Accuracy of gravimetric deflections of the vertical and optimization of gravity networks

In order to solve the problems of determining the shape of a part of the earth of national or continental extent, that is, of rigorous constituting and computing of the astrogeodetic network, it is required to determine gravimetric deflections of the vertical with an accuracy of, say, 0″.3. For this it is adequate to carry out additional gravity surveys in the neighborhoods of computation points, in addition to a given uniform gravity survey (normal density gravity survey). The study offers a method to determine the optimal distribution of gravity stations in such a gravity survey, which guarantees a given accuracy of computed gravimetric deflections of the vertical for a given statistical condition which characterizes the variation of the gravity field. The approach used here is based on the concept of the error of representation and the error propagation of Vening Meinesz integrals.     

Vening Meinesz inverse isostatic problem with local and global Bouguer anomalies

Inverse problems in isostasy will consist in making the isostatic anomalies to be zero under a certain isostatic hypothesis. In the case of the Vening Meinesz isostatic hypothesis, the density contrast is constant, while the Moho depth (depth of the Mohorovičić discontinuity) is variable. Hence, the Vening Meinesz inverse isostatic problem aims to determine a suitable variable Moho depth for a prescribed constant density contrast. The main idea is easy but the theoretical analysis is somewhat difficult. Moreover, the practical determination of the variable Moho depths based on the Vening Meinesz inverse problem is a laborious and time-consuming task. The formulas used for computing the inverse Vening Meinesz Moho depths are derived. The computational tricks essentially needed for computing the inverse Vening Meinesz Moho depths from a set of local and global Bouguer anomalies are described. The Moho depths for a test area are computed based on the inverse Vening Meinesz isostatic problem. These Moho depths fit the Moho depths derived from seismic observations with a good accuracy, in which the parameters used for the fitting agree well with those determined geophysically. Received: 4 February 1999 / Accepted: 4 October 1999     

根据多卫星高度计海面高数据推算南中国海及菲律宾海域重力异常

t Gravity anomalies on a2.5 ×2.5 arc-minute grid in a non-tidal system were derived over the South China and Philippine Seas from multi-satellite altimetry data. North and east components of deflections of the vertical were computed from altimeter-derived sea surface heights at crossover locations, and gridded onto a 2.5 × 2.5 arc-minute resolution grid. EGM96-derived components of deflections of the vertical and gravity anomalies gridded into 2.5 × 2.5 arc-minute resolutions were then used as reference global geopotential model quantities in a remove-restore procedure to implement the Inverse Vening Meinesz formula via the 1D-FFT technique to predict the gravity anomalies over the South China and Philippine Seas from the gridded altimeter-derived components of deflections of the vertical. Statistical comparisons between the altimeter-derived and the shipboard gravity anomalies showed that there is a root-mean-square agreement of 5.7 mgals between them.     

Determination of deflections of the vertical using a combination of spherical harmonics and gravimetric data for the area of Greece

The determination of gravimetric deflections of the vertical for the area of Greece is attempted by combining a spherical hamonics model and gravity nomalies using the method of least squares collocation. The components of deflections of the vertical are estimated on a grid with spacing 15′ in latitude and 20′ in longitude covering only the continental area of Greece, where a sufficient number of point gravity anomalies is available. In order to test the accuracy of the determination, gravimetric deflections of the vertical are computed at stations where astrogeodetic data are available. The results show that in a large region of rugged topography and irregular potential field, the prediction is possible with a standard deviation of 18% ... 28% of the root mean square variation of the observations, without taking into account the topography. Furthermore, the estimation of some systematic differences between observed and computed deflections of the vertical is attempted.     

Geocentric indian geodetic datum by astrogeodetic — gravimetric method using smoothened geoidal heights

The astrogeodetic—gravimetric method based on the principle of least—squares solution has been used to determine the geocentric Indian geodetic datum making use of the available nongeocentric astrogeodetic data and the gravimetric geocentric geoidal heights in the form of smoothened values. Everett's method of interpolation has been used to obtain the smoothened geoidal heights at the astrogeodetic stations in India from the available generalized values at 1°×1° corners. The values of the geoidal height and deflections of the vertical at the geodetic datum Kalianpur H.S. so obtained have the negligible difference from the values computed earlier by the same method using directly computed gravimetric geoidal heights at the astrogeodetic stations, indicating that the use of the interpolated values in the astrogeodetic—gravimetric method employed would be an economical approach of absolute orientation of a nongeocentric system if the gravimetric geoidal heights are available at 1°×1° corners in the area of interest.     

