Earth Gravity Model Research Articles

A spectral assessment review of current satellite‐only and combined Earth gravity models

AbstractThe realization over the last decade of dedicated gravity field satellite missions enabled the production of a series of new satellite‐only and combined models for the Earth's gravity field. Using different sensors, measurement techniques, and algorithmic procedures, the final product in each case is a set of spherical harmonic coefficients representing the series expansion of the gravitational potential up to a certain maximum degree, depending on the mission characteristics and the range of the available data. The present review performs a detailed quantified analysis of a representative selection of currently available CHAMP (Challenging Minisatellite Payload), GRACE (Gravity Recovery and Climate Experiment), GOCE (Gravity Field and Steady‐State Ocean Circulation Explorer), and combined Earth gravity models. In this comparative analysis, we also include the so‐called topographic/isostatic gravity models that represent the contribution of global digital elevation maps for the topography and ocean bathymetry. Applying a range of available spatial and spectral accuracy and assessment measures, such as correlation per degree and order, smoothing per degree and order, signal‐to‐noise ratio, gain, degree variances, and error degree variances, one gains a quantified “peek” inside the quality of these models spanning over their whole spectrum. The applied error and assessment measures are defined both in an absolute and relative manner with respect to other similar models or some reference Earth gravity models. Furthermore, the nature of the performed analysis (degree‐wise, order‐wise, and cumulative) permits the identification of distinct spectral bandwidths in these models, enables the quantification of some standard features of the observed field, such as its “long wavelength”, “short wavelength”, or “very high frequency part”, and specifies the attenuation of the gravity signal with increasing altitude from the Earth's surface. An examination of the assessment quantities reveals certain bandwidths of these models with characteristic statistical features. A band‐limited synthesis of these bandwidths in the space domain quantifies the corresponding contributions in terms of selected gravity field functionals, including second‐order derivatives at GOCE altitude.

The exploitation of state of the art digital terrain databases and combined or satellite-only Earth gravity models for the estimation of the crust–mantle interface over oceanic regions

SUMMARY Thepredictionofthegeometryofcharacteristiccrustalinterfaces,suchasbathymetryorMoho, can be carried out efficiently by applying well tested estimation procedures. The computation of the gravity response at the Earth’s surface of a characteristic density layer in the Earth’s interior in combination with the statistical tool of Least Squares Collocation have been proven to be a flexible and useful prediction tool, especially at regions with lack in primary crustal observations.Toinvestigatethecontributionofrecentlyreleaseddigitalterraindatabasestothis estimation procedure we use the one-minute global elevation databases GEBCO and TOPO14. Using the combined Earth gravity model EGM2008 and recently released GOCE-only models we produce the observed grid of gravity anomalies over two test areas in the Indian and the Atlantic oceans. The estimation procedure is based on a two-layer model of the upper crust and is driven mainly by the definition and quality of the involved covariance functions. The application leads to independent and much denser estimations of the crust–mantle interface over these regions, in comparison with the available Moho information provided by global databasessuchasCRUST2.0.Apartfromthedirectcomparisonbetweentheobtainedsolutions a further assessment was carried out in terms of computing the corresponding isostatic effects on the available regional altimeter data.

An integral approach to regional gravity field refinement using earth gravity models

The idea of this paper is to refine the terrestrial gravimetric data with the Earth's gravity models (EGMs) and produce a high quality source of gravity data. For this purpose, biased and unbiased integral estimators are presented. These estimators are used to refine gravimetric data over Fennoscandia with the ITG-GRACE2010s and GO_CONS_GCF_2_DIR_R2 EGMs, which are the recent products of the gravity field and climate experiment (GRACE) and the gravity field and steady-state ocean circulation explorer (GOCE) satellite missions. Numerical results show that the biased integral estimator has smaller global root mean square error (RMSE) than the unbiased one. Also a simple strategy is presented to down-weight the low-frequencies the terrestrial data in spectral combination. The gravity anomalies, computed by EGM08, are compared to the refined anomalies for evaluation purpose. In the case of using a cap size of 1° for integration the EGM08 gravity anomalies are more correlated with the refined ones. Also the band-limited kernels can simply be generated to maximum degree of the used EGMs for both estimators. Comparisons of the combined anomalies and those of EGM08 show insignificant differences between the biased and unbiased estimators in practice. However, the biased estimator seems to be proper one for gravity data refinement due to its smaller global RMSE.

