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Since the publication of the Earth gravitational model (EGM)96 considerable improvements in the observation techniques resulted in the development of new improved models. The improvements are due to the availability of data from dedicated gravity mapping missions (CHAMP, GRACE) and to the use of 5′ × 5′ terrestrial and altimetry derived gravity anomalies. It is expected that the use of new EGMs will further contribute to the improvement of the resolution and accuracy of the gravity and geoid modeling in continental and regional scale. To prove this numerically, three representative Earth gravitational models are used for the reduction of several kinds of data related to the gravity field in different places of the Earth. The results of the reduction are discussed regarding the corresponding covariance functions which might be used for modeling using the least squares collocation method. The contribution of the EIGEN-GL04C model in most cases is comparable to that of EGM96. However, the big difference is shown in the case of EGM2008, due not only to its quality but obviously to its high degree of expansion. Almost in all cases the variance and the correlation length of the covariance functions of data reduced to this model up to its maximum degree are only a few percentages of corresponding quantities of the same data reduced up to degree 360. Furthermore, the mean value and the standard deviation of the reduced gravity anomalies in extended areas of the Earth such as Australia, Arctic region, Scandinavia or the Canadian plains, vary between −1 and +1 and between 5 and 10 × 10−5 ms−2, respectively, reflecting the homogenization of the gravity field on a regional scale. This is very important in using least squares collocation for regional applications. However, the distance to the first zero-value was in several cases much longer than warranted by the high degree of the expansion. This is attributed to errors of medium wavelengths stemming from the lack of, e.g., high-quality data in some area.

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.

Digital terrain models are the basis for geodetic, geophysical, oceanographic and other studies in regional or global scale. In gravity field modelling the global DTMs ETOPO5, TerrainBase and JGP95E have been used among other in combination with other data related to the gravity field, for the development of geopotential models. An intercomparison of these DTMs showed considerable differences between them. The standard deviation of these differences are of the order of 100m, while the minimum and the maximum differences range from −3.8km to 4.7km were observed. This statistics shows that the models are affected by considerable errors. The refinement of the DTMs will contribute to the improvemet of the gravity field approximation.

Least squares collocation is a very comprehensive method for gravity field modelling, since it may use known noise characteristics of the data. In many earlier applications the errors affecting the data were considered uncorrelated, mainly due to the difficulty in estimating the systematic character of such kind of errors. In this study, error covariance functions of airborne gravity gradiometer data are estimated by comparing model covariance functions with empirical covariance functions of the gravity gradiometer data. The model covariance functions were estimated from accurate surface gravity data and continuated upward to the height of the airborne measurements using the covariance propagation law. The estimated error covariance functions were modeled as finite ones and used as an additional information for the prediction of gravity anomalies from gravity gradiometer data. The assessment of the prediction results was made by comparing the gravity values predicted from the airborne gradient data and showed up to 25% improvement compared to not using correlated errors.

The contribution of bathymetry to the prediction of quantities related to the gravity field (e.g., gravity anomalies, geoid heights) is discussed in an extended test area of the central Mediterranean Sea. Sea gravity anomalies and a priori statistical characteristics of depths are used in a least-squares collocation procedure in order to produce new depths, giving a better smoothing of the gravity field when using a remove-restore procedure. The effect of the bottom topography on gravity-field modeling is studied using both the original and the new depths through a residual terrain modeling reduction. The numerical tests show a considerable smoothing of the sea gravity anomalies and the available altimeter heights when the new depth information is taken into account according to the covariance analysis performed. Moreover, geoid heights are computed by combining the sea gravity anomalies either with the original depths or with the new ones, using as a reference surface the OSU91A geopotential model. Comparing the computed geoid heights with adjusted altimeter sea-surface heights (SSHs), better results are obtained when subtracting the attraction of the new depth information. Similar results are obtained when predicting gravity anomalies from altimeter SSHs where the terrain effect on altimetry is based on the new bottom topography.

Least Squares Collocation (LSC) is a very flexible method in gravity field modeling, which permits the use of data of different kinds and with different noise-characteristics. It also permits the computation of drift parameters and of error estimates of both computed values and parameters. The method has however the drawback that one needs to solve a system of normal equations involving as many unknowns as the number of data and parameters. The global use of the method have therefore been restricted to the use of block mean values as input data. The combined use of point and mean values could be possible if finite instead of full covariance functions could be used. Preliminary prediction results using LSC with finite covariance functions showed satisfactory agreement with corresponding results obtained with full covariance functions in the cases where the finite covariances fit satisfactory the corresponding full covariances. Some new experiments showed that it is possible to find some general procedure to approximate full by finite covariance functions without significant loss in the quality of the prediction. The computational savings which might be gained in finite covariance functions shows that LSC in the future could be used globally. Iterative solutions based on solutions where finite covariance functions are used may be the way to go.

