How to handle topography in practical geoid determination: three examples

 Three different methods of handling topography in geoid determination were investigated. The first two methods employ the residual terrain model (RTM) remove–restore technique, yielding the quasigeoid, whereas the third method uses the classical Helmert condensation method, yielding the geoid. All three methods were used with the geopotential model Earth Gravity Model (1996) (EGM96) as a reference, and the results were compared to precise global positioning system (GPS) levelling networks in Scandinavia. An investigation of the Helmert method, focusing on the different types of indirect effects and their effects on the geoid, was also carried out. The three different methods used produce almost identical results at the 5-cm level, when compared to the GPS levelling networks. However, small systematic differences existed. Received: 18 March 1999 / Accepted: 21 March 2000     

The northern European geoid: a case study on long-wavelength geoid errors

 The long-wavelength geoid errors on large-scale geoid solutions, and the use of modified kernels to mitigate these effects, are studied. The geoid around the Nordic area, from Greenland to the Ural mountains, is considered. The effect of including additional gravity data around the Nordic/Baltic land area, originating from both marine, satellite and ground-based measurements, is studied. It is found that additional data appear to increase the noise level in computations, indicating the presence of systematic errors. Therefore, the Wong–Gore modification to the Stokes kernel is applied. This method of removing lower-order terms in the Stokes kernel appears to improve the geoid. The best fit to the global positioning system (GPS) leveling points is obtained with a degree of modification of approximately 30. In addition to the study of modification errors, the results of different methods of combining satellite altimetry gravity and other gravimetry are presented. They all gave comparable results, at the 6-cm level, when evaluated for the Nordic GPS networks. One dimensional (1-D) and 2-D fast Fourier transform (FFT) methods are also compared. It is shown that even though methods differ by up to 6 cm, the fit to the GPS is essentially the same. A surprising conclusion is that the addition of more data does not always produce a better geoid, illustrating the danger of systematic errors in data. Received: 4 July 2001 / Accepted: 21 February 2002     

The AUSGeoid98 geoid model of Australia: data treatment, computations and comparisons with GPS-levelling data

 The AUSGeoid98 gravimetric geoid model of Australia has been computed using data from the EGM96 global geopotential model, the 1996 release of the Australian gravity database, a nationwide digital elevation model, and satellite altimeter-derived marine gravity anomalies. The geoid heights are on a 2 by 2 arc-minute grid with respect to the GRS80 ellipsoid, and residual geoid heights were computed using the 1-D fast Fourier transform technique. This has been adapted to include a deterministically modified kernel over a spherical cap of limited spatial extent in the generalised Stokes scheme. Comparisons of AUSGeoid98 with GPS and Australian Height Datum (AHD) heights across the continent give an RMS agreement of ±0.364 m, although this apparently large value is attributed partly to distortions in the AHD. Received: 10 March 2000 / Accepted: 21 February 2001     

Geoid determination using adapted reference field, seismic Moho depths and variable density contrast

 The traditional remove-restore technique for geoid computation suffers from two main drawbacks. The first is the assumption of an isostatic hypothesis to compute the compensation masses. The second is the double consideration of the effect of the topographic–isostatic masses within the data window through removing the reference field and the terrain reduction process. To overcome the first disadvantage, the seismic Moho depths, representing, more or less, the actual compensating masses, have been used with variable density anomalies computed by employing the topographic–isostatic mass balance principle. In order to avoid the double consideration of the effect of the topographic–isostatic masses within the data window, the effect of these masses for the used fixed data window, in terms of potential coefficients, has been subtracted from the reference field, yielding an adapted reference field. This adapted reference field has been used for the remove–restore technique. The necessary harmonic analysis of the topographic–isostatic potential using seismic Moho depths with variable density anomalies is given. A wide comparison among geoids computed by the adapted reference field with both the Airy–Heiskanen isostatic model and seismic Moho depths with variable density anomaly and a geoid computed by the traditional remove–restore technique is made. The results show that using seismic Moho depths with variable density anomaly along with the adapted reference field gives the best relative geoid accuracy compared to the GPS/levelling geoid. Received: 3 October 2001 / Accepted: 20 September 2002 Correspondence to: H.A. Abd-Elmotaal     

Local geoid determination combining gravity disturbances and GPS/levelling: a case study in the Lake Nasser area, Aswan, Egypt

 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     

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     

Downward continuation and geoid determination based on band-limited airborne gravity data

