Determination of the degree 120 time-variable gravity field in the Sichuan-Yunnan region using Slepian functions and terrestrial measurements

The terrestrial time-variable gravity measurements are characterized by a high signal-to-noise ratio and sensitivity to the sources of mass change in the Earth's crust.These gravity data have many applications,such as surface deformation,groundwater storage changes,and mass migration before and after earthquakes.Based on repeated terrestrial gravity measurements at 198 gravity stations in the Sichuan-Yunnan region(SYR)from 2015 to 2017,we determine a time series of degree 120 gravity fields using the localized spherical harmonic(Slepian)basis functions.Our results show that adopting the first 6 Slepian basis functions is sufficient for effective localized Slepian modeling in the SYR.The differences between two gravity campaigns at the same time of year show an obvious correlation with tectonic features.The degree 120 timevariable gravity models presented in this paper will benefit the study of the regional mass migration inside the crust of the SYR and supplement the existing geophysical models for the China Seismic Experimental Site.

GRACE time-variable gravity and its application to geoscience:Quantitative analysis of relevant literature

The Gravity Recovery and Climate Experiment(GRACE)is the most important gravity satellite to date in human history.Since its launch in 2002,GRACE time-varying gravity has had an unprecedented impact on earth science and has generated revolutionary changes.Because of natural phenomena such as climate warming,glacial melting,sea level rise,and earthquakes,earth science research has become an increasingly popular discipline in recent years.This article summarizes the importance of GRACE time-varying gravity,its application to geoscience,and its development.We analyzed the historical development and current status of GRACE time-varying gravity as well as research hotspots by searching the literature in the core collection databases of the China National Knowledge Infrastructure and the Web of Science over the past 20 years.The CiteSpace and VOSviewer software packages were applied with reference to the principle of literature metrology.Our investigation and analysis of characteristic indexes,such as the numbers of publications,co-occurrence of keywords,and co-citation of documents,uncovered the wide application and promotion of gravity satellites,especially GRACE time-varying gravity,in earth science.The results showed that the number of publications addressing GRACE data and time-varying gravity theory is increasing annually and that the USA,China,and Germany are the main producers.The Chinese Academy of Sciences,the National Aeronautics and Space Administration of the United States,and the Helmholtz Association of German Research Centres rank among the top three institutions in the world in terms of producing the most publications on this topic.We found that GRACE time-varying gravity plays unique roles in measuring changes in terrestrial water storage changes,ice and snow melting and sea level changes,and(co)seismic gravity changes,as well as in promoting other disciplines.

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.

Time-varying gravity field model of Sichuan-Yunnan region based on the equivalent mass source model

High-precision time-varying gravity field is an effective way to study the internal mass movement and understanding the spatio-temporal evolution process of the geodynamic system.Compared to the satellite gravity measurement,the repeated terrestrial gravity observation can provide a more high-order signal related to the shallow crust and subsurface.However,the suitable and unified method for gravity model estimation is a key problem for further applications.In this study,we introduce the spherical hexahedron element to simulate the field source mass and forward model the change of gravity field located at the Sichuan-Yunnan region(99—104°E,23—29°N)in the four epochs from 2015 to 2017.Compared to the experimental results based on Slepian or spherical harmonics frequency domain method,this alternative approach is suitable for constructing the equivalent mass source model of regional-scale gravity data,by introducing the first-order smooth prior condition of gravity time-varying signal to suppress the high-frequency component of the signal.The results can provide a higher spatial resolution reference for regional gravity field modeling in the Sichuan-Yunnan region.

