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Highlights from the article: The third contribution 'Ill-Posed Problems: Operator Methodologies of Resolution and Regularization', with 114 pages was written by Will Freedmen and M. Zuhair Nashed. The sixth contribution "Oblique Stochastic Boundary Value Problem" with 26 pages was written by Martin Grothaus and Thomas Raskop. After the preliminaries, the multipole methods for splines are presented, the fast multipole method for splines with illustrations, and the parameter choice methods for spline approximation. To work with inverse gravimetry, the Newton volume integral is presented first, then the ill-posedness of the gravimetry problem, the mollifier methods, and the reproducing kernel Hilbert space methods.

Abstract: This paper focuses on studying long distance height datum transfer across seas by a method, using GPS/leveling and gravimetric geoid fitting with an additional parameter. The method combines earth gravitational model EGM2008 with ellipsoidal height derived from GPS, to determine the height datum difference between a mainland and an island. Fitting function is given as a polynomial surface. An iteration algorithm by using different regularization parameters is adopted to solve the ill-conditioned matrix in the normal equation. The precision of the additional parameter is estimated. The feasibility of the proposed method is validated by practical data of two bridge constructions. It can obtain the height datum parameter and quasi-geoid model successively. The objective of long distance height datum transfer is then achieved. Comparing the computed value with known value, the results show that the inconsistencies of the height datum difference are small. No matter how many the differences are, they can be computed precisely. Precision of the computed height datum can satisfy the engineering requirement of the height. Therefore it can be applied in practical engineering, or to validate the reliability of other methods for long distance leveling. This method is low cost, efficient and reliable, and can be generally used in geodesy. [Copyright &y& Elsevier]

Abstract: Current and new applications of 2D storm surge models such as the Dutch Continental Shelf Model (DCSM) require that the models provide proper estimates of the instantaneous water levels expressed relative to a particular geoid, rather than only the tide and surge components expressed relative to the ill-defined model''s zero height surface. For DCSM, this is realized by adding the depth-averaged horizontal baroclinic pressure gradients to the model equations, which are derived from 4D salinity and temperature fields provided by the Proudman Oceanographic Laboratory hydrodynamic model (POL''s hindcast). The vertical datum of the extended model is fixed to that of the European Gravimetric Geoid model 2008 (EGG08). This is done by an adjustment of the model parameters that depend on the choice of the reference surface (e.g., bathymetry) and by referring the water levels along the open boundaries to this reference surface. Using different numerical experiments we investigate the effects on the water levels of several approximations we have made during the implementation. The ability of the model to reproduce both the mean sea level (MSL) and instantaneous water levels is assessed by a comparison with the MSL derived from POL''s hindcast as well as with instantaneous water levels acquired by various radar altimeter satellites. From this comparison we conclude that our modeled MSL is in good agreement with the MSL derived from POL''s hindcast; the standard deviation of the differences is below 2cm. However, larger differences in MSL are observed when comparing the model output with the MSL derived from radar altimeter data. They are attributed to either geoid errors or errors in the used salinity and temperature fields. The root mean squared (rms) differences between observed and modeled instantaneous water levels over the entire model domain varies from 9cm for data acquired by the TOPEX satellite to 11cm for data acquired by the GFO-1 satellite. These numbers improve to 8–10cm on the North Sea, for data acquired by the TOPEX and ERS-2 satellites, respectively. These numbers are a factor two to three larger than the expected accuracy of water levels derived from radar altimeter data (which is ). About 25% of these differences can be explained by a bias between the modeled and observed water levels of a single satellite pass. These biases are attributed to errors in the applied correction for the net steric expansion/contraction of the global oceans as this is not captured by DCSM that makes use of the Boussinesq approximation, as well as other errors in model and data. [Copyright &y& Elsevier]

Abstract: In mountainous regions, a gravimetric geoid determination is a difficult task which needs special attention to obtain reliable results satisfying the demands, e.g. of the engineering infrastructure. The present study compares the KTH (Royal Institute of Technology) method with the remove–compute–restore (RCR) technique to attain a precise geoid model for the Konya Closed Basin (KCB) and thus aims to clarify the theoretical and numerical differences between these techniques. The KCB located in the central part of Turkey is a mountainous area with a very limited amount of the ground gravity data. Various data sources, such as the Turkish Digital Elevation Model with resolution (TRDEM3), a recently published satellite-only global geopotential model and terrestrial gravity observations are combined in the gravimetric geoid models of the KCB by applying the KTH and RCR techniques, separately. The both gravimetric geoid models (KG-KTH11 and KG-RCR11) are compared with GNSS/levelling data at the control points in the target area, resulting the RMS (Root Mean Square Error) differences of ±6.7cm and ±9.8cm in the absolute sense, respectively. These results show that the KTH method gives more reasonable accuracy than the RCR technique in a mountainous area with scarce terrestrial gravity data. [Copyright &y& Elsevier]

SUMMARY The aim of this work is to propose a first-order estimate of the crustal and lithospheric mantle geometry of the Arabia-Eurasia collision zone and to separate the measured Bouguer anomaly into its regional and local components. The crustal and lithospheric mantle structure is calculated from the geoid height and elevation data combined with thermal analysis. Our results show that Moho depth varies from ∼42 km at the Mesopotamian-Persian Gulf foreland basin to ∼60 km below the High Zagros. The lithosphere is thicker beneath the foreland basin (∼200 km) and thinner underneath the High Zagros and Central Iran (∼140 km). Most of this lithospheric mantle thinning is accommodated under the Zagros mountain belt coinciding with the suture between two different mantle domains on the Sanandaj-Sirjan Zone. The regional gravity field is obtained by calculating the gravimetric response of the 3-D crustal and lithospheric mantle structure obtained by combining elevation and geoid data. The calculated regional Bouguer anomaly differs noticeably from those obtained by filtering or just isostatic methods. The residual gravity anomaly, obtained by subtraction of the regional components to the measured field, is analyzed in terms of the dominating upper crustal structures. Deep basins and areas with salt deposits are characterized by negative values (∼−20 mGal), whereas the positive values are related to igneous and ophiolite complexes and shallow basement depths (∼20 mGal). [ABSTRACT FROM AUTHOR]

