Bearing capacity of surface circular footings on granular material under low gravity fields

Low gravity fields have been simulated through magnetic acceleration to conduct experimental study on bearing capacity of circular footings on a type of crushable planetary regolith simulant,which has comparable density and particle size distribution of lunar soil.The loadesettlement responses of surface spread footings are obtained by investigating the relative density,footing size and gravity effects.Applying the hyperbolic asymptote method,normalised foundation stiffness and ultimate bearing capacity are obtained by curve fitting and predicted by power functions using multivariate nonlinear regression.The results show that the nonlinear gravity effect is not negligible,related to stress condition,soil dilatancy and mobilised friction angle.A cone penetration test(CPT)-based method for prediction of bearing capacity is proposed with correlations between ultimate bearing capacity of footings and shallow penetration stiffness of CPTs,avoiding the uncertainties of soil property estimations.Analyses of allowable bearing capacity and footing influence zone in consideration of footing size and gravity effects could therefore improve the design of shallow foundations on the Moon and Mars,and provide new understandings and potential implications to the bearing capacity of shallow foundations on crushable granular material in both terrestrial and extraterrestrial geotechnical engineering.

Characteristics of gravity fields in the Jinggu M6. 6 earthquake

Based on the study of high-precision gravity data obtained from recent studies and the regional gravi- ty network for Yunnan province, a variation in the regional gravity field was identified before the occurrence of the Yunnan Jinggu M6.6 earthquake.

Dynamic changes of gravity fields before and after the 2008 Wenchuan earthquake(Ms8.0)

The pattern evolution and dynamic mechanism of the dynamic changes of regional gravity fields occurring before and after the Wenchuan Ms8.0 earthquake are analyzed, based on five epochs of 1998 -2007 mobile gravity data from the middle-south section of the north-south seismic belt, and two epochs of field research data collected after the 2008 Wenchuan earthquake in combination with GPS data, leveling observations, and geotectonic environment data. The regional dynamic gravity changes demonstrate the effects of the eastward flow of solid matter in the Qinghai-Tibetan plateau and the preparation of the 2008 Wenchuan earthquake （2- 10 yr）. The two most meaningful gravity indicators of the Wcnchuan earthquake preparation are the positive （increasing） gravity changes occurring over many years in the southwest epicenter and the largescale gradient zone of gravity variation, with the cumulative difference between the two sides of the gradient zone of gravity exceeding 200 μGal. The positive gravity changes may facilitate a constant energy accumulation and the gradient belt may support seismic shear breakage. Overall, the gravity changes associated with the earthquake preparation indicate a pattern of accelerating increase-decelerating increase-earthquake occurrence. The Songpan-Ganzi block generally displays a negative gravity change, providing evidence for a local upwarp- ing of the deep crust-mantle and an interior expansion of the deep crust attributable to high temperatures. The viewpoint is consistent with the dilatant mechanism for earthquake preparation.

Modeling of Earth’s Gravity Fields Visualization Based on Quad Tree

The problems of the earth's gravity fields' visualization are both focus and puzzle currently. Aiming at multiresolution rendering, modeling of the Earth's gravity fields' data is discussed in the paper by using LOD algorithm based on Quad Tree. First, this paper employed the method of LOD based on Quad Tree to divide up the regional gravity anomaly data, introduced the combined node evaluation system that was composed of viewpoint related and roughness related systems, and then eliminated the T-cracks that appeared among the gravity anomaly data grids with different resolutions. The test results demonstrated that the gravity anomaly data grids' rendering effects were living, and the computational power was low. Therefore, the proposed algorithm was a suitable method for modeling the gravity anomaly data and has potential applications in visualization of the earth's gravity fields.

Moho depth inversion in the Tibetan Plateau from high-precision gravity data

The Tibetan Plateau(TP)is the youngest orogenic belt resulting from a continental collision on the Earth.It is a natural laboratory for studying continental dynamics,such as continental convergence,plate subduction,and plateau uplift.Investigating the deep structure of the TP has always been a popular issue in geological research.The Moho is the boundary between the crust and the mantle and therefore plays a crucial role in the Earth’s structure.Parameters such as depth and lateral variation,as well as the fine structure of the crust-mantle interface,reveal the lithospheric dynamics in the TP.Two methods are generally employed to study the Moho surface:seismic detection and gravity inversion.Seismic detection has the characteristic of high precision,but it is limited to a few cross-sectional lines and is quite costly.It is not suitable for and cannot be carried out over a large area of the TP.The Moho depth over a large area can be obtained through gravity inversion,but this method is affected by the nature of gravity data,and the accuracy of the inversion method is lower than that of seismic detection.In this work,a high-precision gravity field model was selected.The Parker-Oldenburg interface inversion method was used,within the constraints of seismic observations,and the Bott iteration method was introduced to enhance the inversion efficiency.The Moho depth in the TP was obtained with high precision,consistent with the seismic detection results.The research results showed that the shape of the Moho in the TP is complex and the variation range is large,reaching 60−80 km.In contrast with the adjacent area,a clear zone of sharp variation appears at the edge of the plateau.In the interior of the TP,the buried depth of the Moho is characterized by two depressions and two uplifts.To the south of the Yarlung Zangbo River,the Moho inclines to the north,and to the north,the Moho depresses downward,which was interpreted as the Indian plate subducting to the north below Tibet.The Moho depression on the north side of the Qiangtang block,reaching 72 km deep,may be a result of the southward subduction of the lithosphere.The Moho uplift of the Qiangtang block has the same strike as the Bangong−Nujiang suture zone,which may indicate that the area is compensated by a low-density and low-velocity mantle.

