Earth Gravity Field Models Research Articles

Gravimetric inversion applied to the study of the basement relief of the Recôncavo-Tucano-Jatobá Rift System in Northeast Brazil

This article presents results from the implementation of an inverse modeling algorithm that considers the Bouguer anomaly signal to estimate and gain a better understanding of the boundary between the sedimentary package and the basement of the sub-basins within the Recôncavo-Tucano-Jatobá rift system. The algorithm for modeling the basement relief is considered a medium constituted by a set of discrete two-dimensional prisms with fixed density contrasts between sediment and basement and variable depth. Initially, it was applied to synthetic models, resulting in depth variation curves of the basement top that fit well with the true model. Subsequently, this methodology was applied to satellite gravity data obtained from the high-resolution Earth gravity field model called SGG-UGM-2. The results of this application were compared with the depth of the Moho interface estimated beneath the studied units and obtained through the Parker-Oldenburg methodology. The information produced here allowed the interpretation of the gravity anomaly over the depositional space of the studied basins and the geometry of their basement. Together with the knowledge of their stratigraphy, this interpretation may be analyzed to characterize the region's reservoirs. By interpreting the basement relief in conjunction with the Moho topography, this study portrays the geotectonic evolution of the aulacogen and its association with estimating the depocenters of the Recôncavo, Tucano, and Jatobá sub-basins.

Combined monthly GRACE-FO gravity fields for a Global Gravity-based Groundwater Product

SUMMARY The Combination Service for Time-variable Gravity fields (COST-G) operationally provides combinations of monthly Earth gravity field models derived from observations of the microwave ranging instrument of the GRACE Follow-on (GRACE-FO) satellite mission, applying the quality control and combination methodology originally developed by the Horizon 2020 project European Gravity Service for Improved Emergency Management for the data of the GRACE satellites. In the frame of the follow-up Horizon 2020 project Global Gravity-based Groundwater Product (G3P), the GRACE-FO combination is used to derive global grids of groundwater storage anomalies. To meet the user requirements and achieve optimal signal-to-noise ratio, the combination has been further developed and extended to incorporate: • new time-series based on the alternative accelerometer transplant product generated in the frame of the project by the Institute of Geodesy at the Graz University of Technology, which specifically improves the estimation of the C30 coefficient and also reduces the noise at medium to short wavelengths, and • the new time-series AIUB–GRACE-FO–RL02 of monthly GRACE-FO gravity fields, which is derived at the Astronomical Institute of the University of Bern by applying empirical noise modelling techniques. The COST-G quality control confirms the consistency of the contributing GRACE-FO time-series concerning the signal amplitude of seasonal hydrology in large river basins and the secular mass change in polar regions, but it also indicates rather diverse noise characteristics. The difference in the noise levels is taken into account in the combination process by relative weights derived by variance component estimation on the solution level. The weights are expected to be inverse proportional to the noise levels of the individual gravity field solutions. However, this expectation is violated when applying the weighting scheme as developed for the GRACE combination. The reason is found in the high-order coefficients of the gravity field, which are poorly determined from the low–low range-rate observations due to the observation geometry and suffer from aliasing due to the malfunctioning accelerometer onboard one of the GRACE-FO satellites. Hence, for the final G3P-combination a revised weighting scheme is applied where the gravity field coefficients beyond order 60 are excluded from the determination of the weights. The quality of the combined gravity fields is assessed by comparison of the noise content and the signal-to-noise ratio with the individual time-series. Independent validation is provided by the COST-G validation centre at the GFZ German Research Centre for Geosciences, where orbit fits of the low-flying Gravity and steady-state Ocean Circulation Explorer satellite are performed that confirm the high quality of the combined GRACE-FO gravity fields. By the end of the G3P project, the new combination scheme is implemented by COST-G as the new COST-G–GRACE-FO–RL02 and continued to be used for the operational GRACE-FO combination.