Artifacts in regional gravity representations with spherical radial basis functions

With the increasing number of high-resolution gravity observations, which became available in the recent years, global Earth gravity models can be regionally refined. While global gravity models are usually represented in spherical harmonic basis functions with global support, a very promising option to model the regional refinements is the use of spherical radial basis functions with quasi-compact support. These functions are not necessarily orthogonal on a sphere, and usually, more functions are used in regional modelling than minimally needed from a global point of view. This makes the modelling more difficult. Furthermore, no techniques or choices of radial basis functions and other parameters in the regional modelling approach are established so far, as it is the case for global gravity modelling in spherical harmonics. In this article, a closed-loop simulation is used to investigate the mathematical modelling accuracy of different radial basis functions, which are compared to each other. Furthermore, artificial effects, which occur in the modelling results with very low levels of noise on the observations, are investigated. The whole study is performed on synthetic observations of a residual gravitational potential signal for the Himalaya area with different levels of noise. Spherical radial basis functions are a compromise between spatial and frequency localization, which are mutually exclusive. We show that spatial localization properties are even more important than frequency localization in the regional case, even though a band-limited signal is modeled.

High-resolution residual geoid and gravity anomaly data of the northern Indian Ocean – An input to geological understanding

Geoid data are more sensitive to density distributions deep within the Earth, thus the data are useful for studying the internal processes of the Earth leading to formation of geological structures. In this paper, we present much improved version of high resolution (1′×1′) geoid anomaly map of the northern Indian Ocean generated from the altimeter data obtained from Geodetic Missions of GEOSAT and ERS-1 along with ERS-2, TOPEX/POSIDEON and JASON satellites. The geoid map of the Indian Ocean is dominated by a significant low of −106m south of Sri Lanka, named as the Indian Ocean Geoid Low (IOGL), whose origin is not clearly known yet. The residual geoid data are retrieved from the geoid data by removing the long-wavelength core–mantle density effects using recent spherical harmonic coefficients of Earth Gravity Model 2008 (EGM2008) up to degree and order 50 from the observed geoid data. The coefficients are smoothly rolled off between degrees 30–70 in order to avoid artifacts related to the sharp truncation at degree 50. With this process we observed significant improvement in the residual geoid data when compared to the previous low-spatial resolution maps. The previous version was superposed by systematic broad regional highs and lows (like checker board) with amplitude up to ±12m, though the trends of geoid in general match in both versions. These methodical artifacts in the previous version may have arisen due to the use of old Rapp’s geo-potential model coefficients, as well as sharp truncation of reference model at degree and order 50. Geoid anomalies are converted to free-air gravity anomalies and validated with cross-over corrected ship-borne gravity data of the Arabian Sea and Bay of Bengal. The present satellite derived gravity data matches well with the ship-borne data with Root Mean Square Error (RMSE) of 5.1–7.8mGal, and this is found to be within the error limits when compared with other globally available satellite data. Spectral analysis of ship-borne and satellite data suggested that the satellite gravity data have a resolution down to 16–18km. Further, the geoid, residual geoid and gravity anomalies are integrated with seismic data along two profiles in the Bay of Bengal and Arabian Sea, and inferences have been made in terms of density distributions at different depths. The new residual geoid anomaly map shows excellent correlation with regional tectonic features such as Sunda subduction zone, volcanic traces (Chagos–Laccadive, Ninetyeast and 85°E ridges) and mid-ocean ridge systems (Central Indian and Carlsberg ridges).

A strategy towards an EGM08-based Fennoscandian geoid model

Today, the recent global Earth's gravity model, EGM08, is successfully utilised for different purposes in geosciences. Here, EGM08 is used to compute a geoid model for Fennoscandia and since it is restricted to degree and order 2160, the higher frequencies of the geoid, or the truncation bias, is recovered directly from terrestrial gravity anomalies using a simple formula. The total topographic and atmospheric effects are computed and added to the derived geoid as well. A very simple EGM08-based non-integral geoid estimator is developed and applied for computing the geoid of Fennoscandia. The outcome of the estimator is compared with the Global Positioning System (GPS)/levelling data of Sweden, Denmark, Finland and Norway. Numerical results show the successful performance of the presented estimator as the geoid become closer to GPS/levelling data than the one computed solely with EGM08. This study will show that considering the truncation bias of EGM08 will reduce the root mean square error (RMSE) of the differences between the geoid and GPS/levelling data by about 1.3 cm and the additive topographic and atmospheric corrections by 1 cm further. It is shown that the correlations among the data have no significant influence on the estimated geoid.

