Earth Geopotential Model Research Articles

Определение гравитационных аномалий в акватории Вьетнама и на прилегающей территории по данным спутниковой альтиметрии CryoSat-2

The purpose of this study is to determine marine gravity anomalies with high accuracy using the CryoSat-2 satellite altimetry data in the Middle Vietnamese Sea and the adjacent territory. For this, the mentioned data and the process of determining gravity anomalies from the CryoSat-2 data were analyzed. The record from 818 GNSS-leveling points in Vietnam were used to evaluate and select the best global gravity field model for use in the calculation process. The findings from 31 gauge tidal were also used to assess and select the best mean dynamic topography model for the study area. Marine gravity anomalies in the study area were calculated using data from the CryoSat-2 satellite altimeter at 72 483 measurement points. The calculated results are compared with the ship-derived gravity anomalies at 1025 measurement points for assessment. The research results show that of the Earth Geopotential Model EIGEN6C4 and the Mean Dynamic Topography Model DTU15MDT are the most suitable for the research area. The accuracy of satellite-derived gravity anomalies, calculated from the CryoSat-2 satellite altimetry data in the study area, reaches ±1,18 mGal. Compared to the available results in the study area, this accuracy is the highest.

DIGITAL ELEVATION MODEL ALTERNATIVES ASSESSMENT FOR DEFORMATION ANALYSIS PURPOSES USING GNSS AND INSAR

<p><em>Digital Elevation Model (DEM) is the starting point in the analysis performed to explain the deformation pattern changes from the Earth's surface. The estimated value of deformation based on point-wise GPS and InSAR data with a better spatial resolution must be defined in a reference frame system that reflects the phenomenon of deformation of the real physical world, e.g., orthometric height for the vertical component. Therefore, this study aims to provide alternative DEM models based on a suitable combination between the Global Geopotential Model of Earth Geopotential Model 2008 (EGM2008) and global terrain models, providing position changes with respect to the orthometric height. The alternative DEM models are (i) the global elevation model of ETOPO1 (DEM1), (ii) the modified global elevation model of SRTM30_PLUS (DEM2), and (iii) the regional elevation model of DEMNAS (DEM3). These alternative models comply with each other for the land areas with mean difference values lower than 1 meter. While for the ocean areas, we found that DEM1 and DEM2 have apparent differences due to the different types of data used. However, a similar assessment could not be performed for DEM3 as it only covers the land areas. Additionally, we compared the orthometric height from these terrain models with leveling observations for the coinciding locations. DEM3 achieves the highest accuracy with the estimated standard deviation of 11.2745 meters and is followed by DEM2 and DEM1 with the respective standard deviation of 29.4498 and 37.6872 meters. We found that these models can be used as a starting position determination for horizontal and vertical deformation analysis.</em></p>

Marine gravity anomaly mapping for the Gulf of Tonkin area (Vietnam) using Cryosat-2 and Saral/AltiKa satellite altimetry data

Marine gravity anomalies are essential data for determining coastal geoid, investigating tectonics and crustal structures, and offshore explorations. The objective of this study is to present a methodology for estimating marine gravity anomalies from CryoSat-2 and Saral/AltiKa satellite altimeter data for the Gulf of Tonkin of Vietnam with a high-resolution 2′ × 2′ grid. A total of 15,665 sea surface height (SSH) grid points, including derived from the Cryosat-2 (6842 grid points) and Saral/AltiKa (8823 grid points) satellite altimeter data were used. Then, the remove-restore technique and the crossover adjustment algorithm were used to remove the long-wavelength geoid height, the mean dynamic topography hMDT, and time-varying sea-surface topography ht. The residual geoid heights ΔN were used to determine the residual gravity anomalies δg using the Least-Squares Collocation method, whereas the Earth Geopotential Model was employed to restore the long-wavelength gravity anomalies ΔgEIGEN. GPS/leveling and tidal gauge of 31 tidal stations were used for assessing and choosing the best Earth Geopotential Model and Mean Dynamic Topography models for the study area (EIGEN6C4 and DTU15MDT models). The accuracy of the final marine gravity anomaly result was assessed using 56,978 marine gravity points, which were distributed in the study area. The result showed that the standard deviation between the satellite-derived gravity anomalies and checked points is ± 3.36 mGal, indicating good accuracy. After improving with ship-measured gravity anomalies, the accuracy of satellite-derived marine gravity anomaly improves to ± 2.63 mGal. The results of this research are useful for geodetic and geophysical applications in the region.