A parallel numerical algorithm by combining MPI and OpenMP programming models with applications in gravity field recovery

Satellite gravimetry missions have enabled the calculation of high-accuracy and high-resolution Earth gravity field models from satellite-to-satellite tracking data and gravitational gradients. However, calculating high maximum degree/order (e.g., 240 or even higher) gravity field models using the least squares method is time-consuming due to the vast amount of gravimetry observations. To improve calculation efficiency, a parallel algorithm has been developed by combining Message Passing Interface (MPI) and Open Multi-Processing (OpenMP) programming models to calculate and invert normal equations for the Earth gravity field recovery. The symmetrical feature of normal equations has been implemented to speed up the calculation progress and reduce computation time. For example, the computation time to generate the normal equation of an IGGT_R1 test version of degree/order 240 was reduced from 88 h to 27 h by considering the symmetrical feature. Here, the calculation was based on the high-performance computing cluster with 108 cores in the School of Geodesy and Geomatics, at Wuhan University. Additionally, the MPI parallel Gaussian-Jordan elimination method was modified to invert normal equation matrices and scaled up to 100 processor cores in this study while the traditional method was limited in a certain number of processors. Furthermore, the Cholesky decomposition from the ScaLAPACK library was used to compare with the parallel Gauss-Jordan elimination method. The numerical algorithm has effectively reduced the amount of calculation and sped up the calculation progress, and has been successfully implemented in applications such as building the gravity field models IGGT_R1 and IGGT_R1C.

Assessment of gravity field recovery from a quantum satellite mission with atomic clocks and cold atom gradiometers

The aim of the MOCAST+ (MOnitoring mass variations by Cold Atom Sensors and Time measures) project, which was carried out during the years 2020–2022, was the investigation of the performance of a gravity field mission based on the integration of atomic clocks and cold atom interferometers. The idea was that the combined observations of the two sensors would be beneficial for the detection and monitoring of geophysical phenomena which have an impact on the time-variable part of the Earth gravity field models. Several different mission scenarios were simulated, considering different satellite configurations such as a Gravity Recovery and Climate Experiment (GRACE)-class formation and a Bender-class formation with either two or three in-line satellites along each orbit. Moreover, different atomic species (rubidium and strontium), different inter-satellite distances, different noise power spectral densities, and different observation rates were taken into account. For the gravity field estimation from the simulated data, the space-wise approach was exploited. The results showed that, as it could be expected, the Bender configuration provides significantly better monthly gravity field solutions, as compared to a ‘nominal’ configuration with two or three satellites in a GRACE-class formation. In this way, and pushing the quantum sensors technology to its limits, it is in fact possible to obtain results which are comparable with those from GRACE at low harmonic degrees, and are better at higher degrees with positive effects in the detectability of localized time variable phenomena, as well as in the determination of the static gravity field at a higher maximum spherical harmonic degree than the one achieved by Gravity Field and Steady-State Ocean Circulation Explorer (of course considering an equivalent mission life-time).

Analysis of the coseismic slip behavior for the MW = 9.1 2011 Tohoku-Oki earthquake from satellite GOCE vertical gravity gradient

