Global Gravity Field Models Research Articles

The development and evaluation of the Earth Gravitational Model 2008 (EGM2008)

EGM2008 is a spherical harmonic model of the Earth's gravitational potential, developed by a least squares combination of the ITG‐GRACE03S gravitational model and its associated error covariance matrix, with the gravitational information obtained from a global set of area‐mean free‐air gravity anomalies defined on a 5 arc‐minute equiangular grid. This grid was formed by merging terrestrial, altimetry‐derived, and airborne gravity data. Over areas where only lower resolution gravity data were available, their spectral content was supplemented with gravitational information implied by the topography. EGM2008 is complete to degree and order 2159, and contains additional coefficients up to degree 2190 and order 2159. Over areas covered with high quality gravity data, the discrepancies between EGM2008 geoid undulations and independent GPS/Leveling values are on the order of ±5 to ±10 cm. EGM2008 vertical deflections over USA and Australia are within ±1.1 to ±1.3 arc‐seconds of independent astrogeodetic values. These results indicate that EGM2008 performs comparably with contemporary detailed regional geoid models. EGM2008 performs equally well with other GRACE‐based gravitational models in orbit computations. Over EGM96, EGM2008 represents improvement by a factor of six in resolution, and by factors of three to six in accuracy, depending on gravitational quantity and geographic area. EGM2008 represents a milestone and a new paradigm in global gravity field modeling, by demonstrating for the first time ever, that given accurate and detailed gravimetric data, asingleglobal model may satisfy the requirements of a very wide range of applications.

GOCE gravity gradients versus global gravity field models

The GOCE mission, a part of ESA′s Living Planet Programme, aims at improved gravity field modelling at high spatial resolution. On-board GOCE a gradiometer, in combination with a scientific on-board GPS receiver, measures the earth′s gravity field with unprecedented accuracy. These measurements have been used to compute GOCE gravity field solutions and combined GOCE/GRACE solutions. The main difference between the solutions is how they incorporate the required a priori information, which consists either of existing gravity field models or Kaula′s rule for the signal variances of the gravity field. We assessed four series of models by comparing the gravity gradients they predict with the measured GOCE gradients. The analysis of the gravity gradients fits may reveal differences between the different solutions that can be attributed to the solution strategy, assuming that the measurement errors are homogeneous. We compared the GOCE gradients with existing state-of-the-art global gravity field models and conclude that the gradient errors are indeed globally homogeneous, with the exception of the cross-track gradient especially south of Australia. Furthermore, we find that the use of existing global gravity field models as a priori information should be avoided because this may increase the gradient residuals in spatial and spectral domain. Finally, we may conclude that the GOCE and GRACE data are compatible and complementary.

The Gauss–Listing geopotential value W 0 and its rate from altimetric mean sea level and GRACE

In geopotential space, the fundamental geodetic parameter W 0 defines the Gauss–Listing geoid which can be used to best represent the Earth’s mean sea level (MSL) and hence specifies a conventional zero height level to unify vertical datums employed by mapping agencies throughout the world. Further, W 0 cannot be considered invariant as the parameter varies temporally as a direct response to sea level change and mass redistributions. This study determines W 0 and its rate, dW 0/dt, by utilizing altimetric MSL models and an independent mean dynamic topography (MDT) model to define points on the geoid. W 0 and dW 0/dt are estimated by two approaches: (i) by means of a global gravity field model (GGM) and (ii) within normal gravity field space as the geopotential value of the best fitting reference ellipsoid. The study shows that uncertainty in W 0 is mainly influenced by MDT while the choice of methodology, GGM and MSL data coverage are not significant within reason. Our estimate W 0 = 62636854.2 ± 0.2 m2 s−2 at epoch 2005.0 differs by 1.8 m2s−2 from the International Astronomical Union reference value. This study shows that, at a sub-decadal time scale, the time variation dW 0/dt stems mainly from sea level change with negligible effect from gravity field variations. dW 0/dt = (−2.70 ± 0.03) × 10−2 m2 s−2 year−1, corresponding to a MSL rise of 2.9 mm year−1, is evaluated from sea level change based on 16 years of TOPEX and Jason-1 data.

