A contrastive study on the influences of radial and three-dimensional satellite gravity gradiometry on the accuracy of the Earth's gravitational field recovery

The accuracy of the Earth＇s gravitational field measured from the gravity field and steady-state ocean circulation explorer（GOCE）,up to 250 degrees,influenced by the radial gravity gradient V zz and three-dimensional gravity gradient V ij from the satellite gravity gradiometry（SGG） are contrastively demonstrated based on the analytical error model and numerical simulation,respectively.Firstly,the new analytical error model of the cumulative geoid height,influenced by the radial gravity gradient V zz and three-dimensional gravity gradient V ij are established,respectively.In 250 degrees,the GOCE cumulative geoid height error measured by the radial gravity gradient V zz is about 2 1/2 times higher than that measured by the three-dimensional gravity gradient V ij.Secondly,the Earth＇s gravitational field from GOCE completely up to 250 degrees is recovered using the radial gravity gradient V zz and three-dimensional gravity gradient V ij by numerical simulation,respectively.The study results show that when the measurement error of the gravity gradient is 3×10 12 /s 2,the cumulative geoid height errors using the radial gravity gradient V zz and three-dimensional gravity gradient V ij are 12.319 cm and 9.295 cm at 250 degrees,respectively.The accuracy of the cumulative geoid height using the three-dimensional gravity gradient V ij is improved by 30%-40% on average compared with that using the radial gravity gradient V zz in 250 degrees.Finally,by mutual verification of the analytical error model and numerical simulation,the orders of magnitude from the accuracies of the Earth＇s gravitational field recovery make no substantial differences based on the radial and three-dimensional gravity gradients,respectively.Therefore,it is feasible to develop in advance a radial cold-atom interferometric gradiometer with a measurement accuracy of 10 13 /s 2-10 15 /s 2 for precisely producing the next-generation GOCE Follow-On Earth gravity field model with a high spatial resolution.

Earth Gravity Field Recovered from CHAMP Science Orbit and Accelerometer Data

The earth gravity field model CDS01S of degree and order 36 has been recovered from the post processed Science Orbits and on-board accelerometer data of GFZ’s CHAMP satellite. The model resolves the geoid with an accuracy of better than 4 cm at a resolution of 700 km half-wavelength. By using the degree difference variances of geopotential coefficients to compare the model CDS01S with EIGEN3P, EIGEN1S and EGM96, the result indicates that the coefficients of CDS01S are most close to those of EIGEN3P. The result of the comparison between the accuracies of geopotential coefficients in the above models, indicates that the accuracy of coefficients in CDS01S is higher than that in EGM96.The geoid undulations of CDS01S and GGM01C up to 30 degrees are calculated and the standard deviation is 4.7 cm between them.

Review of the Research Progress on Static Earth Gravity Field and Vertical Datum in China during 2019—2023

The contribution presents the representative research progress on global static gravity field modeling,regional geoid/quasigeoid determination,vertical datum study,as well as the theory,algorithm and software for gravity field study in China from 2019 to 2023,which are the highlights of the chapter 6“Progress in Earth Gravity Model and Vertical Datum”in the“2019—2023 China National Report on Geodesy”that submitted to the International Association of Geodesy(IAG).In addition,suggestions are proposed to promote the research in the fields of earth gravity field,geoid/quasigeoid and vertical datumin China according to trends of international geodesy and related disciplines.

GRACE detection of the medium-to far-field coseismic gravity changes caused by the 2004 Mw9.3 Sumatra-Andaman earthquake

Large earthquakes cause observable changes in the Earth’s gravity field, which have been detected by the Gravity Recovery and Climate Experiment (GRACE). Since most previous studies focus on the detection of near-field gravity effects, this study provides the results from the medium- to far-field gravity changes caused by the 2004 Sumatra-Andaman earthquake that are recorded within GRACE monthly solutions. Utilizing a spherical-earth dislocation model we documented that large-scale signals predominate in the global field of the coseismic gravity changes caused by the earthquake. After removing the near-field effects, the coseismic gravity changes show a negative anomaly feature with an average magnitude of -0.18×10-8 m·s-2 in the region ranging ～40° from the epicenter, which is considered as the 'medium ffield' in this study. From the GRACE data released by Center for Space Research from August 2002 to December 2008, we retrieved the large-scale gravity changes smoothed with 3 000 km Gaussian ffilter. The results show that the coseismic gravity changes detected by GRACE in the medium field have an average of (-0.20±0.06)×10-8 m·s-2, which agrees with the model prediction. The detection confirms that GRACE is sensitive to large-scale medium-field coseismic gravitational effects of mega earthquakes, and also validates the spherical-earth dislocation model in the medium field from the perspective of satellite gravimetry.