Spatial distribution characteristics and mechanism of nonhydrological time-variable gravity in China's Mainland

The purpose of this study is to explore nonhydrological mass transfer in China's Mainland.For this purpose,gravity recovery and climate experiment(GRACE)data were obtained to study the spatial distribution of time variant gravity signals in China's Mainland.Then,from auxiliary hydrological data processed according to the current hydrological model,a new more comprehensive hydrological model of China's Mainland was constructed.Finally,the time variant signals of this new hydrological model were removed from the time variant gravity field computed from GRACE data,thus obtaining a description of the nonhydrological mass transfer of China's Mainland.The physical sources and mechanisms of the resulting mass transfer are then discussed.The improved,more realistic,hydrological model used here was created by selecting the hydrological components with the best correlations in existing hydrological models,by use of correlation calculation,analysis,and comparison.This improved model includes water in soils and deeper strata,in the vegetation canopy,in lakes,snow,and glaciers,and in other water components(mainly reservoir storage,swamps,and rivers).The spatial distribution of the transfer signals due to nonhydrological mass in China's Mainland was obtained by subtracting the combined hydrological model from the GRACE time-variable gravity field.The results show that the nonhydrological signals in China's Mainland collected in GRACE data were mainly positive signals,and were distributed in the Bohai Rim and the northern and eastern parts of the Tibetan Plateau.The above nonhydrological mass transfer signals have been studied further and are discussed.The results show that the nonhydrological mass migration signals in the Bohai Rim region originate primarily from sea level change and marine sediment accumulation.The mass accumulation from Indian plate collision in the Tibetan Plateau appears to be the main reason for the increase in the residual gravity field in that region.

Seismologic applications of GRACE time-variable gravity measurements

The Gravity Recovery and Climate Experiment（GRACE） has been measuring temporal and spatial variations of mass redistribution within the Earth system since2002. As large earthquakes cause significant mass changes on and under the Earth＇s surface,GRACE provides a new means from space to observe mass redistribution due to earthquake deformations. GRACE serves as a good complement to other earthquake measurements because of its extensive spatial coverage and being free from terrestrial restriction. During its over 10 years mission,GRACE has successfully detected seismic gravitational changes of several giant earthquakes,which include the 2004 Sumatra–Andaman earthquake,2010 Maule（Chile） earthquake,and 2011 Tohoku-Oki（Japan） earthquake. In this review,we describe by examples how to process GRACE timevariable gravity data to retrieve seismic signals,and summarize the results of recent studies that apply GRACE observations to detect co- and post-seismic signals and constrain fault slip models and viscous lithospheric structures. We also discuss major problems and give an outlook in this field of GRACE application.

Quintessence anisotropic stellar models in quadratic and Born-Infeld modified teleparallel Rastall gravity

This study aims to discuss anisotropic solutions that are spherically symmetric in the quintessence field,which describe compact stellar objects in the modified Rastall teleparallel theory of gravity.To achieve this goal,the Krori and Barua arrangement for spherically symmetric components of the line element is incorporated.We explore the field equations by selecting appropriate off-diagonal tetrad fields.Born-Infeld function of torsion f(T)=β√λT+1-1 and power law form h(T)=δTn are used.The Born-Infeld gravity was the first modified teleparallel gravity to discuss inflation.We use the linear equation of state pr=ξρto separate the quintessence density.After obtaining the field equations,we investigate different physical parameters that demonstrate the stability and physical acceptability of the stellar models.We use observational data,such as the mass and radius of the compact star candidates PSRJ 1416-2230,Cen X-3,&4U 1820-30,to ensure the physical plausibility of our findings.

The Nature of Inertia Explained Using the Field Theory

Analysis of free fall and acceleration of the mass on the Earth shows that using abstract entities such as absolute space or inertial space to explain mass dynamics leads to the violation of the principle of action and reaction. Many scientists including Newton, Mach, and Einstein recognized that inertial force has no reaction that originates on mass. Einstein calls the lack of reaction to the inertial force a serious criticism of the space-time continuum concept. Presented is the hypothesis that the inertial force develops in an interaction of two masses via the force field. The inertial force created by such a field has reaction force. The dynamic gravitational field predicted is strong enough to be detected in the laboratory. This article describes the laboratory experiment which can prove or disprove the hypothesis of the dynamic gravitational field. The inertial force, calculated using the equation for the dynamic gravitational field, agrees with the behavior of inertial force observed in the experiments on the Earth. The movement of the planets in our solar system calculated using that equation is the same as that calculated using Newton’s method. The space properties calculated by the candidate equation explain the aberration of light and the results of light propagation experiments. The dynamic gravitational field can explain the discrepancy between the observed velocity of stars in the galaxy and those predicted by Newton’s theory of gravitation without the need for the dark matter hypothesis.

