A solution to the downward continuation effect on the geoid determined by Stokes' formula

 The analytical continuation of the surface gravity anomaly to sea level is a necessary correction in the application of Stokes' formula for geoid estimation. This process is frequently performed by the inversion of Poisson's integral formula for a sphere. Unfortunately, this integral equation corresponds to an improperly posed problem, and the solution is both numerically unstable, unless it is well smoothed, and tedious to compute. A solution that avoids the intermediate step of downward continuation of the gravity anomaly is presented. Instead the effect on the geoid as provided by Stokes' formula is studied directly. The practical solution is partly presented in terms of a truncated Taylor series and partly as a truncated series of spherical harmonics. Some simple numerical estimates show that the solution mostly meets the requests of a 1-cm geoid model, but the truncation error of the far zone must be studied more precisely for high altitudes of the computation point. In addition, it should be emphasized that the derived solution is more computer efficient than the detour by Poisson's integral. Received: 6 February 2002 / Accepted: 18 November 2002 Acknowledgements. Jonas ?gren carried out the numerical calculations and gave some critical and constructive remarks on a draft version of the paper. This support is cordially acknowledged. Also, the thorough work performed by one unknown reviewer is very much appreciated.     

Local geoid determination from airborne vector gravimetry

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

Geoid determination using one-step integration

A residual (high-frequency) gravimetric geoid is usually computed from geographically limited ground, sea and/or airborne gravimetric data. The mathematical model for its determination from ground gravity is based on the transformation of observed discrete values of gravity into gravity potential related to either the international ellipsoid or the geoid. The two reference surfaces are used depending on height information that accompanies ground gravity data: traditionally orthometric heights determined by geodetic levelling were used while GPS positioning nowadays allows for estimation of geodetic (ellipsoidal) heights. This transformation is usually performed in two steps: (1) observed values of gravity are downward continued to the ellipsoid or the geoid, and (2) gravity at the ellipsoid or the geoid is transformed into the corresponding potential. Each of these two steps represents the solution of one geodetic boundary-value problem of potential theory, namely the first and second or third problem. Thus two different geodetic boundary-value problems must be formulated and solved, which requires numerical evaluation of two surface integrals. In this contribution, a mathematical model in the form of a single Fredholm integral equation of the first kind is presented and numerically investigated. This model combines the solution of the first and second/third boundary-value problems and transforms ground gravity disturbances or anomalies into the harmonically downward continued disturbing potential at the ellipsoid or the geoid directly. Numerical tests show that the new approach offers an efficient and stable solution for the determination of the residual geoid from ground gravity data.     

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.     

利用空中平均重力异常确定区域大地水准面

提出了直接利用空中平均重力异常计算区域大地水准面的方法。模拟计算的结果表明, 该方法与传统的利用地面平均空间重力异常确定的大地水准面精度相当, 但其显著优点是勿需空中重力异常的向下解析延拓, 从而可以避免延拓误差对大地水准面精化的影响。     

The topographic bias by analytical continuation in physical geodesy

This study emphasizes that the harmonic downward continuation of an external representation of the Earth’s gravity potential to sea level through the topographic masses implies a topographic bias. It is shown that the bias is only dependent on the topographic density along the geocentric radius at the computation point. The bias corresponds to the combined topographic geoid effect, i.e., the sum of the direct and indirect topographic effects. For a laterally variable topographic density function, the combined geoid effect is proportional to terms of powers two and three of the topographic height, while all higher order terms vanish. The result is useful in geoid determination by analytical continuation, e.g., from an Earth gravity model, Stokes’s formula or a combination thereof.     

