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Based on a satellite constellation composed of two GRACE-type satellite formations with different inclinations (near polar orbit + low inclination) and the theory of repeat orbit cycle, we discuss the methods for selecting medium-low inclinations for global and local gravity fields. The effects of this constellation configuration on gravity field inversion are comparatively analyzed using a whole-course dynamics simulation. The results show that compared with the single GRACE-type satellite formation, the use of satellite constellations with different inclination configurations improves the gravity solution precision by 34%. The inclusion of multi-directional observations can improve the spatio-temporal resolution of the satellite missions, and yield gravity field solutions with higher isotropic sensitivity. Furthermore, it is necessary to select the optimal low inclination according to the study area, which will have a significant influence on the gravity field solution.

The regions around the world need to perform their results based on the local geoid. However, each region has different ground topography based on the amount of gravity in this region. Nowadays, the recent global Earth's gravity model of 2008 is successfully used for different purposes in geosciences research. This research presents an overview of the preliminary evaluation results of the new Earth Gravitation Model (EGM08) in the middle of Iraq. For completeness, the evaluation tests were also performed for EGM96 by examining 31 stations distributed over four Iraqi provinces. The national orthometric heights were compared with the GPS /leveling data obtained from these stations. This study illustrated that the GPS /leveling based on EGM08 data was better than that based on EGM96 data in terms of reducing the root mean square error (RMSE) of the differences between the orthometric heights and GPS/leveling data. The standard deviation (SD) values for the national orthometric heights and GPS heights were about 4 and 26cm, respectively. The results also show that there is a small difference in hight ranged (0.0013 - 0.1333 m) in Karbala, (0.0023 – 0.0062 m), in Najaf and (0.0173 – 0.0703 m), in Babylon. Due to the flat area, better results were obtained in Karbala and Najaf than Babylon. The EGM08 geoid method has shown to yield very close results to reality for various projects, thus its accuracy is acceptable.

—The empirical results on estimating the resolution and high-frequency noise in the Earth’s gravity models are presented. The Schmidt Institute of Physics of the Earth of the Russian Academy of Sciences participated in a marine gravity survey in the Indian Ocean. Based on the survey data, gravity maps were constructed for ten regions with a total of 611 986 gravimetric points. The root mean square measurement error is at most 0.27 mGal. The differences between measured gravity and predictions by the EGM2008 gravity model are calculated. The statistical processing of the differences shows that they vary from –25 to 44 mGal on the shelf and from –7.4 to 6.2 mGal on the abyssal plain with a root mean square value of 1.33 mGal. The results of the survey indicate that the anomalies in the ultra-high-degree model can only be used for constructing gravity maps on the scale of 1 : 500 000 or smaller.

Global monitoring of time-variable Earth gravity field from space – at first, observation of secular changes in the Earth flattening by SLR satellites and in recent years especially the regional hydrology variations revealed by the GRACE mission – has revolutionized the exploration of the Earth in many fields. To be able to obtain this rather weak part of the measured signal, it is necessary to measure/model and to subtract all other gravitational and nongravitational forces whose magnitude is not negligible compared to that of the time-variable gravity. In our paper, we analyze the influence of the perturbations arising from the general theory of relativity. Using the modelled data over one year of three example satellites at altitudes 250 km (GOCE), 450 km (Swarm A) and 5900 km (LAGEOS-1), we characterized the magnitude and frequency of occurrence of the three main general relativistic perturbing terms and compared their action to other relevant accelerations. Namely, for GOCE and Swarm A the accelerations due to time-varying gravity reach 10−9 − 10−8 m s−2, for LAGEOS-1 10−11 − 10−10 m s−2; relative to the preceding, the dominant relativistic term is greater by one order of magnitude in the radial direction, the relativistic precession terms happen to be comparable in the cross-track direction. Therefore, for a proper extraction of the time-varying part of the gravity signal, all three relativistic terms have to be taken into account. This conclusion is validated by the real monthly gravity fields obtained through the inversion of GPS orbits of the Swarm A satellite.

The geodetic and geophysical applications of Earth Gravity Field parameters computed from Global Geopotential Models (GGMs) are quite on the increase despite the inherent commission and omission errors of these models. In view of this, this study focuses on refining and quantifying terrain-induced effects on Bouguer gravity anomalies computed directly from a total of seven recent GGMs. In the study, the Residual Terrain Model (RTM) technique was used to estimate the residual terrain effects that were added to the GGM-computed Bouguer gravity anomalies at the sixty test points in Enugu State, Nigeria. The computed residual terrain effects range from -24.6 to 37.5 mgal while the percentage of the omission errors of the GGMs based on their Root-Mean-Square (RMS) differences ranges from 7.8% to 44.7%. It can be concluded that GGM-refined Bouguer gravity anomalies are better in accuracy than the unrefined GGM-computed Bouguer gravity anomalies and hence there is need for accurate height information in the development of GGMs. We, therefore, recommend that refined Bouguer gravity anomalies obtained from HUST-Grace2016s, EIGEN-6C4 and GECO that gave best improvement amongst the seven GGMs under consideration should be used to supplement the available terrestrial Bouguer anomalies for geodetic and geophysical applications within the study area.