Precise information of geoid undulations is essential for obtaining the orthometric heights from Global Positioning System (GPS) measurements over any region; apart from providing the information of subsurface density distribution. This paper presents computation of geoid undulations over a part of southern Indian region from terrestrial gravity and elevation data using remove-restore technique that involves spherical Fast Fourier Transform (FFT) to compute 'Stokes' coefficients. Computed geoid undulations are compared with geoid obtained from global geopotential models such as EGM2008 and EIGEN-GRACE02S and measured GPS-levelling records at 67 locations. Statistical analysis of comparison suggests that the computed gravimetric geoid model has a good match with the geoid determined from GPS-levelling with rms of 0.1 m whereas EGM2008 has 0.46 m. The differences of GPS-levelling with EGM2008 at majority of stations fall in the range of ±0.5 m, which indicates that EGM2008 may be used for orthometric height determination with an accuracy of <0.5 m in the south Indian region and offers a reasonably good transformation platform from ellipsoid to local datum. However, local determination of geoid is necessary for better accuracy of orthometric height from GPS. The gravimetric geoid calculated from the available gravity data shows considerable improvement to the global model and can be used to achieve orthometric height with an accuracy of 0.1 m. [ABSTRACT FROM AUTHOR]

The mean dynamic topography (MDT) can be computed as the difference between the mean sea level (MSL) and a gravimetric geoid. This requires that both data sets are spectrally consistent. In practice, it is quite common that the resolution of the geoid data is less than the resolution of the MSL data, hence, the latter need to be low-pass filtered before the MDT is computed. For this purpose conventional low-pass filters are inadequate, failing in coastal regions where they run into the undefined MSL signal on the continents. In this paper, we consider the use of a bandlimited, spatially concentrated Slepian basis to obtain a low-resolution approximation of the MSL signal. We compute Slepian functions for the oceans and parts of the oceans and compare the performance of calculating the MDT via this approach with other methods, in particular the iterative spherical harmonic approach in combination with Gaussian low-pass filtering, and various modifications. Based on the numerical experiments, we conclude that none of these methods provide a low-resolution MSL approximation at the sub-decimetre level. In particular, we show that Slepian functions are not appropriate basis functions for this problem, and a Slepian representation of the low-resolution MSL signal suffers from broadband leakage. We also show that a meaningful definition of a low-resolution MSL over incomplete spherical domains involves orthogonal basis functions with additional properties that Slepian functions do not possess. A low-resolution MSL signal, spectrally consistent with a given geoid model, is obtained by a suitable truncation of the expansions of the MSL signal in terms of these orthogonal basis functions. We compute one of these sets of orthogonal basis functions using the Gram-Schmidt orthogonalization for spherical harmonics. For the oceans, we could construct an orthogonal basis only for resolutions equivalent to a spherical harmonic degree 36. The computation of a basis with a higher resolution fails due to inherent instabilities. Regularization reduces the instabilities but destroys the orthogonality and, therefore, provides unrealistic low-resolution MSL approximations. More research is needed to solve the instability problem, perhaps by finding a different orthogonal basis that avoids it altogether. [ABSTRACT FROM AUTHOR]

The forthcoming GRAV-D gravimetric geoid model over the United States is to be updated regularly to account for changes in geoid height. Its baseline precision is to be at the 10-20 mm level over non-mountainous regions. The aim of this study is to provide an estimate of the magnitude, time scale, and spatial footprint of geoid height change over North America, from mass redistribution processes of hydrologic, cryospheric and solid Earth nature. Geoid height changes from continental water storage changes over the past 50 years and predicted over the next century are evaluated and are highly dependent on the used model. Groundwater depletion from anthropogenic pumping in regional scale aquifers may lead to geoid changes of 10 mm magnitude every 50-100 years. The GRACE time varying gravity fields are used to (i) assess the errors in a glacial isostatic adjustment model, which, if used to correct the GRAV-D model, may induce errors at the 10 mm geoid height level after ~20 years, (ii), evaluate geoid height change over ice mass loss regions of North America, which, if they remain unchanged in the future, may lead to geoid height changes at the 10 mm level in under a decade and (iii), compute sea level rise and its effect on the geoid, which is found to be negligible. Coseismic gravitational changes from past North American earthquakes are evaluated, and lead to geoid change at the 10-mm level for only the largest thrust earthquakes. Finally, geoid change from volcanic processes are assessed and found to be significant with respect to the GRAV-D geoid model baseline precision for cataclysmic events, such as that of the 1980 Mt. St. Helens eruption. Recommendations on how to best monitor geoid change in the future are given. [ABSTRACT FROM AUTHOR]