The Origin of Quarks in Quantum Gravity

A theory of quantum gravity has recently been developed by the author based on the concept that all forces converge to one at the moment of Creation. This primordial field can only interact with itself, as no other field exists, contrasting with the Standard Model of Particle Physics in which each elementary particle is an excitation in its own quantum field. The primordial field theory of quantum gravity has produced a model of a fermion with a mass gap, ½-integral spin, discrete charge, and magnetic moment. The mass gap is based on an existence theorem that is anchored in Yang-Mills, while Calabi-Yau anchors ½-integral spin, with charge and magnetic moment based on duality. Based on N-windings, this work is here extended to encompass fractional charge, with the result applied to quarks, yielding fermion mass and charge in agreement with experiment and novel size correlations and a unique quantum gravity-based ontological understanding of quarks.

Ashtekar-Kodama Gravity as a Classical and Quantum Extension of Loop Quantum Gravity

This paper presents a new theory of gravity, called here Ashtekar-Kodama (AK) gravity, which is based on the Ashtekar-Kodama formulation of loop quantum gravity (LQG), yields in the limit the Einstein equations, and in the quantum regime a full renormalizable quantum gauge field theory. The three fundamental constraints (hamiltonian, gaussian and diffeomorphism) were formulated in 3-dimensional spatial form within LQG in Ashtekar formulation using the notion of the Kodama state with positive cosmological constant Λ. We introduce a 4-dimensional covariant version of the 3-dimensional (spatial) hamiltonian, gaussian and diffeomorphism constraints of LQG. We obtain 32 partial differential equations for the 16 variables Emn (E-tensor, inverse densitized tetrad of the metric) and 16 variables Amn (A-tensor, gravitational wave tensor). We impose the boundary condition: for large distance the E-generated metric g(E) becomes the GR-metric g (normally Schwarzschild-spacetime). The theory based on these Ashtekar-Kodama (AK) equations, and called in the following Ashtekar-Kodama (AK-) gravity has the following properties. • For Λ = 0 the AK equations become Einstein equations, A-tensor is trivial (constant), and the E-generated metric g(E) is identical with the GR-metric g. • When the AK-equations are developed into a Λ-power series, the Λ-term yields a gravitational wave equation, which has only at least quadrupole wave solutions and becomes in the limit of large distance r the (normal electromagnetic) wave equation. • AK-gravity, as opposed to GR, has no singularity at the horizon: the singularity in the metric becomes a (very high) peak. • AK-gravity has a limit scale of the gravitational quantum region 39 μm, which emerges as the limit scale in the objective wave collapse theory of Gherardi-Rimini-Weber. In the quantum region, the AK-gravity becomes a quantum gauge theory (AK quantum gravity) with the Lie group extended SU(2) = ε-tensor-group(four generators) as gauge group and a corresponding covariant derivative. • AK quantum gravity is fully renormalizable, we derive its Lagrangian, which is dimensionally renormalizable, the normalized one-graviton wave function, the graviton propagator, and demonstrate the calculation of cross-section from Feynman diagrams.

The Origin of Electric Charge in Quantum Gravity

A model fermion has been produced in a theory of quantum gravity that establishes the existence of a mass gap and half-integral spin. The third requirement for the fermion is electric charge. We herein develop a dualism-based analysis that explains the origin of charge at the fermion scale in a primordial field theory of quantum gravity.

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.