Modelling Orthometric Heights from a Combination of Ellipsoidal Heights and Gravimetric Geoid Model in Rivers State, Nigeria

Many applications in geodesy, hydrography and engineering require geoid-related heights. Spirit leveling which is the traditional means of obtaining geoid- or mean sea level-related heights is slow, time-consuming and costly. Global Navigation Satellite Systems (GNSS) offer faster and relatively cheaper way of obtaining geoid-related heights when geoidal undulation is applied to ellipsoidal heights. However, difficulties involved in determining acceptable geoid height have seriously hampered the application of GNSS for leveling in Rivers State, thus necessitating the need to develop an acceptable geoid model which will serve as a means of conversion of GNSS-delivered ellipsoidal heights to their orthometric heights equivalent. In pursuance of this objective, a detailed gravimetric geoid has been evaluated for Rivers State, Nigeria. The computation of the geoid was carried out by the traditional remove-restore procedure. The Earth Geopotential Model 2008 (EGM08) was applied as the reference field for both the remove and restore parts of the procedures; spherical Fast Fourier Transform (FFT) was employed for the evaluation of the Molodenskii’s integral formula for the height anomaly, (ζ) to yield the quasi-geoid; while the Residual Terrain Modelling (RTM) was done by prism integration. The classical gravimetric geoid over Rivers State was obtained from the rigorously evaluated quasi-geoid by adding the quasi-geoid to geoid (N - ζ) correction it. The minimum and maximum geoid height values are 18.599 m and 20.114 m respectively with standard deviation of 0.345 m across the study area. Comparison of the gravimetric geoidal heights with the GPS/Leveling-derived geoidal heights of 13 stations across Rivers State, Nigeria showed that the absolute agreement with respect to the GPS/leveling datum is generally better than 7 cm root mean squares (r.m.s) error. Results also showed that combining both GPS heights and the computed Rivers State geoid model can give orthometric heights accurate to 3 cm post-fit using a 4-parameter empirical model. The geoid model can thus serve as a good alternative to traditional leveling when used with GPS leveling, particularly for third order leveling in the study area.

Accuracy assessment of the PGM17 global geopotential model: a case study of Egypt and Northeast Africa

Global geopotential models play a fundamental role in height transformation to convert the Global Navigation Satellite Systems (GNSS)-based ellipsoidal heights into orthometric heights in several geosciences applications. The U.S. National Geospatial-Intelligence Agency (NGA) is preparing to release the Earth Geopotential Model 2020 (EGM2020) in the near future. Prior to that official release, two primary geopotential models (PGM17 and PGM18) will be distributed to selective members for assessment in different regions worldwide. Within the activities of that working group, the new PGM17 geopotential model has been obtained and judged over a regional scale in northeast Africa, and over a national basis of Egypt. The assessment has been carried out using terrestrial gravity and GPS/leveling datasets within a GIS environment. The utilized dataset consists of 231 terrestrial gravity points in northeast Africa, 247 measured gravity points in Egypt, and 978 GNSS/leveling stations in Egypt. Three other GGMs have been included in the comparison stage, namely GECO, EIGEN-6C4, and EGM2008. For each model, grids of gravity anomalies and geoidal undulations have been computed and, then, compared against the corresponding values at known terrestrial data. Next, statistical measures of mean, range, and standard deviations of the discrepancies have been analyzed. For northeast Africa area, the accomplished results indicated that the PGM17 attains 100% improvements over the old EGM2008, while the GECO, and EIGEN-6C4 models resulted in improvements of 11%, and 1% respectively. Accordingly, it can be concluded that PGM17 represents the gravitational field over northeast Africa in a more enhanced manner. On a national scale, the GGMs assessment has been performed, over Egypt, in two steps: using gravity data, and using GNSS/leveling data. The findings emphasize that PGM17 noticeably enhance the performance of EGM2008, when compared against terrestrial gravity stations in Egypt, by almost 55%. Furthermore, the assessment over GPS/leveling stations concluded that the overall improvements of PGM17 are approximately 54%. Other GGMs did not produce a comparable improvement level in both case studies. Thus, it can be concluded that PGM17 is significantly precise than the other investigated GGMs in representing the gravitational field over Egypt in particular and northeast Africa in general.

