Geoid undulation computations at laser tracking stations

This paper studies the use of two new methods for gravimetric geoid undulation computations: The Molodenskii's and Sjöberg's methods that both modify the original Stokes'function so that certainrms errors are minimized. These new methods were checked against the traditional methods of Stokes' and Meissl's modification with the criterion of the globalrms undulation error that each method implies. Sjöberg's method gave consistently the smallest globalrms undulation error of all the other methods for capsizes 0° to 10°. However with the exception of Stokes' method, for capsizes between 0° to 5°, all the methods gave approximately (within±5cm) the same globalrms undulation error. Actual gravity data within a cap of 2° and potential coefficient information were then combined to compute the undulation of 39 laser stations distributed around the world. Therms discrepancy between the gravimetric undulations using all the four methods and the undulations computed as the ellipsoidal minus the orthometric height of 28 at the above stations was±1.70,±1.65,±1.66,±1.65m for the Stokes', Meissl's, Molodenskii's and Sjöberg's method respectively. For five oceanic laser stations where no terrestrial gravity data was available, theGEOS-3/SEASAT altimeter sea surface heights were used to compute the undulations of these stations in a collocation method. Therms discrepancy between the altimeter derived undulation and the ellipsoidal mirus orthometric value of the undulation was ±1.30m for the above five laser stations.     

Undulation and anomaly estimation using Geos-3 altimeter data without precise satellite orbits

The paper describes results obtained from the processing of 53 Geos-3 arcs of altimeter data obtained during the first weeks after the launch of the satellite in April, 1975. The measurement from the satellite to the ocean surface was used to obtain an approximate geoid undulation which was contaminated by long wavelength errors caused primarily by altimeter bias and orbit error. This long wavelength error was reduced by fitting with a low degree polynomial the raw undulation data to the undulations implied by the GEM 7 potential coefficients, in an adjustment process that included conditions on tracks that cross. The root mean square crossover discrepancy before this adjustment was ±12.4 meters while after the adjustment it was ±0.9 m. These adjusted undulations were used to construct a geoid map in the Geos-3 calibration area using a least squares filter to remove remaining noise in the undulations. Comparing these undulations to ones computed from potential coefficients and terrestrial gravity data indicates a mean difference of 0.25 m and a root mean square difference of ±1.92 m. The adjusted undulations were also used to estimate several 5^o, 2^o, and 1^o anomalies using the method of least squares collocation. The resulting predictions agreed well with known values although the 1^o x 1^o anomalies could not be considered as reliably determined.     

Minimization and estimation of geoid undulation errors

The objective of this paper is to minimize the geoid undulation errors by focusing on the contribution of the global geopotential model and regional gravity anomalies, and to estimate the accuracy of the predicted gravimetric geoid.The geopotential model's contribution is improved by (a) tailoring it using the regional gravity anomalies and (b) introducing a weighting function to the geopotential coefficients. The tailoring and the weighting function reduced the difference (1) between the geopotential model and the GPS/levelling-derived geoid undulations in British Columbia by about 55% and more than 10%, respectively.Geoid undulations computed in an area of 40° by 120° by Stokes' integral with different kernel functions are analyzed. The use of the approximated kernels results in about 25 cm () and 190 cm (maximum) geoid errors. As compared with the geoid derived by GPS/levelling, the gravimetric geoid gives relative differences of about 0.3 to 1.4 ppm in flat areas, and 1 to 2.5 ppm in mountainous areas for distances of 30 to 200 km, while the absolute difference (1) is about 5 cm and 20 cm, respectively.A optimal Wiener filter is introduced for filtering of the gravity anomaly noise, and the performance is investigated by numerical examples. The internal accuracy of the gravimetric geoid is studied by propagating the errors of the gravity anomalies and the geopotential coefficients into the geoid undulations. Numerical computations indicate that the propagated geoid errors can reasonably reflect the differences between the gravimetric and GPS/levelling-derived geoid undulations in flat areas, such as Alberta, and is over optimistic in the Rocky Mountains of British Columbia.Paper presented at the IAG General Meeting, Beijing, China, August 8–13, 1993.     