A new mission called GRACE Follow-On is now flying to continue the measurements started by the GRACE mission, and to test a laser interferometry system for making more accurate measurements of the ...

High-accuracy and homogeneously distributed dense terrestrial gravity observations are essential data for detailed analysis of the Earth gravity field as well as for precise modeling of the geoid, which is a natural datum for point heights in geodetic and surveying applications. Precise modeling of the regional geoid and the related quality assessments of the terrestrial data are still substantial issues in many countries. The terrestrial gravity measurements in Turkey have been carried out by different institutions; therefore, the collected data have different qualities and standards. For this reason, detailed analysis of the archival data quality and datum consistency is required for regional geoid modeling using these data. In this article, the accuracy of a gridded gravity anomaly dataset was validated using pointwise gravity values with known accuracy in a limited area in northwest Turkey. Hence, this study aimed to test the contribution of the regridded gravity anomalies in combination with two different gravity datasets to determine a precise geoid model for the region. The assessments showed that the accuracy of the grid gravity anomaly dataset was approximately 5.0 mGal in the study area. Although the pointwise data used as control data in this study have high accuracy (similar to 3.0 mGal), they exhibit nonhomogeneous and striped distribution in the area, which is not appropriate for geoid modeling purposes. The calculations using regridded data with a combination of the two datasets resulted in a geoid model accuracy of 9.4 cm. The calculated accuracy was compared with the official Turkey Geoid model (TG) and Earth Gravitational Model 2008 (EGM2008) in its full spectrum, and the improvements were reported.

The Earth’s time-variable gravity field is of great significance to study mass change within the Earth’s system. Since 2002, the NASA-DLR Gravity Recovery and Climate Experiment (GRACE) and its successor GRACE follow-on mission provide observations of monthly changes in the Earth gravity field with unprecedented accuracy and resolution by employing low-low satellite-to-satellite tracking (LLSST) measurements. In addition to LLSST, monthly gravity field models can be acquired from satellite laser ranging (SLR) and high-low satellite-to-satellite tracking (HLSST). The monthly gravity field solutions HLSST+SLR were derived by combining HLSST observations of low earth orbiting (LEO) satellites with SLR observations of geodetic satellites. Bandpass filtering was applied to the harmonic coefficients of HLSST+SLR solutions to reduce noise. In this study, we analyzed the performance of the monthly HLSST+SLR solutions in the spectral and spatial domains. The results show that: (1) the accuracies of HLSST+SLR solutions are comparable to those from GRACE for coefficients below degree 10, and significantly improved compared to those of SLR-only and HLSST-only solutions; (2) the effective spatial resolution could reach 1000 km, corresponding to the spherical harmonic coefficient degree 20, which is higher than that of the HLSST-only solutions. Compared with the GRACE solutions, the global mass redistribution features and magnitudes can be well identified from HLSST+SLR solutions at the spatial resolution of 1000 km, although with much noise. In the applications of regional mass recovery, the seasonal variations over the Amazon Basin and the long-term trend over Greenland derived from HLSST+SLR solutions truncated to degree 20 agree well with those from GRACE solutions without truncation, and the RMS of mass variations is 282 Gt over the Amazon Basin and 192 Gt in Greenland. We conclude that HLSST+SLR can be an alternative option to estimate temporal changes in the Earth gravity field, although with far less spatial resolution and lower accuracy than that offered by GRACE. This approach can monitor the large-scale mass transport during the data gaps between the GRACE and the GRACE follow-on missions.

There is considerable evidence that gravitational acceleration is taken into account in the interaction with falling targets through an internal model of Earth gravity. Here we asked whether this internal model is accessed also when target motion is imagined rather than real. In the main experiments, naïve participants grasped an imaginary ball, threw it against the ceiling, and caught it on rebound. In different blocks of trials, they had to imagine that the ball moved under terrestrial gravity (1g condition) or under microgravity (0g) as during a space flight. We measured the speed and timing of the throwing and catching actions, and plotted ball flight duration versus throwing speed. Best-fitting duration-speed curves estimate the laws of ball motion implicit in the participant's performance. Surprisingly, we found duration-speed curves compatible with 0g for both the imaginary 0g condition and the imaginary 1g condition, despite the familiarity with Earth gravity effects and the added realism of performing the throwing and catching actions. In a control experiment, naïve participants were asked to throw the imaginary ball vertically upwards at different heights, without hitting the ceiling, and to catch it on its way down. All participants overestimated ball flight durations relative to the durations predicted by the effects of Earth gravity. Overall, the results indicate that mental imagery of motion does not have access to the internal model of Earth gravity, but resorts to a simulation of visual motion. Because visual processing of accelerating/decelerating motion is poor, visual imagery of motion at constant speed or slowly varying speed appears to be the preferred mode to perform the tasks.