One technique for refining the global Earth gravity models

The results of the theoretical and experimental research on the technique for refining the global Earth geopotential models such as EGM2008 in the continental regions are presented. The discussed technique is based on the high-resolution satellite data for the Earth’s surface topography which enables the allowance for the fine structure of the Earth’s gravitational field without the additional gravimetry data. The experimental studies are conducted by the example of the new GGMplus global gravity model of the Earth with a resolution about 0.5 km, which is obtained by expanding the EGM2008 model to degree 2190 with the corrections for the topograohy calculated from the SRTM data. The GGMplus and EGM2008 models are compared with the regional geoid models in 21 regions of North America, Australia, Africa, and Europe. The obtained estimates largely support the possibility of refining the global geopotential models such as EGM2008 by the procedure implemented in GGMplus, particularly in the regions with relatively high elevation difference.

On the Analysis of Multistep-Out-of-Grid Method for Celestial Mechanics Tasks

Abstract Occasionally, there is a necessity in high-accurate prediction of celestial body trajectory. The most common way to do that is to solve Kepler’s equation analytically or to use Runge-Kutta or Adams integrators to solve equation of motion numerically. For low-orbit satellites, there is a critical need in accounting geopotential and another forces which influence motion. As the result, the right side of equation of motion becomes much bigger, and classical integrators will not be quite effective. On the other hand, there is a multistep-out-of-grid (MOG) method which combines Runge-Kutta and Adams methods. The MOG method is based on using m on-grid values of the solution and n × m off-grid derivative estimations. Such method could provide stable integrators of maximum possible order, O (hm + mn + n− 1). The main subject of this research was to implement and analyze the MOG method for solving satellite equation of motion with taking into account Earth geopotential model (ex. EGM2008 (Pavlis at al., 2008)) and with possibility to add other perturbations such as atmospheric drag or solar radiation pressure. Simulations were made for satellites on low orbit and with various eccentricities (from 0.1 to 0.9). Results of the MOG integrator were compared with results of Runge-Kutta and Adams integrators. It was shown that the MOG method has better accuracy than classical ones of the same order and less right-hand value estimations when is working on high orders. That gives it some advantage over ”classical” methods.

Combination of GPS and Leveling Observations and Geoid Models Using Least-Squares Variance Component Estimation

AbstractIn the last few years, satellite-based positioning techniques, such as the Global Positioning System (GPS), proved their abilities in various geodetic fields. Height determination, as one of such applications, requires a more likely correct geoid model to provide reliable geoid heights for transformation of the ellipsoidal heights to orthometric heights. An important step is then to establish such a model by optimal combination of the available geoid models. This can be achieved via variance component estimation (VCE) methods, which provide appropriate weights to GPS and leveling observations as well as the geoid models. The authors demonstrate the efficacy of the least-squares VCE (LS-VCE) to this problem. The algorithm is applied to real data sets in Shahin-Shahr, Isfahan, Iran, to evaluate the EGM2008 and GGMplus Earth geopotential models and a regional geoid model (named IRGeoid10) in terms of agreement to the GPS/leveling observations and introduce the more likely correct model over the case-...

Evolution of geoids in recent years and its impact on oceanography

Mean surface geostrophic ocean currents may be calculated from the Mean Dynamic Topography ( MDT ), estimated as the difference between a mean sea surface height ( MSS ) calculated from radar altimeters and a reference geoid height. A review of the most widely used geoids is presented. The difference between the third release of the Gravity field and steady-state Ocean Circulation Explorer (GOCE) geoid and three earlier geoids (the Earth Geopotential Model 1996 [EGM96], one of the geoids obtained by the Gravity Recovery and Climate Experiment [GRACE05], and the Earth Gravitational Model 2008 [EGM2008]) is computed and interpreted as an ‘artefact’ MDT , i.e. a misfit when non-accurate geoid models are used to calculate the ocean MDT and related geostrophic currents. These results are contrasted with the MDT computed by comparing the GOCE geoid with the MSS distributed by Collecte Localisation Satellites in 2001 (CLS01). The comparison shows that there was a strong influence of altimetry measurements in the construction of the EGM96 geoid, i.e. the artefact MDT calculated using EGM96 shows a high resemblance to the MDT computed using the MSS CLS01 field, both considering GOCE as the reference geoid. The correlation disappears largely, but not completely, for the two most recent geoids; in particular, the MSS has greater global influence on GRACE05 than on EGM2008 although the latter does better at latitudes of less than 60° and is more useful for reproducing the intense western boundary currents. The results show that EGM96 may lead to significant errors in the spatial gradients of MDT (for latitudes of less than 60° the global root mean square is 0.2422 m) and therefore in the geostrophic surface velocities. When the spatially averaged GRACE and EGM2008 geoids are used for latitudes of less than 60°, the global MDT root mean square is substantially reduced.