Integrated gravity and magnetic study on patterns of petroleum basin occurrence in the China seas and adjacent areas

The China seas and adjacent areas contain numerous petroleum basins.One of the main challenges for future oil and gas exploration is to identify the inherent patterns of petroleum basin distribution.The formation and evolution of petroleum basins along with the migration and accumulation of oil and gas are often closely related to the tectonic environment.The gravity and magnetic fields with high lateral resolution and wide coverage provide important data for regional tectonic research.Based on the gravity data in the Global Satellite Gravity Anomaly Database(V31.1)and magnetic data from the Earth Magnetic Anomaly Grid(2-arc-minute resolution)(V2),this study uses integrated gravity and magnetic field technique to obtain integrated gravity and magnetic field result for the China seas and adjacent areas,and then adopts the normalized vertical derivative of the total horizontal derivative technique to conduct partition.Finally,it identifies the relationship between the partition characteristics and tectonics as well as the patterns of petroleum basin occurrence.The research shows that the partition of gravity and magnetic field integrated result has a good correlation with the Neo-Cathaysian tectonic system and tectonic units.The petroleum basins are characterized according to three blocks arranged from north to south and four zones arranged from east to west.The north-south block structure causes the uneven distribution of oil and gas resources in the mainland area and the differences in the hydrocarbon-bearing strata.Petroleum basins are more abundant in the north than in the south.The ages of the main oil-and gas-bearing strata are“Paleozoic–Mesozoic,Paleozoic–Mesozoic–Cenozoic,and Paleozoic–Mesozoic”,in order from north to south.The difference in the overall type of oil and gas resources in all basins is controlled by the east–west zonation.From east to west,the oil and gas resource type exhibits a wave-like pattern of“oil and gas,gas,oil and gas,gas”.The vertical distribution is characterized by an upper oil(Mesozoic–Cenozoic)and lower gas(Mesozoic–Paleozoic)structure.Within the study area,the Paleozoic marine strata should be the main strata of future natural gas exploration.

Characteristics of gravity and magnetic fields and deep structural responses in the southern part of the Kyushu-Palau Ridge

The southern part of the Kyushu-Palau Ridge(KPR)is located at the conjunction of the West Philippine Basin,the Parece Vela Basin,the Palau Basin,and the Caroline Basin.This area has extremely complex structures and is critical for the research on the tectonic evolution of marginal seas in the Western Pacific Ocean.However,only few studies have been completed on the southern part,and the geophysical fields and deep structures in this part are not well understood.Given this,this study finely depicts the characteristics of the gravity and magnetic anomalies and extracts information on deep structures in the southern part of the KPR based on the gravity and magnetic data obtained from the 11th expedition of the deep-sea geological survey of the Western Pacific Ocean conducted by the Guangzhou Marine Geological Survey,China Geological Survey using the R/V Haiyangdizhi 6.Furthermore,with the data collected on the water depth,sediment thickness,and multichannel seismic transects as constraints,a 3D density model and Moho depths of the study area were obtained using 3D density inversion.The results are as follows.(1)The gravity and magnetic anomalies in the study area show distinct zoning and segmentation.In detail,the gravity and magnetic anomalies to the south of 11°N of the KPR transition from high-amplitude continuous linear positive anomalies into low-amplitude intermittent linear positive anomalies.In contrast,the gravity and magnetic anomalies to the north of 11°N of the KPR are discontinuous and show alternating positive and negative anomalies.These anomalies can be divided into four sections,of which the separation points correspond well to the locations of deep faults,thus,revealing different field-source attributes and tectonic genesis of the KPR.(2)The Moho depth in the basins in the study area is 6-12 km.The Moho depth in the southern part of KPR show segmentation.Specifically,the depth is 10‒12 km to the north of 11°N,12‒14 km from 9.5°N to 11°N,14-16 km from 8.5°N to 9.5°N,and 16‒25 km in the Palau Islands.(3)The KPR is a remnant intra-oceanic arc with the oceanic-crust basement.which shows noticeably discontinuous from north to south in geological structure and is intersected by NEE-trending lithospheric-scale deep faults.With large and deep faults F3 and F1(the Mindanao fault)as boundaries overall,the southern part of the KPR can be divided into three zones.In detail,the portion to the south of 8.5°N(F3)is a tectonically active zone,the KPR portion between 8.5°N and 11°N is a tectonically active transition zone,and the portion to the north of 11°N is a tectonically inactive zone.(4)The oceanic crust in the KPR is slightly thicker than that in the basins on both sides of the ridge,and it is inferred that the KPR formed from the thickening of the oceanic crust induced by the upwelling of deep magma in the process of rifting of remnant arcs during the Middle Oligocene.In addition,it is inferred that the thick oceanic crust under the Palau Islands is related to the constant upwelling of deep magma induced by the continuous northwestward subduction of the Caroline Plate toward the Palau Trench since the Late Oligocene.This study provides a scientific basis for systematically understanding the crustal attributes,deep structures,and evolution of the KPR.