Fitting gravimetric geoid models to vertical deflections

Regional gravimetric geoid and quasigeoid models are now commonly fitted to GPS-levelling data, which simultaneously absorbs levelling, GPS and quasi/geoid errors due to their inseparability. We propose that independent vertical deflections are used instead, which are not affected by this inseparability problem. The formulation is set out for geoid slopes and changes in slopes. Application to 1,080 astrogeodetic deflections over Australia for the AUSGeoid98 model shows that it is feasible, but the poor quality of the historical astrogeodetic deflections led to some unrealistic values.     

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     

Quasigeoid and deflection determinations by astrogravimetric methods

The main objective of the present work is to present methods to obtain detailed surveys of the shape of the quasigeoid and of deflections of the vertical from the point of view of three-dimensional constituting and rigorous computing of the astrogeodetic network. The error of an astrogravimetric leveling line in the most general case, i.e., in the shape of a polygon has been estimated. This error can be tested and checked by comparison of gravimetric deflections of the vertical with astrogeodetic deflections, i.e., by computation of the error of astrogeodetic gravimetric deflection of the vertical. The astrogeodetic deflections of the vertical required for the horizontal angle correction in triangulation and traverse are easily obtained by interpolation. An example of astrogravimetric leveling demonstrates the possibility to carry out an astrogravimetric leveling with any required accuracy, for example, with the accuracy of ±1 ml/1000 km. In connection with height determination from PGS a procedure of constituting a well-distributed set of fiducial ground stations by using high-precision astrogravimetric methods together with millimeter-level accuracy astrogravimetric leveling to test various space systems observations has been suggested.     

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。     

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.     

Truncation error formulas for the geoidal height and the deflection of the vertical

A general formula giving Molodenskii coefficientsQ _n of the truncation errors for the geoidal height is introduced in this paper. A relation betweenQ _n andq _n, Cook’s truncation function, is also obtained. Cook (1951) has treated the truncation errors for the deflection of the vertical in the Vening Meinesz integration. Molodenskii et al. (1962) have also derived the truncation error formulas for the deflection of the vertical. It is proved in this paper that these two formulas are equivalent.     

Evaluation of deflections of the vertical using topographic-isostatic and astrogeodetic data for the area of Greece

The evaluation of deflections of the vertical for the area of Greece is attempted using a combination of topographic and astrogeodetic data. Tests carried out in the area bounded by 35°≤ϕ≤42°, 19°≤λ≤27° indicate that an accuracy of ±3″.3 can be obtained in this area for the meridian and prime vertical deflection components when high resolution topographic data in the immediate vicinity of computation points are used, combined with high degree spherical harmonic expansions of the geopotential and isostatic reduction potential. This accuracy is about 25% better than the corresponding topographic-Moho deflection components which are evaluated using topographic and Moho data up to 120 km around each station, without any combination with the spherical harmonic expansion of the geopotential or isostatic reduction potential. The accuracy in both cases is increased to about 2″.6 when the astrogeodetic data available in the area mentioned above are used for the prediction of remaining values. Furthermore the estimation of datum-shift parameters is attempted using least squares collocation.     

Investigations on the accuracy concerning the derivation of absolute gravimetric height anomalies and deflections of the vertical based on the theory of Molodensky

The investigations refer to the compartment method by using mean terrestrial free air anomalies only. Three main error influences of remote areas (distance from the fixed point >9°) on height anomalies and deflections of the vertical are being regarded:

1. The prediction errors of mean terrestrial free air anomalies have the greatest influence and amount to about ±0″.2 in each component for deflections of the vertical and to ±3 m for height anomalies;
2. The error of the compartment method, which originates from converting the integral formulas of Stokes and Vening-Meinesz into summation formulas, can be neglected if the anomalies for points and gravity profiles are compiled to 5°×5° mean values.
3. The influences of the mean gravimetric correction terms of Arnold—estimated for important mountains of the Earth by means of an approximate formula—on height anomalies may amount to 1–2 m and on deflections of the vertical to 0″0.5–0″.1, and, therefore, they have to be taken into account for exact calculations.