A new gravimetric geoid model, USGG2009 (see Abbreviations), has been developed for the United States and its territories including the Conterminous US (CONUS), Alaska, Hawaii, Guam, the Commonwealth of the Northern Mariana Islands, American Samoa, Puerto Rico and the US Virgin Islands. USGG2009 is based on a 1′ × 1′ gravity grid derived from the NGS surface gravity data and the DNSC08 altimetry-derived anomalies, the SRTM-DTED1 3′′ DEM for its topographic reductions, and the global geopotential model EGM08 as a reference model. USGG2009 geoid heights are compared with control values determined at 18,398 Bench Marks over CONUS, where both the ellipsoidal height above NAD 83 and the Helmert orthometric height above NAVD 88 are known. Correcting for the ellipsoidal datum difference, this permits a comparison of the geoid heights to independent data. The standard deviation of the differences is 6.3 cm in contrast to 8.4 cm for its immediate predecessor- USGG2003. To minimize the effect of long-wavelength errors that are known to exist in NAVD88, these comparisons were made on a state-by-state basis. The standard deviations of the differences range from 3-5 cm in eastern states to about 6-9 cm in the more mountainous western states. If the GPS/Bench Marks-derived geoid heights are corrected by removing a GRACE-derived estimate of the long-wavelength NAVD88 errors before the comparison, the standard deviation of their differences from USGG2009 drops to 4.3 cm nationally and 2-4 cm in eastern states and 4-8 in states with a maximum error of 26.4 cm in California and minimum of −32.1 cm in Washington. USGG2009 is also compared with geoid heights derived from 40 tide-gauges and a physical dynamic ocean topography model in the Gulf of Mexico; the mean of the differences is 3.3 cm and their standard deviation is 5.0 cm. When USGG2009-derived deflections of the vertical are compared with 3,415 observed surface astro-geodetic deflections, the standard deviation of the differences in the N-S and E-W components are 0.87′′ and 0.94′′, respectively. [ABSTRACT FROM AUTHOR]

Abstract: Gravimetric geoid computation is often based on modified Stokes''s integration, where Stokes''s integral is evaluated with some stochastic or deterministic kernel modification. Accurate numerical evaluation of Stokes''s integral requires the modified kernel to be integrated across the area of each discretised grid cell (mean kernel). Evaluating the modified kernel at the center of the cell (point kernel) is an approximation, which may result in larger numerical integration errors near the computation point, where the modified kernel exhibits a strongly nonlinear behavior. The present study deals with the computation of whole-of-the-cell mean values of modified kernels, exemplified here with the kernel modification [Featherstone, W.E., Evans, J.D., Olliver, J.G., 1998. A Meissl-modified Vaníček and Kleusberg kernel to reduce the truncation error in gravimetric geoid computations. Journal of Geodesy 72(3), 154–160]. We investigate two approaches (analytical and numerical integration), which are capable of providing accurate mean kernels. The analytical integration approach is based on kernel weighting factors which are used for the conversion of point to mean kernels. For the efficient numerical integration, Gauss–Legendre quadrature is applied. The comparison of mean kernels from both approaches shows a satisfactory mutual agreement at the level of 10−4 and better, which is considered to be sufficient for practical geoid computation requirements. Closed-loop tests based on the EGM2008 geopotential model demonstrate that using mean instead of point kernels reduces numerical integration errors by ∼65%. The use of mean kernels is recommended in remove–compute–restore geoid determination with the kernel or any other kernel modification under the condition that the kernel changes rapidly across the cells in the neighborhood of the computation point. [Copyright &y& Elsevier]

Today air-gravimetry is a versatile technique to quickly collect gravity data over large regions, where terrestrial gravity data are sparse and/or of poor quality. The method requires the data to be downward continued to sea level for use in geoid determination, an inverse problem operation that calls for smoothing of the data and/or the kernel function involved (in either spectral or space domain). In this purely theoretical study we avoid this separate computational step by performing direct geoid estimation by so-called spectral combination/filtering of the data, which includes terrestrial gravimetry, airgravimetry, an Earth Gravitational Model (EGM) as well as their signal and error degree variances. Each derived geoid estimator is presented as the sum of one or two integral formulas and the harmonic series of the EGM together with the expected mean square error of the estimator. The article is limited to a theoretical study, leaving its practical tests for future investigation. [ABSTRACT FROM AUTHOR]

Abstract: Sudan is characterized by vast area with flat terrains in most of the country’s regions and therefore the existence of a high-resolution geoid model is considered very important especially with the widespread of the GPS technology in the country. A few studies were previously conducted to compute a geoid model for Sudan, the current study is the second of its kind to compute the Sudanese geoid model by means of gravimetric method. In this study, a new gravimetric geoid model (KTH-SDG08) is computed for Sudan, we apply the method developed at the Royal Institute of Technology (KTH) Stockholm-Sweden. The method utilizes the least-squares modification (LSM) of Stokes formula providing three stochastic solutions (biased, unbiased and optimum). The three solutions are quite similar however in this study we select the optimum solution which provides the best agreement with GPS-levelling data. The modified Stokes formula combines the regional terrestrial gravity data together with the long-wavelength gravity information from a global gravitational model (GGM). We use two sets of regional gravity data: the first set is provided by the GETECH, the points of this set cover discretely some parts of Sudan while the rest of the country’s area are still not covered, the second set was provided by BGI contains a few points scattered over the neighbouring countries. Both datasets (GETECH and BGI) are unified together into one dataset and evaluated by Cross validation technique. Due to the lack of gravity observations we construct our final grid for geoid computation with spatial resolution of 5×5 arc-min. The long-wavelength contribution is donated by two geopotential models, EIGEN-GRACE02S satellite-only model is employed in the modified Stokes formula whereas the EIGEN-GL04C combined model is used to enrich the local gravity coverage over the areas with missing data. The Digital Elevation Model (DEM), SRTM generated byNASAand theNational Imagery and Mapping Agency (NIMA) is used to compute topography effects on the geoid. Four additive corrections are computed over the entire target area and applied to the approximate geoid heights obtaining the final geoid solution. The new Sudanese gravimetric geoid model KTH-SDG08 is computed on a 5×5 arc-min geographical grid over the computation area bounded by the parallels of 4 and 23arc-deg northern latitude, and the meridians of 22 and 38arc-deg of eastern longitude. The gravimetric geoid is validated using GPS-levelling information at 19 points distributed over the whole country. The results show that the standard deviation (STD) of differences between the gravimetric and geometric geoid heights at 19 GPS-levelling points is about 0.3m. [Copyright &y& Elsevier]

