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.     

High-resolution regional geoid computation without applying Stokes’s formula: a case study of the Iranian geoid

A new theory for high-resolution regional geoid computation without applying Stokess formula is presented. Operationally, it uses various types of gravity functionals, namely data of type gravity potential (gravimetric leveling), vertical derivatives of the gravity potential (modulus of gravity intensity from gravimetric surveys), horizontal derivatives of the gravity potential (vertical deflections from astrogeodetic observations) or higher-order derivatives such as gravity gradients. Its algorithmic version can be described as follows: (1) Remove the effect of a very high degree/order potential reference field at the point of measurement (POM), in particular GPS positioned, either on the Earths surface or in its external space. (2) Remove the centrifugal potential and its higher-order derivatives at the POM. (3) Remove the gravitational field of topographic masses (terrain effect) in a zone of influence of radius r. A proper choice of such a radius of influence is 2r=4×10⁴ km/n, where n is the highest degree of the harmonic expansion. (cf. Nyquist frequency). This third remove step aims at generating a harmonic gravitational field outside a reference ellipsoid, which is an equipotential surface of a reference potential field. (4) The residual gravitational functionals are downward continued to the reference ellipsoid by means of the inverse solution of the ellipsoidal Dirichlet boundary-value problem based upon the ellipsoidal Abel–Poisson kernel. As a discretized integral equation of the first kind, downward continuation is Phillips–Tikhonov regularized by an optimal choice of the regularization factor. (5) Restore the effect of a very high degree/order potential reference field at the corresponding point to the POM on the reference ellipsoid. (6) Restore the centrifugal potential and its higher-order derivatives at the ellipsoidal corresponding point to the POM. (7) Restore the gravitational field of topographic masses ( terrain effect) at the ellipsoidal corresponding point to the POM. (8) Convert the gravitational potential on the reference ellipsoid to geoidal undulations by means of the ellipsoidal Bruns formula. A large-scale application of the new concept of geoid computation is made for the Iran geoid. According to the numerical investigations based on the applied methodology, a new geoid solution for Iran with an accuracy of a few centimeters is achieved.Acknowledgments. The project of high-resolution geoid computation of Iran has been support by National Cartographic Center (NCC) of Iran. The University of Tehran, via grant number 621/3/602, supported the computation of a global geoid solution for Iran. Their support is gratefully acknowledged. A. Ardalan would like to thank Mr. Y. Hatam, and Mr. K. Ghazavi from NCC and Mr. M. Sharifi, Mr. A. Safari, and Mr. M. Motagh from the University of Tehran for their support in data gathering and computations. The authors would like to thank the comments and corrections made by the four reviewers and the editor of the paper, Professor Will Featherstone. Their comments helped us to correct the mistakes and improve the paper.     

Geoid determination in central Spain from gravity and height data

Summary The least-squares collocation method has been used for the computation of a geoid solution in central Spain, combining a geopotential model complete to degree and order 360, gravity anomalies and topographic information. The area has been divided in two 1°× 1° blocks and predictions have been done in each block with gravity data spacing about 5 × 5 within each block, extended 1/2°. Topographic effects have been calculated from 6 × 9 heights using an RTM reduction with a reference terrain model of 30 × 30 mean heights.     

A general model for modifying Stokes’ formula and its least-squares solution

Today the combination of Stokes formula and an Earth gravity model (EGM) for geoid determination is a standard procedure. However, the method of modifying Stokes formula varies from author to author, and numerous methods of modification exist. Most methods modify Stokes kernel, but the most widely applied method, the remove compute restore technique, removes the EGM from the gravity anomaly to attain a residual gravity anomaly under Stokes integral, and at least one known method modifies both Stokes kernel and the gravity anomaly. A general model for modifying Stokes formula is presented; it includes most of the well-known techniques of modification as special cases. By assuming that the error spectra of the gravity anomalies and the EGM are known, the optimum model of modification is derived based on the least-squares principle. This solution minimizes the expected mean square error (MSE) of all possible solutions of the general geoid model. A practical formula for estimating the MSE is also presented. The power of the optimum method is demonstrated in two special cases. AcknowledgementsThis paper was partly written whilst the author was a visiting scientist at The University of New South Wales, Sydney, Australia. He is indebted to Professor W. Kearsley and his colleagues, and their hospitality is acknowledged.     

