Gravity Field Models Research Articles

The updated ESA Earth System Model for future gravity mission simulation studies

A new synthetic model of the time-variable global gravity field is now available based on realistic mass variability in atmosphere, oceans, terrestrial water storage, continental ice-sheets, and the solid Earth. The updated ESA Earth System Model is provided in Stokes coefficients up to degree and order 180 with a temporal resolution of 6 h covering the time period 1995–2006, and can be readily applied as a source model in future gravity mission simulation studies. The model contains plausible variability and trends in both low-degree coefficients and the global mean eustatic sea level. It depicts reasonable mass variability all over the globe at a wide range of frequencies including multi-year trends, year-to-year variability, and seasonal variability even at very fine spatial scales, which is important for a realistic representation of spatial aliasing and leakage. In particular on these small spatial scales between 50 and 250 km, the model contains a range of signals that have not been reliably observed yet by satellite gravimetry. In addition, the updated Earth System Model provides substantial high-frequency variability at periods down to a few hours only, thereby allowing to critically test strategies for the minimization of temporal aliasing.

Gravity gradient distribution in mainland China from GOCE satellite gravity gradiometry data

At present, gravity field and steady-state ocean circulation explorer (GOCE) gravity data are always used to compute regional gravity anomaly and geoid height. In this study, the latest GOCE gravity field model data (from Oct. 2009 to Jul. 2010) are used to compute the gravity gradient of mainland China according to a rigorous recursion formula (in all the six directions). The results show that the numerical values of the gravity gradients are larger in the Trr direction than those in the other directions. They reflect the terrain characteristics in detail and correlate with the regional tectonics; however, in the Tθλ and Trλ directions, the numerical values are relatively smaller and the gravity gradients in the Trλ direction do not reflect the terrain characteristics in detail.

Ocean Surface Geostrophic Circulation Climatology and Annual Variations Inferred from Satellite Altimetry and GOCE Gravity Data

We have studied, for the first time, variations in absolute surface geostrophic currents (SGC) using satellite data only. The proposed approach combines 18 years’ altimetry data, which provide reliable measurements of absolute sea level (ASL), with a gravity field and steady-state ocean circulation explorer geoid model to obtain dynamic topography, and achieves unprecedented precision and accuracy. Our proposal overcomes the main limitations of existing approaches based solely on altimetry data (which suffer from lack of an independent reference for derivation of ASL maps), and approximations based on in-situ data (which are characterized by a sparse and inhomogeneous coverage in time and space). Features of annual variations of SGC are also addressed. As a result of our study we provide new absolute SGC climatology in the form of a 52-week data set of surface current fields, gridded at quarter degree longitude and latitude resolution and resolving spatial scales as short as 140 km. For presentation, this data set is averaged monthly and the results, presented as monthly climatology, are compared with climatology based on in-situ observations from drifter data.

EGM_TIM_RL05: An independent geoid with centimeter accuracy purely based on the GOCE mission

AbstractAfter more than 4.5 years in orbit, the Gravity field and steady‐state Ocean Circulation Explorer (GOCE) mission ended with the reentry of the satellite on 11 November 2013. This publication serves as a reference for the fifth gravity field model based on the time‐wise approach (EGM_TIM_RL05), a global model only determined from GOCE observations. Due to its independence of any other gravity data, a consistent and homogeneous set of spherical harmonic coefficients up to degree and order 280 (corresponding to spatial resolution of 71.5 km on ground) is provided including a full covariance matrix characterizing the uncertainties of the model. The associated covariance matrix realistically describes the model quality. It is the first model which is purely based on GOCE including all observations collected during the entire mission. The achieved mean global accuracy is 2.4 cm in terms of geoid heights and 0.7 mGal for gravity anomalies at a spatial resolution of 100 km.

