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1.
The height datum problem and the role of satellite gravity models   总被引：1，自引：0，他引：1  
Regional height systems do not refer to a common equipotential surface, such as the geoid. They are usually referred to the mean sea level at a reference tide gauge. As mean sea level varies (by ±1 to 2 m) from place to place and from continent to continent each tide gauge has an unknown bias with respect to a common reference surface, whose determination is what the height datum problem is concerned with. This paper deals with this problem, in connection to the availability of satellite gravity missions data. Since biased heights enter into the computation of terrestrial gravity anomalies, which in turn are used for geoid determination, the biases enter as secondary or indirect effect also in such a geoid model. In contrast to terrestrial gravity anomalies, gravity and geoid models derived from satellite gravity missions, and in particular GRACE and GOCE, do not suffer from those inconsistencies. Those models can be regarded as unbiased. After a review of the mathematical formulation of the problem, the paper examines two alternative approaches to its solution. The first one compares the gravity potential coefficients in the range of degrees from 100 to 200 of an unbiased gravity field from GOCE with those of the combined model EGM2008, that in this range is affected by the height biases. This first proposal yields a solution too inaccurate to be useful. The second approach compares height anomalies derived from GNSS ellipsoidal heights and biased normal heights, with anomalies derived from an anomalous potential which combines a satellite-only model up to degree 200 and a high-resolution global model above 200. The point is to show that in this last combination the indirect effects of the height biases are negligible. To this aim, an error budget analysis is performed. The biases of the high frequency part are proved to be irrelevant, so that an accuracy of 5 cm per individual GNSS station is found. This seems to be a promising practical method to solve the problem.  相似文献   

2.
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 εΔg=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 εΔg 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 εΔg=1 mgal (for most separations) and ranging from 93 cm to 160 cm if εΔg=10 mgal. Introducing a perfect 30-degree reference field, the latter errors are reduced to about 110 cm for most separations.  相似文献   

3.
A comparison of methods for the inversion of airborne gravity data   总被引：2，自引：2，他引：0  
Four integral-based methods for the inversion of gravity disturbances, derived from airborne gravity measurements, into the disturbing potential on the Bjerhammar sphere and the Earths surface are investigated and compared with least-squares (LS) collocation. The performance of the methods is numerically investigated using noise-free and noisy observations, which have been generated using a synthetic gravity field model. It is found that advanced interpolation of gravity disturbances at the nodes of higher-order numerical integration formulas significantly improves the performance of the integral-based methods. This is preferable to the commonly used one-point composed Newton–Cotes integration formulas, which intrinsically imply a piecewise constant interpolation over a patch centered at the observation point. It is shown that the investigated methods behave similarly for noise-free observations, but differently for noisy observations. The best results in terms of root-mean-square (RMS) height-anomaly errors are obtained when the gravity disturbances are first downward continued (inverse Poisson integral) and then transformed into potential values (Hotine integral). The latter has a strong smoothing effect, which damps high-frequency errors inherent in the downward-continued gravity disturbances. An integral method based on the single-layer representation of the disturbing potential shows a similar performance. This representation has the advantage that it can be used directly on surfaces with non-spherical geometry, whereas classical integral-based methods require an additional step if gravity field functionals have to be computed on non-spherical geometries. It is shown that defining the single-layer density on the Bjerhammar sphere gives results with the same quality as obtained when using the Earths topography as support for the single-layer density. A comparison of the four integral-based methods with LS collocation shows that the latter method performs slightly better in terms of RMS height-anomaly errors.  相似文献   

4.
Summary The geopotential on and outside the earth is represented as a series in surface harmonics. The principal terms in it correspond to the solid harmonics of the external potential expansion with the coefficients being Stokes’ constantsC nm andS nm . The additional terms which occur near the earth’s surface due to its non-sphericity and topography are expressed in terms of Stokes’ constants too. This allows performing downward continuation of the potential derived from satellite observations. In the boundary condition which correlates Stokes’ constants and the surface gravity anomalies there occur additional terms due to the earth’s non-sphericity and topography. They are expressed in terms of Stokes’ constants as well. This improved boundary condition can be used for upward and downward continuations of the gravity field. Simple expressions are found representingC nm andS nm as explicit functions of the surface anomalies and its derivatives. The formula for the disturbing potential on the surface is derived in terms of the surface anomalies. All the formulas do not involve the earth’s surface in clinations.  相似文献   