Geoid determination in Turkey (TG-91)

It is considered that precise geoid determination is one of the main current geodetic problems in Turkey since GPS defined coordinates require geoidal heights in practice. In order to determine the geoid by least squares collocation (LSC) the area covering Turkey was divided into 114 blocks of size 1° × 1°. LSC approximation to the geoid based upon the tailored geopotential model GPM2-T1 is constructed within each block. The model GPM2-T1 complete to degree and order 200 has been developed by tailoring of the model GPM2 to mean free-air anomalies and mean heights of one degree blocks in Turkey. Terrain effect reduced point gravity data spaced 5 × 5 within each block which the sides extended 0°.5 were used in LSC. Residual terrain model (RTM) depends on point heights at 15×20 griding and 5×5 and 15×15 mean heights has been carried out in terrain effect reduction. Indirect effect of RTM on geoid is also taken into account. The geoid, called Turkish Geoid 1991 (TG-91), referenced to GRS-80 ellipsoid has been computed at 3 × 3 griding nodes within each block. The quality of the TG-91 is also evaluated by comparing computed and GPS derived geoidal height differences, and 2.1 – 2.6 ppm accuracy for average baseline lenght of 45 km is obtained.     

A computational scheme to model the geoid by the modified Stokes formula without gravity reductions

In a modern application of Stokes formula for geoid determination, regional terrestrial gravity is combined with long-wavelength gravity information supplied by an Earth gravity model. Usually, several corrections must be added to gravity to be consistent with Stokes formula. In contrast, here all such corrections are applied directly to the approximate geoid height determined from the surface gravity anomalies. In this way, a more efficient workload is obtained. As an example, in applications of the direct and first and second indirect topographic effects significant long-wavelength contributions must be considered, all of which are time consuming to compute. By adding all three effects to produce a combined geoid effect, these long-wavelength features largely cancel. The computational scheme, including two least squares modifications of Stokes formula, is outlined, and the specific advantages of this technique, compared to traditional gravity reduction prior to Stokes integration, are summarised in the conclusions and final remarks. AcknowledgementsThis paper was written whilst the author was a visiting scientist at Curtin University of Technology, Perth, Australia. The hospitality and fruitful discussions with Professor W. Featherstone and his colleagues are gratefully acknowledged.     

On the ellipsoidal correction to the spherical Stokes solution of the gravimetric geoid

The solutions of four ellipsoidal approximations for the gravimetric geoid are reviewed: those of Molodenskii et al., Moritz, Martinec and Grafarend, and Fei and Sideris. The numerical results from synthetic tests indicate that Martinec and Grafarends solution is the most accurate, while the other three solutions contain an approximation error which is characterized by the first-degree surface spherical harmonic. Furthermore, the first 20 degrees of the geopotential harmonic series contribute approximately 90% of the ellipsoidal correction. The determination of a geoid model from the generalized Stokes scheme can accurately account for the ellipsoidal effect to overcome the first-degree surface spherical harmonic error regardless of the solution used.     

A new strategy for the atmospheric gravity effect in gravimetric geoid determination

Prior to Stokes integration, the gravitational effect of atmospheric masses must be removed from the gravity anomaly g. One theory for the atmospheric gravity effect on the geoid is the well-known International Association of Geodesy approach in connection with Stokes integral formula. Another strategy is the use of a spherical harmonic representation of the topography, i.e. the use of a global topography computed from a set of spherical harmonics. The latter strategy is improved to account for local information. A new formula is derived by combining the local contribution of the atmospheric effect computed from a detailed digital terrain model and the global contribution computed from a spherical harmonic model of the topography. The new formula is tested over Iran and the results are compared with corresponding results from the old formula which only uses the global information. The results show significant differences. The differences between the two formulas reach 17 cm in a test area in Iran.     

Marine gravity surveying line system adjustment

The general theories and methods of marine surveying line system adjustment were introduced in (1979) and Tang (1991) . According to the characteristics of marine gravity measurement, this paper presents a new method of combined adjustment which takes into account both direct and indirect influence of position errors. The method is particularly suitable to be used in the post- processing of marine gravity observation data. With some practical applications, it is proved to be effective in improving the quality of marine gravity data.     