Abstract: A new geoid computed with high-resolution for Italy, is presented in this paper. The computation of this gravimetric geoid is based on the most recent geopotential model: EGM2008 (Earth Gravity Model released in 2008). The method used in the computation of this new gravimetric geoid has been the Stokes integral in convolution form. The terrain correction has been applied to the gridded gravity anomalies, to obtain the corresponding reduced anomalies. Also the indirect effect has been taken into account. Thus, a new geoid model has been calculated and it is provided as a data grid in the Geodetic Reference System of 1980 (GRS80), distributed for the study area from 37° to 48° of latitude and 6° to 19° of longitude, on a 441×521 regular grid with a mesh size of 1.5′×1.5′. This new high-resolution geoid and the global geoid EGM2008; are compared with the geoid undulations measured for 11 points of the EUropean Vertical Network (EUVN) on Italy. The new geoid shows an improvement in precision and reliability, fitting the geoidal heights of these EUVN points with more accuracy than the global geoid. Moreover, this new geoid has a smaller standard deviation (12.5cm) than that obtained by any previous geoid developed for Italy (and the surrounding area) up to date (). This new model will be useful for orthometric height determination by GPS over this study area, because it will allow orthometric height determination in the mountains and remote areas, in which levelling has many logistic problems. This new model can be also interesting for other geophysical purposes, other than geodesy and the height measurements, because it can provide a constraint for the density distribution and the thermal state of lithosphere and the viscosity in the mantle. Such details can be inferred from a geoid model and the seismic velocity structure. [ABSTRACT FROM AUTHOR]

Gravity data in Estonia have been collected by different institutions over many decades. This study assesses the suitability of available gravity data for ensuring a 1 cm geoid modelling accuracy over Estonia and in the Baltic Sea region in general. The main focus of this study is on the determination and elimination of discrepancies between three nationwide datasets. It was detected that one tested historic gravity dataset contained inadmissible systematic biases with respect to other tested datasets. Possible ways of gravity data improvement are discussed. More specifically, new field observation campaigns and aspects of using their outcomes in subsequent regional geoid modelling are suggested. [ABSTRACT FROM AUTHOR]

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Abstract: A statistical comparison has been made between gravity field parameters derived from different global geopotential models (GGMs) and observed gravity anomalies, gravimetric geoid and GPS-Leveling data. The motivation behind this study is the selection of best possible global geopotential model that best matches statistically with the local observed data in Pakistan. This will facilitate in decreasing the load on observed data for the development of regional gravimetric geoid in remove-compute-restore technique when used in the Stokes’s integral for computation of the residual part. It is observed that combined geopotential models such as EGM96 and PGM200A, EIGEN-GL04C and EIGEN-CG03C reflect the better match in the total spectral range of gravity and GPS-Leveling data. Results of the precise local geoid model also indicate similar characteristics. A very-high-degree model “EGM2008” (degree/order 2160) exhibits relatively superior statistical fit with observed ground data in Pakistan region. For satellite-only models an increasing trend in the standard deviation can be seen with maximum of about ∼4m in difference between GPS-Leveling and corresponding GGM’s geoid with increase in the order from 50 to 120 and then it decreases afterwards. However, for the EIGEN-CHAMP03SP, standard deviation saturates to a value of 3.4m. This is an indication of contamination in the long to medium wavelength part, i.e. 50–100° for the satellite-only models. Moreover, the models DEOS-CHAMP-01C, GGM02C and then ITG-GRACE03 appear to have better fit for medium to long wavelength and can possibly be recommended for use as long wavelength part with the local observed data. While a hybrid geopotential model selection can be achieved through the selection from either of DEOS-CHAMP-01C, GGM02C, GGM02S, EIGEN-GRACE02S or ITG-GRACE03 in the long wavelength (to degree and order 40) and EGM96, PGM200A, EIGEN-GL04C, EIGEN-CG03C or even EGM2008 in medium to short wavelength, i.e. from degree 41 to maximum degree and order. [Copyright &y& Elsevier]

In geoid computation, effects of real three-dimensional topographic masses on the Earth’s gravity field must be accurately quantified and, in the Stokes–Helmert scheme, replaced with effects of those masses condensed on the geoid. The most comprehensive modern schemes for evaluation of topographical effects account for terrain effects, use a spherical model of topography, and incorporate two-dimensionally varying models of topographical mass density. In this contribution, we employ a three-dimensionally varying model of topographical density. We use Newton’s integration to determine the direct topographical effect (DTE) on gravity and primary indirect topographical effect (PITE) on gravity potential. Lastly, we apply Stokes’ integration to calculate the DTE, PITE, and secondary indirect topographical effect (SITE) on geoidal height. We focus here on validation of our results and demonstration of our software’s capabilities. We present results for the simple geometrical shape of a disc under various rotations and for the anomalous density of lake waters. Effects on geoidal height for these simulations reach centimetre level, up to 2.2 cm in magnitude. For a simulation of the effects of neglected mass anomalies of the lakes, we find results reaching 0.8 cm in magnitude. We examine the behavior of our results as calculated using various step sizes for numerical integration and by comparing numerical results with analytical results for the specific case of a disc. These results suggest that the maximum percent error of our results is about 23.5% for the DTE on gravity and 7.6% for the PITE on gravity potential. [ABSTRACT FROM AUTHOR]