Satellite gravity data fusion with multi-type and huge-amount is one of the hot topics in physical geodesy. After a brief review of dynamic approach, the CHAMP-only and GRACE-only gravity fields by using HL-SST and LL-SST data from 2003 to 2009 are recovered respectively. An combination strategy of CHAMP and GRACE data by using Helmert variance component estimation (VCE) is proposed based on normal equation level fusion. Three gravity field models with 150° and order by CHAMP-only data, GRACE-only data and combining CHAMP and GRACE data from 2003 to 2009 are recovered. The comparisons between our recovered models and those latest released models were performed. The external accuracy validations using marine gravity anomalies from DTU13 products and height anomalies from GPS/leveling data are also conducted in this paper. The results show that long-term CHAMP data do contribute to the accuracy improvement of gravity field solution. The accuracy of the combined model using CHAMP and GRACE data is better than those of the individuals and comparative to the models published by international groups.

We present the results of photometric observations of the TOPEX/Poseidon satellite performed during 2008–2016. The satellite become space debris after a failure in January, 2006, in a low Earth orbit. In the Laboratory of Space Research of Uzhhorod National University 73 light curves of the spacecraft were obtained. Standardization of photometric light curves is briefly explained. We have calculated the color indices of reflecting surfaces and the spin rate change. The general tendency of the latter is described by an exponential decay function. The satellite spin periods based on 126 light curves (including 53 light curves from the MMT-9 project operating since 2014) were taken into account. In 2016 the period of its own rotation reached its minimum of 10.6 s. A method to derive the direction of the spin axis of an artificial satellite and the angles of the light scattered by its surface has been developed in the Laboratory of Space Research of Uzhhorod National University. We briefly describe the “Orientation” program used for these purposes. The orientation of the TOPEX/Poseidon satellite in mid-2016 is given. The angle of precession β = 45°–50° and period of precession P pr = 141.5 s have been defined. The reasons for the identified nature of the satellite’s own rotation have been found. They amount to the perturbation caused by a deviation of the Earth gravity field from a central-symmetric shape and the presence of moving parts on the satellite.

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

The global geopotential models (GGMs) derived from the gravity field and steady-state ocean circulation explorer (GOCE) mission provide important information about Earth gravity functionals (e.g., geoid heights, gravity anomalies, and disturbances). Among gravity functionals, we utilize geoid heights which have been determined from several recent GOCE-based GGMs and validate them against 5187 collocated Global Navigation Satellite System (GNSS)/leveling observations over a network of dedicated benchmarks in Saudi Arabia. Our aim is to consider the spectral consistency between GOCE-based GGMs and ground-based data. Accordingly, we incorporate high/very high frequencies of gravity functionals, i.e., the gravity signal beyond the maximum d/o of GOCE-based GGMs, using EGM2008 and a high-resolution digital terrain model based on the Shuttle Radar Topography Mission (SRTM). This investigation indicates that completing the missing high-frequency component of geoid heights in GOCE-based GGMs, using EGM2008 and SRTM data, results in an improvement of about 16% in the reduction in the standard deviation (SD) of the differences. This is provided by DIR_R5 at SH d/o 230, which shows improvement from 37.5 cm, without applying the spectral enhancement method (SEM), compared to 31.4 cm when applying the SEM. Finally, three types of transformation models, namely four-, five-, and seven-parameter transformations, are examined to deal with the data biases and to provide a better fitting of geoid heights obtained from the studied GOCE-based GGMs to those from GNSS/leveling data. These models reveal that the SD of vertical datum over the region of Saudi Arabia is at the level of about 22 cm.

The Gauss–Helmert observational model is derived in this paper for total station positioning directly in geocentric reference frame including Global Navigation Satellite Systems (GNSS) meas...

This paper describes the sensitivity of both the orbital frame domain selection and the gravity model on the performance of on-board real-time orbit determination. Practical error sources, which affect the navigation solution of spaceborne global positioning system (GPS) receivers, are analyzed first. Then, a reasonable orbital frame (radial, in-track, cross-track (RIC)) is proposed to clearly represent the characteristics of the error in order to improve the performance of the orbit determination (OD) logic. In addition, the sensitivity of the gravity model affecting the orbit determination logic is analyzed by comparison with the precise orbit ephemeris (POE) of the Challenging Minisatellite Payload (CHAMP) satellite, and it is confirmed that the Gravity Recovery And Climate Experiment (GRACE) Gravity Model 03 (GGM03) outperforms the Earth Gravity Model 1996 (EGM96). The effects of both proposed orbit frames and the gravity model on the orbit determination logic are verified using a GPS simulator and observation data from the CHAMP satellite. Moreover, the practical performance of on-board real-time orbit determination logic is verified by updating the software of the spaceborne GPS receiver, GPS-12, on DubaiSat-2 operating at low Earth orbit (LEO). The results show that the position accuracy of on-board real-time orbit determination logic in GPS-12 is improved by 59%, from 12.6 m (1 &sigma, ) to 5.1 m (1 &sigma, ), after applying the proposed methods. The velocity accuracy is also improved by 57%, from 13.7 mm/s (1 &sigma, ) to 5.9 mm/s (1 &sigma, ).