Global Earth’s gravity field solution with GRACE orbit and range measurements using modified short arc approach

Traditionally, the Earth’s gravity field model is computed from GRACE orbit and range rate measurements, e.g., in a short arc approach where both the position and the velocity vectors are integrated from a force model. In this contribution, we use the GRACE orbit and range measurements to recover the Earth’s gravity field model, thus we only need to integrate the position vectors. We use the range differences between two adjacent epochs to eliminate the range ambiguities. Using GRACE Level-1B RL02 data released by Jet Propulsion laboratory, the gravity field model TJGRACE02O complete to degree and order 90 is developed from 7 years of reduced dynamic orbits covering the period 2004–2010, and the gravity field model TJGRACE02K complete to degree and order 120 is computed from 1 month of kinematic orbits and K-band range data of January. Comparing the degree geoid errors of our new models with recent gravity field models such as the CHAMP-only models EIGEN-CHAMP05S, AIUB-CHAMP03S, ULUX-CHAMP2013S and the GRACE-only models GGM05S, Tongji-GRACE01 as well as a monthly model from the ITG-GRACE2010 time series, and validating these models with GPS-leveling data sets in the USA, we can conclude that the TJGRACE02O model is more accurate than all the CHAMP-only models and TJGRACE02K is comparable in quality to the corresponding GRACE monthly model from ITG-GRACE2010.

ESA's satellite‐only gravity field model via the direct approach based on all GOCE data

AbstractGravity field and steady state Ocean Circulation Explorer (GOCE) gravity gradient data of the entire science mission and data from LAGEOS 1/2 and Gravity Recovery and Climate Experiment (GRACE) were combined in the construction of a satellite‐only gravity field model to maximum degree 300. When compared to Earth Gravitational Model 2008, it is more accurate at low to medium resolution, thanks to GOCE and GRACE data. When compared to earlier releases of European Space Agency GOCE models, it is more accurate at high degrees owing to the larger amount of data ingested, which was moreover taken at lower altitude. The impact of orbiting at lower altitude in the last year of the mission is large: a model based on data of the last 14 months is significantly more accurate than the release 4 model constructed with the first 28 months. The (calibrated) cumulated geoid error estimate at 100 km resolution is 1.7 cm. The optimal resolution of the GOCE model for oceanographic application is between 100 and 125 km.

Errors of Mean Dynamic Topography and Geostrophic Current Estimates in China’s Marginal Seas from GOCE and Satellite Altimetry

AbstractThe Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) and satellite altimetry can provide very detailed and accurate estimates of the mean dynamic topography (MDT) and geostrophic currents in China’s marginal seas, such as, the newest high-resolution GOCE gravity field model GO-CONS-GCF-2-TIM-R4 and the new Centre National d’Etudes Spatiales mean sea surface model MSS_CNES_CLS_11 from satellite altimetry. However, errors and uncertainties of MDT and geostrophic current estimates from satellite observations are not generally quantified. In this paper, errors and uncertainties of MDT and geostrophic current estimates from satellite gravimetry and altimetry are investigated and evaluated in China’s marginal seas. The cumulative error in MDT from GOCE is reduced from 22.75 to 9.89 cm when compared to the Gravity Recovery and Climate Experiment (GRACE) gravity field model ITG-Grace2010 results in the region. The errors of the geostrophic currents from GRACE are smaller than from GOCE with the truncation degrees 90 and 120. However, when the truncation degree is higher than 150, the GRACE mean errors increase rapidly and become significantly larger than the GOCE results. The geostrophic velocities based on GOCE-TIM4 have higher accuracy and spatial resolution, and the mean error is about 12.6 cm s−1, which is more consistent with the in situ drifter’s results than using GRACE data.

Gravity field processing with enhanced numerical precision for LL-SST missions

On their way to meet the augmenting demands of the Earth system user community concerning accuracies of temporal gravity field models, future gravity missions of low-low satellite-to-satellite tracking (LL-SST) type are expected to fly at optimized formations and make use of the latest technological achievements regarding the on-board sensor accuracies. Concerning the main measuring unit of an LL-SST type gravity mission, the inter-satellite measuring instrument, a much more precise interferometric laser ranging system is planned to succeed the K-band ranging system used by the Gravity Recovery and Climate Experiment (GRACE) mission. This study focuses on investigations concerning the potential performance of new generation sensors such as the laser interferometer within the gravity field processing chain. The sufficiency of current gravity field processing accuracies is tested against the new sensor requirements, via full-scale closed-loop numerical simulations of a GRACE Follow-On configuration scenario. Each part of the processing is validated separately with special emphasis on numerical errors and their impact on gravity field solutions. It is demonstrated that gravity field processing with double precision may be a limiting factor for taking full advantage of the laser interferometer’s accuracy. Instead, a hybrid processing scheme of enhanced precision is introduced, which uses double and quadruple precision in different parts of the processing chain, leading to system accuracies of only 17 nm in terms of geoid height reconstruction errors. Simulation results demonstrate the ability of enhanced precision processing to minimize the processing errors and thus exploit the full precision of a laser interferometer, when at the same time the computational times are kept within reasonable levels.