5.
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6.
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).  相似文献   

7.
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  相似文献   

8.
We propose a methodology for local gravity field modelling from gravity data using spherical radial basis functions. The methodology comprises two steps: in step 1, gravity data (gravity anomalies and/or gravity disturbances) are used to estimate the disturbing potential using least-squares techniques. The latter is represented as a linear combination of spherical radial basis functions (SRBFs). A data-adaptive strategy is used to select the optimal number, location, and depths of the SRBFs using generalized cross validation. Variance component estimation is used to determine the optimal regularization parameter and to properly weight the different data sets. In the second step, the gravimetric height anomalies are combined with observed differences between global positioning system (GPS) ellipsoidal heights and normal heights. The data combination is written as the solution of a Cauchy boundary-value problem for the Laplace equation. This allows removal of the non-uniqueness of the problem of local gravity field modelling from terrestrial gravity data. At the same time, existing systematic distortions in the gravimetric and geometric height anomalies are also absorbed into the combination. The approach is used to compute a height reference surface for the Netherlands. The solution is compared with NLGEO2004, the official Dutch height reference surface, which has been computed using the same data but a Stokes-based approach with kernel modification and a geometric six-parameter “corrector surface” to fit the gravimetric solution to the GPS-levelling points. A direct comparison of both height reference surfaces shows an RMS difference of 0.6 cm; the maximum difference is 2.1 cm. A test at independent GPS-levelling control points, confirms that our solution is in no way inferior to NLGEO2004.  相似文献   

9.
The spherical harmonic coefficients of the Earth’s gravitational potential are conveniently determined by integration of gravity data or potential data (derived from satellite altimetry) over a sphere. The major problem of such a method is that the data, given on the non-spherical surface of the Earth, must be reduced to the sphere. A new integral formula over the surface of the Earth is derived. With this formula improved first order topographic corrections to the spherical formulas are obtained.  相似文献   

10.
 The use of GPS for height control in an area with existing levelling data requires the determination of a local geoid and the bias between the local levelling datum and the one implicitly defined when computing the local geoid. If only scarse gravity data are available, the heights of new data may be collected rapidly by determining the ellipsoidal height by GPS and not using orthometric heights. Hence the geoid determination has to be based on gravity disturbances contingently combined with gravity anomalies. Furthermore, existing GPS/levelling data may also be used in the geoid determination if a suitable general gravity field modelling method (such as least-squares collocation, LSC) is applied. A comparison has been made in the Aswan Dam area between geoids determined using fast Fourier transform (FFT) with gravity disturbances exclusively and LSC using only the gravity disturbances and the disturbances combined with GPS/levelling data. The EGM96 spherical harmonic model was in all cases used in a remove–restore mode. A total of 198 gravity disturbances spaced approximately 3 km apart were used, as well as 35 GPS/levelling points in the vicinity and on the Aswan Dam. No data on the Nasser Lake were available. This gave difficulties when using FFT, which requires the use of gridded data. When using exclusively the gravity disturbances, the agreement between the GPS/levelling data were 0.71 ± 0.17 m for FFT and 0.63 ± 0.15 for LSC. When combining gravity disturbances and GPS/levelling, the LSC error estimate was ±0.10 m. In the latter case two bias parameters had to be introduced to account for a possible levelling datum difference between the levelling on the dam and that on the adjacent roads. Received: 14 August 2000 / Accepted: 28 February 2001  相似文献   

11.
The formulas of the ellipsoidal corrections to the gravity anomalies computed using the inverse Stokes integral are derived. The corrections are given in the integral formulas and expanded in the spherical harmonics series. If a coefficient model such as the OSU91A is given, the corrections can be easily computed. Received: 19 August 1996 / Accepted: 28 September 1998  相似文献   

12.
Satellite gravity missions, such as CHAMP, GRACE and GOCE, and airborne gravity campaigns in areas without ground gravity will enhance the present knowledge of the Earths gravity field. Combining the new gravity information with the existing marine and ground gravity anomalies is a major task for which the mathematical tools have to be developed. In one way or another they will be based on the spectral information available for gravity data and noise. The integration of the additional gravity information from satellite and airborne campaigns with existing data has not been studied in sufficient detail and a number of open questions remain. A strategy for the combination of satellite, airborne and ground measurements is presented. It is based on ideas independently introduced by Sjöberg and Wenzel in the early 1980s and has been modified by using a quasi-deterministic approach for the determination of the weighting functions. In addition, the original approach of Sjöberg and Wenzel is extended to more than two measurement types, combining the Meissl scheme with the least-squares spectral combination. Satellite (or geopotential) harmonics, ground gravity anomalies and airborne gravity disturbances are used as measurement types, but other combinations are possible. Different error characteristics and measurement-type combinations and their impact on the final solution are studied. Using simulated data, the results show a geoid accuracy in the centimeter range for a local test area.  相似文献   