An attempt to link the Brazilian Height System to a World Height System

This paper deals with the geopotential approach to investigate the present Brazilian Height System (BHS). Geopotential numbers are derived from Global Positioning System (GPS) satellite surveying and disturbing potential on selected benchmarks. A model for the disturbing potential can be obtained by an existing set of spherical harmonic coefficients such as the Earth Gravity Model 2008 (EGM08). The approach provides absolute evaluation of local normal geopotential numbers (aka spheropotential numbers) related to a so-called World Height System (WHS). To test the validity of the proposed methodology, a numerical experiment was carried out related to a test region in Southern Brazil. The accuracy of the derived geopotential numbers was tested versus local normal geopotential numbers based on 262 GPS/leveling points. The root mean square error (RMSE) value for metric offset of BHS derived from geopotential numbers and the disturbing potential modeling in the test area was estimated to be near 0.224 meters in the absolute view. Therefore, since these spheropotential numbers are referred to a local datum, these results of comparisons may be an indicator of the mean bias of local network due to the effect of local Sea Surface Topography (SSTop) and possible offset between the unknown reference for the BHS and the quasigeoid model in the region.

Are higher degree even zonals really harmful for the LARES/LAGEOS frame-dragging experiment?

The low-altitude effects of LARES are examined to determined how they can impact the outcome of the hoped 1% frame-dragging measurement in the LARES–LAGEOS experiment. This analysis, based on a different approach than other studies recently appearing in the literature, shows that the spherical harmonics of the Earth gravity field with degree ℓ > 60 may represent a threat because their errors map significantly into LARES orbital disturbances compared to frame-dragging. The GIF48 model was used. It is questionable whether future Earth gravity models by GRACE and GOCE will be of sufficient accuracy.

GOCE satellite derived gravity and gravity gradient corrected for topographic effect in the South Central Andes region

SUMMARY Global gravity field models, derived from satellite measurements integrated with terrestrial observations, provide a model of the Earth's gravity field with high spatial resolution and accuracy. The Earth Gravity Model EGM08, a spherical harmonic expansion of the geopotential up to degree and order 2159, has been used to calculate two functionals of the geopotential: the gravity anomaly and the vertical gravity gradient applied to the South Central Andes area. The satellite-only field of the highest resolution has been developed with the observations of satellite GOCE, up to degree and order 250. The topographic effect, a fundamental quantity for the downward continuation and validation of satellite gravity gradiometry data, was calculated from a digital elevation model which was converted into a set of tesseroids. This data is used to calculate the anomalous potential and vertical gravity gradient. In the Southern Central Andes region the geological structures are very complex, but not well resolved. The processing and interpreting of the gravity anomaly and vertical gradients allow the comparison with geological maps and known tectonic structures. Using this as a basis, a few features can be clearly depicted as the contact between Pacific oceanic crust and the Andean fold and thrust belt, the seamount chains over the Oceanic Nazca Plate, and the Famatinian and Pampean Ranges. Moreover the contact between the Rio de la Plata craton and the Pampia Terrain is of great interest, since it represents a boundary that has not been clearly defined until now. Another great lineament, the Valle Fertil-Desaguadero mega-lineament, an expression of the contact between Cuyania and Pampia terranes, can also be clearly depicted. The authors attempt to demonstrate that the new gravity fields can be used for identifying geological features, and therefore serve as useful innovative tools in geophysical exploration.

A synthetic Earth gravity model based on a topographic-isostatic model

The Earth’s gravity field is related to the topographic potential in medium and higher degrees, which is isostatically compensated. Hence, the topographic-isostatic (TI) data are indispensable for extending an available Earth Gravitational Model (EGM) to higher degrees. Here we use TI harmonic coefficients to construct a Synthetic Earth Gravitational Model (SEGM) to extend the EGMs to higher degrees. To achieve a high-quality SEGM, a global geopotential model (EGM96) is used to describe the low degrees, whereas the medium and high degrees are obtained from the TI or topographic potential. This study differes from others in that it uses a new gravimetric-isostatic model for determining the TI potential. We test different alternatives based on TI or only topographic data to determine the SEGM. Although the topography is isostatically compensated only to about degree 40–60, our study shows that using a compensation model improves the SEGM in comparison with using only topographic data for higher degree harmonics. This is because the TI data better adjust the applied Butterworth filter, which bridges the known EGM and the new high-degree potential field than the topographic data alone.