A 10 km-resolution synthetic Venus gravity field model based on topography

A high resolution gravity field model is extremely important in the exploration of Venus. In this paper, we present a 3-dimensional Venus gravity field VGM2014 constructed by using the latest gravity and topography models, residual terrain model (RTM) and the Airy–Heiskanen isostatic compensation model. The VGM2014 is the first 10km scale Venus gravity field model; the final results are representations of the 3-dimensional surface gravity accelerations and gravity disturbances for Venus. We found that the optimal global compensation depth of Venus is about 60km, and the crustal density is potentially less than the commonly accepted value of 2700–2900kgm−3. This model will be potentially beneficial for the precise orbit determination and landing navigation of spacecraft around Venus, and may be utilized as a priori model for Venus gravity field simulation and inversion studies. The VGM2014 does not incorporate direct gravity information beyond degree 70 and it is not recommended for small-scale geophysical interpretation.

Investigating the effect of tide parameterization and gravity field models on SLR solutions

Satellite laser ranging (SLR) as a proven geodetic technique provides a wide and significant contribution to scientific studies of the Earth/Atmosphere/Ocean systems. In particular, modelling of the Earth and pole tides plays a very important role when analyzing SLR data. The accuracy of the determined satellite orbits is highly dependent on the models used for dynamic orbit determination. Gravity field models, which are represented by a series of spherical harmonic coefficients, have an impact on the satellite orbit and its precision. In addition, different empirical model parameterization used during SLR data analysis also has significant influence on the computed spherical harmonics, satellite orbits and their accuracies. In this contribution we investigate the impact of different SLR parameterizations on gravity field models used for precise satellite orbit determination. Data from satellites LAGEOS 1 and 2 were analysed using four different parameterization schemes namely: IERS1, IERS2 and IERS3 Earth tide models and pole tides. These are compatible Earth tide models of different complexity derived from IERS2010, a standard model of the International Earth Rotation and Reference Systems Service (IERS) and the standard IERS2010 pole tide model, and they aim to study their impacts on SLR solutions. The results indicate that the combination of IERS3, which is the most complex Earth tide model and the standard IERS2010 pole tide model, has a significant influence on the accuracy of gravity field models in precise orbit determination. In particular almost all the evaluated models give the smallest RMS values when IERS3 and pole tides are jointly selected in the analysis software during SLR data analysis, which indicates that the most complex models are also the most accurate. This work therefore validates the currently accepted IERS2010 Earth tide and pole tide models.

Comparison of global Earth’s gravity field models with the aerogravimetric data obtained during a transcontinental flight

Some methodological features of the airborne gravity surveys of Russia’s territory at the latitude of 60° from Arkhangelsk to Kamchatka are described. The results of these surveys are presented. The data obtained from gravimeters installed aboard an aircraft laboratory are compared with EGM2008 and EIGEN-6C3, the state-of-the-art global Earth gravity field models. It is shown that the present-day models have the highest errors near the coastal tectonic structures of the Kamchatka Peninsula.