形態: カラー図版あり, Physical characteristics: Original contains color illustrations, Accepted: 2018-06-29, 資料番号: PA1910055000

Publisher’s Note : This paper, originally published on 18 January 2019, was retracted from the SPIE Digital Library on 24 April 2019 upon verification that an incomplete draft of the paper was submitted and published in which text and figures were omitted, and figures were adapted or copied from the following publications without attribution: Figure 1 : Christian Rothleitner, “Ultra-high Precision, Absolute, Earth Gravity Measurements,” Dissertation, Universitat Erlangen-Nuremberg, Fig. 2.3, page 28 (June 2008). Figure 4 : S Svitlov, P Maslyk, Ch Rothleitner, H Hu, and L J Wang, “Comparison of three digital fringe signal processing methods in a ballistic free-fall absolute gravimeter,” Metrologia, Volume 47, Number 6 (2010); https://doi.org/10.1088/0026-1394/47/6/007 Figures 7 and 8 : S Svitlov, Ch Rothleitner, and L J Wang, “Accuracy assessment of the two-sample zero-crossing detection in a sinusoidal signal,” Metrologia, Volume 49, Number 4 (2012); https://doi.org/10.1088/0026-1394/49/4/413 The authors regret their oversight.

There is evidence that humans rely on an earth gravity (9.81 m/s²) prior for a series of tasks involving perception and action, the reason being that gravity helps predict future positions of moving objects. Eye-movements in turn are partially guided by predictions about observed motion. Thus, the question arises whether knowledge about gravity is also used to guide eye-movements: If humans rely on a representation of earth gravity for the control of eye movements, earth-gravity-congruent motion should elicit improved visual pursuit. In a pre-registered experiment, we presented participants (n = 10) with parabolic motion governed by six different gravities (−1/0.7/0.85/1/1.15/1.3 g), two initial vertical velocities and two initial horizontal velocities in a 3D environment. Participants were instructed to follow the target with their eyes. We tracked their gaze and computed the visual gain (velocity of the eyes divided by velocity of the target) as proxy for the quality of pursuit. An LMM analysis with gravity condition as fixed effect and intercepts varying per subject showed that the gain was lower for −1 g than for 1 g (by −0.13, SE = 0.005). This model was significantly better than a null model without gravity as fixed effect (p

We report the improved test of frame-dragging, an intriguing phenomenon predicted by Einstein's General Relativity, obtained using 7 years of Satellite Laser Ranging (SLR) data of the satellite LARES (ASI, 2012) and 26 years of SLR data of LAGEOS (NASA, 1976) and LAGEOS 2 (ASI and NASA, 1992). We used the static part and temporal variations of the Earth gravity field obtained by the space geodesy mission GRACE (NASA and DLR) and in particular the static Earth's gravity field model GGM05S augmented by a model for the 7-day temporal variations of the lowest degree Earth spherical harmonics. We used the orbital estimator GEODYN (NASA). We measured frame-dragging to be equal to 0.9910 +/- 0.02, where 1 is the theoretical prediction of General Relativity normalized to its frame-dragging value and +/- 0.02 is the estimated systematic error due to modelling errors in the orbital perturbations, mainly due to the errors in the Earth's gravity field determination. Therefore, our measurement confirms the prediction of General Relativity for frame-dragging with a few percent uncertainty., Comment: Submitted on September 27, 2019; accepted for publication on October 8, 2019 and published on October 23, 2019 in The European Physical Journal C: Eur. Phys. J. C (2019) 79:872 https://doi.org/10.1140/epjc/s10052-019-7386-z

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.

We estimate the signature of the climate-induced mass transfers in repeated absolute gravity measurements based on satellite gravimetric measurements from the Gravity Recovery and Climate Experiment (GRACE) mission. We show results at the globe scale and compare them with repeated absolute gravity (AG) time behavior in three zones where AG surveys have been published: Northwestern Europe, Canada, and Tibet. For 10 yearly campaigns, the uncertainties affecting the determination of a linear gravity rate of change range 3–4 nm/s^2/a in most cases, in the absence of instrumental artifacts. The results are consistent with what is observed for long-term repeated campaigns. We also discuss the possible artifact that can result from using short AG survey to determine the tectonic effects in a zone of high hydrological variability. We call into question the tectonic interpretation of several gravity changes reported from stations in Tibet, in particular the variation observed prior to the 2015 Gorkha earthquake.

For a stationary system, coarse alignment is used to initially determine the orientation of the INS by utilizing the Earth gravity and rotation vectors. Yet, a gyro-free INS has only accelerometers and hence cannot measure the Earth rotation vector and thus cannot deduce its yaw angle. In this paper, we formulate the coarse-alignment procedure and derive error characteristics for a gyro-free system. In addition, a comparison between different configurations of accelerometers is being conducted.

By combining a Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) Earth Gravity Model (EGM) and in situ gravity data obtained from the Japanese Antarctic Research Expedition (JARE) surveys, we estimated the regional gravity field in the area of Syowa Station, a Japanese research station located in Lutzow-Holm Bay, East Antarctica. In situ data sets that were used consisted of land gravity data collected since 1967, shipborne data collected since 1985 and airborne gravity data collected in 2006. The GOCE direct (DIR) solution release 5 (R5) model was used as the long-wavelength reference of the gravity field. Using these data sets, we calculated gravity anomalies and geoid heights at 1-by-1′ grid by means of least-squares collocation. The resulting geoid height at Syowa Station was compared with a local height based on GPS, spirit leveling and tide gauge data. The result suggests that the sea surface height at Syowa Station is −1.57 m, which is consistent with a dynamic ocean topography model. During this investigation, we also evaluated GOCE EGMs and other recent EGMs by comparing them with the airborne gravity data. The results indicate that the GOCE DIR R5 produced the smallest RMS (Root Mean Square) differences and that the newer models performed nearly as well. These comparisons demonstrate the importance of using reliable in situ data when evaluating satellite-only EGMs.