Over the past decade, the three largest and most destructive earthquakes in recent history with associated tsunamis occurred: the Mw = 9.2 Sumatra-Andamam in 2004, then the Mw = 8.8 Maule in 2010, and finally the Mw = 9.1 Tohoku- Oki in 2011. Due to the technological and scientific developments achieved in recent decades, it has been possible to study and model these phenomena with unprecedented resolution and precision. In addition to the coseismic slip models, for which joint inversions of data from various sources are carried out (e.g., teleseismic data, GNSS, INSAR, and Tsunami, among others), depicting the space-time evolution of the rupture, we have high-resolution models of the degree of interseismic coupling (based on GNSS) and also maps of seismic b-value changes. Among these advances, new Earth gravity field models allow mapping densities distribution homogeneously and with a resolution (in wavelengths) of approximately the large rupture areas of great megathrust earthquakes. In this regard, the maximum resolution of GOCE-derived static models is in the order of λ/2≈66 km, while GRACE monthly solutions are in the order of λ/2≈300 km. From the study of the static and dynamic gravitational field, it has been possible to infer mass displacements associated with these events, which have been modeled and compared to the deformation inferred using other methods, yielding very good results. In this work we study the kinematic behavior of the rupture process for one of these largest events, the Mw = 9.1 Tohoku-Oki 2011 earthquake, employing the vertical gradient of gravity derived from the GOCE satellite, finding that the maximum slip occurred close to a lobe of minimum Tzz, as was observed for other case-studies in other subduction-related settings studied in previous works (e.g., the Maule earthquake and the Sumatra-Andaman earthquake, among others). In addition, from the rupture propagation using kinematic models, it can be observed that the rupture is arrested when it approaches high-density structures and, it is enhanced when connecting with lobes of low vertical gravity gradient. We also mapped a block expressed as a low Tzz lobe, developed along the marine forearc, which is controlled by a parallel-to-the-trench normal fault that accommodates subsidence during the interseismic period, as it is coupled with the subducted slab. Then, after rupturing the plate interface, this block is decoupled promoting tectonic inversion and uplift. In this way, the hypothesis that the density structure along the forearc is the ultimate first-order factor that governs the rupture process is reinforced.

Optimization of earth gravity field model and its application in elevation fitting

Based on the non-statistical theory, this paper proposes a new technique for the optimization of the earth gravity field model based on the maximum membership degree and the principle of information entropy, combined with the “removal and recovery” algorithm. On the basis of theoretical analysis, eight earth gravity field models were selected and 33 elevation points in the same region were selected. Combined with the new technology adopted in this paper, model optimization and precision analysis were carried out, and it was found that the model based on the optimal model was more suitable for elevation fitting. Further data analysis shows that the remove-recovery algorithm based on the optimal model can improve the feasibility of GPS elevation measurement in high-precision measurement.

Support for two subglacial impact craters in northwest Greenland from Earth gravity model EIGEN 6C4 and other data

We support the very recent discovery of two impact craters under the ice of northwest Greenland (Hiawatha Glacier and Paterson). These discoveries are based mainly on geology and bedrock topography. We added an analysis of gravity field aspects (descriptors) in addition to the traditional gravity and magnetic anomalies. The gravity aspects (the Marussi tensor of the second derivatives, the gravity invariants and their special ratio, strike angles and virtual deformations) provide more complex and comprehensive information about the underground density variations due to a causative body than ordinary gravity anomalies. They show signals typical for the individual geological features like a mountain/volcano, fault, (river)valley, (paleo)lake, (ground)water, hydrocarbon/mineral deposits, etc., as well as for the targets known as impact craters. Our method has been tested on various geological features on the Earth and the Moon. The gravity aspects are, in our case, derived from the recent global static combined Earth gravity field model EIGEN 6C4 with a ground resolution ~9 km and a precision ~10 mGal. A further data come from the digital magnetic field database EMAG 2 with resolution ~5 km. Our method is novel and independent of anything which led to discoveries of these craters in Greenland.

An Assessment of the GOCE High-Level Processing Facility (HPF) Released Global Geopotential Models with Regional Test Results in Turkey