Abstract  This paper deals with the analysis of gravity anomaly and precise levelling in conjunction with GPS-Levelling data for the computation of a gravimetric geoid and an estimate of the height system bias parameter No for the vertical datum in Pakistan by means of least squares collocation technique. The long term objective is to obtain a regional geoid (or quasi-geoid) modeling using a combination of local data with a high degree and order Earth gravity model (EGM) and to determine a bias (if there is one) with respect to a global mean sea surface. An application of collocation with the optimal covariance parameters has facilitated to achieve gravimetric height anomalies in a global geocentric datum. Residual terrain modeling (RTM) technique has been used in combination with the EGM96 for the reduction and smoothing of the gravity data. A value for the bias parameter No has been estimated with reference to the local GPS-Levelling datum that appears to be 0.705 m with 0.07 m mean square error. The gravimetric height anomalies were compared with height anomalies obtained from GPS-Levelling stations using least square collocation with and without bias adjustment. The bias adjustment minimizes the difference between the gravimetric height anomalies with respect to residual GPS-Levelling data and the standard deviation of the differences drops from 35 cm to 2.6 cm. The results of this study suggest that No adjustment may be a good alternative for the fitting of the final gravimetric geoid as is generally done when using FFT methods. [ABSTRACT FROM AUTHOR]

Abstract: A new gravimetric geoid model (ICAGM07) has been determined for the North-East Atlantic Ocean, Iberia, and the Macaronesian Islands using multi-sensor gravity data and a GRACE derived Earth geopotential model. A high resolution gravity model, determined using least squares optimal interpolation of marine, land, and satellite derived gravity anomalies, was used to resolve the medium and short wavelengths of the geoid. Long wavelengths of the geoid were provided by the GRACE derived Earth geopotential model. The topographic effects were computed in the spectral domain using a high resolution (100m) digital terrain model derived from SRTM mission data and cartographic charts. The remove–restore technique was used to compute the geoid model on a 1.5′ grid, and the residual geoid height was computed using spherical FFT and a modified Stokes’ kernel. The effects of different Earth tide models on the geoid were computed and analysed. Comparison over the sea with an oceanographic geoid determined from CLS01 MSSH and Rio05 MDT yield a relative accuracy of 8cm, with larger differences close to the shoreline. Over land, comparisons with 1646 GPS/levelling marks indicate an overall precision of 8–10cm and relative vertical datum offsets of up to 2m. This new geoid model represents a significant improvement over existing geoid models, with a homogeneous relative accuracy of 8–10cm over both marine and land areas, and exposes the inaccuracies of local vertical datums as references for studies of vertical deformation. [Copyright &y& Elsevier]

Airborne gravimetry is an important method for measuring gravity over large unsurveyed areas. This technology has been widely applied in Canada, Antarctica and Greenland to map the gravity fields of these regions and in recent years, in the oil industry. In 2005, two tests in the Italian area were performed by ENI in cooperation with the Politecnico di Milano and the Danish National Space Center. To the knowledge of the authors, these were the first experiments of this kind in Italy and were performed over the Ionian coasts of Calabria and the Maiella Mountains. The Calabria test field is characterized by strong gravity variations due to the geophysical and topographic structure of the area. The ground gravity coverage is also quite dense. It was thus possible to compare airborne gravity with the ground observed values in order to check the precision of the airborne gravimetry. The second campaign was performed in an unsurveyed area centred on the Maiella Mountains, thus filling the data gap of this zone. Comparisons with existing ground data were also carried out in this area. After smoothing, the collected data have an accuracy of 2–3 mgal, as derived by cross-over analysis. Moreover, the collocation method applied to compare and merge ground-based and airborne data proved to be efficient and reliable. The standard deviation of the discrepancies between airborne data and collocation upward continued gravity is, in both cases, less than 8 mgal. In the Maiella test, the gravity field obtained by merging airborne and ground data using collocation also provides a more detailed description of the high-frequency pattern of the geopotential field in this area. [ABSTRACT FROM AUTHOR]

Abstract: In Algeria and since 2000, two published geoid models have been produced by the Geodetic Laboratory of the National Centre of Space Techniques using different data sets and techniques. Although these results were satisfactory and internally consistent they do not have the required accuracy to be able to transform a GPS ellipsoidal height to an orthometric height. Recently, the quantity and quality of terrestrial gravity data increased slightly and, especially, several new geopotential models from the recent satellite missions (e.g., GRACE) were released. At the same time, the new high-resolution SRTM (Shuttle Radar Topography Mission) global DEM was constructed. The main purpose of this paper is to study the impact of the above new data on geoid heights in Algeria. For this reason, a new gravimetric geoid determination has been carried out including these new data. This solution was based on the land gravity data supplied by the BGI (Bureau Gravimétrique International), Digital Elevation model derived from SRTM for topographic correction and the optimal gravity field model GRACE (EIGEN-GL04C), which were combined using the remove–restore technique in connection with the Fast Fourier transformation technique. This gravimetric geoid and previous geoids existing for this study area are compared to the geoid undulations corresponding to 62GPS/levelling points located in northern part of Algeria. The study shows that the new gravimetric geoid model agrees considerably better with GPS/levelling than any of the other local geoid models in the tested area. Its standard deviation fit with GPS/levelling data are ±27 and ±25.0cm before and after fitting using the four-parametric model as corrector surface in minimizing the long-wavelength geoid errors and the datum inconsistencies between our height data and GPS. The new geoid model will be used in low accuracy scientific applications and in low-order levelling network densification with regard to the national levelling network coverage considered as good in the north and becomes poor in the south and west of the country. It is an important contribution to geophysics, because it can provide a constraint for density distribution, thermal state of the lithosphere and viscosity in the mantle. [Copyright &y& Elsevier]