The Unified Field

This is a Unified Field description based on the holographic Time Dilation Cosmology, TDC, model, which is an eternal continuum evolving forward in the forward direction of time, at the speed of light, c, at an invariant 1 s/s rate of time. This is the Fundamental Direction of Evolution, FDE. There is also an evolution down time dilation gradients, the Gravitational Direction of Evolution, GDE. These evolutions are gravity, which is the evolutionary force in time. Gravitational velocities are compensation for the difference in the rate of time, dRt, in a dilation field, and the dRtis equal to the compensatory velocity’s percentage of c, and is a measure of the force in time inducing the velocity. In applied force induced velocities, the dRt is a measure of the resistance in time to the induced velocity, which might be called “anti-gravity” or “negative gravity”. The two effects keep the continuum uniformly evolving forward at c. It is demonstrated that gravity is already a part of the electromagnetic field equations in way of the dRt element contained in the TDC velocity formula. Einstein’s energy formula is defined as a velocity formula and a modified version is used for charged elementary particle solutions. A time dilation-based derivation of the Lorentz force ties gravity directly to the electromagnetic field proving the unified field of gravity and the EMF. It is noted how we could possibly create gravity drives. This is followed by a discussion of black holes, proving supermassive objects, like massive black hole singularities, are impossible, and that black holes are massless Magnetospheric Eternally Collapsing Objects (MECOs) that are vortices in spacetime. .

A New Perspective on Time and Gravity

This paper presents a hypothesis regarding the existence of time fused in spacetime, assuming that time possesses the properties of both a particle and a field. This duality is referred to as the field-particle of time (FPT). The analysis shows that when the FPT moves through matter, it causes time dilation. The FPT is also a significant element that appears in relativistic kinetic energy (KE = (γ - 1) · mc2). Accelerating matter to near the speed of light requires relativistic energy approaching infinity, which corresponds to the relativistic kinetic energy. Meanwhile, the potential energy (PE = mc2) from the rest mass remains constant. Then, the mass-energy equation can be rearranged in terms of PE and KE, as shown in E = (1 + (γ - 1)) · mc2. The relativistic energy of the FPT also directly affects the gravitational attraction of matter. It transfers energy to each other through spacetime. The analysis demonstrates that the gravitational force is inversely proportional to the distance squared, following Newton’s law of gravity, and it varies with the relative velocity of matter. The relationship equation between relative time and the gravitational constant indicates that a higher intensity of the gravitational field leads to a slower reference time for matter, in accordance with the general theory of relativity. A thought experiment presents a comparison of two atomic clocks placed in different locations. The first one is placed in a room temperature, around 25°C, on the surface of the Earth, and the second one is placed in high-density areas. The analysis, considering the presence of the FPT, shows that the reference time slows down in high-density areas. Therefore, the second clock must be noticeably slower than the first one, indicating the existence of the FPT passing through both atomic clocks at different speeds.

Analysis of limitations on recovery of gravity field based on satellite gravity gradient data

Although satellite gravity gradient data plays a great role in determining short-wavelength part of static gravity field model,accuracy of the long-wavelength part of gravity field model recovered by them are poor,which leads to only a few applications in time-variable gravity investigation.The reason is that some factors limit the accuracy of the gravity field recovered using gradient data,including accuracy of the gravity gradient observations,measurement bandwidth(MBW)of gradiometer,satellite inclination,etc.This paper aims at analyzing the influence of these limitations on gravity field recovery and discusses the possibility of time-variable gravity field detection by using gravity gradient observations.Firstly,for arbitrary satellite orbit inclination,we give the frequency distributions of all the components of gravity gradients(i.e.Txx;Tyy;Tzz;Txy;Txz and Tyz,).The results show that the maximum frequency of each component of the gravity gradients is the same,i.e.l=Ts(l is degree of the gravity field model,Ts is the orbital periods),and it is not influenced by the inclination of the satellite orbits.Secondly,the paper gives a theory proof to explain why only the low orders of the coefficients are influenced by polar gaps.Big polar gaps are experimented by a numerical test with inclination of 45°.Finally,considering that the measurement bandwidth can be expanded and accuracy of gradient observations can be improved by superconducting gravity gradiometer(SGG)compared to gradiometer used in Gravity field and steadystate Ocean Circulation Explorer(GOCE),the possibility of detecting time-variable gravity using gravity gradient observations is discussed.The results show that the SGG creates errors in MBW with magnitude of 0.014 m E,which is smaller than the magnitude of the time-variable gravity gradient signals(i.e.,0.02 m E)derived from Gravity Recovery and Climate Experiment(GRACE)gravity field models.This indicates the potential of SGG in time-variable gravity detection.