Requirements Analysis for Future Satellite Gravity Mission Improved-GRACE

The Earth’s gravitational field from the Next-Generation Gravimetry Mission (NGGM) and the Improved-Gravity Recovery and Climate Experiment (Improved-GRACE) complete up to degree and order 120 is recovered by a closed-loop numerical simulation using different orbital altitudes of 325 and 300 km, different orbital inclinations of 96.78° and 89° and different inter-satellite ranges of 10 and 50 km. The preferred orbit parameters of the future twin Improved-GRACE satellites are proposed based on the results of the simulations in this study. The research results show: (1) In order to achieve the scientific objectives, which require that the accuracy of the next-generation Earth gravity field models is at least one order of magnitude better than that of the current gravity models, the orbit design at an altitude of 300 ± 50 km is recommended for the future Improved-GRACE mission. This altitude is determined by a trade-off analysis between the recovery accuracy of the gravity field and the operational lifetime of the satellite system. (2) Because the accuracy of the Earth’s gravitational field from NGGM with an orbital inclination of 96.78° will be decreased due to a lack of the observation data in the polar areas, we propose that a near-polar orbit (inclination of 89° ± 2°) is a preferable selection for the future twin Improved-GRACE satellites. (3) The future Improved-GRACE mission has to adopt an inter-satellite range of 50 ± 10 km, because the common signals of the Earth’s gravitational field between the twin NGGM satellites will be substantially eliminated with a shorter inter-satellite range of 10 km. With these orbit design parameters, the Earth’s gravitational field from the Improved-GRACE mission is precisely recovered complete up to degree and order 120 with a cumulative geoid height error of about 0.7 mm.

Gravity Matching Aided Navigation Using a New Mixed Filter Algorithm

To solve low precision of basic gravity aided matching algorithm caused by the influence of gravimetric map error, observed model linearization error, insufficiency observed quantity and so on, a novel algorithm of square root KF+CKF mixed filter algorithm based on Rao-Blackwellizationptive is proposed. The calculation accurary,speed, noise immunity, and application scope are improved by approximation local gravity field model using gauss spline interpolation,establishing nonliear mixed model viewing at gravity and position error, model decomposition using Rao-Blackwellization thoery,improving KF and CKF by changing error covariance matrix to square root model, adaptively adjusting filter renewal process.The result shows that it is less sensitive to INS initial error and nonlinearized degree of navigation system model with higher speed,better accurary and stronger robustness.

Physical analysis on improving the recovery accuracy of the Earth's gravity field by a combination of satellite observations in along-track and cross-track directions

The physical investigations on the accuracy improvement to the measurement of the Earth's gravity field recovery are carried out based on the next-generation Pendulum-A/B out-of-plane twin-satellite formation in this paper. Firstly, the Earth's gravity field complete up to degree and order 100 is, respectively, recovered by the collinear and pendulum satellite formations using the orbital parameters of the satellite and the matching accuracies of key payloads from the twin GRACE satellites. The research results show that the accuracy of the Earth's gravity field model from the Pendulum-A/B satellite formation is about two times higher than from the collinear satellite formation, and the further improvement of the determination accuracy of the Earth's gravity field model is feasible by the next-generation Pendulum-A/B out-of-plane twin-satellite formation. Secondly, the Earth's gravity field from Pendulum-A/B complete up to degree and order 100 is accurately recovered based on the orbital parameters of the satellite (e.g., an orbital altitude of 400 km, an intersatellite range of 100 km, an orbital inclination of 89° and an orbital eccentricity of 0.001), the matching accuracies of space-borne instruments (e.g. 10−6 m in the intersatellite range, 10−3 m in the orbital position, 10−6 m/s in orbital velocity, and 10−11 m/s2 in non-conservative force), an observation time of 30 days and a sampling interval of 10 s. The measurement accuracy of the Earth's gravity field from the next-generation Pendulum-A/B out-of-plane twin-satellite formation is full of promise for being improved by about 10 times compared with that from the current GRACE satellite formation. Finally, the physical requirements for the next-generation Pendulum-A/B out-of-plane twin-satellite formation are analyzed, and it is proposed that the satellite orbital altitude be preferably designed to be close to 400±50 km and the matching precision of key sensors from the Pendulum-A/B mission be about one order of magnitude higher than from the GRACE program.