The selection of an appropriate global gravity field model and refinement method can effectively improve the accuracy of the refined regional geoid model in a certain research area. We analyzed the accuracy of Experimental Geopotential Model (XGM2016) based on the GPS-leveling data and the modification parameters of the global mean square errors in the KTH geoid refinement in Jilin Province, China. The regional geoid was refined based on Earth Gravitational Model (EGM2008) and XGM2016 using both the Helmert condensation method with an RCR procedure and the KTH method. A comparison of the original two gravity field models to the GPS-leveling benchmarks showed that the accuracies of XGM2016 and EGM2008 in the plain area of Jilin Province are similar with a standard deviation (STD) of 5.8 cm, whereas the accuracy of EGM2008 in the high mountainous area is 1.4 cm better than that of XGM2016, which may be attributed to its low resolution. The modification parameters between the two gravity field models showed that the coefficient error of XGM2016 model was lower than that of EGM2008 for the same degree of expansion. The different modification limits and integral radii may produce a centimeter level difference in global mean square error, while the influence of the truncation error caused by the integral was at the millimeter level. The terrestrial gravity data error accounted for the majority of the global mean square error. The optimal least squares modification obtained the minimum global mean square error, and the global mean square error calculated based on XGM2016 model was reduced by about 1~3 cm compared with EGM2008. The refined geoid based on the two gravity field models indicated that both KTH and RCR method can effectively improve the STD of the geoid model from about six to three centimeters. The refined accuracy of EGM2008 model using RCR and KTH methods is slightly better than that of XGM2016 model in the plain and high mountain areas after seven-parameter fitting. EGM2008 based on the KTH method was the most precise at ± 2.0 cm in the plain area and ± 2.4 cm in the mountainous area. Generally, for the refined geoid based on the two Earth gravity models, KTH produced results similar to RCR in the plain area, and had relatively better performance for the mountainous area where terrestrial gravity data is sparse and unevenly distributed.

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.

In high-precision GNSS/INS integrated navigation systems, the effect of gravity disturbance to the position, velocity and attitude estimation should be considered. The velocity and position errors induced by gravity disturbance can be corrected directly by GNSS position and velocity through Kalman filtering. The horizontal components of gravity disturbance defined as DOV can affect the roll and pitch estimation and should be compensated by gravity map products, such as EGM2008 or other earth gravity models. While the effect of gravity anomaly to the attitude estimation is usually ignored, since it is much smaller than normal gravity and can be absorbed by the vertical accelerometer’s bias. However, in some special cases, such as IMU rotating around one certain horizontal axis, the gravity anomaly can transfer to horizontal accelerometers, and introduce attitude estimation error. In this paper, we analyze the effect of gravity anomaly on attitude estimation in GNSS/INS integrated navigation systems under overturning cases. A theoretical analysis in the case that IMU rotate around east-axis and north-axis to the attitude estimation is derived based on IMU error propagation functions. It is indicated that the rotation around east-axis (north-axis) can introduce pitch (roll) estimation error proportional to gravity anomaly, the yaw estimation is also be affected due to the intercoupling characteristic of attitude errors, and which more sensitive to the rotation around east-axis while not north-axis. Then, the simulation is conducted and agrees well with the theoretical results. Finally, an RLG (Ring Laser Gyroscope) IMU, which has similar quality to Honeywell HG9900, is fixed on a dual-axis turntable to access the effect of gravity anomaly. The experimental result shows that the yaw error induced by gravity anomaly with magnitude 100mGal can reach 15 arcsecs when IMU rotation around a certain horizontal axis at an angle rate of 1.5°/s°.

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 &plusmn, 2.0 cm after correction with the deflections of the vertical, which meets the design requirement of &plusmn, 3.0 cm according to the Technical Specification for Concrete Structures of Tall Buildings in China.

In 2016, we published "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 [1]", a measurement of frame-dragging, a fundamental prediction of Einstein's theory of General Relativity, using the laser-ranged satellites LARES, LAGEOS and LAGEOS 2. The formal error, or precision, of our test was about 0.2% of frame-dragging, whereas the systematic error was estimated to be about 5%. In the 2017 paper "A comment on "A test of general relativity using the LARES and LAGEOS satellites and a GRACE Earth gravity model by I. Ciufolini et al." "by L. Iorio [2] (called I2017 in the following), it was incorrectly claimed that, when comparing different Earth gravity field models, the systematic error in our test due to the Earth's even zonal harmonics of degree 6, 8, 10 could be aslarge as 15%, 6% and 36%, respectively. Furthermore, I2017 contains other, also incorrect, claims about the number of necessary significant decimal digits of the coefficients used in our test (claimed to be nine), in order to eliminate the largest uncertainties in the even zonals of degree 2 and 4, and about the non-repeatability of our test. Here we analyze and rebut those claims in I2017., Comment: Published on The European Physical Journal C 78, no. 11 (2018): 880

The paper focuses on the problem of gravity anomaly determination by airborne gravimetry data. To solve the problem, nonlocal information on the Earth gravity field is needed, usually provided by one of the Earth’s global gravity field models. The possibility of using spherical wavelet expansion for combining airborne gravimetry and global data is studied. Combination algorithms based on optimal estimation methods are developed and tested using Arctic airborne gravity survey data and EGM2008 global model.