On Helmert’s methods of condensation

Helmerts first and second method of condensation are reviewed and generalized in two respects: First, the point at which the effects of topographical and condensation masses are calculated may be situated on or outside the topographical surface; second, the depth of the condensation layer below the geoid is arbitrary. While the first extension permits the application of the generalized model to the evaluation of airborne and satellite data, the second one gives an additional degree of freedom which can be used to provide a smooth gravity field after reducing the observation data. The respective formulae are derived for the generalized condensation model in both planar and spherical approximation. A comparison of the planar and the spherical model shows some structural differences, which are primarily visible in the out-of-integral terms. Considering the respective formulae for the combined topographic–condensation reduction on the background of the density structure of the Earths lithosphere, the consequences for the residual gravity field are investigated; it is shown that the residual field after applying Helmerts second model of reduction is very rough, making this procedure unfavourable for downward continuation. Further considerations refer to the question of which sets of formulae should be used in geoid and quasigeoid determination. It is concluded that for high-precision applications the generalized spherical model, involving a depth of the condensation layer of between 20 and 30 km, should be superior to Helmerts second model of condensation, although it requires the direct calculation of the indirect effect, which is larger than in the case of Helmerts second method of condensation.     

A numerical investigation on height anomaly prediction in mountainous areas

This paper provides numerical examples for the prediction of height anomalies by the solution of Molodensky's boundary value problem. Computations are done within two areas in the Canadian Rockies. The data used are on a grid with various grid spacings from 100 m to 5 arc-minutes. Numerical results indicate that the Bouguer or the topographicisostatic gravity anomalies should be used in gravity interpolation. It is feasible to predict height anomalies in mountainous areas with an accuracy of 10 cm (1) if sufficiently dense data grids are used. After removing the systematic bias, the differences between the geoid undulations converted from height anomalies and those derived from GPS/levelling on 50 benchmarks is 12 cm (1) when the grid spacing is 1km, and 50 cm (1) when the grid spacing is 5. It is not necessary, in most cases, to require a grid spacing finer than 1 km, because the height anomaly changes only by 3 cm (1) when the grid spacing is increased from 100 m to 1000 m. Numerical results also indicate that, only the first two terms of the Molodensky series have to be evaluated in all but the extreme cases, since the contributions of the higher order terms are negligible compared to the objective accuracy.     

A new,high-resolution geoid for Canada and part of the U.S. by the 1D-FFT method

A new, high-resolution and high-precision geoid has been computed for the whole of Canada and part of the U.S., ranging from 35°N to about 90°N in latitude and 210°E to 320°E in longitude. The OSU91A geopotential model complete to degree and order 360 was combined with a 5 × 5 mean gravity anomaly grid and 1km × 1km topographical information to generate the geoid file. The remove-restore technique was adopted for the computation of terrain effects by Helmert's condensation reduction. The contribution of the local gravity data to the geoid was computed strictly by the 1D-FFT technique, which allows for the evaluation of the discrete spherical Stokes integral without any approximation, parallel by parallel. The indirect effects of up to second order were considered. The internal precision of the geoid, i.e. the contribution of the gravity data and the model coefficients noise, was also evaluated through error propagation by FFT. In a relative sense, these errors seem to agree quite well with the external errors and show clearly the weak areas of the geoid which are mostly due to insufficient gravity data coverage. Comparison of the gravimetric geoid with the GPS/levelling-derived geoidal heights of eight local GPS networks with a total of about 900 stations shows that the absolute agreement with respect to the GPS/levelling datum is generally better than 10 cm RMS and the relative agreement ranges, in most cases, from 4 to 1 ppm over short distances of about 20 to 100km, 1 to 0.5 ppm over distances of about 100 to 200 km, and 0.5 to 0.1 ppm for baselines of 200 to over 1000 km. Other existing geoids, such as UNB90, GEOID90 and GSD91, were also included in the comparison, showing that the new geoid achieves the best agreement with the GPS/levelling data.Presented at theIAG General Meeting, Beijing, P.R. China, Aug. 6–13, 1993     