The method of least squares collocation has been used to investigate the regional recovery of the gravity field from satellite gravity gradiometer (SGG) data and gravity vector data derived from, e.g., satellite-to-satellite tracking (SST). We chose a region centered over the European Alps where the gravity anomalies showed a (large) standard deviation of 60 mGal. Mean gravity anomalies were used to generate SGG data at a satellite altitude of 200 km. As SGG data we used the second-order derivatives in radial direction, across-track, and the mixed radial/cross-track derivative, all assumed to have an associated noise equal to 0.01 EU (Eotvos unit =EU, 1 EU=10−9 s−2). As gravity data we used the three components of the gravity vector, assuming that the associated errors were uncorrelated noise with standard deviation equal to 1 mGal. The satellite data in the test area were used in combination to predict 0.5° mean gravity anomalies and geoid heights. Ground truth data were used to investigate the quality of the recovery. The difference between observed and computed values have a standard deviation equal to 22 mGal in the best case. The use of the gravity vector data gave no improvement when added to the SGG data. The use of topographic information with 5 arc min resolution (ETOPO5U) gave, after removal of severe errors, a 17-mGal standard deviation of observed minus computed values. This is a much smaller improvement than expected but is due to errors in the topographic values.

Terrestrial gravity anomalies selected from three extended continental regions having a smooth gravity field were used in order to determine the appropriate size of the area for gravity data collection as well as the required data-sampling for calibration of the GOCE satellite gravity gradient (SGG) data. Using Least Square Collocation (LSC), prediction of gravity gradient components was carried out at points on a realistic orbit. Based on the mean error estimation it was shown that up to 80% of the signal of the gravity gradient components, as it is expressed through the covariance function of the terrestrial gravity data, can be recovered in the case of an optimal size of the collection area and of the optimum resolution of the data. These optimal conditions e.g. for the Australian gravity field, correspond to an 10° × 12° area extend and a 5′ data-sampling. It was also numerically demonstrated that it is possible to calibrate the GOCE SGG data for systematic errors such as bias and tilt.

When planning a satellite gravity gradiometer (SSG) mission, it is important to know the quality of quantities to be recovered at ground level as a function of e.g. satellite altitude, data type and sampling rate, and signal variance and noise. This kind of knowledge may be provided either using the formal error estimates of wanted quantities using least-squares collocation (LSC) or by comparing simulated data at ground level with results computed by methods like LSC or FFT.

The new satellite gravity missions (CHAMP, GRACE and GOCE) will all bring substantial improvements to our knowledge of the gravity field and thereby of the (quasi-) geoid. One of the aims of the Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) is to determine the geoid to within 1 cm at wavelengths down to 100 km.

From the separate processing and crossover analysis of repeat ERS-1 and TOPEX altimeter data in the Mediterranean basin was assessed that the TOPEX data are characterized by higher accuracy than that of ERS-1 and other previous missions data. It is also known from the coverage of both missions that the resolution of ERS-1 data is superior to that of the TOPEX data, since the distance across track in TOPEX mission is about four times the corresponding distance in ERS-1 mission. Moreover, numerical tests carried out to recover the gravity anomalies in the Mediterranean sea showed that the prediction results are affected by the data accuracy in the case of ERS-1 and by the poor coverage in the case of TOPEX. Therefore, taking into account the advantages and drawbacks of both missions we carried out a common adjustment of both altimeter data sources in order to improve the prediction accuracy of gravity anomalies and geoid heights in the Mediterranean sea.

The optimal combination of surface gravity data with the OSU86F geopotential model (GM) complete to degree and order 360 is studied in a North-American test area. Experiments using gravity data and different truncations of the geopotential expansion are carried out in order to obtain for absolute geoidal heights an accuracy close to 0.40m and for relative geoidal heights 2–3 ppm. These results have been derived by comparing the computed gravimetric geoidal heights and corresponding heights derived from GPS. The same GM is employed with and without the additional use of gravity and height data in a second test area (Greece) in order to predict components of the deflections of the vertical. The results indicate an accuracy better than 2“. Conclusions for optimal combination of the 360x360 GM with regional gravity data and topographic heights are drawn and are verified numerically by the prediction of different quantities related to the gravity field.

The gravity field approximation for the Central Mediterranean is attempted using (a) seagravimetry, (b) bias and tilt adjusted satellite altimetry, (c) bathymetry and mean heights and (d) a “tailored” spherical harmonic coefficient set complete to degree 360 (IfE88). A number of numerical tests is carried out to check the compatibility of the different data types. The Least Squares Collocation (LSC) and the parametric LSC methods are used to predict subsets of the data types(a) and for (b) from subsets of the same data types. The above tests show that the combination of such different data sets gives the optimal approximation of the gravity field in terms of the standard deviation (s.d.) of the observed minus predicted values. Using as input data in the parametric LSC method the data types (a) and (b) (in un-adjusted form) and after the removal of the terrain effect and the contribution of the geopotential model used, an accuracy ±4.3 mgal for gravity anomalies and ±0.04 m for geoidal heights is achieved.