Earth`s gravity field recovery was improved after the pre-processing upgrade of the Gravity Recovery And Climate Experiments (GRACE) satellite`s star trackers. The star tracker measurements were filtered with a tighter low-pass filtering of 0.025Hz cutoff frequency, instead of a nominal filtering of 0.1Hz cutoff frequency. In addition, a jump removal algorithm was applied to remove discontinuities, due to direct Sun and/or Moon interventions, in the star tracker measurements. During the K-Band Ranging (KBR) calibration maneuvers, large attitude variations could be detected concurrently by both of the star trackers and the accelerometer. The misalignment angles of star trackers between the true frame and the normal frame could be determined by comparing measurements from these sensors. In this paper, new Earth` gravity field maps were obtained using above improvement. Based on comparisons to nominal Earth`s gravity field maps, the new Earth`s gravity field maps were found better than the nominal ones. Among the applied methods, the misalignment calibration of the star trackers had a major impact on the improvement of the new Earth`s gravity field maps.

Moving-base gravity gradiometry has proven to be a convenient method to determine the Earth’s gravity field. The ESA mission GOCE (Gravity field and steady-state Ocean Circulation Explorer) has enabled to map the Earth gravity field and its gradients with a resolution of 80 km, leading to significant advances in physical oceanography and solid Earth physics. At smaller scales, airborne gravity gradiometry has been increasingly used during the past decade in mineral and hydrocarbon exploration. In both cases the sensitivity of gradiometers to the short wavelengths of the gravity field is of crucial interest. Here, we quantify and characterize the error on the gravity gradients estimated from measurements performed with a new instrument concept, called GREMLIT, for typical airborne conditions. GREMLIT is an ultra-sensitive planar gravitational gradiometer which consists in a planar acceleration gradiometer together with 3 gyroscopes. To conduct this error analysis, a simulation of a realistic airborne survey with GREMLIT is carried out. We first simulate realistic GREMLIT synthetic data, taking into account the acceleration gradiometer and gyroscope noises and biases and the variation of orientation of the measurement reference frame. Then, we estimate the gravity gradients from these data. Special attention is paid to the processing of the gyroscopes measurements whose accuracy is not commensurate with the ultra-sensitive gradiometer. We propose a method to calibrate the gyroscopes biases with a precision of the order $$10^{-8}$$ rad/s. In order to transform the tensor from the measurement frame to the local geodetic frame, we estimate the error induced when replacing the non-measured elements of the gravity gradient tensor by an a priori model. With the appropriate smoothing, we show that it is possible to achieve a precision better than 2E for an along-track spatial resolution of 2 km.

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.

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.

Since 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.

In Geodesy, the heights of points are normally orthometric heights measured above the geoid (an equipotential surface created by the earth masses and rotation which approximately coincides with the mean sea level) or the normal heights. It is necessary to transform the GNSS/GPS measured ellipsoidal heights (h) to classical physical heights (orthometric H/Normal H). The total gravity potential of the earth (W) is the summation of two components; gravitational potential (V) by earth masses and the centrifugal potential (Ω). The centrifugal potential is directly calculated, while the gravitational potential (V) needs to be modeled globally or locally using given measurements. The global models of the earth gravitational potential/gravity models (or so-called geoid models) are mostly given using spherical harmonics (SH). A modified approach of SH was defined to fit the use of SH for regional gravity/potential modeling called spherical cap harmonics (SCH). Due to the numerical difficulties of SCH, a simplified approach of SCH is selected to be used for a combined modeling of the earth potential using a variety of observations. This approach is called the Adjusted Spherical Cap harmonics.

The rate of change of surface gravity, dg/dt, and vertical deformation rate of the solid surface, du/dt, are two observables of glacial isostatic adjustment (GIA). They contribute with different information on the same phenomenon. Their relation contains information of the underlying physics and a trustworthy relation allows to combine these observations to strengthen the overall observational accuracy of the phenomenon. In this paper we investigate the predicted relation between dg/dt and du/dt in previously glaciated areas. We use the normal mode approach for one dimensional earth models and solutions of the sea level equation with time-dependent coastline geometry. Numerical predictions of dg/dt and du/dt are computed for Laurentia, Fennoscandia and the British Isles respectively, using six different earth models. Within each region a linear trend is then fitted using the relation dg/dt = C du/dt + dg_0/dt. The estimated C and dg_0/dt differ more between the regions than between different earth models within each region. For Fennoscandia C ≈ −0.163 μGal/mm and for Laurentia C ≈ −0.152 μGal/mm. Maximum residuals between the linear trend and spatially varying model predictions of dg/dt are 0.04 μGal/yr in Fennoscandia and 0.17 μGal/yr in Laurentia. For the British Isles the results are harder to interpret, mainly since this region is located on the zero uplift isoline of Fennoscandia. In addition, we show temporal variation of the relation since the last glacial maximum till present-day. The temporal and spatial variation of the relation between dg/dt and du/dt can be explained by (i) the elastic respectively viscous proportion of the total signal and (ii) the spectral composition of the regional signal. Additional local effects, such as the Newtonian attraction and elastic deformation from local sea level changes, are examined in a case study for six stations in the Nordic absolute gravity network. The influence of these local effects on the relation between View the dg/dt and du/dt is negligible except for extreme locations close to the sea.