Detailed gravimetric geoid for the North Atlantic

A detailed gravimetric geoid in the North Atlantic Ocean, named DGGNA-77, has been computed, based on a satellite and gravimetry derived earth potential model (consisting in spherical harmonic coefficients up to degree and order 30) and mean free air surface gravity anomalies (35180 1°×1° mean values and 245000 4′×4′ mean values). The long wavelength undulations were computed from the spherical harmonics of the reference potential model and the details were obtained by integrating the residual gravity anomalies through the Stokes formula: from 0 to 5° with the 4′×4′ data, and from 5° to 20° with the 1°×1° data. For computer time reasons the final grid was computed with half a degree spacing only. This grid extends from the Gulf of Mexico to the European and African coasts. Comparisons have been made with Geos 3 altimetry derived geoid heights and with the 5′×5′ gravimetric geoid derived byMarsh andChang [8] in the northwestern part of the Atlantic Ocean, which show a good agreement in most places apart from some tilts which porbably come from the satellite orbit recovery.     

利用低阶大地水准面异常反演大尺度核幔边界起伏

核幔边界（core-mantle boundary,CMB）是地球内部最重要的物理化学界面之一,地核和地幔通过核幔边界发生多种相互作用,这对地球重力场、地球自转及地磁场等都能产生重要影响。大地水准面异常是地球重力场的重要观测量,反映了地球内部的物质密度异常及界面变化等重要信息。推导了通过大地水准面异常反演核幔边界起伏的公式,利用2~4阶大地水准面异常反演了大尺度核幔边界起伏形态。结果显示,核幔边界起伏的径向幅度达±5 km、与Morelli的地震层析成像结果的幅度接近,但在形态上略有差异。以高为5 km、底边长为1 000 km的棱柱体模型模拟计算了核幔边界密度异常引起的大地水准面异常响应,结果与观测大地水准面异常比较接近。     

A comparison of altimeter and gravimetric geoids in the Tonga Trench and Indian Ocean areas

A gravimetric geoid computed using different techniques has been compared to a geoid derived from Geos-3 altimeter data in two 30°×30° areas: one in the Tonga Trench area and one in the Indian Ocean. The specific techniques used were the usual Stokes integration (using 1°×1° mean anomalies) with the Molodenskii truncation procedure; a modified Stokes integration with a modified truncation method; and computations using three sets of potential coefficients including one complete to degree 180. In the Tonga Trench area the standard deviation of the difference between the modified Stokes’ procedure and the altimeter geoid was ±1.1 m while in the Indian Ocean area the difference was ±0.6 m. Similar results were found from the 180×180 potential coefficient field. However, the differences in using the usual Stokes integration procedure were about a factor of two greater as was predicted from an error analysis. We conclude that there is good agreement at the ±1 m level between the two types of geoids. In addition, systematic differences are at the half-meter level. The modified Stokes procedure clearly is superior to the usual Stokes method although the 180×180 solution is of comparable accuracy with the computational effort six times less than the integration procedures.     

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     

Regional geopotential model improvement for the Iranian geoid determination

Spherical harmonic expansions of the geopotential are frequently used for modelling the earth’s gravity field. Degree and order of recently available models go up to 360, corresponding to a resolution of about50 km. Thus, the high degree potential coefficients can be verified nowadays even by locally distributed sets of terrestrial gravity anomalies. These verifications are important when combining the short wavelength model impact, e.g. for regional geoid determinations by means of collocation solutions. A method based on integral formulae is presented, enabling the improvement of geopotential models with respect to non-global distributed gravity anomalies. To illustrate the foregoing, geoid computations are carried out for the area of Iran, introducing theGPM2 geopotential model in combination with available regional gravity data. The accuracy of the geoid determination is estimated from a comparison with Doppler and levelling data to ±1.4m.     