13.
The principle and method for solving three types of satellite gravity gradient boundary value problems by least-squares are discussed in detail. Also, kernel function expressions of the least-squares solution of three geodetic boundary value problems with the observations {Гzz}, {Гxz, Гyz} and {Гxx ? Гyy, 2Гxy} are presented. From the results of recovering gravity field using simulated gravity gradient tensor data, we can draw a conclusion that satellite gravity gradient integral formulas derived from least-squares are valid and rigorous for recovering the gravity field.  相似文献   

14.
重力归算及其对大地水准面及外部重力场的影响   总被引：2，自引：1，他引：2  
申文斌 《测绘科学》2006,31(6):30-32
本文论述并探讨了各种不同的重力归算方法对大地水准面以及地球外部重力场的影响,讨论了与确定cm级大地水准面相关的一些问题,特别阐述了高程误差对大地水准面的影响。  相似文献   

15.
 The downward continuation of the harmonic disturbing gravity potential, derived at flight level from discrete observations of airborne gravity by the spherical Hotine integral, to the geoid is discussed. The initial-boundary-value approach, based on both the direct and inverse solution to Dirichlet's problem of potential theory, is used. Evaluation of the discretized Fredholm integral equation of the first kind and its inverse is numerically tested using synthetic airborne gravity data. Characteristics of the synthetic gravity data correspond to typical airborne data used for geoid determination today and in the foreseeable future: discrete gravity observations at a mean flight height of 2 to 6 km above mean sea level with minimum spatial resolution of 2.5 arcmin and a noise level of 1.5 mGal. Numerical results for both approaches are presented and discussed. The direct approach can successfully be used for the downward continuation of airborne potential without any numerical instabilities associated with the inverse approach. In addition to these two-step approaches, a one-step procedure is also discussed. This procedure is based on a direct relationship between gravity disturbances at flight level and the disturbing gravity potential at sea level. This procedure provided the best results in terms of accuracy, stability and numerical efficiency. As a general result, numerically stable downward continuation of airborne gravity data can be seen as another advantage of airborne gravimetry in the field of geoid determination. Received: 6 June 2001 / Accepted: 3 January 2002  相似文献   

16.
The European Space Agency’s Gravity field and steady-state ocean circulation explorer mission (GOCE) was launched on 17 March 2009. As the first of the Earth Explorer family of satellites within the Agency’s Living Planet Programme, it is aiming at a better understanding of the Earth system. The mission objective of GOCE is the determination of the Earth’s gravity field and geoid with high accuracy and maximum spatial resolution. The geoid, combined with the de facto mean ocean surface derived from twenty-odd years of satellite radar altimetry, yields the global dynamic ocean topography. It serves ocean circulation and ocean transport studies and sea level research. GOCE geoid heights allow the conversion of global positioning system (GPS) heights to high precision heights above sea level. Gravity anomalies and also gravity gradients from GOCE are used for gravity-to-density inversion and in particular for studies of the Earth’s lithosphere and upper mantle. GOCE is the first-ever satellite to carry a gravitational gradiometer, and in order to achieve its challenging mission objectives the satellite embarks a number of world-first technologies. In essence the spacecraft together with its sensors can be regarded as a spaceborne gravimeter. In this work, we describe the mission and the way it is operated and exploited in order to make available the best-possible measurements of the Earth gravity field. The main lessons learned from the first 19 months in orbit are also provided, in as far as they affect the quality of the science data products and therefore are of specific interest for GOCE data users.  相似文献   