On computing ellipsoidal harmonics using Jekeli’s renormalization

Gravity data observed on or reduced to the ellipsoid are preferably represented using ellipsoidal harmonics instead of spherical harmonics. Ellipsoidal harmonics, however, are difficult to use in practice because the computation of the associated Legendre functions of the second kind that occur in the ellipsoidal harmonic expansions is not straightforward. Jekeli’s renormalization simplifies the computation of the associated Legendre functions. We extended the direct computation of these functions—as well as that of their ratio—up to the second derivatives and minimized the number of required recurrences by a suitable hypergeometric transformation. Compared with the original Jekeli’s renormalization the associated Legendre differential equation is fulfilled up to much higher degrees and orders for our optimized recurrences. The derived functions were tested by comparing functionals of the gravitational potential computed with both ellipsoidal and spherical harmonic syntheses. As an input, the high resolution global gravity field model EGM2008 was used. The relative agreement we found between the results of ellipsoidal and spherical syntheses is 10−14, 10−12 and 10−8 for the potential and its first and second derivatives, respectively. Using the original renormalization, this agreement is 10−12, 10−8 and 10−5, respectively. In addition, our optimized recurrences require less computation time as the number of required terms for the hypergeometric functions is less.

Local lunar gravity field analysis over the South Pole‐Aitken basin from SELENE farside tracking data

We present a method with which we determined the local lunar gravity field model over the South Pole‐Aitken (SPA) basin on the farside of the Moon by estimating adjustments to a global lunar gravity field model using SELENE tracking data. Our adjustments are expressed in localized functions concentrated over the SPA region in a spherical cap with a radius of 45° centered at (191.1°E, 53.2°S), and the resolution is equivalent to a 150th degree and order spherical harmonics expansion. The new solution over SPA was used in several applications of geophysical analysis. It shows an increased correlation with high‐resolution lunar topography in the frequency band l = 40–70, and admittance values are slightly different and more leveled when compared to other, global gravity field models using the same data. The adjustments expressed in free‐air anomalies and differences in Bouguer anomalies between the local solution and the a priori global solution correlate with topographic surface features. The Moho structure beneath the SPA basin is slightly modified in our solution, most notably at the southern rim of the Apollo basin and around the Zeeman crater.

Validation of GOCE global gravity field models using terrestrial gravity data in Norway

Validation of GOCE global gravity field models using terrestrial gravity data in NorwayThe GOCE (Gravity field and steady-state Ocean Circulation Explorer) satellite gravity gradiometry mission maps the Earth's gravity field. Harmonic analysis of GOCE observations provides a global gravity field model (GGFM). Three theoretical strategies, namely the direct, the space-wise and the time-wise approach, have been proposed for GOCE harmonic analysis. Based on these three methods, several GGFMs have been provided to the user community by ESA. Thereby different releases are derived from different periods of GOCE observations and some of the models are based on combinations with other sources of gravity field information. Due to the multitude of GOCE GGFMs, validation against independent data is a crucial task for the quality description of the different models.In this study, GOCE GGFMs from three releases are validated with respect to terrestrial free-air gravity anomalies in Norway. The spectral enhancement method is applied to avoid spectral inconsistency between the terrestrial and the GOCE free-air gravity anomalies.The results indicate that the time-wise approach is a reliable harmonic analysis procedure in all three releases of GOCE models. The space-wise approach, available in two releases, provides similar results as the time-wise approach. The direct approach seems to be highly affected by a-priori information.