A study of different wavelength spectral components of the gravity field derived from various terrestrial data sets

In the general scheme of gravity field modelling long-, medium- and short-wavelength constituents of the gravity field derived from e.g. geopotential model, terrestrial data and digital terrain model respectively, are routinely combined. In this study, spectral characteristics of terrestrial data sets are investigated. The estimation of spectral sensitivity of gravity related quantities such as gravity anomaly, vertical deflections and gravity gradients was accomplished through Fourier PSD and covariance analysis depending on the spatial distribution of data points. The information content of the estimated spectra were validated on global and local levels to access their further utilization. The spectra were compared to the 1D spectrum of the gravitational field derived from spherical harmonic coefficients using a high resolution global gravitational model as well as to an analytical approximation. Besides the frequency domain investigations the information content regarding the different wavelength structure comprised in terrestrial and EGM2008 model is investigated in the space domain based on covariance analysis. As a combined validation process the gravity degree variances were transformed to the necessary auto- and cross covariance functions to predict geoid height from gravity anomaly, which ensures an independent validation process of the computed spectrum. Based on the spectral characteristics of terrestrial measurement spectral weights for spectral combination were derived involving global gravity field model, gravity and gravity gradient data in gravity field modelling. To determine the geoid in the whole spectral band the specific integral kernels in the spectral domain should be modified using the suggested spectral weights.

High-resolution global gravity field modelling by the finite volume method

We discuss the parallel computational solution to the modified fixed gravimetric boundary-value problem (MFGBVP). In our approach, the computational domain is a finite space bounded by two spatial boundaries. The boundaries represent an approximation of the Earth’s surface and an approximation of the chosen satellite orbit. Then the MFGBVP consists of the Laplace equation for unknown disturbing potential with the Neumann and Dirichlet boundary conditions. Solution of such elliptic boundary-value problem is understood in a weak sense, so it always exists and is unique. As a numerical method for our parallel approach, the finite volume method (FVM) has been designed and implemented. The FVM is a method for solving elliptic equations and it leads to a solution of the sparse linear system of equations with an appropriate structure for parallel implementation concerning memory costs. The parallel implementation of FVM algorithms using MPI and NUMA procedures is also described. Several numerical experiments are discussed. In the first testing experiment, we show that the proposed approach is second-order accurate. Then we test a convergence of the FVM solution to the EGM2008 Earth gravitational model when refining the grid. In this case all boundary conditions (BCs) are generated from this model. Finally we present high-resolution global gravity field modelling using input data generated from the DTU10 gravity field model and the GOCO03S satellite-only geopotential model. It combines information from the GRACE and GOCE satellite misions prescribed on the upper boundary with the altimetryderived and terrestrial gravity data available on the Earth’s surface. The obtained global gravity field model is compared with the EGM2008.

Improvement of the GRACE star camera data based on the revision of the combination method

The new release of the sensor and instrument data (Level-1B release 02) of the Gravity Recovery and Climate Experiment (GRACE) had a substantial impact on the improvement of the overall accuracy of the gravity field models. This has implied that improvements on the sensor data level can still significantly contribute to arriving closer to the GRACE baseline accuracy. The recent analysis of the GRACE star camera data (SCA1B RL02) revealed their unexpectedly higher noise. As the star camera (SCA) data are essential for the processing of the K-band ranging data and the accelerometer data, thorough investigation of the data set was needed. We fully reexamined the SCA data processing from Level-1A to Level-1B with focus on the combination method of the data delivered by the two SCA heads. In the first step, we produced and compared our own combined attitude solution by applying two different combination methods on the SCA Level-1A data. The first method introduces the information about the anisotropic accuracy of the star camera measurement in terms of a weighing matrix. This method was applied in the official processing as well. The alternative method merges only the well determined SCA boresight directions. This method was implemented on the GRACE SCA data for the first time. Both methods were expected to provide optimal solution characteristic by the full accuracy about all three axes, which was confirmed. In the second step, we analyzed the differences between the official SCA1B RL02 data generated by the Jet Propulsion Laboratory (JPL) and our solution. SCA1B RL02 contains systematically higher noise of about a factor 3–4. The data analysis revealed that the reason is the incorrect implementation of algorithms in the JPL processing routines. After correct implementation of the combination method, significant improvement within the whole spectrum was achieved. Based on these results, the official reprocessing of the SCA data is suggested, as the SCA attitude data are one of the key observations needed for the gravity field recovery.