Comparison of Several Techniques for Fitting of the EGM08 to GPS/Leveling Datum

This paper aims to provide a precise transformation model for the geoid heights between earth geopotential models and regional vertical datum in geodetic applications. With this purpose, Earth Gravitational Model 2008 (EGM08), which is ultra-high-resolution geopotential model, that is complete spherical harmonic degree and order 2190 is considered and fitted to the local vertical datum in three local test areas in west and south of Turkey. The observations at GPS/leveling benchmarks were used for fitting the EGM08 to the local height reference surface and a couple of mathematical methods were evaluated, namely similarity transformation model (STM), polynomial regression model (PRM), least squares collocation (LSC), spline interpolation model (SIM). They are commonly used for corrector surface fitting approaches in height datum studies. Their performances were assessed depending on the statistics at test benchmarks. The results show that fitting of the differences between the GPS/leveling, EGM08-derived geoid heights with LSC, and SIM supplies more accurate solutions than STM and PRM. The accuracies of the achieved models (improved from EGM08 global model) are tested in the several GPS/leveling benchmarks throughout Turkey; the external model accuracy has been estimated as ±4–5 cm. In this aspect, this study will provide benefits as both representing the performance of EGM08 model in given local areas and clarifying the modeling performances of the applied transformation approaches for fitting of EGM08 geoid models to GPS/leveling defined vertical datum.

Ellipsoidal topographic potential: New solutions for spectral forward gravity modeling of topography with respect to a reference ellipsoid

AbstractForward gravity modeling in the spectral domain traditionally relies on spherical approximation. However, this level of approximation is insufficient for some present day high‐accuracy applications. Here we present two solutions that avoid the traditional spherical approximation in spectral forward gravity modeling. The first solution (the extended integration method) applies integration over masses from a reference sphere to the topography and applies a correction for the masses between ellipsoid and sphere. The second solution (the harmonic combination method) computes topographic potential coefficients from a combination of surface spherical harmonic coefficients of topographic heights above the ellipsoid, based on a relation among spherical harmonic functions introduced by Claessens (2005). Using a degree 2160 spherical harmonic model of the topographic masses, both methods are applied to derive the Earth's ellipsoidal topographic potential in spherical harmonics. The harmonic combination method converges fastest and—akin to the EGM2008 geopotential model—generates additional spherical harmonic coefficients in spectral band 2161 to 2190 which are found crucial for accurate evaluation of the ellipsoidal topographic potential at high degrees. Therefore, we recommend use of the harmonic combination method to model ellipticity in spectral‐domain forward modeling. The method yields ellipsoidal topographic potential coefficients which are “compatible” with global Earth geopotential models constructed in ellipsoidal approximation, such as EGM2008. It shows that the spherical approximation significantly underestimates degree correlation coefficients among geopotential and topographic potential. The topographic potential model is, for example, of immediate value for the calculation of Bouguer gravity anomalies in fully ellipsoidal approximation.

New Technology Going Up from the GPS Measurements, Known W0 and Its Possible Utilization in Praxis

New height precise Earth Geopotential Model EGM08 made it possible to develop the value of the geoidal geopotential W0. This value is independent on Earth’s tides and others disturbances, therefore it can be used as basic quantity for many applications. Value W0 represents mean sea surface. W0 was developed from data of satellite altimetry which monitores actual ocean’s surface. This value is one of the four fundamental geodetic constants. This value has been accepted worldwide and is included to the International Earth Rotation and Reference Systems Service Conventions (IERS Conventions) 2003, 2007. This value defines in principle parameters of level ellipsoid which best fit to the surface defined by W0. The body of this level ellipsoid produces normal gravity. We developed other important parameters for determination of the density deviations (VTT technology), i.e. values of normal gravity ge at the equator and gp at the pole on the elipsoid E0 and the value of parameter β. All parameters above used for development of VTT technology are supplied by GPS/leveling data that enable to create so-called density deviation maps at any place in the World. In comparison with gravity anomalies, the density deviations represent more precise density anomalous places in the Earth’s crust. A construction of density deviation map is applied at two from geologically different regions—the Western Carpathians and the Central African Rifts.