The Earth’s gravity field tends to change due to various reasons. These are diverse processes occurring inside the Earth or changes initiated by human activity. The increasing accuracy of measurements has enabled to take into consideration fluctuations in the gravity field. The article presents research study on the effect of gravity potential describing variations in the gravity field affecting gravity field elements related to the performed measurements. Due to the effect of the disturbing potential, a change in gravity and the deviation of vertical and equipotential surface deformation have been determined. The paper has analyzed the disturbing effect of the material point or homogeneous sphere and has specified the possibilities of assessing the disturbing effect of any form of the homogeneous body. Based on the tidal potential induced by the celestial body, the disturbing effect of the Moon and Sun onto the Earth’s gravity field has been assessed. The carried out research indicates that the range of deformation changes in the equipotential surface of the Earth’s gravity field induced by the effect of the Moon is equal to 0.4824 m, whereas that of the Sun makes 0.1861 m. The conducted studies prove that, due to the effect of the Moon, the direction of the vertical, in terms of the Earth’s surface, tends to change up to 0.0541", and as a result of the Sun’s effect it could reach 0.0204". Having assessed the Lunisolar effect in the first order vertical network measurements in Lithuania, in a polygon with the perimeter of 451 km, it has been determined that the closing error of normal height difference has decreased by the factor of 1.3.

ESA’s Gravity field and steady-state Ocean Circulation Explorer (GOCE) mission and the American-German Gravity Recovery and Climate Experiment (GRACE) mission map the Earth’s gravity field and deliver valuable data for climate research.

The national/regional unified height datum is of great importance for the establishment of geospatial information infrastructure and the corresponding surveying applications. The geopotential-difference method is proposed for long-distance height-datum connection across seas based on global positioning system (GPS)/leveling, global geopotential model (GGM), and satellite altimetry data sets. For a case study, the Chinese height datum 1985 (CNHD85) in the mainland was connected to the local vertical datum (LVD) in Hainan across Qiongzhou Strait up to a distance over 30 km. The results show that this method is independent of the selection of virtual leveling lines. Furthermore, evaluation with an indirect approach shows that the choice of GGM affects the quality of the normal height connection, where the relative accuracies of the solutions based on the Earth Gravity Model 2008 (EGM2008), European Improved Gravity Model of the Earth by New Techniques 6C (EIGEN-6C), EIGEN-6C3STAT, and EIGEN-6C4 model...

Laser ranging satellites have proved their efficiency for high precision testing of the effect of frame-dragging, one of remarkable predictions of the General Relativity. The analysis of the randomness properties of the residuals of LAGEOS and LAGEOS 2 satellites reveals the role of the thermal thrust -- Yarkovsky effect -- on the satellite which was in the orbit for longer period (LAGEOS). We also compute Earth's tidal modes affecting the satellite LARES. The recently obtained 5\% accuracy limit reached for the frame dragging effect based on the 3.5 year data of LARES analysed together with those of LAGEOS satellites and using the Earth gravity model of GRACE satellite, is also represented., 7 pages 3 figures

This chapter discusses fundamentals of orbital dynamics and provides a description of key perturbations affecting global navigation satellite system (GNSS ) satellites along with their impact on the orbits. Models for perturbing accelerations including Earth gravity, third body perturbations, surface forces, and relativistic corrections are described with emphasis on empirical and semiempirical solar radiation pressure models. Long-term evolution of GNSS orbits and orbit keeping maneuvers are discussed. The concepts of broadcast orbit models such as almanac models, analytical ephemeris models and numerical ephemeris models used by current GNSS systems are presented along with cook book algorithms and a summary of their performance. Complementary to the discussion of GNSS satellite orbits, the chapter introduces the basic concepts of GNSS satellite attitude, which are, for example, required to describe the antenna location relative to the center-of-mass.