GOCE: precise orbit determination for the entire mission

The Gravity field and steady-state Ocean Circulation Explorer (GOCE) was the first Earth explorer core mission of the European Space Agency. It was launched on March 17, 2009 into a Sun-synchronous dusk-dawn orbit and re-entered into the Earth’s atmosphere on November 11, 2013. The satellite altitude was between 255 and 225 km for the measurement phases. The European GOCE Gravity consortium is responsible for the Level 1b to Level 2 data processing in the frame of the GOCE High-level processing facility (HPF). The Precise Science Orbit (PSO) is one Level 2 product, which was produced under the responsibility of the Astronomical Institute of the University of Bern within the HPF. This PSO product has been continuously delivered during the entire mission. Regular checks guaranteed a high consistency and quality of the orbits. A correlation between solar activity, GPS data availability and quality of the orbits was found. The accuracy of the kinematic orbit primarily suffers from this. Improvements in modeling the range corrections at the retro-reflector array for the SLR measurements were made and implemented in the independent SLR validation for the GOCE PSO products. The satellite laser ranging (SLR) validation finally states an orbit accuracy of 2.42 cm for the kinematic and 1.84 cm for the reduced-dynamic orbits over the entire mission. The common-mode accelerations from the GOCE gradiometer were not used for the official PSO product, but in addition to the operational HPF work a study was performed to investigate to which extent common-mode accelerations improve the reduced-dynamic orbit determination results. The accelerometer data may be used to derive realistic constraints for the empirical accelerations estimated for the reduced-dynamic orbit determination, which already improves the orbit quality. On top of that the accelerometer data may further improve the orbit quality if realistic constraints and state-of-the-art background models such as gravity field and ocean tide models are used for the reduced-dynamic orbit determination.

Systematic effects in LOD from SLR observations

Beside the estimation of station coordinates and the Earth’s gravity field, laser ranging observations to near-Earth satellites can be used to determine the rotation of the Earth. One parameter of this rotation is ΔLOD (excess Length Of Day) which describes the excess revolution time of the Earth w.r.t. 86,400s. Due to correlations among the different parameter groups, it is difficult to obtain reliable estimates for all parameters. In the official ΔLOD products of the International Earth Rotation and Reference Systems Service (IERS), the ΔLOD information determined from laser ranging observations is excluded from the processing. In this paper, we study the existing correlations between ΔLOD, the orbital node Ω, the even zonal gravity field coefficients, cross-track empirical accelerations and relativistic accelerations caused by the Lense–Thirring and deSitter effect in detail using first order Gaussian perturbation equations. We found discrepancies due to different a priories by using different gravity field models of up to 1.0ms for polar orbits at an altitude of 500km and up to 40.0ms, if the gravity field coefficients are estimated using only observations to LAGEOS 1. If observations to LAGEOS 2 are included, reliable ΔLOD estimates can be achieved. Nevertheless, an impact of the a priori gravity field even on the multi-satellite ΔLOD estimates can be clearly identified. Furthermore, we investigate the effect of empirical cross-track accelerations and the effect of relativistic accelerations of near-Earth satellites on ΔLOD. A total effect of 0.0088ms is caused by not modeled Lense–Thirring and deSitter terms. The partial derivatives of these accelerations w.r.t. the position and velocity of the satellite cause very small variations (0.1μs) on ΔLOD.

The Research of GPS Elevation Fitting Considering the Influence of Covariance Function

The common method to determine Quasi-Geoids is GPS leveling however the Quasi-Geoid of this method determined is a kind of trend surface which not take the physical property of geoid into consideration, and the fitting method is surface fitting which only consider the surveying error, lead to inaccurate fitting result. In allusion to these problems, Remove-restore method is used to remove the long wave information of earth gravity field model to get more smooth residual gravity height anomaly, then compared the influence of different covariance function to the fitting result of least square collocation which take surveying error and model error into account. The results show that Gaussian and resemble Gaussian function can achieve higher fitting precision to the large area with height anomaly value changes significance; the Remove-restore method can effectively improve the fitting precision to least square collocation method which depend on the covariance value of each points.