源重力数据和地形特点,通过理论分析和试算,对若干影响厘米级似大地水准面确定的关键问题进行了剖析,得出一些有益的结论。我国海岸带Molodensky一阶项对高程异常的贡献在10～30cm,需在Molodensky框架中精化重力似大地水准面;精细处理地形影响是提升多源重力场数据处理水平的重要途径;地球外空间不同高度、任意类型重力场参数的地形影响、地形补偿和地形Helmert凝聚算法可以统一;重力场数据处理中大地测量基准不一致的影响会随数据处理算法的不同而变化,在多源重力数据处理时此类影响易变得不可预测和控制;将地形Helmert凝聚理论引入Molodensky框架,可以解决以其他重力场参数（如扰动重力、垂线偏差等）为边界条件的似大地水准面精化问题。     

Solution of the geodetic boundary value problem

Summary The possibility of improving the convergence of Molodensky’s series is considered. Then new formulas are derived for the solution of the geodetic boundary value problem. One of them presents an expression for the boundary condition which involves a linear combination of Stokes’ constants and surface gravity anomalies. This differs from the usually used relation by the appearance of additional terms dependent on second order terns with respect to the elevations of the earth’s surface. The formulas are derived for Stokes’ constants and the anomalous potential in terms of surface anomalies. As compared to the Taylor’s series of Molodensky, they are presented in the form of surface harmonic series. Due regard is made to the earth’s oblateness, in addition to the terrain topography.     

Prediction of vertical deflections from high-degree spherical harmonic synthesis and residual terrain model data

This study demonstrates that in mountainous areas the use of residual terrain model (RTM) data significantly improves the accuracy of vertical deflections obtained from high-degree spherical harmonic synthesis. The new Earth gravitational model EGM2008 is used to compute vertical deflections up to a spherical harmonic degree of 2,160. RTM data can be constructed as difference between high-resolution Shuttle Radar Topography Mission (SRTM) elevation data and the terrain model DTM2006.0 (a spherical harmonic terrain model that complements EGM2008) providing the long-wavelength reference surface. Because these RTM elevations imply most of the gravity field signal beyond spherical harmonic degree of 2,160, they can be used to augment EGM2008 vertical deflection predictions in the very high spherical harmonic degrees. In two mountainous test areas—the German and the Swiss Alps—the combined use of EGM2008 and RTM data was successfully tested at 223 stations with high-precision astrogeodetic vertical deflections from recent zenith camera observations (accuracy of about 0.1 arc seconds) available. The comparison of EGM2008 vertical deflections with the ground-truth astrogeodetic observations shows root mean square (RMS) values (from differences) of 3.5 arc seconds for ξ and 3.2 arc seconds for η, respectively. Using a combination of EGM2008 and RTM data for the prediction of vertical deflections considerably reduces the RMS values to the level of 0.8 arc seconds for both vertical deflection components, which is a significant improvement of about 75%. Density anomalies of the real topography with respect to the residual model topography are one factor limiting the accuracy of the approach. The proposed technique for vertical deflection predictions is based on three publicly available data sets: (1) EGM2008, (2) DTM2006.0 and (3) SRTM elevation data. This allows replication of the approach for improving the accuracy of EGM2008 vertical deflection predictions in regions with a rough topography or for improved validation of EGM2008 and future high-degree spherical harmonic models by means of independent ground truth data.     

Terrain corrections to power 3 in gravimetric geoid determination

In precise geoid determination by Stokes formula, direct and primary and secondary indirect terrain effects are applied for removing and restoring the terrain masses. We use Helmert's second condensation method to derive the sum of these effects, together called the total terrain effect for geoid. We develop the total terrain effect to third power of elevation H in the original Stokes formula, Earth gravity model and modified Stokes formula. It is shown that the original Stokes formula, Earth gravity model and modified Stokes formula all theoretically experience different total terrain effects. Numerical results indicate that the total terrain effect is very significant for moderate topographies and mountainous regions. Absolute global mean values of 5–10 cm can be reached for harmonic expansions of the terrain to degree and order 360. In another experiment, we conclude that the most important part of the total terrain effect is the contribution from the second power of H, while the contribution from the third power term is within 9 cm. Received: 2 September 1996 / Accepted: 4 August 1997     

Far-zone effects for different topographic-compensation models based on a spherical harmonic expansion of the topography

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.     

Downward continuation of the free-air gravity anomalies to the ellipsoid using the gradient solution and terrain correction—An attempt of global numerical computations

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).     