Abstract  New Zealand uses 13 separate local vertical datums (LVDs) based on geodetic levelling from 12 different tide-gauges. We describe their unification using a regional gravimetric quasigeoid model and GPS-levelling data on each LVD. A novel application of iterative quasigeoid computation is used, where the LVD offsets computed from earlier models are used to apply additional gravity reductions from each LVD to that model. The solution converges after only three iterations yielding LVD offsets ranging from 0.24 to 0.58 m with an average standard deviation of ±0.08 m. The so-computed LVD offsets agree, within expected data errors, with geodetically levelled height differences at common benchmarks between adjacent LVDs. This shows that iterated quasigeoid models have a role in vertical datum unification. [ABSTRACT FROM AUTHOR]

The idea was to compare the geoid of sea areas by an independent method, like GPS levelling, on the mainland. On the earth surface we can compare the gravimetric geoid with GPS levelling to get an accuracy estimation and tilt information. On the sea we can do it by the GPS methodology and eliminating the current water tilt corrections and the sea surface topography effect. A modern GPS device on board a ferry can store data every second and determine heights with an accuracy of a few centimetres (using the kinematic method with the post-processing of data obtained from several base stations close to the ferry line). As a result, it is possible to observe the current water level's relative profile in reference to the ellipsoid. Some areas close to Estonia, such as the eastern part of the Gulf of Finland, are not completely covered by gravity measurements. The Baltic Sea has been measured using airborne gravimetry with the accuracy of about 2 mGal. Therefore, the gravimetric geoid is not fully reliable for the region either. If we take into account the tilt of the water level at the moment of measurement, we can observe the relative change of the geoid using an independent methodology, which serves as a comparison to the gravimetric geoid solution. The main problem during the measurement campaign, of course, was how to eliminate a water tilt. Water placement in relation to level surface is a very complex issue; special studies of that were conducted as well. [ABSTRACT FROM AUTHOR]

Abstract: A new gravimetric geoid is computed for South Spain and the Gibraltar Strait area. This geoid is located just in the junction between two tectonic plates (Euro-Asiatic and African plates) and in the junction of two gravimetric geoids: IGG2005 (the Iberian Gravimetric Geoid obtained in 2005) and MORGEO (the MORoccan GEOid). IGG2005 is the Iberian geoid and MORGEO is the Moroccan geoid, both geoids have been previously obtained. The new geoid is the gravimetric geoid solution that connects the two above-mentioned geoids, getting a more accurate and reliable picture of this area than the other previous geoids. The method used is the Stokes integral in convolution form, which shows to be an efficient method to reach the proposed objective. The terrain correction and the indirect effect have been taken into account. The new geoid is obtained as a regular grid (with a mesh size of 1.5′×1.5′) in the GRS80 reference system, covering the study area from 34° to 40° of latitude and from −8° to 0° of longitude. This gravimetric geoid and the previous geoids: IGG2005 and MORGEO; are compared to the geoid undulations derived at the validation points located on the study area (four GPS/levelling points measured on Morocco and five points of the European vertical reference network (EUVN) measured on Iberia). As it is expected, the new geoid is a more precise and reliable model, fitting the geoidal heights of these validation points with more accuracy than the other previous geoids. This new model will be useful for orthometric height determination by GPS in the mountains and remote areas, where levelling has many logistic problems. Also, it can be interesting for other geophysical purposes different to the height measurements, because it can provide a constraint for the density distribution, the thermal state of Lithosphere and the viscosity in the mantle. Such details can be inferred from a geoid model and the seismic velocity structure. [Copyright &y& Elsevier]

Abstract: The choice of the best geopotential model to reduce geodetic data is one of the critical steps in computing the geoid. Several studies have shown that the geopotential models tailored to regional or local gravity data are best suited for high precision geoid computations. Since 2000 a number of geoid models for Algeria have been produced by Geodetic Laboratory of the National Centre of Space Techniques. In particular 5′×5′ geoid models were generated in 2000, [] and in 2004 [ using different data sets and techniques. Although these results were satisfactory and internally consistent they do no have the required accuracy to be able to transform a GPS ellipsoidal height to an orthometric height. During the same time and with the recent satellite missions CHAMP and GRACE several new global gravity models were released. These lead to substantial improvements of our knowledge of the long-wavelength part of the Earth''s gravity field, and thereby of the long-wavelengths of the geoid. For the computation of a new gravimetric geoid model for Algeria we need a new investigation on the choice of the best and optimal geopotential model for the combined solution with local gravimetric and topographic data using the remove-restore technique. In this paper, an analysis was carried out to define the geopotential model, which fits best the local gravity field in Algeria. Six global geopotential models are used in this study: The new GRACE satellite-only and combined models EIGEN-GRACE02S and GGM02C, combined CHAMP and GRACE model EIGEN-CG01C, combined CHAMP and LAGEOS model EIGEN-GL04C, OSU91A and EGM96. The test of the fitting of these high order geopotential models to the gravity field in Algeria is based on the gravity data supplied by the B.G.I. and GETECH, and some of the precise GPS data collected from the international TYRGEONET (TYRhenian GEOdynamical NETwork), ALGEONET (ALGerian GEOdynamical NETwork) projects with baseline length ranging from about 1 to 1000km. The comparisons were made at all gravity and GPS levelled data by computation of the residual data (i.e. observed data minus model). The geopotential model that provides the closest statistical fit to these data can be assumed to be the most suitable model to adopt for the determination of the new Algerian gravimetric geoid. The study shows that the newly released combined model (EIGEN-GL04C) is consistently superior to other models in the Algerian region. Its standard deviation fit with GPS/levelling data is 30cm and 27.5cm before and after fitting. Hence, we strongly recommend the use of this new model in the computation of the new gravimetric geoid model for Algeria. [Copyright &y& Elsevier]