Sensitivity of Lageos Orbits to Global Gravity Field Models

Sensitivity of Lageos Orbits to Global Gravity Field ModelsPrecise orbit determination is an essential task when analyzing SLR data. The quality of the satellite orbits strongly depends on the models used for dynamic orbit determination. The global gravity field model used is one of the crucial elements, which has a significant influence on the satellite orbit and its accuracy. We study the impact of different gravity field models on the determination of the LAGEOS-1 and -2 orbits for data of the year 2008. Eleven gravity field models are compared, namely JGM3 and EGM96 based mainly on SLR, terrestrial and altimetry data, AIUB-CHAMP03S based uniquely on GPS-measurements made by CHAMP, AIUB-GRACE03S, ITG-GRACE2010 based on GRACE data, and the combined gravity field models based on different measurement techniques, such as EGM2008, EIGEN-GL04C, EIGEN51C, GOCO02S, GO-CONS-2-DIR-R2, AIUB-SST. The gravity field models are validated using the RMS of the observation residuals of 7-day LAGEOS solutions. The study reveals that GRACE-based models have the smallest RMS values (i.e., about 7.15 mm), despite the fact that no SLR data were used to determine them. The coefficient C20is not always well estimated in GRACE-only models. There is a significant improvement of the gravity field models based on CHAMP, GRACE and GOCE w.r.t. models of the pre-CHAMP era. The LAGEOS orbits are particularly sensitive to the long wavelength part of the gravity fields. Differences of the estimated orbits due to different gravity field models are noticeable up to degree and order of about 30. The RMS of residuals improves from about 40 mm for degree 8, to about 7 mm for the solutions up to degrees 14 and higher. The quality of the predicted orbits is studied, as well.

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.

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.

GOCE gravitational gradients along the orbit

GOCE is ESA’s gravity field mission and the first satellite ever that measures gravitational gradients in space, that is, the second spatial derivatives of the Earth’s gravitational potential. The goal is to determine the Earth’s mean gravitational field with unprecedented accuracy at spatial resolutions down to 100 km. GOCE carries a gravity gradiometer that allows deriving the gravitational gradients with very high precision to achieve this goal. There are two types of GOCE Level 2 gravitational gradients (GGs) along the orbit: the gravitational gradients in the gradiometer reference frame (GRF) and the gravitational gradients in the local north oriented frame (LNOF) derived from the GGs in the GRF by point-wise rotation. Because the V XX , V YY , V ZZ and V XZ are much more accurate than V XY and V YZ , and because the error of the accurate GGs increases for low frequencies, the rotation requires that part of the measured GG signal is replaced by model signal. However, the actual quality of the gradients in GRF and LNOF needs to be assessed. We analysed the outliers in the GGs, validated the GGs in the GRF using independent gravity field information and compared their assessed error with the requirements. In addition, we compared the GGs in the LNOF with state-of-the-art global gravity field models and determined the model contribution to the rotated GGs. We found that the percentage of detected outliers is below 0.1% for all GGs, and external gravity data confirm that the GG scale factors do not differ from one down to the 10?3 level. Furthermore, we found that the error of V XX and V YY is approximately at the level of the requirement on the gravitational gradient trace, whereas the V ZZ error is a factor of 2–3 above the requirement for higher frequencies. We show that the model contribution in the rotated GGs is 2–35% dependent on the gravitational gradient. Finally, we found that GOCE gravitational gradients and gradients derived from EIGEN-5C and EGM2008 are consistent over the oceans, but that over the continents the consistency may be less, especially in areas with poor terrestrial gravity data. All in all, our analyses show that the quality of the GOCE gravitational gradients is good and that with this type of data valuable new gravity field information is obtained.

Fitting the GPS/Leveling Quasi-Geoid Using Bayesian-Regulation BP Neural Network

The 2.5′×2.5′resolution local quasi-geoid is calculated using the global gravity field model and GPS/leveling data of region which points spacing is about 10km with the Bayesian- regulation BP neural network in this paper. The inner and outer precision of quasi-geoid are both superior 0.05m.The result indicat that the Bayesian regulation BP neural network could improve the precision of fitting and restrain the over-fitting in fitting. The region quasi-geoid excelled than 0.05m can be computed using the global gravity field model and about 10km baseline GPS/leveling data in smoothness region.

A crustal thickness map of Africa derived from a global gravity field model using Euler deconvolution

We develop a new continental scale crustal model for Africa by modelling the free-air gravity anomaly EIGEN-GL04C, which was developed from 30 months of GRACE Level 1B data covering the period from 2003 February to 2005 July, and surface gravity data from seven different sources. From this gravity model, crustal thickness is estimated using 3-D Euler deconvolution, a method that does not rely on a priori depth and density constraints. The results are in good agreement (i.e. within 5 km) of seismically determined Moho depth estimates from across the continent, except for narrow tectonic regions, such as rift valleys, and areas where seismic velocity models of the crust indicate a gradational Moho. The results show that crustal thickness is fairly homogeneous, with an average crustal thickness for the whole continent of 39 ± 2(SD) km. The average Moho depth for most terrains is within 5 km of the continental average, and there is little variability between terrains of different age. The average thickness for Archean, Proterozoic and Palaeozoic crust is 39, 39 and 41 km, respectively. Crustal thickness in sedimentary basins across northern and central Africa varies between 33 and 36 km. Through comparison with global averages for similar-aged terrains, we find that African crustal thickness does not deviate significantly from the thickness of crust in other parts of the world.