Dependence of Gravity Induced Absorption Changes on the Earth’s Magnetic Field as Measured during Parabolic Flight Campaigns

Various spectroscopic experiments performed on the AIRBUS ZERO G—located in Bordeaux, France—in the years 2002 to 2012 exhibit minute optical reflection/absorption changes (GIACs) as a result of gravitational changes between 0 and 1.8 g in various biological species such as maize, oats, Arabidopsis and particularly Phycomyces sporangiophores. During a flight day, the AIRBUS ZERO G conducts 31 parabolas, each of which lasts about three minutes including a period of 22 s of weightlessness. So far, we participated in 11 parabolic flight campaigns including more than 1000 parabolas performing various kinds of experiments. During our campaigns, we observed an unexplainable variability of the measuring signals (GIACs). Using GPS-positioning systems and three dimensional magnetic field sensors, these finally were traced back to the changing earth’s magnetic field associated with the various flight directions. This is the first time that the interaction of gravity and the Earth’ magnetic field in the primary induction process in living system has been observed.

Discovering Crustal Deformation Bands by Processing Regional Gravity Field

Objectives： This article presents a new computational procedure to discover scratches buried in the earth＇s crust. We also validate this new interdisciplinary analysis method with regional gravity data located in a well-known Dabie orogenic zone for test. Methods： Based on the scratch analysis method evolved with mathematical morphology of surfaces, we present a procedure that extracts information of the crustal scratches from regional gravity data. Because the crustal scratches are positively and highly correlated to crustal deformation bands, it can be used for delineation of the crustal deformation belts. The scratches can be quantitatively characterized by calculation of the ridge coefficient function, whose high value traces delineate the deformation bands hidden in the regional gravity field. In addition, because the degree of crustal deformation is an important indicator of tectonic unit divisions, so the crust can be further divided according to the degree of crustal deformation into some tectonic units by using the ridge coefficient data, providing an objective base map for earth scientists to build tectonic models with quantitative evidence. Results： After the ridge coefficients are calculated, we can further enhance the boundary of high ridge-coefficient blocks, resulting in the so-called ridge-edge coefficient function. The high-value ridge-edge coefficients are well correlated with the edge faults of tectonic units underlay, providing accurate positioning of the base map for compilation of regional tectonic maps. In order to validate this new interdisciplinary analysis method, we select the Dabie orogenic zone as a pilot area for test, where rock outcrops are well exposed on the surface and detailed geological and geophysical surveys have been carried out. Tests show that the deformation bands and the tectonic units, which are conformed by tectonic scientists based on surface observations, are clearly displayed on the ridge and ridge-edge coefficient images obtained in this article. Moreover, these computer-generated images provide more accurate locations and geometric details. Conclusions： This work demonstrates that application of modern mathematical tools can promote the quantitative degree in research of modern geosciences, helping to open a door to develop a new branch of mathematical tectonics.

Efficient and rapid accuracy estimation of the Earth's gravitational field from next-generation GOCE Follow-On by the analytical method

Firstly, a new analytical error model of the cumulative geoid height using the three-dimensional diagonal tensors of satellite gravity gradiometry （SGG） is introduced based on the variance-covariance matrix principle. Secondly, a study for the requirements demonstration on the next-generation GOCE Follow-On satellite gravity gradiometry system is developed using different satellite orbital altitudes and measurement accuracies of satellite gravity gradiometer by the new analytical error model of SGG. The research results show that it is preferable to design satellite orbital altitudes of 300 km–400km and choose the measurement accuracies of 10-13/s2 –10-15/s2 from satellite gravity gradiometer. Finally, the complementarity of the four-stage satellite gravity missions, including past CHAMP, current GRACE, and GOCE, and next-generation GOCE Follow-On, is contrastively demonstrated for precisely recovering the Earth’s full-frequency gravitational field with high spatial resolution.