The pre-processing and the crossover adjustment of GEOSAT altimeter data are carried out in order to obtain a precise approximation of the gravity field in the Mediterranean sea. Altimeter data from the U.S. National Oceanographic Data Center (NODC) have been used covering a time period of about one year. With the rejection of outliers and sub-satellite points whose standard deviation (s.d.) exceeds ±0.09 m, the s.d. of the crossover differences was ±3.5 m. After applying a fit and crossover adjustment model, the s.d. of the crossover differences decreased to ±0.16 m. In order to test the efficiency of the sea surface heights performed, a number of predictions was carried out using as data: (a) adjusted satellite altimetry and (b) sea-gravimetry. Least squares collocation was then used to predict subsets of the data types (a) and (b) from subsets of the same data types. The results given here are in terms of s.d. of observed minus predicted values. Using adjusted altimeter data for gravity prediction a s.d. of ±9.36 mgal was found. When gravity values spaced 10′ apart were used to predict, (1) gravity values in a grid with the same mesh width shifted 5′ and (2) altimeter observations, a s.d. of ±5.08 mgal and ±0.18m, respectively, were found. A combination of gravity and altimeter data made the first s.d. of ±5.08 mgal decrease to ±4.34 mgal showing the impact of combining the two data types. In all the above prediction computations the use of topographic-isostatic effects has been taken into account.

A set of airborne Gravity Gradiometer Survey System (GGSS) data was used in combination with the OSU89B geopotential model to predict gravity anomalies and deflections of the vertical. The gradiometer data were collected during an airborne survey of the GGSS in the Texas/Oklahoma area. Nineteen flight tracks were in our disposal for further analysis. These tracks resulted from 54 initial distinct tracks from which a processing was performed by the Analytic Sciences Corporation. In order to assess the quality of the GGSS measurements, a first crossover analysis was carried out. The results showed large discrepancies at the crossing points between East-West (EW) and North-South (NS) tracks. The least squares collocation method and the fast fourier transform technique were used to produce gravity anomalies and deflections of the vertical referred to geoid surface from second-order derivatives measured at the flight level. The predicted gravity values were then compared to “truth-based” point free air gravity anomalies with a standard deviation (s.d.) of the differences at the level of 3 mGal in the best cases. Presumably, improved GGSS data would have given improved predictions of surface gravity values. The predicted deflections of the vertical were compared to “truth-based” values at a small number of ground stations. The very low accuracy of this prediction (s.d.=5.62 arc sec for ξ and 6.78 for η) should be studied in combination with the inherent characteristics of the gravity field and the validation of GGSS data in the test area.

SUMMARY Using data from the Eastern Mediterranean it has been shown how different data types may improve the quality of a gravity field approximation, and a detailed geoid has been computed for the area. The following data types were available: (a) bias-adjusted satellite altimetry, (b) sea-gravimetry, (c) bathymetry and heights and (d) the spherical harmonic coefficient set OSU81, complete to degree 180. A comparison of the satellite altimetry with geoid heights computed from the OSU81 coefficients with and without the additional use of gravity data revealed substantial errors due to unmodelled orbit tilts. A trend analysis of the altimeter data showed a tilt of 0.0026 m km−1 in eastern direction and 0.0067 m km−1 in northern direction. The standard deviation (s.d.) of observed altimeter values minus OSU81 geoid heights decreased from ±1.53 m to ±1.07 m after a correction for the tilt. Least-squares collocation was then used to predict subsets of the data types (a) and/or (b) from subsets of the same data types. The results given here are in terms of the s.d. of observed minus computed values. Using every third value of the corrected altimeter data for gravity prediction a s.d. of ±17 mGal was found. When gravity values spaced 10′ apart were used to predict, (1) gravity values in a grid with the same mesh width shifted 5′ and (2) altimeter observations, a s.d. of ±10 mGal and ±0.42 m, respectively, were found. A combination of gravity and altimeter data made the first s.d. of ±10 mGal decrease to ±7 mGal, showing the impact of combining the two data types. The use of topographic information to smooth the locally strongly-varying gravity field gave substantial improvements in the gravity values predicted from either altimeter observations or gravity data. Error estimates of geoid heights predicted in between the altimeter tracks showed that a 65 per cent improvement had been achieved.

The accuracy of the gravity field approximation depends on the amount of the available data and their distribution as well as on the variation of the gravity field. The variation of the gravity field in the Greek mainland, which is the test area in this study, is very high (the variance of point free air gravity anomalies is 3191.5mgal 2). Among well known reductions used to smooth the gravity field, the complete isostatic reduction causes the best possible smoothing, however remain strong local anomalies which disturb the homogeneity of the gravity field in this area. The prediction of free air gravity anomalies using least squares collocation and regional covariance function is obtained within a ±4 ... ±19mgal accuracy depending on the local peculiarities of the free air gravity field. By taking into account the topography and its isostatic compensation with the usual remove-restore technique, the accuracy of the prediction mentioned obove was increased by about a factor of 4 and the prediction results become quite insensitive to the covariance function used (local or regional). But when predicting geoidal heights, in spite of using the smoothed field, the prediction results remain still depend on the covariance function used in such a way that differences up to about 50cm/100km result between relative geoidal heights computed with regional or local covariance functions.