We derive a number of rough theoretical estimates for the precision of a geoid model computed from a local gravimetric survey combined with global reference model information. Example calculations for Finland and Estonia are presented. [ABSTRACT FROM AUTHOR]

National leveling height networks have recently been set up in Taiwan, and global positioning system (GPS) observations were carried out at the benchmarks of the networks. The purpose of this paper is to develop an improved local geoid model for Taiwan by integrating a local gravimetric geoid model with the GPS and leveling height data. To overcome the problems of datum inconsistencies and possible systematic errors in height data of different types, the least squares collocation technique was applied. The result from fit of the improved geoid model to the control data at 300 benchmarks shows that a mean accuracy of geoidal heights at a 2 cm level is obtainable. [ABSTRACT FROM AUTHOR]

Abstract: The gravity field of the earth is a natural element of the Global Geodetic Observing System (GGOS). Gravity field quantities are like spatial geodetic observations of potential very high accuracy, with measurements, currently at part-per-billion (ppb) accuracy, but gravity field quantities are also unique as they can be globally represented by harmonic functions (long-wavelength geopotential model primarily from satellite gravity field missions), or based on point sampling (airborne and in situ absolute and superconducting gravimetry). From a GGOS global perspective, one of the main challenges is to ensure the consistency of the global and regional geopotential and geoid models, and the temporal changes of the gravity field at large spatial scales. The International Gravity Field Service, an umbrella “level-2” IAG service (incorporating the International Gravity Bureau, International Geoid Service, International Center for Earth Tides, International Center for Global Earth models, and other future new services for, e.g., digital terrain models), would be a natural key element contributing to GGOS. Major parts of the work of the services would, however, remain complementary to the GGOS contributions, which focus on the long-wavelength components of the geopotential and its temporal variations, the consistent procedures for regional data processing in a unified vertical datum and Terrestrial Reference Frame, and the ensuring validations of long-wavelength gravity field data products. [Copyright &y& Elsevier]

It is well known that orthometric heights can be obtained without levelling by using ellipsoidal and geoidal heights. For engineering purposes, these orthometric heights must be determined with high accuracy. For this reason, the determination of a high-resolution geoid is necessary. In Iberia, since the publication of the most recent geoid (IBERGEO95), a new geopotential model has become available (EIGEN-CG01C, released on 2004 October 29) and a new high-resolution digital terrain model (SRTM 90M obtained from the Shuttle Radar Topography Mission) has been developed for the Earth. Logically, these new data represent improvements that must be included in a new geoid of Iberia. With this goal in mind, we have carried out a new gravimetric geoid determination in which these new data are included. The computation of the geoid uses the Stokes integral in convolution form, which has been shown as an efficient method to reach the proposed objective. The terrain correction has been applied to the gridded gravity anomalies to obtain the corresponding reduced anomalies. The indirect effect has also been taken into account. Thus, a new geoid is provided as grid data distributed for Iberia from 35° to 44° latitude and –10° to 4° longitude (extending to ) in a regular grid with a mesh size of and 202 521 points in the GRS80 reference system. This calculated geoid and previous geoids that exist for this study area (IBERGEO95, EGM96, EGG97 and EIGEN-CG01C) are compared to the geoid undulations corresponding to 16 points of the European Vertical Reference Network (EUVN) on Iberia. The new geoid shows an improvement in precision and reliability, fitting the geoidal heights of these EUVN points with more accuracy than the other previous geoids. [ABSTRACT FROM AUTHOR]

Abstract: A new gravimetric, satellite altimetry, astronomical ellipsoidal boundary value problem for geoid computations has been developed and successfully tested. This boundary value problem has been constructed for gravity observables of the type (i) gravity potential, (ii) gravity intensity (i.e. modulus of gravity acceleration), (iii) astronomical longitude, (iv) astronomical latitude and (v) satellite altimetry observations. The ellipsoidal coordinates of the observation points have been considered as known quantities in the set-up of the problem in the light of availability of GPS coordinates. The developed boundary value problem is ellipsoidal by nature and as such takes advantage of high precision GPS observations in the set-up. The algorithmic steps of the solution of the boundary value problem are as follows: [-] Application of the ellipsoidal harmonic expansion complete up to degree and order 360 and of the ellipsoidal centrifugal field for the removal of the effect of global gravity and the isostasy field from the gravity intensity and the astronomical observations at the surface of the Earth. [-] Application of the ellipsoidal Newton integral on the multi-cylindrical equal-area map projection surface for the removal from the gravity intensity and the astronomical observations at the surface of the Earth the effect of the residual masses at the radius of up to 55km from the computational point. [-] Application of the ellipsoidal harmonic expansion complete up to degree and order 360 and ellipsoidal centrifugal field for the removal from the geoidal undulations derived from satellite altimetry the effect of the global gravity and isostasy on the geoidal undulations. [-] Application of the ellipsoidal Newton integral on the multi-cylindrical equal-area map projection surface for the removal from the geoidal undulations derived from satellite altimetry the effect of the water masses outside the reference ellipsoid within a radius of 55km around the computational point. [-] Least squares solution of the observation equations of the incremental quantities derived from aforementioned steps in order to obtain the incremental gravity potential at the surface of the reference ellipsoid. [-] The removed effects at the application points are restored on the surface of reference ellipsoid. [-] Application of the ellipsoidal Bruns’ formula for converting the potential values on the surface of the reference ellipsoid into the geoidal heights with respect to the reference ellipsoid. [-] Computation of the geoid of Iran has successfully tested this new methodology. [Copyright &y& Elsevier]