Geoid, topography, and the Bouguer plate or shell

 Topography plays an important role in solving many geodetic and geophysical problems. In the evaluation of a topographical effect, a planar model, a spherical model or an even more sophisticated model can be used. In most applications, the planar model is considered appropriate: recall the evaluation of gravity reductions of the free-air, Poincaré–Prey or Bouguer kind. For some applications, such as the evaluation of topographical effects in gravimetric geoid computations, it is preferable or even necessary to use at least the spherical model of topography. In modelling the topographical effect, the bulk of the effect comes from the Bouguer plate, in the case of the planar model, or from the Bouguer shell, in the case of the spherical model. The difference between the effects of the Bouguer plate and the Bouguer shell is studied, while the effect of the rest of topography, the terrain, is discussed elsewhere. It is argued that the classical Bouguer plate gravity reduction should be considered as a mathematical construction with unclear physical meaning. It is shown that if the reduction is understood to be reducing observed gravity onto the geoid through the Bouguer plate/shell then both models give practically identical answers, as associated with Poincaré's and Prey's work. It is shown why only the spherical model should be used in the evaluation of topographical effects in the Stokes–Helmert solution of Stokes' boundary-value problem. The reason for this is that the Bouguer plate model does not allow for a physically acceptable condensation scheme for the topography. Received: 24 December 1999 / Accepted: 11 December 2000     

Solving Molodensky’s series by fast Fourier transform techniques

The use of the fast Fourier transform algorithm in the evaluation of the Molodensky series terms is demonstrated in this paper. The solution by analytical continuation to point level has been reformulated to obtain convolution integrals in planar approximation which can be efficiently evaluated in the frequency domain. Preliminary results show that the solution by Faye anomalies is not sufficient for highly accurate deflections of the vertical and height anomalies. The Molodensky solution up to at least the second-order term must be carried out. Part of the unrecovered deflection and height anomaly signal appears to be due to density variations, verifying the essential role of density modelling. A remove-restore technique for the terrain effects can improve the convergence of the series and minimize the interpolation errors. Paper presented at theI Hotine-Marussi Symposium on Mathematical Geodesy, Rome, June 3–6, 1985.     

On the error of analytical continuation in physical geodesy

Analytical continuation of gravity anomalies and height anomalies is compared with Helmert's second condensation method. Assuming that the density of the terrain is constant and known the latter method can be regarded as correct. All solutions are limited to the second power of H/R, where H is the orthometric height of the terrain and R is mean sea-level radius. We conclude that the prediction of free-air anomalies and height anomalies by analytical continuation with Poisson's formula and Stokes's formula goes without error. Applying the same technique for geoid determination yields an error of the order of H², stemming from the failure of analytical continuation inside the masses of the Earth.     

利用卫星测高数据反演海洋重力异常研究

全面研究了利用卫得测高数据反演海洋重力异常3种主要方法（即Stokes数据解析反解以及逆Vening-Meinesz公式）的技术特点，建立了3种算法的数学模型及其谱计算式，在以1440阶次位模型定义的标准场中完成了3种算法的数值比较和内部检核，通过仿真试验实现了3种算法的可靠性和稳定性检验，最后，本文利用卫得测高实测对南中国海地区的海洋重力异常进行了实际反演，并将反演结果同船测数据进行了比较。     

Inverse Vening Meinesz formula and deflection-geoid formula: applications to the predictions of gravity and geoid over the South China Sea

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

Determination of Potential coefficients to degree 52 from 5° Mean Gravity Anomalies

A set of 38406 1°×1° mean free air anomalies were used to derive a set of 1507 5° equal area anomalies that were supplemented by 147 predicted anomalies to form a global coverage of 1654 anomalies. These anomalies were used to derive potential coefficients to degree 52 using the summation formulae. In these computations, a smoothing operator was introduced and found to significantly effect the results at higher degrees. In addition, the effects of the atmosphere, spherical approximation and terrain were studied. It was found that the atmospheric effects and spherical approximation effects were about 0.3% of the actual coefficients. The terrain correction effects amounted to 10 to 25% of the low degree coefficients depending on a specific terrain correction model chosen; however, the correction terms found from the models did not yield solutions that agreed better with current satellite derived potential coefficient determinations. Anomalies were computed from the derived potential coefficients for comparison to the original anomalies. These comparisons showed that the agreement between the two anomalies became significantly better as the degree of expansion increased to the maximum considered. These comparisons shed some doubt on the rule of thumb that a block of size θ^° can be represented by a spherical harmonic expansion to 180^°/θ^°.     