 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     

A new hybrid geoid model for Japan, GSIGEO2000

 The latest gravimetric geoid model for Japan, JGEOID2000, was successfully combined with the nationwide net of GPS at benchmarks, yielding a new hybrid geoid model for Japan, GSIGEO2000. The least-squares collocation (LSC) method was applied as an interpolation for fitting JGEOID2000 to the GPS/leveling geoid undulations. The GPS/leveling geoid undulation data were reanalyzed in advance, in terms of three-dimensional positions from GPS and orthometric heights from leveling. The new hybrid geoid model is, therefore, compatible with the new Japanese geodetic reference frame. GSIGEO2000 was evaluated internally and independently and the precision was estimated at 4 cm throughout nearly the whole region. Received: 15 October 2001 / Accepted: 27 March 2002 Acknowledgments. Messrs. Toshio Kunimi and Tadashi Saito at the Third Geodetic Division of the Geographical Survey Institute (GSI) mainly carried out the computations of most of the updated leveled heights. With regard to the reanalysis of GPS data, the discussions with Messrs. Yuki Hatanaka and Shoichi Matsumura of GSI were of great help in building the analysis strategy. Messrs. Kazuyuki Tanaka and Hiromi Shigematsu collaborated in the preparatory stages of GPS data computation. The authors' thanks are extended to these colleagues. Some plots were made by GMT software (Wessel and Smith 1991). Correspondence to: Y. Kuroishi     

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     

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.     

Geoid determination with density hypotheses from isostatic models and geological information

 Geoid determination by Stokes's formula requires a complete knowledge of the topographical mass density distribution in order to perform gravity reductions to the geoid boundary. However, deeper masses are also of interest, in order to produce a smooth field of gravity anomalies which will improve results from interpolation procedures. Until now, in most cases a constant mass density has been considered, which is a very rough approximation of reality. The influence on the geoid height coming from different mass density hypotheses given by the isostatic models of Pratt/Hayford, Airy/Heiskanen and Vening Meinesz is studied. Apart from a constant mass density value, additional density information deduced from geological maps and thick sedimentary layers is considered. An overview of how mass density distributions act within Stokes's theory is given. The isostatic models are considered in spherical and planar approximation, as well as with constant and lateral variable mass density of the topographical and deeper masses. Numerical results in a test area in south-west Germany show that the differences in the geoid height due to different density hypotheses can reach a magnitude of more than 1 decimetre, which is not negligible in a precise geoid determination with centimetre accuracy. Received: 7 January 2002 / Accepted: 20 September 2002 M. Kuhn now at: Western Australian Centre for Geodesy, Curtin University of Technology, GPO Box U1987, Perth, WA 6845, Australia Acknowledgements. The author would gratefully thank Prof. Dr.-Ing. B. Heck, who was the supervisor of my PhD thesis, and the second examiner Prof. Dr.-Ing. K.H. Ilk, as well as all other colleagues for their support of this work. Particular thanks go to the Landesvermessungsamt Baden–Württemberg (Survey Department of Baden–Württemberg), Bureau Gravimetrique International (BGI, France) for providing the gravity data and the Geologisches Landesamt Baden–Württemberg (Geological Department of Baden–Württemberg) for providing data and maps of the sediment layers within the Rhine Valley. Grateful thanks goes to Prof. W.E. Featherstone and the reviewers Prof. S.D. Pagiatakis, Dr. U. Marti as well as an unknown reviewer for their helpful comments on this paper.     

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     

Improvements in height datum transfer expected from the GOCE mission

 One of the aims of the Earth Explorer Gravity Field and Steady-State Ocean Circulation (GOCE) mission is to provide global and regional models of the Earth's gravity field and of the geoid with high spatial resolution and accuracy. Using the GOCE error model, simulation studies were performed in order to estimate the accuracy of datum transfer in different areas of the Earth. The results showed that with the GOCE error model, the standard deviation of the height anomaly differences is about one order of magnitude better than the corresponding value with the EGM96 error model. As an example, the accuracy of the vertical datum transfer from the tide gauge of Amsterdam to New York was estimated equal to 57 cm when the EGM96 error model was used, while in the case of GOCE error model this accuracy was increased to 6 cm. The geoid undulation difference between the two places is about 76.5 m. Scaling the GOCE errors to the local gravity variance, the estimated accuracy varied between 3 and 7 cm, depending on the scaling model. Received: 1 March 2000 / Accepted: 21 February 2001     

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     

Formation of surface spherical harmonic normal matrices and application to high-degree geopotential modeling