Topographic effects by the Stokes–Helmert method of geoid and quasi-geoid determinations

The topographic potential and the direct topographic effect on the geoid are presented as surface integrals, and the direct gravity effect is derived as a rigorous surface integral on the unit sphere. By Taylor-expanding the integrals at sea level with respect to topographic elevation (H) the power series of the effects is derived to arbitrary orders. This study is primarily limited to terms of order H ². The limitations of the various effects in the frequently used planar approximations are demonstrated. In contrast, it is shown that the spherical approximation to power H ² leads to a combined topographic effect on the geoid (direct plus indirect effect) proportional to H˜² (where terms of degrees 0 and 1 are missing) of the order of several metres, while the combined topographic effect on the height anomaly vanishes, implying that current frequent efforts to determine the direct effect to this order are not needed. The last result is in total agreement with Bjerhammar's method in physical geodesy. It is shown that the most frequently applied remove–restore technique of topographic masses in the application of Stokes' formula suffers from significant errors both in the terrain correction C (representing the sum of the direct topographic effect on gravity anomaly and the effect of continuing the anomaly to sea level) and in the term t (mainly representing the indirect effect on the geoidal or quasi-geoidal height). Received: 18 August 1998 / Accepted: 4 October 1999     

国产零长弹簧原理动态重力仪首次飞行试验及精度评估

CHZ-Ⅱ重力仪是首套完全国产零长弹簧原理航空重力仪,2018年4月在陕西渭南地区进行了首次飞行试验,共完成4个架次24条测线的有效飞行,标志着我国航空重力仪在自主研发的道路上又取得了长足的进步。利用飞行地区地面重力数据对CHZ-Ⅱ重力仪的测线扰动重力和格网扰动重力数据进行精度评估,其中空中测线在10 km分辨率条件下,精度达到1 mGal。采用地形辅助法对测量形成的5'测格网重力数据进行向下延拓,经延拓至地面后精度优于5 mGal,基本满足平原地区的测量需求。     

Downward continuation of Helmert's gravity

. The aim of this contribution is to show that mean Helmert's gravity anomalies obtained at the earth surface on a grid of a `reasonable' step can be transferred to corresponding mean Helmert's anomalies on the geoid. To demonstrate this, we take the by mean Helmert's anomalies from a very rugged region, the south-western corner of Canada which contains the two main chains of the Canadian Rocky Mountains, and formulate the problem of downward continuation of Helmert's anomalies for this region. This can be done exactly because Helmert's disturbing potential is harmonic everywhere outside the geoid, therefore even within the topography. Then we solve the problem numerically by transforming the Poisson integral to a system of 53,856 linear algebraic equations. Since the matrix of this system is well conditioned, there is no theoretical obstacle to the solution. The correctness of the solution is then checked by back substitution and by evaluating the contribution of the downward continuation term to Helmert's co-geoid. This contribution comes out positive for all the points. We thus claim that the determination of the downward continuation of mean Helmert's gravity anomalies on a grid of a `reasonable' step is a well posed problem with a unique solution and can be done routinely to any accuracy desired in the geoid computaion. Received 27 October 1995; Accepted 9 July 1996     

联合使用位模型和地形信息的陆区航空重力向下延拓方法

为了规避传统逆Poisson积分向下延拓解算过程的不适定性问题,借鉴导航定位中的"差分"概念,利用超高阶位模型直接计算海域航空重力测量向下延拓改正数的方法。本文在此基础上提出联合使用重力位模型和地形高数据,计算陆部航空重力向下延拓总改正数的改进方案,以飞行高度面与地面对应点的位模型差分信息表征总改正数的中长波分量,以相对应的局部地形改正差分修正量表征总改正数的中高频成分,从而实现航空重力数据点对点向地面的全频段延拓。在地形变化不同区域,联合使用EGM2008位模型、地面实测重力和高分辨率高程数据进行了实际数值计算和精度评估,验证了该方法的有效性。     

A formula for computing the gravity disturbance from the second radial derivative of the disturbing potential

 A formula for computing the gravity disturbance and gravity anomaly from the second radial derivative of the disturbing potential is derived in detail using the basic differential equation with spherical approximation in physical geodesy and the modified Poisson integral formula. The derived integral in the space domain, expressed by a spherical geometric quantity, is then converted to a convolution form in the local planar rectangular coordinate system tangent to the geoid at the computing point, and the corresponding spectral formulae of 1-D FFT and 2-D FFT are presented for numerical computation. Received: 27 December 2000 / Accepted: 3 September 2001     