Improvement of the geoid in local areas by satellite gradiometry

Summary Satellite gradiometry is studied as a means to improve the geoid in local areas from a limited data coverage. Least-squares collocation is used for this purpose because it allows to combine heterogeneous data in a consistent way and to estimate the integrated effect of the attenuated spectrum. In this way accuracy studies can be performed in a general and reliable manner. It is shown that only three second-order gradients contribute significantly to the estimation of the geoidal undulations and that it is sufficient to have gradiometer data in a 5°×5° area around the estimation point. The accuracy of the geoid determination is strongly dependent on the degree and order of the reference field used. An accuracy of about ±1 m can be achieved with a reference field of (12, 12). There is an optimal satellite altitude for each reference field and this altitude may be higher than 300 km for a field of low degree and order. The influence of measuring errors is discussed and it is shown that only gradiometer data with accuracies better than ±0.05 E will give a significant improvement of the geoid. Finally, some results on the combination of satellite gradiometry and terrestrial gravity measurements are given. The proposed method seems to be well suited for local geoid determinations down to the meter range. It is especially interesting for unsurveyed and difficult areas because no terrestrial measurements are necessary. Furthermore, it has the practical advantage that only a local data coverage is needed.     

Updating and computing the geoid using two dimensional fast Hartley transform and fast T transform

In the analyses of 2D real arrays, fast Hartley (FHT), fast T (FTT) and real-valued fast Fourier transforms are generally preferred in lieu of a complex fast Fourier transform due to the advantages of the former with respect to disk storage and computation time. Although the FHT and the FTT in one dimension are identical, they are different in two or more dimensions. Therefore, first, definitions and some properties of both transforms and the related 2D FHT and FTT algorithms are stated. After reviewing the 2D FHT and FTT solutions of Stokes' formula in planar approximation, 2D FHT and FTT methods are developed for geoid updating to incorporate additional gravity anomalies. The methods are applied for a test area which includes a 64×64 grid of 3^′×3^′ point gravity anomalies and geoid heights calculated from point masses. The geoids computed by 2D FHT and FTT are found to be identical. However, the RMS value of the differences between the computed and test geoid is ±15 mm. The numerical simulations indicate that the new methods of geoid updating are practical and accurate with considerable savings on storage requirements. Received: 15 February 1996; Accepted: 22 January 1997     

Hotine函数法的椭球面积分解

本文给出了Hotine 函数法的椭球面积分解,以应用于计算精确的大地水准面起伏。计算表明,当积分半径为20°时,我国近海的椭球改正只有10cm,远比stokes公式的椭球改正要小。     

利用Abel-Poisson径向基函数模型化局部重力场

多种类型高分辨率重力场数据的不断增加,使得在局部范围内精化重力场模型成为了可能。本文采用Abel-Poisson核将重力场量表示成有限个径向基函数线性求和的形式,对局部区域的多种重力场数据进行联合建模。为了提高运算速度,运用了基于自适应精化格网算法的最小均方根误差准则(RMS)来求解径向基函数平均带宽。以南海核心地区为例,联合两种不同类型、不同分辨率的重力场资料(大地水准面起伏6'×6'、重力异常2'×2'),构建了局部区域高分辨率的重力场模型。所建模型表示的重力场参量达到了2'×2'的分辨率,对原始的重力异常数据(2'×2')拟合的符合程度达到±0.8×10-5m/s2。结果表明,利用径向基函数方法进行局部重力场建模,避免了球谐函数建模收敛慢的问题,有效提高了模型表示重力场的分辨率。     

A bias-free geodetic boundary value problem approach to height datum unification

A geodetic boundary value problem (GBVP) approach has been formulated which can be used for solving the problem of height datum unification. The developed technique is applied to a test area in Southwest Finland with approximate size of 1.5° × 3° and the bias of the corresponding local height datum (local geoid) with respect to the geoid is computed. For this purpose the bias-free potential difference and gravity difference observations of the test area are used and the offset (bias) of the height datum, i.e., Finnish Height Datum 2000 (N2000) fixed to Normaal Amsterdams Peil (NAP) as origin point, with respect to the geoid is computed. The results of this computation show that potential of the origin point of N2000, i.e., NAP, is (62636857.68 ± 0.5) (m²/s²) and as such is (0.191 ± 0.003) (m) under the geoid defined by W ₀ = 62636855.8 (m²/s²). As the validity test of our methodology, the test area is divided into two parts and the corresponding potential difference and gravity difference observations are introduced into our GBVP separately and the bias of height datums of the two parts are computed with respect to the geoid. Obtaining approximately the same bias values for the height datums of the two parts being part of one height datum with one origin point proves the validity of our approach. Besides, the latter test shows the capability of our methodology for patch-wise application.     