Asymptotic theory for calculating deformations caused by dislocations buried in a spherical earth: geoid change

An asymptotic theory is presented for calculating co-seismic potential and geoid changes, as an approximation of the dislocation theory for a spherical Earth. This theory is given by a closed-form mathematical expression, so that it is mathematically simple and can be applied easily. Moreover, since the asymptotic theory includes sphericity and vertical structure effects, it is physically more reasonable than the flat-Earth theory. A comparison between results calculated by three dislocation theories (the flat-Earth theory, the theory for a spherical Earth and its asymptotic solution) shows that the true co-seismic geoid changes are approximated better by the asymptotic results than by those of a flat Earth. Numerical results indicate that the sphericity effect is obvious large, especially for a tensile source on a vertical fault plane. AcknowledgementsThe author is grateful to Dr S. Okubo for his helpful suggestions and discussions. Comments by anonymous reviewers are also greatly acknowledged. This research was financially supported by JSPS research grants (C13640420) and Basic design and feasibility studies for the future missions for monitoring Earths environment.     

Interannual variability of low-degree gravitational change, 1980–2002

Time variations in the Earths gravity field at periods longer than 1 year, for degree-two spherical harmonics, C₂₁, S₂₁, and C₂₀, are estimated from accurately measured Earth rotational variations. These are compared with predictions of atmospheric, oceanic, and hydrologic models, and with independent satellite laser ranging (SLR) results. There is remarkably good agreement between Earth rotation and model predictions of C₂₁ and S₂₁ over a 22-year period. After decadal signals are removed, Earth-rotation-derived interannual C₂₀ variations are dominated by a strong oscillation of period about 5.6 years, probably due to uncertainties in wind and ocean current estimates. The model-predicted C₂₀ agrees reasonably well with SLR observations during the 22-year period, with the exception of the recent anomaly since 1997/1998.     

The Stokes and Vening-Meinesz functionals in a moving tangent space

The regularized solution of the external sphericalStokes boundary value problem as being used for computations of geoid undulations and deflections of the vertical is based upon theGreen functions S ₁(₀, ₀, , ) ofBox 0.1 (R = R ₀) andV ₁(₀, ₀, , ) ofBox 0.2 (R = R ₀) which depend on theevaluation point {₀, ₀} S _R0 ² and thesampling point {, } S _R0 ² ofgravity anomalies (, ) with respect to a normal gravitational field of typegm/R (free air anomaly). If the evaluation point is taken as the meta-north pole of theStokes reference sphere S _R0 ² , theStokes function, and theVening-Meinesz function, respectively, takes the formS() ofBox 0.1, andV ₂() ofBox 0.2, respectively, as soon as we introduce {meta-longitude (azimuth), meta-colatitude (spherical distance)}, namely {A, } ofBox 0.5. In order to deriveStokes functions andVening-Meinesz functions as well as their integrals, theStokes andVening-Meinesz functionals, in aconvolutive form we map the sampling point {, } onto the tangent plane T₀S _R0 ² at {₀, ₀} by means ofoblique map projections of type(i) equidistant (Riemann polar/normal coordinates),(ii) conformal and(iii) equiareal.Box 2.1.–2.4. andBox 3.1.– 3.4. are collections of the rigorously transformedconvolutive Stokes functions andStokes integrals andconvolutive Vening-Meinesz functions andVening-Meinesz integrals. The graphs of the correspondingStokes functions S ₂(),S ₃(r),,S ₆(r) as well as the correspondingStokes-Helmert functions H ₂(),H ₃(r),,H ₆(r) are given byFigure 4.1–4.5. In contrast, the graphs ofFigure 4.6–4.10 illustrate the correspondingVening-Meinesz functions V ₂(),V ₃(r),,V ₆(r) as well as the correspondingVening-Meinesz-Helmert functions Q ₂(),Q ₃(r),,Q ₆(r). The difference between theStokes functions / Vening-Meinesz functions andtheir first term (only used in the Flat Fourier Transforms of type FAST and FASZ), namelyS ₂() – (sin /2)^–1,S ₃(r) – (sinr/2R ₀)^–1,,S ₆(r) – 2R ₀/r andV ₂() + (cos /2)/2(sin² /2),V ₃(r) + (cosr/2R ₀)/2(sin² r/2R ₀),, illustrate the systematic errors in theflat Stokes function 2/ or flatVening-Meinesz function –2/². The newly derivedStokes functions S ₃(r),,S ₆(r) ofBox 2.1–2.3, ofStokes integrals ofBox 2.4, as well asVening-Meinesz functionsV ₃(r),,V ₆(r) ofBox 3.1–3.3, ofVening-Meinesz integrals ofBox 3.4 — all of convolutive type — pave the way for the rigorousFast Fourier Transform and the rigorousWavelet Transform of theStokes integral / theVening-Meinesz integral of type equidistant, conformal and equiareal.     