After the successful European gravity mission GOCE (Gravity field and steady-state Ocean Circulation Explorer), which provided an unprecedented high resolution static global map of the Earth’s gravity field, the European Space Agency has proposed several preparatory studies for a Next Generation Gravity Mission (NGGM). The NGGM mission objective aims at measuring the temporal variations of the Earth gravity field over a long time span with an unprecedented level of accuracy, both in spatial and temporal resolution. The GOCE technological heritage is leveraged as starting point while defining the NGGM future mission concept. Nonetheless, to accomplish its challenging scientific objective, the NGGM mission concept envisages a wide range of innovations, with respect to the past or flying missions, both on technological and automatic control side. Thus, this paper focuses on the guidance, navigation and control design evolution for the European gravity missions, from GOCE to NGGM. After recalling the GOCE GNC main design concepts, the paper will describe the most important innovation required by NGGM. Indeed, such a future concept will consist of a two-satellite long-distance loose formation, where each satellite is controlled independently to be drag-free, GOCE-like. The satellite-to-satellite distance variations, encoding gravity anomalies, will be then measured by laser heterodyne interferometry for inter-satellite ranging at 20 nm resolution, or better. Hence, an orbit and formation control is now required to counteract bias and drift of the residual drag-free accelerations, in order to reach a bounded orbit/formation long-term stability. Finally, GOCE control flight results as well as NGGM simulated results, via a high-fidelity simulator, will be provided. These results highlight the GOCE GNC in-flight achievements as well as the NGGM concept validity, showing that the expected control performances are in agreement with the consolidated mission requirements, all over the 10-year mission.

This study attempts to develop a strategy of data management, processing, evaluation, and analysis that best suits for the regional gravimetric geoid in Pakistan where gravimetric data is sparse. A high-precision gravimetric geoid has been developed covering the Salt Range and Kohat-Potowar Plateau along with Punjab and KPK plain area south of it bounded by latitude 30–33N and longitude 70–73E and with random data distribution. The study also covers the analysis and comparison of observed gravity data with GETECH Bouguer anomalies. The gravity prediction error study was carried out using least square collocation (LSC) at the data gaps and at GETECH locations for data quality analysis. Covariance analysis of both datasets using LSC was made for error study and compared with covariance function by FFT technique. Variety of datasets, e.g., observed gravity; GETECH Bouguer anomalies, SRTM30 digital elevation model, and observed leveling data were combined with global geopotential model through LSC and FFT techniques. The geoid model was computed by applying high-frequency corrections from observed gravity and topography to the Earth Gravity Model 1996 global geopotential model in a remove-restore procedure. The elevation grids with 30″ 5′ and 18′ resolutions gave a best possible statistics in combination with the EGM96 reference field to order and degree 360. The results of bias parameter N o estimation and adjustment were compared with draping of GPS/leveling height anomalies for final fitting of the geoid; LSC- and FFT-derived height anomalies were found to be comparable after the N o adjustment and draping with respect to GPS/leveling.

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.

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.

The main principle and mathematical model of GOCE kinematic orbit adjustment for Earth gravity field model (EGM) validation and accelerometer calibration are presented. Based on 60 days GOCE kinematic orbits with 1–2 cm accuracy and accelerometer data from 2009-11-02 to 2009-12-31, the RMS-of-fit (ROF) of them using EGM2008, EIGEN-5C, ITG-GRACE2010S and GOCO01S up to 120, 150 and 180 degree and order (d/o) are evaluated and compared. The scale factors and biases of GOCE accelerometer data are calibrated and the energy balance method (EBM) is performed to test the accuracy of accelerometer calibration. The results show that GOCE orbits are also sensitive to EGM from 120 to 150 d/o. The ROFs of EGMs with 150 and 180 d/o are obviously better than those of EGMs with 120 d/o. The ROFs of GOCO01S and ITG-GRACE2010S are almost the same up to 120 and 150 d/o, which are about 3.3 cm and 1.8 cm, respectively. They are far better than those of EGM2008 and EIGEN-5C with the same d/o. The ROF of GOCO01S with 180 d/o is about 1.6 cm, which is the best one among those EGMs. The accelerometer calibration accuracies (ACAs) of ITG-GRACE2010S and GOCO01S are obviously higher that those of EGM2008 and EIGEN-5C. The ACA of GOCO01S with 180 d/o is far higher than that of EGMs with 120 d/o, and a little higher than that of ITG-GRACE2010S with 150 d/o. It is suggested that the newest released EGM such as GOCO01S or GOCO02S till at least 150 d/o should be chosen in GOCE precise orbit determination (POD) and accelerometer calibration.

Elucidating the effects of gravity on rover/rover wheel performance is important because it can provide a basis for predicting their performance on extraterrestrial surfaces based on test results obtained on Earth. Results of previous studies by Wong (2012) and by Wong and Kobayashi (2012) indicate that if in conducting performance testing of rovers/rover wheels on Earth with identical mass to that on the extraterrestrial surface, instead of with identical normal load (force) used in the current practice, their performance on the extraterrestrial surface can be predicted based on test results obtained on Earth. As described herein, the effects of gravity on rigid rover wheel sinkage and motion resistance were investigated using the two-dimensional discrete element method. The rover wheel mass is unchanged under various gravity conditions. The results of this study substantiate the findings presented in Wong (2012) and Wong and Kobayashi (2012). Given identical mass, the sinkage of a rigid rover wheel under Earth gravity g is the same as that under a different gravity g ex : 1/6g–2g. It is also shown that the ratio of the motion resistance of a rigid rover wheel under gravity g ex to that under Earth gravity g is equal to the ratio g ex / g .