IMPLEMENTATION OF THE DISTRIBUTED PARALLEL PROGRAM FOR GEOID HEIGHTS COMPUTATION USING MPI AND OPENMP

Abstract. Much research have been carried out using optimization algorithms for developing high-performance program, under the parallel computing environment with the evolution of the computer hardware technology such as dual-core processor and so on. Then, the studies by the parallel computing in geodesy and surveying fields are not so many. The present study aims to reduce running time for the geoid heights computation and carrying out least-squares collocation to improve its accuracy using distributed parallel technology. A distributed parallel program was developed in which a multi-core CPU-based PC cluster was adopted using MPI and OpenMP library. Geoid heights were calculated by the spherical harmonic analysis using the earth geopotential model of the National Geospatial-Intelligence Agency(2008). The geoid heights around the Korean Peninsula were calculated and tested in diskless-based PC cluster environment. As results, for the computing geoid heights by a earth geopotential model, the distributed parallel program was confirmed more effective to reduce the computational time compared to the sequential program.

Spectral evaluation of Earth geopotential models and an experiment on its regional improvement for geoid modelling

As the number of Earth geopotential models (EGM) grows with the increasing number of data collected by dedicated satellite gravity missions, CHAMP, GRACE and GOCE, measuring the differences among the models and monitoring the improvements in gravity field recovery are required. This study assesses the performance of recent EGMs derived from CHAMP, GRACE, and other data in comparison to the earlier two models in Turkish territory. Also the improvement capacities of these EGMs using local terrestrial data are inspected with two different approaches. In the first approach, the spherical harmonic coefficients of EGMs are modified depending on the local gravity data. In the second part, the original EGMs with their maximum harmonic expansions are employed in Remove Compute Restore algorithm for high resolution local geoid modelling. The assessment results with the local terrestrial data exhibited large disagreements among the models in Turkey. The outputs from regional improvements of EGMs using the gravity observations clarified the important role of the ground truth data contribution in geopotential models. The validations of the computed high resolution geoids rely on the EGMs at the GNSS/levelling networks, having 451 and 309 benchmarks, provided an absolute accuracy by means of standard deviations of the geoid height differences around 20 cm. These results from the final stage of the case study provide a comparison among the EGMs by means of their fit to the local gravity field.

A study of regional gravity field recovery from GOCE vertical gravity gradient data in the Auvergne test area using collocation

Gravity field and steady-state Ocean Circulation Explorer (GOCE) is the first satellite mission that observes gravity gradients from the space, to be primarily used for the determination of high precision global gravity field models. However, the GOCE gradients, having a dense data distribution, may potentially provide better predictions of the regional gravity field than those obtained using a spherical harmonic Earth Geopotential Model (EGM). This is investigated in Auvergne test area using Least Squares Collocation (LSC) with GOCE vertical gravity gradient anomalies (Tzz), removing the long wavelength part from EGM2008 and the short wavelength part by residual terrain modelling (RTM). The results show that terrain effects on the vertical gravity gradient are significant at satellite altitude, reaching a level of 0.11 Eotvos unit (E.U.) in the mountainous areas. Removing the RTM effects from GOCE Tzz leads to significant improvements on the LSC predictions of surface gravity anomalies and quasigeoid heights. Comparison with ground truth data shows that using LSC surface free air gravity anomalies and quasi-geoid heights are recovered from GOCE Tzz with standard deviations of 11 mGal and 18 cm, which is better than those obtained by using GOCE EGMs, demonstrating that information beyond the maximal degree of the GOCE EGMs is present. Investigation of using covariance functions created separately from GOCE Tzz and terrestrial free air gravity anomalies, suggests that both covariance functions give almost identical predictions. However, using covariance function obtained from GOCE Tzz has the effect that the predicted formal average error estimates are considerably larger than the standard deviations of predicted minus observed gravity anomalies. Therefore, GOCE Tzz should be used with caution to determine the covariance functions in areas where surface gravity anomalies are not available, if error estimates are needed.

Discrete Spherical Harmonic Transforms of Nearly Equidistributed Global Data

Discrete Spherical Harmonic Transforms of Nearly Equidistributed Global DataDiscrete Spherical Harmonic Transforms (SHTs) are commonly defined for equiangular grids on the sphere. However, when global array data exhibit near equidistributed patterns rather than equiangular grids, discrete SHTs require appropriate adaptations for analysis and synthesis. Computational efficiency and reliability impose structural constraints on possible equidistribution characteristics of data patterns such as for instance with Chebychev quadratures and Fast Fourier Transforms (FFTs). Following some general introduction to discrete SHTs and equidistributions on the sphere, equitriangular (near equiareal) lattices based on the octahedron and the icosahedron are introduced for SHT analysis and synthesis. The developed formulations are described and implemented using simulated data and geopotential models such as the Earth Geopotential Model EGM 2008. Comparative results for analysis and synthesis at different levels of resolution show the potential of the spherical equitriangular approach for geodetic and other applications with nearly equidistributed global data.