Geoid information by wavelength

Two models describing potential coefficient behavior are used to estimate the root mean square geoid undulation by wavelength. Four wavelength types were defined: long wavelengths: ℓ=2 to 10; intermediate wavelengths: ℓ=11 to 100; short wavelengths: ℓ=101 to 1000; and very short wavelengths: ℓ=1001 to ∞. By one representation of potential coefficient behavior the intermediate wavelength geoid information was ±5.64 m, short wavelength, ±0.66 m, and the very short wavelength, ±0.05 m. The procedures of this paper were applied to an actual residual undulation computation using detailed gravity material.     

Comparison of undulation difference accuracies using gravity anomalies and gravity disturbances

Errors are considered in the outer zone contribution to oceanic undulation differences as obtained from a set of potential coefficients complete to degree 180. It is assumed that the gravity data of the inner zone (a spherical cap), consisting of either gravity anomalies or gravity disturbances, has negligible error. This implies that error estimates of the total undulation difference are analyzed. If the potential coefficients are derived from a global field of 1°×1° mean anomalies accurate to ε_Δg=10 mgal, then for a cap radius of 10°, the undulation difference error (for separations between 100 km and 2000 km) ranges from 13 cm to 55 cm in the gravity anomaly case and from 6 cm to 36 cm in the gravity disturbance case. If ε_Δg is reduced to 1 mgal, these errors in both cases are less than 10 cm. In the absence of a spherical cap, both cases yield identical error estimates: about 68 cm if ε_Δg=1 mgal (for most separations) and ranging from 93 cm to 160 cm if ε_Δg=10 mgal. Introducing a perfect 30-degree reference field, the latter errors are reduced to about 110 cm for most separations.     

Discussion on some FFT problems to determine the geoid

The aim of this investigation is to study some FFT problems related to the application of FFT to gravity field convolution integrals. And the others, such as the effect of spectral leakage, edge effects, cyclic convolution and effect of padding, are also discussed. A numerical test for these problems is made. A large area of Western China selected for the test is located between 30°N～36°N and 96°E～102°E and includes 1 858 gravity observations on land. The results show that the removal of the bias in the residual gravity anomalies is important to avoid spectral leakage. One hundred percent zero padding is highly recommended for further research of the geoid to remove cyclic convolution errors and edge effects. 1-D FFT is recommended for precise local geoid determination because it does not use kernel approximation.     

Comparisons of spectral techniques for geoid computations over large regions

The Stokes formula is efficiently evaluated by the one-and two- dimensional (1D, 2D) fast Fourier transform (FFT) technique in the plane and on the sphere in order to obtain precise geoid determinatiover a large area such as Europe. Using a high-pass filtered spherical harmonic reference model (OSU91A truncated to different degrees), gridded gravity anomalies and geoid heights were produced and the anomalies were used as input in the FFT software. Various tests were performed with respect to the different kernel functions used, to the spherical computations in bands, as well as to windowing, edge effects and extent of the area. It is thus demonstrated that, in geoid computations over large regions, the 1D spherical FFT and the 2D multiband spherical FFT in combination with discrete spectra for the kernel functions and 100% zero-padding give better results than those obtained by the other transform techniques. Additionally, numerical tests were carried out at the same test area using the planar fast Hartley transform (FHT) instead of the FFT and the results obtained by the two attractive alternatives were compared regarding the requirements in both computer time and computer memory needed in geoid height computations.A slightly modified version of the paper has been presented at the XX EGS General Assembly, Hamburg, 3–7 April, 1995     

Discussion on some FFT problems to determine the geoid

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Determination of geoidal heights and deflections of the vertical for the hellenic area using heterogeneous data

Mean gravity anomalies, deflections of the vertical, and a geopotential model complete to degree and order180 are combined in order to determine geoidal heights in the area bounded by [34°≦ϕ≤42°, 18°≦λ≦28°]. Moreover, employing point gravity anomalies simultaneously with the above data, an attempt is made to predict deflections of the vertical in the same area. The method used in the computations is least squares collocation. Using empirical covariance functions for the data, the suitable errors for the different sources of observations, and the optimum cap radius around each point of evaluation, an accuracy better than±0.60m for geoidal heights and±1″.5 for deflections of the vertical is obtained taking into account existing systematic effects. This accuracy refers to the comparison between observed and predicted values.     

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.