 The structure of normal matrices occurring in the problem of weighted least-squares spherical harmonic analysis of measurements scattered on a sphere with random noises is investigated. Efficient algorithms for the formation of the normal matrices are derived using fundamental relations inherent to the products of two surface spherical harmonic functions. The whole elements of a normal matrix complete to spherical harmonic degree L are recursively obtained from its first row or first column extended to degree 2L with only O(L ⁴) computational operations. Applications of the algorithms to the formation of surface normal matrices from geoid undulations and surface gravity anomalies are discussed in connection with the high-degree geopotential modeling. Received: 22 March 1999 / Accepted: 23 December 1999     

Numerical modeling of gravitational field lines – the effect of mass attraction on horizontal coordinates

There are two basic types of geodetic projections by which the points located on the Earth's surface can be projected onto a reference ellipsoid. Because of the different principles of Helmert's and Pizzetti's methods two sets of horizontal and vertical coordinates are obtained for the same set of surface points. The difference is investigated in terms of offsets of horizontal coordinates. For an estimation of the offsets the field lines going through topographic masses are determined by three different numerical methods in `flat Earth approximation' using a volume element model of the density distribution of the lithosphere in a part of central Europe. The maximum horizontal offset reaches 20 cm on the investigated area at the level of the geoid. Received: 28 October 1998 / Accepted: 16 August 1999     

Improved convergence rates for the truncation error in gravimetric geoid determination

 When Stokes's integral is used over a spherical cap to compute a gravimetric estimate of the geoid, a truncation error results due to the neglect of gravity data over the remainder of the Earth. Associated with the truncation error is an error kernel defined over these two complementary regions. An important observation is that the rate of decay of the coefficients of the series expansion for the truncation error in terms of Legendre polynomials is determined by the smoothness properties of the error kernel. Previously published deterministic modifications of Stokes's integration kernel involve either a discontinuity in the error kernel or its first derivative at the spherical cap radius. These kernels are generalised and extended by constructing error kernels whose derivatives at the spherical cap radius are continuous up to an arbitrary order. This construction is achieved by smoothly continuing the error kernel function into the spherical cap using a suitable degree polynomial. Accordingly, an improved rate of convergence of the spectral series representation of the truncation error is obtained. Received: 21 April 1998 / Accepted: 4 October 1999     

Some modifications of Stokes' formula that account for truncation and potential coefficient errors

 Stokes' formula from 1849 is still the basis for the gravimetric determination of the geoid. The modification of the formula, originating with Molodensky, aims at reducing the truncation error outside a spherical cap of integration. This goal is still prevalent among various modifications. In contrast to these approaches, some least-squares types of modification that aim at reducing the truncation error, as well as the error stemming from the potential coefficients, are demonstrated. The least-squares estimators are provided in the two cases that (1) Stokes' kernel is a priori modified (e.g. according to Molodensky's approach) and (2) Stokes' kernel is optimally modified to minimize the global mean square error. Meissl-type modifications are also studied. In addition, the use of a higher than second-degree reference field versus the original (Pizzetti-type) reference field is discussed, and it is concluded that the former choice of reference field implies increased computer labour to achieve the same result as with the original reference field. Received: 14 December 1998 / Accepted: 4 October 1999     

National height datum,the Gauss–Listing geoid level value w0 and its time variation 0 (Baltic Sea Level Project: epochs 1990.8, 1993.8, 1997.4)

 A methodology for precise determination of the fundamental geodetic parameter w ₀, the potential value of the Gauss–Listing geoid, as well as its time derivative ₀, is presented. The method is based on: (1) ellipsoidal harmonic expansion of the external gravitational field of the Earth to degree/order 360/360 (130 321 coefficients; http://www.uni-stuttgard.de/gi/research/ index.html projects) with respect to the International Reference Ellipsoid WGD2000, at the GPS positioned stations; and (2) ellipsoidal free-air gravity reduction of degree/order 360/360, based on orthometric heights of the GPS-positioned stations. The method has been numerically tested for the data of three GPS campaigns of the Baltic Sea Level project (epochs 1990.8,1993.4 and 1997.4). New w ₀ and ₀ values (w ₀=62 636 855.75 ± 0.21 m²/s², ₀=−0.0099±0.00079 m²/s² per year, w ₀/&γmacr;=6 379 781.502 m,₀/&γmacr;=1.0 mm/year, and &γmacr;= −9.81802523 m²/s²) for the test region (Baltic Sea) were obtained. As by-products of the main study, the following were also determined: (1) the high-resolution sea surface topography map for the Baltic Sea; (2) the most accurate regional geoid amongst four different regional Gauss–Listing geoids currently proposed for the Baltic Sea; and (3) the difference between the national height datums of countries around the Baltic Sea. Received: 14 August 2000 / Accepted: 19 June 2001     

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