Development of a Synthetic Earth Gravity Model by 3D mass optimisation based on forward modelling

Several previous Synthetic Earth Gravity Model (SEGM) simulations are based on existing information about the Earth’s internal mass distribution. However, currently available information is insufficient to model the Earth’s anomalous gravity field on a global scale. The low-frequency information is missing when modelling only topography, bathymetry and crust (including the Mohorovicic discontinuity), but the inclusion of information on the mantle and core does not seem to significantly improve this situation. This paper presents a method to determine a more realistic SEGM by considering simulated 3D mass distributions within the upper mantle as a proxy for all unmodelled masses within the Earth. The aim is to improve an initial SEGM based on forward gravity modelling of the topography, bathymetry and crust such that the missing low-frequency information is now included. The simulated 3D mass distribution has been derived through an interactive and iterative mass model optimisation algorithm, which minimises geoid height differences with respect to a degree-360 spherical harmonic expansion of the EGM2008 global external gravity field model. We present the developed optimisation algorithm by applying it to the development of a global SEGM that gives a reasonably close fit to EGM2008, and certainly closer than a SEGM based only on the topography, bathymetry and crust.

Efficient accuracy improvement of GRACE global gravitational field recovery using a new Inter-satellite Range Interpolation Method

New studies on the influence of inter-satellite range on the accuracy of the Earth's gravitational field recovery were carried out based on the Inter-satellite Range Interpolation Method (IRIM). Firstly, the IRIM observation equation by introducing the precise inter-satellite range into the Line-Of-Sight (LOS) component of the relative orbital position vector was established for the first time using the original satellite observations including the orbital position of the spaceborne Global Positioning System (GPS) receiver, the inter-satellite range of the K-Band Ranging (KBR) system and the non-conservative force of the accelerometer (ACC). Secondly, the nine-point IRIM formula is shown by a comparison of multipoint IRIM formulas to be preferable for effectively improving the accuracy of the Earth's gravitational field recovery. Thirdly, the WHIGG-GEGM01S (GRACE Earth Gravity Model developed by Wuhan Institute of Geodesy and Geophysics) global gravity field model, complete up to degree and order 120, is produced using the GRACE Level-1B data products, acquired during 2008-01-01 to 2008-12-31 and released by the American Jet Propulsion Laboratory (JPL). Cumulative geoid height and gravity anomaly errors are 1.098 × 10 −1 m and 1.741 × 10 −6 m/s 2 at degree 120, respectively. Finally, the availability of the WHIGG-GEGM01S model is validated by the GPS/Levelling observations in the U.S., Europe and Australia.

On the application of the coupled finite-infinite element method to geodetic boundary-value problem

Our aim is to introduce the Coupled Finite-Infinite Element Method (CFIEM) as a new alternative approach to the Earth’s gravity field modelling. We show that if the computational domain is large enough in radial direction, one can obtain the qualitatively and quantitatively comparable solution to the solution by the Finite Element Method (FEM). We study the influence of the size of the computational domain on the final CFIEM solution as well as the successive refinement of the discretization and its convergence to the exact solution. As an input data we use the synthetic boundary conditions computed from a Synthetic Earth Gravity Model (SEGM) and we test the CFIEM solution by the data generated directly from SEGM and the solution by the FEM.

Evaluation of EGM 2008 with EGM96 and its Utilization in Topographical Mapping Projects

Precise terrain elevation information is required in various remote sensing and Engineering projects. There are many technologies to derive the terrain elevation information like GPS, ground surveys, LiDAR, Photogrammetry. GPS is the most widely used technology to obtain information due to its ease of operation. However the usage of ellipsoidal heights, i.e. with respect to WGS84 has limited usage in hydrological applications. GPS heights must be converted into orthometric heights for use in hydrological applications, and this requires geoid undulation information. These geoid undulations can be deduced from earth gravity models. There are various earth gravity models available for ready usage like EGM96, EGM2008, GFZ96 in the public domain. This paper discusses the improvements observed in deriving orthometric heights using EGM2008 over its predecessor model EGM96. The utilization of the new model in topographical mapping projects are also presented.

Generation of vertical–horizontal and horizontal–horizontal gravity gradients using stochastically modified integral estimators

The Earth’s gravity field modelling is an ill-posed problem having a sensitive solution to the error of data. Satellite gravity gradiometry (SGG) is a space technique to measure the second-order derivatives of geopotential for modelling this field, but the measurements should be validated prior to use. The existing terrestrial gravity anomalies and Earth gravity models can be used for this purpose. In this paper, the second-order vertical–horizontal (VH) and horizontal–horizontal (HH) derivatives of the extended Stokes formula in the local north-oriented frame are modified using biased, unbiased and optimum types of least-squares modification. These modified integral estimators are used to generate the VH and HH gradients at 250 km level for validation purpose of the SGG data. It is shown that, unlike the integral estimator for generating the second-order radial derivative of geopotential, the system of equations from which the modification parameters are obtained is unstable for all types of modification, with large cap size and high degree, and regularization is strongly required for solving the system. Numerical studies in Fennoscandia show that the SGG data can be estimated with an accuracy of 1 mE using an integral estimator modified by a biased type least-squares modification. In this case an integration cap size of 2.5° and a degree of modification of 100 for integrating 30′ × 30′ gravity anomalies are required.