The launch of dedicated satellite missions at the beginning of the 2000s led to significant improvement in the determination of Earth gravity field models. As a consequence of this progress, both the accuracies and the spatial resolutions of the global geopotential models increased. However, the spectral behaviors and the accuracies of the released models vary mainly depending on their computation strategies. These strategies are briefly explained in this article. Comprehensive quality assessment of the gravity field models by means of spectral and statistical analyses provides a comparison of the gravity field mapping accuracies of these models, as well as providing an understanding of their progress. The practical benefit of these assessments by means of choosing an optimal model with the highest accuracy and best resolution for a specific application is obvious for a broad range of geoscience applications, including geodesy and geophysics, that employ Earth gravity field parameters in their studies. From this perspective, this study aims to evaluate the GOCE High-Level Processing Facility geopotential models including recently published sixth releases using different validation methods recommended in the literature, and investigate their performances comparatively and in addition to some other models, such as GOCO05S, GOGRA04S and EGM2008. In addition to the validation statistics from various countries, the study specifically emphasizes the numerical test results in Turkey. It is concluded that the performance improves from the first generation RL01 models toward the final RL05 models, which were based on the entire mission data. This outcome was confirmed when the releases of different computation approaches were considered. The accuracies of the RL05 models were found to be similar to GOCO05S, GOGRA04S and even to RL06 versions but better than EGM2008, in their maximum expansion degrees. Regarding the results obtained from these tests using the GPS/leveling observations in Turkey, the contribution of the GOCE data to the models was significant, especially between the expansion degrees of 100 and 250. In the study, the tested geopotential models were also considered for detailed geoid modeling using the remove-compute-restore method. It was found that the best-fitting geopotential model with its optimal expansion degree (please see the definition of optimal degree in the article) improved the high-frequency regional geoid model accuracy by almost 15%.

Design and analysis of finite volume methods for elliptic equations with oblique derivatives; application to Earth gravity field modelling

We develop and analyse finite volume methods for the Poisson problem with boundary conditions involving oblique derivatives. We design a generic framework, for finite volume discretisations of such models, in which internal fluxes are not assumed to have a specific form, but only to satisfy some (usual) coercivity and consistency properties. The oblique boundary conditions are split into a normal component, which directly appears in the flux balance on control volumes touching the domain boundary, and a tangential component which is managed as an advection term on the boundary. This advection term is discretised using a finite volume method based on a centred discretisation (to ensure optimal rates of convergence) and stabilised using a vanishing boundary viscosity. A convergence analysis, based on the 3rd Strang Lemma [9], is conducted in this generic finite volume framework, and yields the expected O(h) optimal convergence rate in discrete energy norm.We then describe a specific choice of numerical fluxes, based on a generalised hexahedral meshing of the computational domain. These fluxes are a corrected version of fluxes originally introduced in [29]. We identify mesh regularity parameters that ensure that these fluxes satisfy the required coercivity and consistency properties. The theoretical rates of convergence are illustrated by an extensive set of 3D numerical tests, including some conducted with two variants of the proposed scheme. A test involving real-world data measuring the disturbing potential in Earth gravity modelling over Slovakia is also presented.

Assessment of Earth Gravity Field Models in the Medium to High Frequency Spectrum Based on GRACE and GOCE Dynamic Orbit Analysis

An analysis of current static and time-variable gravity field models is presented focusing on the medium to high frequencies of the geopotential as expressed by the spherical harmonic coefficients. A validation scheme of the gravity field models is implemented based on dynamic orbit determination that is applied in a degree-wise cumulative sense of the individual spherical harmonics. The approach is applied to real data of the Gravity Field and Steady-State Ocean Circulation (GOCE) and Gravity Recovery and Climate Experiment (GRACE) satellite missions, as well as to GRACE inter-satellite K-band ranging (KBR) data. Since the proposed scheme aims at capturing gravitational discrepancies, we consider a few deterministic empirical parameters in order to avoid absorbing part of the gravity signal that may be included in the monitored orbit residuals. The present contribution aims at a band-limited analysis for identifying characteristic degree ranges and thresholds of the various GRACE- and GOCE-based gravity field models. The degree range 100–180 is investigated based on the degree-wise cumulative approach. The identified degree thresholds have values of 130 and 160 based on the GRACE KBR data and the GOCE orbit analysis, respectively.