The spherical horizontal and spherical vertical boundary value problem – vertical deflections and geoidal undulations – the completed Meissl diagram

 In a comparison of the solution of the spherical horizontal and vertical boundary value problems of physical geodesy it is aimed to construct downward continuation operators for vertical deflections (surface gradient of the incremental gravitational potential) and for gravity disturbances (vertical derivative of the incremental gravitational potential) from points on the Earth's topographic surface or of the three-dimensional (3-D) Euclidean space nearby down to the international reference sphere (IRS). First the horizontal and vertical components of the gravity vector, namely spherical vertical deflections and spherical gravity disturbances, are set up. Second, the horizontal and vertical boundary value problem in spherical gravity and geometry space is considered. The incremental gravity vector is represented in terms of vector spherical harmonics. The solution of horizontal spherical boundary problem in terms of the horizontal vector-valued Green function converts vertical deflections given on the IRS to the incremental gravitational potential external in the 3-D Euclidean space. The horizontal Green functions specialized to evaluation and source points on the IRS coincide with the Stokes kernel for vertical deflections. Third, the vertical spherical boundary value problem is solved in terms of the vertical scalar-valued Green function. Fourth, the operators for upward continuation of vertical deflections given on the IRS to vertical deflections in its external 3-D Euclidean space are constructed. Fifth, the operators for upward continuation of incremental gravity given on the IRS to incremental gravity to the external 3-D Euclidean space are generated. Finally, Meissl-type diagrams for upward continuation and regularized downward continuation of horizontal and vertical gravity data, namely vertical deflection and incremental gravity, are produced. Received: 10 May 2000 / Accepted: 26 February 2001     

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

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

Harmonic continuation and gridding effects on geoid height prediction

Least-squares collocation and Stokes integral formula, as implemented using the Fast Fourier Technique, handle the harmonic downward continuation problem quite differently. FFT furthermore requires gridded data, amplifying the difference of methods.We have in this paper studied numerically the effects of downward continuation and gridding in a mountainous area in central Norway. Topographically smoothed data were used in order to reduce these effects. Despite the smoothing, it was found that the vertical gravity gradient had values up to -11 mgal/km. The corresponding differences between geoid heights and the height anomalies at altitude reached 12 cm.The differences between geoid heights obtained using collocation or FFT with gravity data at terrain level or sea level showed differences between the values of up to 10 cm r.m.s. A part of this difference was a consequence of different data areas used in the FFT and collocation solution, though.Major discrepancies between the solutions were found in areas where the topographic smoothing could not be applied (deep fjords with no depth information in the used DTM) or where there seemed to be gross errors in the data.We conclude that proper handling of harmonic continuation is important, even when we as here have used a 1 km resolution DTM for the calculation of topographic effects. The effect of data gridding, required for the FFT method, seems not to be as serious as the need to limit the data distribution area, required when least squares collocation is used with randomly distributed data.     

Two different methods of geoidal height determinations using a spherical harmonic representation of the geopotential, topographic corrections and the height anomaly–geoidal height difference

 It is suggested that a spherical harmonic representation of the geoidal heights using global Earth gravity models (EGM) might be accurate enough for many applications, although we know that some short-wavelength signals are missing in a potential coefficient model. A `direct' method of geoidal height determination from a global Earth gravity model coefficient alone and an `indirect' approach of geoidal height determination through height anomaly computed from a global gravity model are investigated. In both methods, suitable correction terms are applied. The results of computations in two test areas show that the direct and indirect approaches of geoid height determination yield good agreement with the classical gravimetric geoidal heights which are determined from Stokes' formula. Surprisingly, the results of the indirect method of geoidal height determination yield better agreement with the global positioning system (GPS)-levelling derived geoid heights, which are used to demonstrate such improvements, than the results of gravimetric geoid heights at to the same GPS stations. It has been demonstrated that the application of correction terms in both methods improves the agreement of geoidal heights at GPS-levelling stations. It is also found that the correction terms in the direct method of geoidal height determination are mostly similar to the correction terms used for the indirect determination of geoidal heights from height anomalies. Received: 26 July 2001 / Accepted: 21 February 2002     