17.
R. H. Rapp  Y. Yi 《Journal of Geodesy》1997,71(10):617-629
Procedures to calculate mean sea surface heights and gravity anomalies from altimeter-derived sea surface heights and along-track sea surface slopes using the least-squares collocation procedure are derived. The slope data is used when repeat track averaging is not possible to reduce ocean variability effects. Tests were carried out using Topex, Geosat, ERS-1 [35-day and 168-day (2 cycle)] data. Calculations of gravity anomalies in the Gulf Stream region were made using the sea surface height and slope data. Tests were also made correcting the sea surface heights for dynamic ocean topography calculated from a degree 360 expansion of data from the POCM-4B global ocean circulation model. Comparisons of the anomaly predictions were carried out with ship data using anomalies calculated for this paper as well as others. Received: 19 August 1996 / Accepted: 14 April 1997  相似文献   

18.
球近似下地球外空间任意类型场元的地形影响   总被引：1，自引：0，他引：1  
传统的重力归算方法只适用于地球表面上的重力异常，不能用于扰动重力、垂线偏差、重力梯度等其他类型扰动重力场元，不适合处理除地面外其他高度上场元的地形影响问题。当前，地球重力场探测的场元类型越来越丰富，探测的高度也逐渐转向航空和卫星高度，精确处理地球外空间各种类型重力场元的地形影响已成为地球重力场领域面临的重要课题。本文通过直接分解由地形生成的具有调和性质的引力场，从而导出地球外空间任意高度、任意类型扰动重力场元的地形影响，在此基础上给出在球近似下地形影响的严密算法和高精度快速算法。利用本文推荐的地形影响计算方案，可以方便地处理各种类型地面重力、海洋重力、航空重力、卫星重力、卫星测高数据的地形影响，从而丰富重力场数据处理的内涵，改善地球重力场算法的性能。  相似文献   

19.
The availability of high-resolution global digital elevation data sets has raised a growing interest in the feasibility of obtaining their spherical harmonic representation at matching resolution, and from there in the modelling of induced gravity perturbations. We have therefore estimated spherical Bouguer and Airy isostatic anomalies whose spherical harmonic models are derived from the Earth’s topography harmonic expansion. These spherical anomalies differ from the classical planar ones and may be used in the context of new applications. We succeeded in meeting a number of challenges to build spherical harmonic models with no theoretical limitation on the resolution. A specific algorithm was developed to enable the computation of associated Legendre functions to any degree and order. It was successfully tested up to degree 32,400. All analyses and syntheses were performed, in 64 bits arithmetic and with semi-empirical control of the significant terms to prevent from calculus underflows and overflows, according to IEEE limitations, also in preserving the speed of a specific regular grid processing scheme. Finally, the continuation from the reference ellipsoid’s surface to the Earth’s surface was performed by high-order Taylor expansion with all grids of required partial derivatives being computed in parallel. The main application was the production of a 1′ × 1′ equiangular global Bouguer anomaly grid which was computed by spherical harmonic analysis of the Earth’s topography–bathymetry ETOPO1 data set up to degree and order 10,800, taking into account the precise boundaries and densities of major lakes and inner seas, with their own altitude, polar caps with bedrock information, and land areas below sea level. The harmonic coefficients for each entity were derived by analyzing the corresponding ETOPO1 part, and free surface data when required, at one arc minute resolution. The following approximations were made: the land, ocean and ice cap gravity spherical harmonic coefficients were computed up to the third degree of the altitude, and the harmonics of the other, smaller parts up to the second degree. Their sum constitutes what we call ETOPG1, the Earth’s TOPography derived Gravity model at 1′ resolution (half-wavelength). The EGM2008 gravity field model and ETOPG1 were then used to rigorously compute 1′ × 1′ point values of surface gravity anomalies and disturbances, respectively, worldwide, at the real Earth’s surface, i.e. at the lower limit of the atmosphere. The disturbance grid is the most interesting product of this study and can be used in various contexts. The surface gravity anomaly grid is an accurate product associated with EGM2008 and ETOPO1, but its gravity information contents are those of EGM2008. Our method was validated by comparison with a direct numerical integration approach applied to a test area in Morocco–South of Spain (Kuhn, private communication 2011) and the agreement was satisfactory. Finally isostatic corrections according to the Airy model, but in spherical geometry, with harmonic coefficients derived from the sets of the ETOPO1 different parts, were computed with a uniform depth of compensation of 30?km. The new world Bouguer and isostatic gravity maps and grids here produced will be made available through the Commission for the Geological Map of the World. Since gravity values are those of the EGM2008 model, geophysical interpretation from these products should not be done for spatial scales below 5 arc minutes (half-wavelength).  相似文献   

20.
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.  相似文献   

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