Static Stress Increase in the Outer Forearc Produced by MW 8.2 September 8, 2017 Mexico Earthquake and its Relation to the Gravity Signal

An Mw 8.2 earthquake affected on September 8, 2017 the Cocos plate beneath the Caribbean plate next to the transform contact with the North American plate across the Tehuantepec gap, one of the two gaps that exist along the Mexican subduction zone offshore Chiapas. The epicenter occurred at a depth of 47 km and was associated with a highly steep normal fault parallel to the trench. The origin can be related to slab pull forces, dehydration processes and reactivation of outer rise faults. This earthquake produced positive changes in the Coulomb static stress, which would favor the development of thrust displacements along the interplate contact across the Tehuantepec gap instead of liberating tensions accumulated on it for decades. In the case that stresses were not released steadily in the future, a thrusting-related event with a magnitude of Mw 7.9 or higher could be expected after the September 8, 2017 Chiapas earthquake. However, the differential weight of the forearc offshore (that contains a region with a high positive gravity gradient) could inhibit rupture propagation to the coast precluding higher magnitudes. Combined and satellite only earth gravity field models show a highly heterogeneous density structure along the forearc of the North America-Caribbean plates and in particular at the studied zone. Maximum displacements in the rupture zone correlate to minimum gravity-derived anomalies showing a probable relationship between the differential weight of the forearc structure and the extensional faulting affecting the bending of the Cocos plate at depth. In this sense, variable sediment thicknesses over the forearc, and a high gravity gradient zone parallel to the trench in the outer forearc, could be reflecting physical heterogeneities that influence the propagation of the rupture zone. Additionally, to the north, a highly positive gradient signal related to the subducted Tehuantepec Fz projection beneath the North American plate marks the ending of the main slip patch acting as a barrier to the seismic rupture propagation. This event exemplifies (1) how extensional ruptures in the downgoing slab of a subduction zone can increment accumulated elastic strain across a seismic gap instead of liberating tensions, (2) that rupture propagation for this intraplate event was probably controlled (at shallow depths) by the density structure of the forearc, similarly to recent interplate earthquakes along the Chilean margin; (3) how a low gravity gradient signal and low density structures identified from the inversion model in the region of the outer forearc, where the normal stress increased after the main seismic event, suggest a higher risk of occurrence of a great megathrust earthquake.

Tongji‐Grace02s and Tongji‐Grace02k: High‐Precision Static GRACE‐Only Global Earth's Gravity Field Models Derived by Refined Data Processing Strategies

AbstractIn order to derive high‐precision static Gravity Recovery and Climate Experiment (GRACE)‐only gravity field solutions, the following strategies were implemented in this study: (1) a refined accelerometer calibration model that treats monthly accelerometer scales as a third‐order polynomial and daily accelerometer biases as a fifth‐order polynomial was developed to calibrate accelerometer measurements; (2) the errors of the acceleration and attitude data were estimated together with the geopotential coefficients and accelerometer parameters on the basis of the weighted least squares adjustments; (3) a nearly complete observation series of GRACE mission was used to decrease the condition number of normal equation; and (4) the GRACE data collected in lower orbit altitude were also included to decrease the condition number. Our results show that (1) the refined accelerometer calibration model with much less parameters performs as well as previous methods (i.e., solving daily scales and hourly biases or estimating biases along with bias rates every 2 hr). However, it provides a system of more stable normal equation and less high‐frequency noise in gravity field solutions; (2) high‐frequency noise in the gravity field solution is reduced by modeling the errors of the acceleration and attitude data; (3) the geopotential coefficients at all degrees is greatly enhanced by using longer GRACE time series (especially the data by the end of 2010); and (4) due to lower orbit altitude, the GRACE data collected since 2014 lead to a significant improvement of the gravity field solution as the satellites are more sensitive to higher‐frequency signal. Using the refined strategies, an unconstrained static solution (named Tongji‐Grace02s) up to degree and order 180 was derived. For further suppressing the high‐frequency noise, a regularization strategy based on the Kaula rule is applied to the degrees and orders beyond 80, leading to a regularized model Tongji‐Grace02k. To validate the quality of the derived models, both Tongji‐Grace02s and Tongji‐Grace02k were compared to the latest GRACE‐only models (i.e., GGM05S, ITU_GRACE16, ITSG‐Grace2014s, and ITSG‐Grace2014k) and validated using independent data (i.e., Global Navigation Satellite Systems (GNSS)/Leveling data and DTU13 oceanic gravity data). Compared to other models, much less spatial noise in terms of global gravity anomalies with respect to the state‐of‐the‐art model EIGEN6C4 and far higher accuracy at high degrees are achieved by Tongji‐Grace02s. The same conclusions can be drawn for Tongji‐Grace02k when the same analyses were applied to the regularized solutions ITSG‐Grace2014k and Tongji‐Grace02k. Validations with independent data confirm that Tongji‐Grace02s has the least noise among the unconstrained GRACE‐only models and Tongji‐Grace02k is the one with the best accuracy among the regularized GRACE‐only solutions. For the tests up to degree and order 180 using GNSS/Leveling data, the improvements of Tongji‐Grace02s with respect to ITSG‐Grace2014s reach 13% over Canada and 23% in Mexico. Even better, no less than 58% of improvement is achieved by both Tongji‐Grace02s relative to ITSG‐Grace2014s and Tongji‐Grace02k with respect to ITSG‐Grace2014k in the validation based on DTU13 data.