Effect of different inter-satellite range on measurement precision of Earth’s gravitational field from GRACE

The precision of Earth＇s gravitational field from GRACE up to degree and order 120 was studied for different inter-satellite ranges using the improved energy conservation principle. Our simulated result shows that： For long wavelength （L≤20） at degree 20, the cumulative geoid-height error gradually decreased with increasing range, from 0. 052 cm for 110 km to 1. 156 times and 1. 209 times as large for 220 km and 330 kin, respectively. For medium-wavelength （ 100 ≤ L ≤ 120） at degree 120, the cumulative geoid-height error de- creased from 13. 052 cm for 110 km, to 1. 327 times and 1. 970 times as large for the ranges of 220 km and 330 km, respectively; By adopting an optimal range of 220 ± 50 km, we can suppress considerably the loss of precision in the measurement of the Earth＇ s long-wavelength and medium-wavelength gravitational field.

An Improved Energy Balance Approach and Its Application in CHAMP Gravity Field Recovery

An efficient method for gravity field determination from CHAMP orbits and accelerometer data is referred to as the energy balance approach. A new CHAMP gravity field recovery strategy based on the improved energy balance approach IS developed in this paper. The method simultaneously solves the spherical harmonic coefficients, daily Integration constant, scale and bias parameters. Two 60 degree and order gravitational potential models, XISM-CHAMPO1S from the classical energy balance approach, and XISM-CHAMPO2S from the improved energy balance, are determined using about one year＇s worth of CHAMP kinematic orbits from TUM and accelerometer data from GFZ. Comparisons among XISM-CHAMPO1S, XISM-CHAMPO2S, EIGEN-CGO3C, EIGEN-CHAMPO3S, EIGEN2, ENIGNIS and EGM96 are made. The results show that the XISM-CHAMPO2S model is more accurate than EGM96, EIGENIS, EIGEN2 and XISM-CHAMPO1S at the same degree and order, and has almost the same accuracy as EIGEN-CHAMPO3S.

Earth’s Magnetic Field—The Key to Global Warming

It is commonly known that the climate debate suffers due to a lack of knowledge about the cause and effect relationship between a number of climatic temperature variations that have occurred in history without being able to blame human emission of greenhouse gas in any way. Only when we are willing to give up the idea that there is a geodynamo deep inside of the Earth being responsible for the Earth’s magnetic field and when we get back to the idea that the origin of the magnetic field is simply ferromagnetic, will it be possible to establish two different cause and effect connections that are suitable to explain why there is an acknowledged coincidence between climatic temperature variations and an intensive, proportional variation in the strength of the Earth’s magnetic field. Such insight may easily prove to be decisive at a time when many people can no longer differentiate between politics, mass hysteria, presumptions and actual knowledge. When there are requirements that a solution to climatic temperature variations must contain the solution to the coincidence mentioned, two possible scenarios exist. The one possibility (although not very likely) that is suitable to solve the mysterious coincidence is that mainly the northern part of the Atlantic Ocean is heated from within (from the interior of the Earth) and that variations in the Earth’s emission of heat cause primarily all of Europe to have witnessed warm winters for decades. The one possible cause and effect connection may (in theory) be that inner heat in the Earth’s crust can loosen frozen, ferromagnetic structures, thereby drive the Earth’s ferromagnetic, magnetic field to restructure and be reorganised from periodically being a chaotic, magnetic field to periodically being a well-structured, ferromagnetic field. The connection between magnetism and thermal impact is already commonly known. The other and somewhat more likely cause and effect connection is building on Henrik Svensmark’s (and teams) theory that says that variations in the cosmic radiation reaching the Earth depend on the strength of the Sun’s magnetic field and that this radiation contributes to creating aerosols, thereby variations in the cloud formation. Solar storms contribute to temporarily strengthening the Earth’s magnetic field. The question is whether these contributions could also periodically have a long-term effect on the Earth’s magnetic field. In that case, this may explain the reason for the above-mentioned coincidence.

Cosmic Dark Energy from ‘t Hooft’s Dimensional Regularization and Witten’s Topological Quantum Field Pure Gravity

We utilize two different theories to prove that cosmic dark energy density is the complimentary Legendre transformation of ordinary energy and vice versa as given by E(dark) = mc2 (21/22) and E(ordinary) = mc2/22. The first theory used is based on G ‘t Hooft’s remarkably simple renormalization procedure in which a neat mathematical maneuver is introduced via the dimensionality of our four dimensional spacetime. Thus, ‘t Hooft used instead of D = 4 and then took at the end of an intricate and subtle computation the limit to obtain the result while avoiding various problems including the pole singularity at D = 4. Here and in contradistinction to the classical form of dimensional and renormalization we set and do not take the limit where and is the theoretically and experimentally well established Hardy’s generic quantum entanglement. At the end we see that the dark energy density is simply the ratio of and the smooth disentangled D = 4, i.e. (dark) = (4 -k)/4 = 3.8196011/4 = 0.9549150275. Consequently where we have ignored the fine structure details by rounding 21 + k to 21 and 22 + k to 22 in a manner not that much different from of the original form of dimensional regularization theory. The result is subsequently validated by another equally ingenious approach due mainly to E. Witten and his school of topological quantum field theory. We notice that in that theory the local degrees of freedom are zero. Therefore, we are dealing essentially with pure gravity where are the degrees of freedom and is the corresponding dimension. The results and the conclusion of the paper are summarized in Figure 1-3, Table 1 and Flow Chart 1.