The computations of errors are carried out using a global covariance function of point free air anomalies.     

The AUSGeoid09 model of the Australian Height Datum

AUSGeoid09 is the new Australia-wide gravimetric quasigeoid model that has been a posteriori fitted to the Australian Height Datum (AHD) so as to provide a product that is practically useful for the more direct determination of AHD heights from Global Navigation Satellite Systems (GNSS). This approach is necessary because the AHD is predominantly a third-order vertical datum that contains a ~1 m north-south tilt and ~0.5 m regional distortions with respect to the quasigeoid, meaning that GNSS-gravimetric-quasigeoid and AHD heights are inconsistent. Because the AHD remains the official vertical datum in Australia, it is necessary to provide GNSS users with effective means of recovering AHD heights. The gravimetric component of the quasigeoid model was computed using a hybrid of the remove-compute-restore technique with a degree-40 deterministically modified kernel over a one-degree spherical cap, which is superior to the remove-compute-restore technique alone in Australia (with or without a cap). This is because the modified kernel and cap combine to filter long-wavelength errors from the terrestrial gravity anomalies. The zero-tide EGM2008 global gravitational model to degree 2,190 was used as the reference field. Other input data are ~1.4 million land gravity anomalies from Geoscience Australia, 1′ × 1′ DNSC2008GRA altimeter-derived gravity anomalies offshore, the 9′′ × 9′′ GEODATA-DEM9S Australian digital elevation model, and a readjustment of Australian National Levelling Network (ANLN) constrained to the CARS2006 mean dynamic ocean topography model. To determine the numerical integration parameters for the modified kernel, the gravimetric component of AUSGeoid09 was compared with 911 GNSS-observed ellipsoidal heights at benchmarks. The standard deviation of fit to the GNSS-AHD heights is ±222 mm, which dropped to ±134 mm for the readjusted GNSS-ANLN heights showing that careful consideration now needs to be given to the quality of the levelling data used to assess gravimetric quasigeoid models. The publicly released version of AUSGeoid09 also includes a geometric component that models the difference between the gravimetric quasigeoid and the zero surface of the AHD at 6,794 benchmarks. This a posteriori fitting used least-squares collocation (LSC) in cross-validation mode to determine a correlation length of 75 km for the analytical covariance function, whereas the noise was taken from the estimated standard deviation of the GNSS ellipsoidal heights. After this LSC surface fitting, the standard deviation of fit reduced to ±30 mm, one-third of which is attributable to the uncertainty in the GNSS ellipsoidal heights.     

Local characteristics of the gravity anomaly covariance function

Summary Using a data set of 260 000 gravity anomalies it is shown that common characteristics for a local covariance function exist in an area as large as Canada excluding the Rocky Mountains. After eliminating global features by referencing the data to the GEM-10 satellite solution, the shape of the covariance function is remarkably consistent from one sample area to the next. The determination of the essential parameters and the fitting of the covariance function are discussed in detail. To test the reliability of the derived function, deflections of the vertical are estimated at about 230 stations where astrogeodetic data are available. Results show that the standard error obtained from the discrepancies is about1″ for each component and that the error covariance matrix of least-squares collocation reflects this accuracy remarkably well.     

Astronomical-topographic levelling using high-precision astrogeodetic vertical deflections and digital terrain model data

At the beginning of the twenty-first century, a technological change took place in geodetic astronomy by the development of Digital Zenith Camera Systems (DZCS). Such instruments provide vertical deflection data at an angular accuracy level of 0.̋1 and better. Recently, DZCS have been employed for the collection of dense sets of astrogeodetic vertical deflection data in several test areas in Germany with high-resolution digital terrain model (DTM) data (10–50 m resolution) available. These considerable advancements motivate a new analysis of the method of astronomical-topographic levelling, which uses DTM data for the interpolation between the astrogeodetic stations. We present and analyse a least-squares collocation technique that uses DTM data for the accurate interpolation of vertical deflection data. The combination of both data sets allows a precise determination of the gravity field along profiles, even in regions with a rugged topography. The accuracy of the method is studied with particular attention on the density of astrogeodetic stations. The error propagation rule of astronomical levelling is empirically derived. It accounts for the signal omission that increases with the station spacing. In a test area located in the German Alps, the method was successfully applied to the determination of a quasigeoid profile of 23 km length. For a station spacing from a few 100 m to about 2 km, the accuracy of the quasigeoid was found to be about 1–2 mm, which corresponds to a relative accuracy of about 0.05−0.1 ppm. Application examples are given, such as the local and regional validation of gravity field models computed from gravimetric data and the economic gravity field determination in geodetically less covered regions.     