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.

External calibration of GOCE accelerations to improve derived gravitational gradients

The aim of ESA's satellite mission GOCE is to determine the Earth's gravity field with high accuracy and resolution. To achieve this aim, GOCE carries a gravitational gradiometer that needs calibration. Existing global gravity fieldmodels in combinationwith GOCE star sensor data may be used to synthesize reference differential accelerations with which the common and differential accelerations, as derived from the gradiometer measurements, can be calibrated. We present a new method in which the data are transformed from the time to the frequency domain, which allows accounting for the coloured noise on the measurements. The weight matrix is iteratively adjusted and we apply our method to real GOCE data. With our method, the gravitational gradient trace significantly reduces as compared with the currently available in-flight calibrated measurements.

Combined satellite gravity field model GOCO01S derived from GOCE and GRACE

The satellite‐only gravity field model GOCO01S is a combination solution based on 61 days of GOCE gravity gradient data, and 7 years of GRACE GPS and K‐band range rate data, resolved up to degree/order 224 of a harmonic series expansion. The combination was performed consistently by addition of full normal equations and stochastic modeling of GOCE and GRACE observations. The model has been validated against external global gravity models and regional GPS/leveling observations. While low to medium degrees are mainly determined by GRACE, significant contributions by the new measurement type of GOCE gradients can already be observed at degree 100. Beyond degree 150, GOCE becomes the dominant contributor. Correspondingly, with GOCO01S a global gravity field model with high performance for the complete spectral range up to degree/order 224 is now available. This new gravity model will be beneficial for many applications in geophysics, oceanography, and geodesy.

Combining EGM2008 and SRTM/DTM2006.0 residual terrain model data to improve quasigeoid computations in mountainous areas devoid of gravity data

A global geopotential model, like EGM2008, is not capable of representing the high-frequency components of Earth’s gravity field. This is known as the omission error. In mountainous terrain, omission errors in EGM2008, even when expanded to degree 2,190, may reach amplitudes of 10 cm and more for height anomalies. The present paper proposes the utilisation of high-resolution residual terrain model (RTM) data for computing estimates of the omission error in rugged terrain. RTM elevations may be constructed as the difference between the SRTM (Shuttle Radar Topography Mission) elevation model and the DTM2006.0 spherical harmonic topographic expansion. Numerical tests, carried out in the German Alps with a precise gravimetric quasigeoid model (GCG05) and GPS/levelling data as references, demonstrate that RTM-based omission error estimates improve EGM2008 height anomaly differences by 10 cm in many cases. The comparisons of EGM2008-only height anomalies and the GCG05 model showed 3.7 cm standard deviation after a bias-fit. Applying RTM omission error estimates to EGM2008 reduces the standard deviation to 1.9 cm which equates to a significant improvement rate of 47%. Using GPS/levelling data strongly corroborates these findings with an improvement rate of 49%. The proposed RTM approach may be of practical value to improve quasigeoid determination in mountainous areas without sufficient regional gravity data coverage, e.g., in parts of Asia, South America or Africa. As a further application, RTM omission error estimates will allow refined validation of global gravity field models like EGM2008 from GPS/levelling data.

Recent developments in high-resolution global altimetric gravity field modeling

In recent years, dedicated effort has been made to improve high-resolution global marine gravity fields. One new global field is the Danish National Space Center (DNSC) 1-minute grid called DNSC08GRA, released in 2008. DNSC08GRA was derived from double-retracked satellite altimetry, mainly from the ERS-1 geodetic mission data, augmented with new retracked GEOSAT data which have significantly enhanced the range and hence the gravity field accuracy.