A Simple Mechanism for Generating a Geomagnetic Field

On the basis of the ideal gas model, the polarization of charges in the mantle was obtained, a physical and mathematical model was constructed, and estimated calculations of the dipole mode of the Earth’s magnetic field were performed, taking into account the speed of its angular rotation, the parameters of density and temperature, the chemical composition, the ionization potential, the dielectric constant and the percentage of the main chemical compounds of the mantle substance.

Reciprocity Relation between Alternative Gravity Formulas

We compare Newton’s force law of universal gravitation with a corrected simple approach based on Bhandari’s recently presented work, where the gravitation constant G is maintained. A reciprocity relation exists between both alternative gravity formulas with respect to the distances between mass centers. We conclude a one-to-one mapping of the two gravitational formulas. We don’t need Einstein’s construct of spacetime bending by matter.

Construction of Earth's gravitational field model from CHAMP,GRACE and GOCE data

The basic principle of spectral combination method is discussed,and the general expressions of the spectral weight and spectral combination of the united-processing of various types of gravimetric data are shown.What＇s more,based on degree error RMS of potential coefficients,the detailed expressions of spectral combination formulae and the corresponding spectral weights in the Earth＇s gravitational field model（EGM） determination using GOCE ＋ GRACE and CHAMP ＋ GRACE ＋ GOCE are derived.The fundamental situation that ulux-champ2013 s,tongji-GRACE01,go-cons-gcf-2-tim-r5 constructed respectively by CHAMP,GRACE,GOCE data and go-cons-gcf-2-dir-r5 constructed by syncretic processing of GRACE,GOCE and LAGEOS data are explained briefly,the degree error RMS,cumulative geoid height error and cumulative gravity anomaly error of these models are calculated.A syncretic model constructed from CHAMP,GRACE and GOCE data,which is expressed by champ ＋ grace ＋ goce,is obtained based on spectral combination method.Experimentation results show that the precision of CHAMP data model is the lowest in satellite-only models,so it is not needed in the determination of syncretic models.The GRACE data model can improve the GOCE data model in medium-long wavelength,so the overall precision of syncretic model can be improved.Consequently,as many types of gravimetric data as possible should be combined together in the data processing in order to strengthen the quality and reliability with widening scope and improve the precision and spatial resolution of the computational results.

Spectral Representation and Analysis of the KBR Precision Based on the Earth Gravity Spectrum

Satellite-to-Satellite Tracking in low-low model （SST-Ⅱ） is a new technique to resolve the series of problems met in the determination of the earth＇s gravity field. As the key technique of SST-Ⅱ, KBR can get SST-Ⅱ measurements directly. So the KBR performance analysis is the first step in SST-Ⅱ design. In this paper, assuming that the satellite pairs of SST-Ⅱ are in near circle polar orbits, the spectrum relationship between the earth gravity field and KBR is established using analytic method. And then some examples are analyzed, the suggestions and conclusions are drawn from these examples. The research results could be taken as a reference for future satellite gravity project of China.

Equation of Motion of a Mass Point in Gravitational Field and Classical Tests of Gauge Theory of Gravity

A systematic method is developed to studY the classical motion of a mass point in gravitational gauge field. First, by using Mathematica, a spherical symmetric solution of the field equation of gravitational gauge field is obtained, which is just the traditional Schwarzschild solution. Combining the principle of gauge covariance and Newton＇s second law of motion, the equation of motion of a mass point in gravitational field is deduced. Based on the spherical symmetric solution of the field equation and the equation of motion of a mass point in gravitational field, we can discuss classical tests of gauge theory of gravity, including the deflection of light by the sun, the precession of the perihelia of the orbits of the inner planets and the time delay of radar echoes passing the sun. It is found that the theoretical predictions of these classical tests given by gauge theory of gravity are completely the same as those given by general relativity.