Software for generating gravity gradients using a geopotential model based on an irregular semivectorization algorithm

The spherical harmonic synthesis of second-order derivatives of geopotential is a task of major concern when the spatial resolution of synthesis is high and/or a high-resolution Earth's gravity model is used. Here, a computational technique is presented for such a process. The irregular semivectorization is introduced as a vectorization technique in which one loop is excluded from matrix-vector products of mathematical models in order to speed up the computation and manage the computer memory. The proposed technique has the capability of considering heights of computation points on a regular grid. MATLAB-based software is developed, which can be used for generating gravity gradients on an ordinary personal computer. The numerical results show that irregular semivectorization significantly reduces the computation time to 1 h for synthesis of these data with global coverage and resolution of 5′×5′ on an elevation model. In addition, a numerical example is presented for testing satellite gravity gradiometry data of the recent European Space Agency satellite mission, the gravity field and steady-state ocean circulation explorer (GOCE), using an Earth's gravity model.

Numerical study on the mixed model in the GOCE polar gap problem

Gravity gradients acquired by the Gravity field and steady-state Ocean Circulation Explorer (GOCE) do not cover the entire earth because of its sun-synchronous orbit leaving data gaps with a radius of about 6.5° in the polar regions. Previous studies showed that the loss of data in the polar regions deteriorates the accuracy of the low order (or near zonal) coefficients of the earth gravity model, which is the so-called polar gap problem in geodesy. In order to find a stable solution for the earth gravity model from the GOCE gravity gradients, three models, i.e. the Gauss-Markov model, light constraint model and the mixed model, are compared and evaluated numerically with the gravity gradient simulated with the EGM2008. The comparison shows that the Best Linear Uniformly Unbiased Estimation (BLUUE) estimator of the mixed model can solve the polar gap problem as effectively as the light constraint model; furthermore, the mixed model is more rigorous in dealing with the supplementary information and leads to a better accuracy in determining the global geoid.

Considering data gaps in geoid modelling by modifying Stokes’ formula

There are numerous methods to modify Stokes’ formula with the usually common feature of reducing the truncation error committed by the lack of gravity data in the far-zone, resulting in an integral formula over the near-zone combined with an Earth Gravity Model that mainly contributes with the long-wavelength information. Here we study the reverse problem, namely to estimate the geoid height with data missing in a cap around the computation point but available in the far-zone outside the cap. Secondly, we study also the problem with gravity data available only in a spherical ring around the computation point. In both cases the modified Stokes formulas are derived using Molodensky and least squares types of solutions. The numerical studies show that the Molodensky type of modification is useless, while the latter method efficiently depresses the various errors contributing to the geoid error. The least squares methods can be used for estimating geoid heights in regions with gravity data gaps, such as in Polar Regions, over great lakes and in some developing countries with lacking gravity data.

The high-resolution gravimetric geoid of Italy: ITG2009

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 (12.5 cm) than that obtained by any previous geoid developed for Italy (and the surrounding area) up to date ( Pavlis et al., 2008). 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.

On the use of global potential field models for regional interpretation of the West and Central African Rift System

The use in regional interpretations of the Earth Gravity Model (EGM08) and Earth magnetic model (EMAG2) is evaluated by comparison to ground gravity and aeromagnetic data in the central sector of the West and Central African Rift System (WCARS). The comparison includes upward continuation, spectral analysis and pseudogravity calculation and statistical evaluation. A correlation between EMAG2 (which contains roughly 25 km resolution aeromagnetic data in the region) and near-surface aeromagnetic data over WCARS is only true for the very low wavelength part but a strong similarity between EGM08 and ground gravity data can be confirmed. Interpretation of the EGM08 data allows identifying and confirming the position of major structural trends, and provides new information on the crustal architecture. The lineaments limiting different grabens forming the Logone Birni basin and in the southern Chad basin are identified. The results presented here show the use of EGM08 data for regional interpretations over Cameroon and adjacent countries, which can overcome the absence and sparseness of data in developing countries and remote areas.