On the combination of global and local data in collocation theory

Due to the successful operation of dedicated satellite gravity missions, nowadays high-accuracy global gravity field models have become available. This triggers the challenge to optimally combine this long to medium wavelength gravity field information derived from space-borne data with high-resolution terrestrial gravity data. In this paper, the least squares collocation concept is revised with the attempt to consistently unify the combination procedure in such a way that the full information contained in both data sets is merged. For example, in local or regional geoid determination the remove-restore method is usually applied only partially taking into account the accuracy of the global model coefficients used for the long-wavelength reduction. The key advantage of the extended formulation is the fact that it automatically accounts for the error covariance of all data types involved. The applicability, feasibility and performance of the proposed method is investigated in the frame of numerical closed-loop simulations. The two main fields of application, i.e., the improvement of a global gravity field model by terrestrial gravity field data, and, vice versa, the support to a regional geoid solution by the incorporation of a global gravity field model, have been analyzed and assessed. Although applied under simplified conditions, it could be shown that the method works and is practically applicable.

Direct BEM for high-resolution global gravity field modelling

The paper presents a high-resolution global gravity field modelling by the boundary element method (BEM). A direct BEM formulation for the Laplace equation is applied to get a numerical solution of the linearized fixed gravimetric boundary-value problem. The numerical scheme uses the collocation method with linear basis functions. It involves a discretization of the complicated Earth’s surface, which is considered as a fixed boundary. Here 3D positions of collocation points are simulated from the DNSC08 mean sea surface at oceans and from the SRTM30PLUS_V5.0 global topography model added to EGM96 on lands. High-performance computations together with an elimination of the far zones’ interactions allow a very refined integration over the all Earth’s surface with a resolution up to 0.1 deg. Inaccuracy of the approximate coarse solutions used for the elimination of the far zones’ interactions leads to a long-wavelength error surface included in the obtained numerical solution. This paper introduces an iterative procedure how to reduce such long-wavelength error surface. Surface gravity disturbances as oblique derivative boundary conditions are generated from the EGM2008 geopotential model. Numerical experiments demonstrate how the iterative procedure tends to the final numerical solutions that are converging to EGM2008. Finally the input surface gravity disturbances at oceans are replaced by real data obtained from the DNSC08 altimetryderived gravity data. The ITG-GRACE03S satellite geopotential model up to degree 180 is used to eliminate far zones’ interactions. The final high-resolution global gravity field model with the resolution 0.1 deg is compared with EGM2008.

AIUB-CHAMP02S: The influence of GNSS model changes on gravity field recovery using spaceborne GPS

The gravity field model AIUB-CHAMP02S, which is based on six years of CHAMP GPS data, is presented here. The gravity field parameters were derived using a two step procedure: In a first step a kinematic trajectory of a low Earth orbiting (LEO) satellite is computed using the GPS data from the on-board receiver. In this step the orbits and clock corrections of the GPS satellites as well as the Earth rotation parameters (ERPs) are introduced as known. In the second step this kinematic orbit is represented by a gravitational force model and orbit parameters. In order to ensure full model consistency the GPS satellite orbits and clock corrections, which have been used for the generation of the kinematic LEO trajectories, were taken from the Center for Orbit Determination in Europe (CODE), located at AIUB ( Dach et al., 2009). In recent years many changes have taken place in the processing chain of global navigation satellite system (GNSS) data, e.g., the implementation of absolute antenna phase center modeling. Therefore a reprocessing of the GPS data to obtain state-of-the-art GPS satellite orbits and clock corrections was performed. From these updated GPS products new kinematic orbits of the CHAMP satellite were derived for the years 2002–2007. From the updated CHAMP trajectories spherical harmonic (SH) coefficients of the Earth’s gravity field were determined in exactly the same way as from the original LEO orbit. This allowed us to study the impact of the improved LEO orbits on the derived gravity field parameters and the generation of the multi-year gravity field model AIUB-CHAMP02S. The change of the IGS standards creates an inconsistency to existing global gravity field models, which mainly affects the zonal coefficients of low even degrees. The inconsistency is caused by the change to the absolute antenna phase center model and can be reduced by estimating the phase center variation of the CHAMP GPS antenna.