Gravity Field Variations Associated with the Buried Geological Structures: San Marcos Fault (NE Mexico) Case Study

Gravity data are sensitive to local vertical offsets across high-angle faults, where rocks with different densities are juxtaposed. Yet high densities in some Mesozoic sedimentary rocks just above the basement may smear out the subtle gravity signatures of basement faults. At this study the gravity data processing tends to avoid ill-described “black-box” techniques. The study area is situated in the Palomas site, Cuatrociénegas region, Coahuila, NE Mexico. The San Marcos Fault is at least 300 km long and has WNW-ESE trend from the central part of Nuevo León State through Coahuila, and finally to the eastern part of Chihuahua State. Gravimetric data shows that the lowest values of free air and Bouguer anomalies are in the southern part of the area, and the highest values are in the western and central part of the area. Between these parts exists a zone of high horizontal gravity gradient. Configuration of linear elements of gravity field (gradient zones) delimited the San Marcos Fault in the San Marcos valley below thickness of recent sedimentary cover. Two density models were carried out, which showed that the Cretaceous rocks are in discordant contact with the Paleo- zoic rocks that can be related to the San Marcos Fault. The density was determinate using to Nettleton’s method, which results highlight the presence of the San Marcos Fault. Density models showed that the Quaternary sediments are in direct contact with the San Marcos Fault.

Feasible Multiple Satellite Mission Scenarios Flying in a Constellation for Refinement of the Gravity Field Recovery

Improving the gravity field recovery in terms of error levels and more isotropic noise distribution by adding cross-track and radial information to the satellite observables has been investigated through a number of studies by a variety of satellite constellations, i.e. satellite pairs that orbit the Earth in alternative configurations than the current GRACE (Gravity Recovery And Climate Experiment) gravity mission. This contribution gives for the first time a comparative study considering the recovery of the global gravity field from three constellations flying in satellite pairs in different directions (i.e. along-track, cross-track and radial). The three constellations include: 1) Foursatellite Bender configuration (flying in two pairs) of type along-track observations, 2) Three-satellite GRAPEN (combined GRACE with Pendulum formations) configuration of type cross-alongtrack observations, 3) Four-satellite Cartwheel configuration (flying in two pairs) of type radialalong-track observations. Additionally, a GRACE mission scenario is added as a reference “comparative” mission. The orbits of all satellites are considered to fly with drag-free system, however, realistic white noise has been added to the simulated observations to mimic the error associated with the drag-free measurement. The results are analyzed in the spectral wavelength spectrum of the gravity field up to a spherical harmonics degree of n = 100 and are plotted spatially on earth maps. The results show that the Three-satellite GRAPEN constellation provides, besides its low economically launches, an improved gravity field solution with respect to the Four-satellite Bender and the Four-satellite Cartwheel constellations.

Secular Effect of Geomagnetic Field and Gravitational Waves on Earth’s Satellite Orbits

In this work we study the perturbation and the change in the orbital elements due to the earth’s magnetic field and the gravitational waves. The acceleration components are derived in the radial, transverse to it and normal to the orbital plane. The equation for the rates of variation of the elements is formed and solved to find the secular variation in the element for polar and equatorial satellites.

Theory of Electromagnetism and Gravity —Modeling Earth as a Rotating Solenoid Coil

Presented in this manuscript are conventional electrical engineering tools to model the earth as a rotating electrical machine. Calculations using known parameters of the earth and measured field data has resulted in new understanding of the earth’s electrical system and gyroscopic rotation. The material makeup of the inner earth is better understood based on derived permeability and permittivity constants. The planet has been modeled as simple coils and then as a parallel impedance circuit which has led to fundamental insight into planetary speed control and RLC combination for Schumann Resonance of 7.83 Hz. Torque and Voltage Constants and the inverse Speed Constant are calculated using three methods and all compare favorably with Newton’s Gravitational Constant. A helical resonator is referenced and Schumann’s Resonant ideal frequency is calculated and compared with others idealism. A new theory of gravity based on particle velocity selector at the poles is postulated. Two equations are presented as the needed links between Faraday’s electromagnetism and Newtonian physics. Acceleration and Deceleration of earth is explained as a centripetal governor. A new equation for planetary attraction and the attraction of atomic matter is theorized. Rotation of the earth’s electrical coil is explained in terms of the Richardson effect. Electric power transfer from the sun to the planets is proposed via Flux Transfer Events. The impact of this evolving science of electromagnetic modeling of planets will be magnified as the theory is proven, and found to be useful for future generations of engineers and scientists who seek to discover our world and other planets.