Gravity gradient modeling using gravity and DEM

A model of the gravity gradient tensor at aircraft altitude is developed from the combination of ground gravity anomaly data and a digital elevation model. The gravity data are processed according to various operational solutions to the boundary-value problem (numerical integration of Stokes’ integral, radial-basis splines, and least-squares collocation). The terrain elevation data are used to reduce free-air anomalies to the geoid and to compute a corresponding indirect effect on the gradients at altitude. We compare the various modeled gradients to airborne gradiometric data and find differences of the order of 10–20 E (SD) for all gradient tensor elements. Our analysis of these differences leads to a conclusion that their source may be primarily measurement error in these particular gradient data. We have thus demonstrated the procedures and the utility of combining ground gravity and elevation data to validate airborne gradiometer systems.     

An evaluation of some systematic error sources affecting terrestrial gravity anomalies

Terrestrial free-air gravity anomalies form a most essential data source in the framework of gravity field determination. Gravity anomalies depend on the datums of the gravity, vertical, and horizontal networks as well as on the definition of a normal gravity field; thus gravity anomaly data are affected in a systematic way by inconsistencies of the local datums with respect to a global datum, by the use of a simplified free-air reduction procedure and of different kinds of height system. These systematic errors in free-air gravity anomaly data cause systematic effects in gravity field related quantities like e.g. absolute and relative geoidal heights or height anomalies calculated from gravity anomaly data. In detail it is shown that the effects of horizontal datum inconsistencies have been underestimated in the past. The corresponding systematic errors in gravity anomalies are maximum in mid-latitudes and can be as large as the errors induced by gravity and vertical datum and height system inconsistencies. As an example the situation in Australia is evaluated in more detail: The deviations between the national Australian horizontal datum and a global datum produce a systematic error in the free-air gravity anomalies of about −0.10 mgal which value is nearly constant over the continent     

DEM-induced errors in developing a quasi-geoid model for Africa

Errors in digital elevation models (DEMs) will introduce errors in geoid and quasi-geoid models, via their use in interpolating free-air gravity anomalies and (in the case of the quasi-geoid) their use in computing the Molodensky G ₁ term. The effects of these errors and those of datum shifts are assessed using three independent DEMs for a test region in South Africa. It is shown that these effects are significant and that it is important to choose the best-possible DEM for use in geoid and quasi-geoid modelling. Acknowledgments.The land gravity data used for this research were provided by the South African Council for Geoscience. Marine gravity anomalies were provided by the Danish National Survey and Cadastre (Kort & Matrikelstyrelsen). The GLOBE DEM was provided by the US National Geophysical Data Centre, and the CDSM DEM was provided by the South African Chief Directorate for Surveying and Mapping. The constructive comments of the reviewers are gratefully acknowledged.     

Improvements on existing geopotential coefficients models for precise determination of 1° global geoid and mean gravity anomalies

The concept of an idealised earth having 1° averaged heights over its land surface is introduced as a means to improve upon the existing geopotential coefficient solutions without the use of additional observed data, in order to provide more precise knowledge of the earth’s gravity field in the form of 1° global geoid and 1° mean free-air gravity anomalies especially over the mountainous regions with the visible topography condensed into the actual geoid, first by referring them to the idealised earth and then by reducing the same to the actual earth on applying appropriate corrections for the differences between the two earths.     

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.     

The effect of topography on the determination of the geoid using analytical downward continuation

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.     

A new gravity map, a new marine geoid around Japan and the detection of the Kuroshio current

About half a million marine gravity measurements over a 30^∘×30^∘ area centered on Japan have been processed and adjusted to produce a new free-air gravity map from a 5′×5′ grid. This map seems to have a better resolution than those previously published as measured by its correlation with bathymetry. The grid was used together with a high-degree and -order spherical harmonics geopotential model to compute a detailed geoid with two methods: Stokes integral and collocation. Comparisons with other available geoidal surfaces derived either from gravity or from satellite altimetry were made especially to test the ability of this new geoid at showing the sea surface topography as mapped by the Topex/Poseidon satellite. Over 2 months (6 cycles) the dynamic topography at ascending passes in the region (23^∘47^∘N and 123^∘147^∘E) was mapped to study the variability of the Kuroshio current. Received: 15 July 1994 / Accepted: 17 February 1997