Validation of ETOPO5U in the Hellenic area

Mean 5 × 5 heights and depths from ETOPO5U (Earth Topography at 5 spacing Updated) Digital Terrain Model (DTM) were compared with corresponding quantities of a local DTM in the test area [38° 40°, 21° 24°]. From this comparison a shift of ETOPO5U with respect to the local DTM in the longitudinal direction equal to 5 min was found after applying an efficient fast Fourier transform (FFT) technique. Furthermore, sparse mean height differences larger than 1,000 m were observed between ETOPO5U and the local DTM due rather to errors of ETOPO5U. The effect of these errors on gravity and height anomalies was computed in a subregion of the area under consideration.     

The ellipsoidal corrections to the topographic geoid effects

The topographic effects by Stokes formula are typically considered for a spherical approximation of sea level. For more precise determination of the geoid, sea level is better approximated by an ellipsoid, which justifies the consideration of the ellipsoidal corrections of topographic effects for improved geoid solutions. The aim of this study is to estimate the ellipsoidal effects of the combined topographic correction (direct plus indirect topographic effects) and the downward continuation effect. It is concluded that the ellipsoidal correction to the combined topographic effect on the geoid height is far less than 1 mm. On the contrary, the ellipsoidal correction to the effect of downward continuation of gravity anomaly to sea level may be significant at the 1-cm level in mountainous regions. Nevertheless, if Stokes formula is modified and the integration of gravity anomalies is limited to a cap of a few degrees radius around the computation point, nor this effect is likely to be significant.AcknowledgementsThe author is grateful for constructive remarks by J Ågren and the three reviewers.     

Combined regional geoid determination for the ERS-1 radar altimeter calibration

Summary A local model of the geoid in NE Italy and its section along the Venice ground track of the ERS-1 satellite of the European Space Agency is presented. The observational data consist of geoid undulations determined with a network of 25 stations of known orthometric (by spirit leveling) and ellipsoidal (by GPS differential survey) and of 13 deflections of the vertical measured at sites of the network for which, besides the ellipsoidal (WGS84) coordinates, also astronomic coordinates were known. The network covers an area of 1×1 degrees and is tied to a vertical and horizontal datum: one vertex of the network is the tide gauge of Punta Salute, in Venice, providing a tie to a mean sea level; a second vertex is the site for mobile laser systems at Monte Venda, on the Euganei Hills, for which geocentric coordinates resulted from the analysis of several LAGEOS passes.The interpolation algorithm used to map sparse and heterogeneous data to a regular grid of geoid undulations is based on least squares collocation and the autocorrelation function of the geoid undulations is modeled by a third order Markov process on flat earth. The algorithm has been applied to the observed undulations and deflections of the vertical after subtraction of the corresponding predictions made on the basis of the OSU91A global geoid model of the Ohio State University, complete to degree and order 360. The locally improved geoid results by adding back, at the nodes of a regular grid, the predictions of the global field to the least squares interpolated values. Comparison of the model values with the raw data at the observing stations indicates that the mean discrepancy is virtually zero with a root mean square dispersion of 8 cm, assuming that the ellipsoidal heights and vertical deflections data are affected by a random error of 3 cm and 0.5 respectively. The corrections resulting from the local data and added to the background 360×360 global model are described by a smooth surface with excursions from the reference surface not larger than ±30 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.     

A study on the combination of satellite, airborne, and terrestrial gravity data

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.