Global mean sea surface and marine gravity anomaly from multi-satellite altimetry: applications of deflection-geoid and inverse Vening Meinesz formulae

 Global mean sea surface heights (SSHs) and gravity anomalies on a 2^′×2^′ grid were determined from Seasat, Geosat (Exact Repeat Mission and Geodetic Mission), ERS-1 (1.5-year mean of 35-day, and GM), TOPEX/POSEIDON (T/P) (5.6-year mean) and ERS-2 (2-year mean) altimeter data over the region 0^∘–360^∘ longitude and –80^∘–80^∘ latitude. To reduce ocean variabilities and data noises, SSHs from non-repeat missions were filtered by Gaussian filters of various wavelengths. A Levitus oceanic dynamic topography was subtracted from the altimeter-derived SSHs, and the resulting heights were used to compute along-track deflection of the vertical (DOV). Geoidal heights and gravity anomalies were then computed from DOV using the deflection-geoid and inverse Vening Meinesz formulae. The Levitus oceanic dynamic topography was added back to the geoidal heights to obtain a preliminary sea surface grid. The difference between the T/P mean sea surface and the preliminary sea surface was computed on a grid by a minimum curvature method and then was added to the preliminary grid. The comparison of the NCTU01 mean sea surface height (MSSH) with the T/P and the ERS-1 MSSH result in overall root-mean-square (RMS) differences of 5.0 and 3.1 cm in SSH, respectively, and 7.1 and 3.2 μrad in SSH gradient, respectively. The RMS differences between the predicted and shipborne gravity anomalies range from 3.0 to 13.4 mGal in 12 areas of the world's oceans. Received: 26 September 2001 / Accepted: 3 April 2002 Correspondence to: C. Hwang Acknowledgements. This research is partly supported by the National Science Council of ROC, under grants NSC89-2611-M-009-003-OP2 and NSC89-2211-E-009-095. This is a contribution to the IAG Special Study Group 3.186. The Geosat and ERS1/2 data are from NOAA and CERSAT/France, respectively. The T/P data were provided by AVISO. The CLS and GSFC00 MSS models were kindly provided by NASA/GSFC and CLS, respectively. Drs. Levitus, Monterey, and Boyer are thanked for providing the SST model. Dr. T. Gruber and two anonymous reviewers provided very detailed reviews that improved the quality of this paper.     

Thin-plate spline quadrature of geodetic integrals

Thin-plate splines — well known for their flexibility and fidelity in representing experimental data — are especially suited for the numerical evaluation of geodetic integrals in the area where these are most sensitive to the data, i.e. in the immediate vicinity of the computation point. Quadrature rules that are exact for thin-plate splines interpolating randomly spaced data are derived for the inner zone contribution (to a planar approximation) to Stokes's formula, to the formulae of Vening Meinesz and to theL ₁ gradient operator in the analytical continuation solution of Molodensky's problem.The quadrature method is demonstrated by calculating the inner zone contribution to height anomalies in a mountainous area of Lesotho and carrying out a comparison with GPS-derived heights. Height anomalies are recovered with an accuracy of 6 cm.     

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     

关于地壳均衡模型的讨论

本文基于３ 个经典地壳均衡模型——普拉特模型、爱黎模型及维宁·迈尼兹模型在其表达的物理含义和方便实际应用等方面所各自具有的特点,提出改化经典模型的方法,给出一个既能反映地壳均衡的区域性补偿,又易于实现的区域性实用模型,并给出平均高的计算方法。最后还给出便于应用的两个近似模型。