Abstract: By delivering monthly maps of the gravity field, the GRACE project allows the determination of tiny time variations of the Earth''s gravity and particularly the effects of fluid mass redistributions at the surface of the Earth. However, GRACE data represent vertically integrated gravity measurements, thus are the sum of all mass redistributions inside the Earth''s system (atmosphere, oceans and continental water storage, plus solid Earth). In this paper, we apply a generalized least-squares inverse approach, previously developed by [1] [G. Ramillien, A. Cazenave, O. Brunau, Global time-variations of hydrological signals from GRACE satellite gravimetry, Geophys. J. Int. 158 (2004) 813–826.], to estimate, from the monthly GRACE geoids, continental water storage variations (and their associated uncertainties) over a 2-year time span (April 2002 to May 2004). Tests demonstrating the robustness of the method are presented, including the separation between liquid water reservoirs (surface waters+soil moisture+groundwaters) and snow pack contributions. Individual monthly solutions of total land water storage from GRACE, with a spatial resolution of ∼660 km, are presented for the 2-year time span. We also derive the seasonal cycle map. We further estimate water volume changes over eight large river basins in the tropics and compare with model predictions. Finally, we attempt to estimate an average value of the evapotranspiration over each river basin, using the water balance equation which links temporal change in water volume to precipitation, evapotranspiration and runoff. Amplitudes of the GRACE-derived evapotranspiration are regionally consistent to the predictions of global hydrological models. [Copyright &y& Elsevier]

In this study, two methodologies are investigated for geoid determination from ground and airborne gravity data. These two methodologies depend on the downward continuation method used. The first is the inverse Poisson integral; the second is the normal free-air gradient. Each of the two methods requires different treatment of the terrain effects and in turn different approaches to determine the geoid. The two geoid solutions, from ground data, are compared with existing GPS/levelling benchmarks and it is found that the second method gives a better fit due to the bias introduced from the inverse Poisson integral. The same process was applied to the airborne data, but with additional processing, that is the filtering of the terrain effects to preserve the consistency of the data due to the filtering of the airborne data. A study on the effect of filtering was also carried out in this paper and it concluded that filtering the terrain effects has no impact on the geoid. In addition, the airborne data, filtered to three different cut-off frequencies, were used to compute the geoid to investigate the possibility of using the denser data, of lower accuracy, to determine a high-resolution geoid. Even though the data filtered to small cut-off frequency have poorer agreement with the ground data, the geoids computed from the different filtered data is the nearly the same. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 Grafarend?s 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. [ABSTRACT FROM AUTHOR]

The problem of global geoid determination is usually solved using satellite altimetry data on the oceans, together with an oceanographic model of sea surface topography, and gravity anomaly data on the continents. Such data, however, enable to obtain only potential differences with respect to a reference surface whose absolute potential is unknown. This situation suggests to modify the classical mixed boundary-value problem of physical geodesy by inserting into the boundary conditions an unknown additive constant, that must be determined by imposing a suitable additional constraint. Yet, such formulation of the boundary-value problem, from the point of view of its mathematical properties, is not unconditionally well-posed, and, furthermore, does not reflect faithfully the available physical model, as the present knowledge of ocean circulation does not allow to connect along coastlines the reference surfaces defined on the oceans and on the continents. The introduction of two different unknown additive constants, one for the oceans and one for the earth, to be determined by imposing two additional constraints, gives rise to a more faithful picture of the present physical knowledge, and, at the same time, to a new well-posed formulation of the boundary-value problem. [ABSTRACT FROM AUTHOR]

The high precision gravity measurements to be made by recently launched (and recently approved) satellites place new demands on models of Earth, atmospheric, and oceanic tides. The latter is the most problematic. The ocean tides induce variations in the Earth's geoid by amounts that far exceed the new satellite sensitivities, and tidal models must be used to correct for this. Two methods are used here to determine the standard errors in current ocean tide models. At long wave- lengths these errors exceed the sensitivity of the GRACE mission. Tidal errors will not prevent the new satellite missions from improving our knowledge of the geopotential by orders of magnitude, but the errors may well contaminate GRACE estimates of temporal variations in gravity. Solar tides are especially problematic because of their long alias periods. The satellite data may be used to improve tidal models once a sufficiently long time series is obtained. Improvements in the long-wavelength components of lunar tides are especially promising. [ABSTRACT FROM AUTHOR]

This paper introduces the first ESA Core Earth Explorer mission, GOCE, in the context of ESA's Living Planet programme. GOCE will measure highly accurate, high spatial resolution differential accelerations in three dimensions along a well characterised orbit: the mission is planned for launch in early 2006. The mission objectives are to obtain gravity gradient data such that new global and regional models of the static Earth's gravity field and of the geoid can be deduced at length scales down to 100 km. These products will have broad application in the fields of geodesy, oceanography, solid-earth physics and glaciology. [ABSTRACT FROM AUTHOR]

As an additional product to centimetric geoid determination for Mexican area, and taking in account that gravimetrical information is useful for other earth sciences, the National Institute of Statistics, Geography and Informatics (INEGI), take the task to generate an assortment of cartographical products that show variations of terrestrial gravity field at interested region. In this paper are shown the theoretical aspect, as well as employed methodology. [ABSTRACT FROM AUTHOR]

Investigates the reduction of aliasing in gravity anomalies and geoid heights using digital terrain data in Australia. Effect of irregular sampling geometry on gravimetry; Reduction the amount of aliasing; Determination of gravimetric geoid.

The article discusses a U.S. National Oceanic and Atmospheric Administration (NOAA) project called Gravity for the Redefinition of the American Vertical Datum (GRAV-D). The purpose of the project is to measure gravity with the hope of improving coastal monitoring. A NOAA Cessna airplane will fly around the U.S. with a gravimeter, surveying and siting National Geodetic Survey markers which will help scientists update their model of the geoid which is a warped, fictional surface on Earth.