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.

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.

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 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 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.

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.

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.

Gravity and gravity gradient changes caused by a point dislocation

In this paper we studied gravitational potential, gravity and its gradient changes, which are caused by a pointdislocation, and gave the concise mathematical deduction with definite physical implication in dealing with thesingular integral at a seismic source. We also analysed the features of the fields of gravity and gravity gradient,gravity-vertical-displacement gradient. The conclusions a re: (1) Gravity and gravit y gradient changes are verysmall with the change of vertical positionl (2) Gravity change is much greater than the gravity gradient changewhich is not so distinct; (3) The gravity change due to redistribution of mass accounts for 10-50 percent of thetotal gravity change caused by dislocation. The signs (positive or negative) of total gravity change and verticaldisplacement are opposite each other at the same point for strike slip and dip slip 1 (4) Gravity-vertical displacement-gradient is not constantl it manifests a variety of patterns for different dislocation modelsl (5) Gravityvertical-displacement-gradient is approximately equal to a pparent gra vit y-- vert ical- displacement -gradient.

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

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

A GOCE only gravity model GOSG01S and the validation of GOCE related satellite gravity models

We compile the GOCE-only satellite model GOSG01S complete to spherical harmonic degree of 220 using Satellite Gravity Gradiometry （SGG） data and the Satellite-to-Satellite Tracking （SST） observations along the GOCE orbit based on applying a least-squares analysis. The diagonal components （Vxx, Vyy, Vzz） of the gravitational gradient tensor are used to form the system of observation equations with the band-pass ARMA filter. The point-wise acceleration observations （ax, ay, az） along the orbit are used to form the system of observation equations up to the maximum spherical harmonic degree/order 130. The analysis of spectral accuracy characteristics of the newly derived gravitational model GOSG01S and the existing models GOTIM04S, GODIR04S, GOSPW04S and JYY_GOCE02S based on their comparison with the ultrahigh degree model EIGEN-6C2 reveals a significant consistency at the spectral window approximately between 80 and 190 due to the same period SGG data used to compile these models. The GOCE related satellite gravity models GOSG01S, GOTIM05S, GODIR05S, GOTIM04S, GODIR04S, GOSPW04S, JYY_- GOCE02S, EIGEN-6C2 and EGM2008 are also validated by using GPS-leveling data in China and USA. According to the truncation at degree 200, the statistic results show that all GGMs have very similar differences at GPS-leveling points in USA, and all GOCE related gravity models have better performance than EGM2008 in China. This suggests that all these models provide much more information on the gravity field than EGM2008 in areas with low terrestrial gravity coverage. And STDs of height anomaly differences in China for the selected truncation degrees show that GOCE has improved the accuracy of the global models beyond degree 90 and the accuracies of the models improve from 24 cm to 16 cm. STDs of geoid height differences in USA show that GOSG01S model has best consistency comparing with GPSleveling data for the frequency band of the degree between 20 and 160.

Simulation of earth gravity field using satellite constellation with variable inclination configuration

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 average acceleration approach applied to gravity coefficients recovery based on GOCE orbits

The average acceleration approach was applied to recover a gravity field model Model;CA from GOCE precise science orbits from September 2 to November 2, 2010, and furthermore a so called sequential least square adjustment was used. The model was compared with other gravity field models based on CHAMP, GRACE and GOCE. The result shows that the model is superior to gravity field based on CHAMP, and with higher accuracy than other international gravity field models based on only GOCE data before 80 degree. The degree geoid height of Model;CA reaches 3cm up to 90 degree and order.

Construction of nonsingular formulae of variance and covariance function of disturbing gravity gradient tensors

When the computational point is approaching the poles, the variance and covariance formulae of the disturbing gravity gradient tensors tend to be infinite, and this is a singular problem. In order to solve the problem, the authors deduced the practical non-singular computational formulae of the first- and second-order derivatives of the Legendre functions and two kinds of spherical harmonic functions, and then constructed the nonsingular formulae of variance and eovarianee function of disturbing gravity gradient tensors.