GNSS-Based Verticality Monitoring of Super-Tall Buildings

In the construction of super-tall buildings, it is rather important to control the verticality. In general, a laser plummet is used to transmit coordinates of reference points from the ground layer-by-layer, which can effectively control the verticality of super-tall buildings. However, the errors in transmission will accumulate with increasing height and motion of the buildings in construction. This paper presents a global navigation satellite system (GNSS)-based method to check the results of laser plumbing. The method consists of four steps: (1) Computing the coordinate time series of monitoring points by adjusting the GNSS monitoring network observations at each epoch; (2) Analyzing the horizontal motion of super-tall buildings and its effect on vertical reference transmission; (3) Calculating the deflections of the vertical at the monitoring point using an Earth gravity field model and a geoid model. With deflections of the vertical, the static GNSS-measured coordinates are aligned to the same datum as used by the laser plummet; and (4) Finally, validating/checking the result of laser plumbing by comparing it with static GNSS results corrected by deflections of the vertical. A case study of a 438-m high building is tested in Guangzhou, China. The result demonstrates that the gross errors of baseline vectors can be eliminated effectively by GNSS network adjustment of the first step. The two-dimensional displacements can be measured at millimeter-level accuracy; the difference between the coordinates of the static GNSS measurement and laser plumbing is less than ±2.0 cm after correction with the deflections of the vertical, which meets the design requirement of ±3.0 cm according to the Technical Specification for Concrete Structures of Tall Buildings in China.

Research on the impact factors of GRACE precise orbit determination by dynamic method

Abstract With the successful use of GPS-only-based POD (precise orbit determination), more and more satellites carry onboard GPS receivers to support their orbit accuracy requirements. It provides continuous GPS observations in high precision, and becomes an indispensable way to obtain the orbit of LEO satellites. Precise orbit determination of LEO satellites plays an important role for the application of LEO satellites. Numerous factors should be considered in the POD processing. In this paper, several factors that impact precise orbit determination are analyzed, namely the satellite altitude, the time-variable earth’s gravity field, the GPS satellite clock error and accelerometer observation. The GRACE satellites provide ideal platform to study the performance of factors for precise orbit determination using zero-difference GPS data. These factors are quantitatively analyzed on affecting the accuracy of dynamic orbit using GRACE observations from 2005 to 2011 by SHORDE software. The study indicates that: (1) with the altitude of the GRACE satellite is lowered from 480 km to 460 km in seven years, the 3D (three-dimension) position accuracy of GRACE satellite orbit is about 3∼4 cm based on long spans data; (2) the accelerometer data improves the 3D position accuracy of GRACE in about 1 cm; (3) the accuracy of zero-difference dynamic orbit is about 6 cm with the GPS satellite clock error products in 5 min sampling interval and can be raised to 4 cm, if the GPS satellite clock error products with 30 s sampling interval can be adopted. (4) the time-variable part of earth gravity field model improves the 3D position accuracy of GRACE in about 0.5∼1.5 cm. Based on this study, we quantitatively analyze the factors that affect precise orbit determination of LEO satellites. This study plays an important role to improve the accuracy of LEO satellites orbit determination.