Monthly gravity field solution from GRACE range measurements using modified short arc approach

In this paper we present a series of monthly gravity field solutions from Gravity Recovery and Climate Experiment（GRACE） range measurements using modified short arc approach,in which the ambiguity of range measurements is eliminated via differentiating two adjacent range measurements.The data used for developing our monthly gravity field model are same as Tongji-GRACEOl model except that the range measurements are used to replace the range rate measurements,and our model is truncated to degree and order 60,spanning Jan.2004 to Dec.2010 also same as Tongji-GRACE01 model.Based on the comparison results of the C_（2,0）,C_（2,1）,S_（2,1）,and C_（15,15）,S_（15,15）,time series and the global mass change signals as well as the mass change time series in Amazon area of our model with those of Tongji-GRACE01 model,we can conclude that our monthly gravity field model is comparable with Tongji-GRACE01 monthly model.

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.

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.

A High-Resolution Earth’s Gravity Field Model SGG-UGM-2 from GOCE,GRACE,Satellite Altimetry,and EGM2008

This paper focuses on estimating a new high-resolution Earth’s gravity field model named SGG-UGM-2 from satellite gravimetry,satellite altimetry,and Earth Gravitational Model 2008(EGM2008)-derived gravity data based on the theory of the ellipsoidal harmonic analysis and coefficient transformation(EHA-CT).We first derive the related formulas of the EHA-CT method,which is used for computing the spherical harmonic coefficients from grid area-mean and point gravity anomalies on the ellipsoid.The derived formulas are successfully evaluated based on numerical experiments.Then,based on the derived least-squares formulas of the EHA-CT method,we develop the new model SGG-UGM-2 up to degree 2190 and order 2159 by combining the observations of the Gravity Field and Steady-State Ocean Circulation Explorer(GOCE),the normal equation of the Gravity Recovery and Climate Experiment(GRACE),marine gravity data derived from satellite altimetry data,and EGM2008-derived continental gravity data.The coefficients of degrees 251–2159 are estimated by solving the block-diagonal form normal equations of surface gravity anomalies(including the marine gravity data).The coefficients of degrees 2–250 are determined by combining the normal equations of satellite observations and surface gravity anomalies.The variance component estimation technique is used to estimate the relative weights of different observations.Finally,global positioning system(GPS)/leveling data in the mainland of China and the United States are used to validate SGG-UGM-2 together with other models,such as European improved gravity model of the earth by new techniques(EIGEN)-6C4,GECO,EGM2008,and SGG-UGM-1(the predecessor of SGG-UGM-2).Compared to other models,the model SGG-UGM-2 shows a promising performance in the GPS/leveling validation.All GOCE-related models have similar performances both in the mainland of China and the United States,and better performances than that of EGM2008 in the mainland of China.Due to the contribution of GRACE data and the new marine gravity anomalies,SGG-UGM-2 is slightly better than SGG-UGM-1 both in the mainland of China and the United States.

Analysis of the impact on the gravity field determination from the data with the ununiform noise distribution using block-diagonal least squares method

The block-diagonal least squares method, which theoretically has specific requirements for the observation data and the spatial distribution of its precision, plays an important role in ultra-high degree gravity field determination. On the basis of block-diagonal least squares method, three data processing strategies are employed to determine the gravity field models using three kinds of simulated global grid data with different noise spatial distri- bution in this paper. The numerical results show that when we employed the weight matrix corresponding to the noise of the observation data, the model computed by the least squares using the full normal matrix has much higher precision than the one estimated only using the block part of the normal matrix. The model computed by the block-diagonal least squares method without the weight matrix has slightly lower precision than the model computed using the rigorous least squares with the weight matrix. The result offers valuable reference to the using of block-diagonal least squares method in ultra-high gravity model determination.