Discrete Spherical Harmonic Transforms for Equiangular Grids of Spatial and Spectral Data

Discrete Spherical Harmonic Transforms for Equiangular Grids of Spatial and Spectral DataSpherical Harmonic Transforms (SHTs) which are non-commutative Fourier transforms on the sphere are critical in global geopotential and related applications. Among the best known global strategies for discrete SHTs of band-limited spherical functions are Chebychev quadratures and least squares for equiangular grids. With proper numerical preconditioning, independent of latitude, reliable analysis and synthesis results for degrees and orders over 3800 in double precision arithmetic have been achieved and explicitly demonstrated using white noise simulations. The SHT synthesis and analysis can easily be modified for the ordinary Fourier transform of the data matrix and the mathematical situation is illustrated in a new functional diagram. Numerical analysis has shown very little differences in the numerical conditioning and computational efforts required when working with the two-dimensional (2D) Fourier transform of the data matrix. This can be interpreted as the spectral form of the discrete SHT which can be useful in multiresolution and other applications. Numerical results corresponding to the latest Earth Geopotential Model EGM 2008 of maximum degree and order 2190 are included with some discussion of the implications when working with such spectral sequences of fast decreasing magnitude.

Mapping the geoid for Iberia and the Macaronesian Islands using multi-sensor gravity data and the GRACE geopotential model

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 (100 m) 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 8 cm, with larger differences close to the shoreline. Over land, comparisons with 1646 GPS/levelling marks indicate an overall precision of 8–10 cm and relative vertical datum offsets of up to 2 m. This new geoid model represents a significant improvement over existing geoid models, with a homogeneous relative accuracy of 8–10 cm over both marine and land areas, and exposes the inaccuracies of local vertical datums as references for studies of vertical deformation.

Circulation of the North Atlantic Ocean from altimetry and the Gravity Recovery and Climate Experiment geoid

We discuss the ocean circulation derived from the temporally averaged sea surface height, which is referenced to the recently released geoid from the Gravity Recovery and Climate Experiment (GRACE) mission (GRACE Gravity Model 02 (GGM02)). The creation of a precise, independent geoid allows for the calculation of the reference gravitational potential undulation surface, which is associated with the resting ocean surface height. This reference height is then removed from the temporally averaged sea surface height, leaving the dynamic ocean topography. At its most basic level the dynamic ocean topography can be related to the ocean's surface circulation through geostrophy. This has previously been impracticable because of large uncertainties in previous estimates of the Earth's geoid. Prior geoids included the temporally averaged sea surface from altimeters as a proxy for the geoid and therefore were unsuitable for calculations of the ocean's circulation. Geoid undulations are calculated from the GRACE geoid and compared to those from the NASA Goddard Space Flight Center and National Imagery and Mapping Agency Joint Earth Geopotential Model (EGM96) geoid. Error estimates are made to assess the accuracy of the new geoid. The deep ocean pressure field is also estimated by combining the calculated dynamic ocean topography with hydrography. Finally, the derived circulation is compared to independent observations of the circulation from sea surface drifters and subsurface floats. It is shown that the GGM02 geoid is significantly more accurate for use in estimating the ocean's circulation.

Regularized wavelet-based multiresolution recovery of the harmonic mass density distribution from data of the Earth's gravitational field at satellite height

The inverse problem of recovering the Earth's density distribution from data of the first or second derivative of the gravitational potential at satellite orbit height is discussed for a ball-shaped Earth. This problem is exponentially ill-posed. In this paper, a multiscale regularization technique using scaling functions and wavelets constructed for the corresponding integro-differential equations is introduced and its numerical applications are discussed. In the numerical part, the second radial derivative of the gravitational potential at 200 km orbit height is calculated on a point grid out of the NASA/GSFC/NIMA Earth Geopotential Model (EGM96). Those simulated derived data out of SGG (satellite gravity gradiometry) satellite measurements are taken for convolutions with the introduced scaling functions yielding a multiresolution analysis of harmonic density variations in the Earth's crust. Moreover, the noise sensitivity of the regularization technique is analysed numerically.