PRECISE ESTABLISHMENT OF THE NEXT‐GENERATION EARTH GRAVITY FIELD MODEL FROM HIP‐3S BASED ON COMBINATION OF INLINE AND PENDULUM SATELLITE FORMATIONS

AbstractBecause the sensitivity of the disturbing intersatellite range measurements to the recovery accuracy of the Earth's gravitational field is superior to that of the intersatellite range measurements, the observation equation of a new Disturbing Intersatellite Range Method (DIRM) is created in this study. Then the orbital stability of the next‐generation Hybrid‐Inline‐Pendulum‐Three‐Satellite (HIP‐3S) formation is efficiently verified. The research results indicate that the HIP‐3S formation is sufficiently steady for further enhancing the accuracy of the future Earth gravity field model. Finally, the Earth's gravitational field complete up to degree and order 120 is precisely recovered by the current GRACE‐2S inline formation and the next‐generation HIP‐3S combined formation based on the new DIRM, and the cumulative geoid height errors are 2.271×10−1 m and 1.923×10−3 m at degree 120, respectively. The study results show that the future HIP‐3S formation is helpful for producing the next‐generation Earth gravity field model with higher accuracy and spatial resolution.

Accurate Establishment of Error Models for the Satellite Gravity Gradiometry Recovery and Requirements Analysis for the Future GOCE Follow-On Mission

Firstly, the new single and combined error models applied to estimate the cumulative geoid height error are efficiently produced by the dominating error sources consisting of the gravity gradient of the satellite-equipped gradiometer and the orbital position of the space-borne GPS/GLONASS receiver using the power spectral principle. At degree 250, the cumulative geoid height error is 1.769 × 10−1 m based on the new combined error model, which preferably accords with a recovery accuracy of 1.760 ×10−1 m from the GOCE-only Earth gravity field model GO_CONS_GCF_2_TIM_R2 released in Germany. Therefore, the new combined error model of the cumulative geoid height is correct and reliable in this study. Secondly, the requirements analysis for the future GOCE Follow-On satellite system is carried out in respect of the preferred design of the matching measurement accuracy of key payloads comprising the gravity gradient and orbital position and the optimal selection of the orbital altitude of the satellite. We recommend the gravity gradient with an accuracy of 10−13−10−15 /s2, the orbital position with a precision of 1-0.1 cm and the orbital altitude of 200-250 km in the future GOCE Follow-On mission.

A test of general relativity using the LARES and LAGEOS satellites and a GRACE Earth gravity model: Measurement of Earth's dragging of inertial frames.

We present a test of general relativity, the measurement of the Earth’s dragging of inertial frames. Our result is obtained using about 3.5 years of laser-ranged observations of the LARES, LAGEOS, and LAGEOS 2 laser-ranged satellites together with the Earth gravity field model GGM05S produced by the space geodesy mission GRACE. We measure mu = (0.994 pm 0.002) pm 0.05, where mu is the Earth’s dragging of inertial frames normalized to its general relativity value, 0.002 is the 1-sigma formal error and 0.05 is our preliminary estimate of systematic error mainly due to the uncertainties in the Earth gravity model GGM05S. Our result is in agreement with the prediction of general relativity.