Gravity/magnetic potential of uneven shell topography

 A fast spherical harmonic approach enables the computation of gravitational or magnetic potential created by a non-uniform shell of material bounded by uneven topographies. The resulting field can be evaluated outside or inside the sphere, assuming that density of the shell varies with latitude, longitude, and radial distance. To simplify, the density (or magnetization) source inside the sphere is assumed to be the product of a surface function and a power series expansion of the radial distance. This formalism is applied to compute the gravity signal of a steady, dry atmosphere. It provides geoid/gravity maps at sea level as well as satellite altitude. Results of this application agree closely with those of earlier studies, where the atmosphere contribution to the Earth's gravity field was determined using more time-consuming methods. Received: 14 August 2000 / Accepted: 19 March 2001     

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.     

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

 Two numerical techniques are used in recent regional high-frequency geoid computations in Canada: discrete numerical integration and fast Fourier transform. These two techniques have been tested for their numerical accuracy using a synthetic gravity field. The synthetic field was generated by artificially extending the EGM96 spherical harmonic coefficients to degree 2160, which is commensurate with the regular 5^′ geographical grid used in Canada. This field was used to generate self-consistent sets of synthetic gravity anomalies and synthetic geoid heights with different degree variance spectra, which were used as control on the numerical geoid computation techniques. Both the discrete integration and the fast Fourier transform were applied within a 6^∘ spherical cap centered at each computation point. The effect of the gravity data outside the spherical cap was computed using the spheroidal Molodenskij approach. Comparisons of these geoid solutions with the synthetic geoid heights over western Canada indicate that the high-frequency geoid can be computed with an accuracy of approximately 1 cm using the modified Stokes technique, with discrete numerical integration giving a slightly, though not significantly, better result than fast Fourier transform. Received: 2 November 1999 / Accepted: 11 July 2000     

Determination of the boundary values for the Stokes–Helmert problem

 The definition of the mean Helmert anomaly is reviewed and the theoretically correct procedure for computing this quantity on the Earth's surface and on the Helmert co-geoid is suggested. This includes a discussion of the role of the direct topographical and atmospherical effects, primary and secondary indirect topographical and atmospherical effects, ellipsoidal corrections to the gravity anomaly, its downward continuation and other effects. For the rigorous derivations it was found necessary to treat the gravity anomaly systematically as a point function, defined by means of the fundamental gravimetric equation. It is this treatment that allows one to formulate the corrections necessary for computing the `one-centimetre geoid'. Compared to the standard treatment, it is shown that a `correction for the quasigeoid-to-geoid separation', amounting to about 3 cm for our area of interest, has to be considered. It is also shown that the `secondary indirect effect' has to be evaluated at the topography rather than at the geoid level. This results in another difference of the order of several centimetres in the area of interest. An approach is then proposed for determining the mean Helmert anomalies from gravity data observed on the Earth's surface. This approach is based on the widely-held belief that complete Bouguer anomalies are generally fairly smooth and thus particularly useful for interpolation, approximation and averaging. Numerical results from the Canadian Rocky Mountains for all the corrections as well as the downward continuation are shown. Received: 9 March 1998 / Accepted: 16 November 1998     

海空重力测量及应用技术研究若干进展

总结分析评述了海军重力研究团队在海空重力测量及应用技术研究领域取得的一些有理论意义和实用价值的研究成果,主要涵盖需求论证与顶层设计、观测数据归算与精度评估、测量误差分析处理与分离补偿、地表观测重力向上延拓、航空观测重力向下延拓、海域重力数据模型构建、地球外部重力场逼近和大地水准面精化等8个研究方向,重点从各项关键技术的研究背景、研究思路、难点突破、成果应用前景等几个方面进行了分析和总结,回答了该研究领域涉及理论方法和工程应用的一系列科学问题,为该领域未来发展提供借鉴和参考。