Progress in satellite gravity recovery from implemented CHAMP, GRACE and GOCE and future GRACE Follow-On missions

Firstly, the Earth's gravitational field from the past Challenging Minisatellite Payload (CHAMP) mission is determined using the energy conservation principle, the combined error model of the cumulative geoid height influenced by three instrument errors from the current Gravity Recovery and Climate Experiment (GRACE) and future GRACE Follow-On missions is established based on the semi-analytical method, and the Earth's gravitational field from the executed Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) mission is recovered by the space-time-wise approach. Secondly, the cumulative geoid height errors are 1.727 × 10−1 m, 1.839 × 10−1 m and 9.025 × 10−2 m at degrees 70,120 and 250 from the implemented three-stage satellite gravity missions consisting of CHAMP, GRACE and GOCE, which preferably accord with those from the existing earth gravity field models involving EIGEN-CHAMP03S, EIGEN-GRACE02S and GO_CONS_GCF_2_DIR_R1. The cumulative geoid height error is 6.847 × 10−2 m at degree 250 from the future GRACE Follow-On mission. Finally, the complementarity among the four-stage satellite gravity missions including CHAMP, GRACE, GOCE and GRACE Follow-On is demonstrated contrastively.

Analyzing spectral characteristics of the global Earth gravity field models obtained from the CHAMP, GRACE and GOCE space missions

Spectral characteristics of modern models of the Earth’s gravity field obtained from the CHAMP, GRACE and GOCE space gravity missions are considered. The resolution is estimated for a specified accuracy.

A Study on the Improvement in Spatial Resolution of the Earth's Gravitational Field by the Next‐Generation ACR‐Cartwheel‐A/B Twin‐Satellite Formation

AbstractSince the Gravity Recovery and Climate Experiment (GRACE) collinear formation has some fatal drawbacks including north‐south striping error, etc., this study carries out the feasibility demonstrations on the next‐generation Along‐Cross‐Radial(ACR)‐Cartwheel twin‐satellite formation for improving the spatial resolution of the Earth's gravitational field by the intersatellite range‐rate interpolation method. Firstly, the Earth's gravitational field from ACR‐Cartwheel‐A/B complete up to degree and order 120 is precisely recovered using the satellite orbital parameters and the measurement precision of key payloads from the twin GRACE satellites. The research results show that the accuracy of the Earth's gravitational field determination from the next‐generation ACR‐Cartwheel twin‐satellite formation is averagely improved by 2.6 times than that from the EIGEN‐GRACE02S model released by the German GeoForschungsZentrum (GFZ) in Potsdam, which sufficiently verifies that the next‐generation ACR‐Cartwheel twin‐satellite formation is obviously better than the current GRACE collinear formation. Secondly, the Earth's gravitational field complete up to degree and order 120 is accurately measured based on the satellite orbital parameters (e.g. orbital altitude of 350 km, intersatellite range of 100 km, orbital inclination of 89° and orbital eccentricity of 0.0046), the measurement precision of key payloads (e.g. 10−7 m·s−1 in intersatellite range‐rate, 10−3 m in orbital position, 10−6 m·s−1 in orbital velocity and 10−11 m·s−2 in non‐conservative force), an observation time of 30 days and a sampling interval of 10 s by the Longitudinal‐Along‐Radial (Lo‐AR)‐Cartwheel‐A/B, Latitudinal‐Along‐Radial (La‐AR)‐Cartwheel‐A/B and ACR‐Cartwheel‐A/B twin‐satellite formations using the intersatellite range‐rate interpolation method, and the cumulative geoid height errors are 5.115×10−4 m, 4.923×10−4 m and 3.488×10−4 m, respectively. The results of this study are as follows. (1) Since the orbital stability of the La‐AR‐Cartwheel‐A/B formation is superior to the Lo‐AR‐Cartwheel‐A/B formation, the accuracy of satellite gravity recovery from the La‐AR‐Cartwheel‐A/B formation is higher than that from the Lo‐AR‐Cartwheel‐A/B formation. (2) Because the ACR‐Cartwheel‐A/B twin‐satellite formation can synchronously obtain gravity information in the along‐track, cross‐track and radial‐track directions, satellite observation data are provided with some strong points consisting of homogeneity, isotropy, and so on. In a word, the next‐generation ACR‐Cartwheel twin‐satellite formation will prospectively play a significant role in producing the Earth gravity field model with higher accuracy and resolution.