Forward Modeling of Gravity,Gravity Gradients,and Magnetic Anomalies due to Complex Bodies

On the basis of the results of improved analytical expression of computation of gravity anomalies due to a homogeneous polyhedral body composed of polygonal facets, and applying the forward theory with the coordinate transformation of vectors and tensors, we deduced both the analytical expressions for gravity gradient tensors and for magnetic anomalies of a polygon, and obtained new analytical expressions for computing vertical gradients of gravity anomalies and vertical component of magnetic anomalies caused by a polyhedral body. And also we developed explicitly the complete unified expressions for the calculation of gravity anomalies, gravity gradient, and magnetic anomalies due to the homogeneous polyhedron. Furthermore, we deduced new analytical expressions for computing vertical gradients of gravity anomalies due to a finite rectangular prism by applying the newly obtained expressions for gravity gradient tensors due to a polyhedral target body. Comparison with forward calculation of models shows the correctness of these new expressions. It will reduce forward calculation time of gravity-magnetic anomalies and improve computational efficiency by applying our unified expressions for joint forward modeling of gravity-magnetic anomalies due to homogeneous polyhedral bodies.

Fast modeling of gravity gradients from topographic surface data using GPU parallel algorithm

The gravity gradient is a secondary derivative of gravity potential,containing more high-frequency information of Earth’s gravity field.Gravity gradient observation data require deducting its prior and intrinsic parts to obtain more variational information.A model generated from a topographic surface database is more appropriate to represent gradiometric effects derived from near-surface mass,as other kinds of data can hardly reach the spatial resolution requirement.The rectangle prism method,namely an analytic integration of Newtonian potential integrals,is a reliable and commonly used approach to modeling gravity gradient,whereas its computing efficiency is extremely low.A modified rectangle prism method and a graphical processing unit(GPU)parallel algorithm were proposed to speed up the modeling process.The modified method avoided massive redundant computations by deforming formulas according to the symmetries of prisms’integral regions,and the proposed algorithm parallelized this method’s computing process.The parallel algorithm was compared with a conventional serial algorithm using 100 elevation data in two topographic areas(rough and moderate terrain).Modeling differences between the two algorithms were less than 0.1 E,which is attributed to precision differences between single-precision and double-precision float numbers.The parallel algorithm showed computational efficiency approximately 200 times higher than the serial algorithm in experiments,demonstrating its effective speeding up in the modeling process.Further analysis indicates that both the modified method and computational parallelism through GPU contributed to the proposed algorithm’s performances in experiments.

Predicting bathymetry based on vertical gravity gradient anomaly and analyses for various influential factors

The prediction of bathymetry has advanced significantly with the development of satellite altimetry.However,the majority of its data originate from marine gravity anomaly.In this study,based on the expression of vertical gravity gradient(VGG)of a rectangular prism,the governing equations for determining sea depths to invert bathymetry.The governing equation is solved by linearization through an iterative process,and numerical simulations verify its algorithm and its stability.We also study the processing methods of different interference errors.The regularization method improves the stability of the inversion process for errors.A piecewise bilinear interpolation function roughly replaces the low-frequency error,and numerical simulations show that the accuracy can be improved by 41.2%after this treatment.For variable ocean crust density,simulation simulations verify that the root-mean-square(RMS)error of prediction is approximately 5 m for the sea depth of 6 km if density is chosen as the average one.Finally,two test regions in the South China Sea are predicted and compared with ship soundings data,RMS errors of predictions are 71.1 m and 91.4 m,respectively.

Autonomous orbit determination using epoch-differenced gravity gradients and starlight refraction

Autonomous orbit determination via integration of epoch-differenced gravity gradients and starlight refraction is proposed in this paper for low-Earth-orbiting satellites operating in GPSdenied environments. Starlight refraction compensates for the significant along-track position error that occurs from only using gravity gradients and benefits from integration in terms of improved accuracy in radial and cross-track position estimates. The between-epoch differencing of gravity gradients is employed to eliminate slowly varying measurement biases and noise near the orbit revolution frequency. The refraction angle measurements are directly used and its Jacobian matrix derived from an implicit observation equation. An information fusion filter based on a sequential extended Kalman filter is developed for the orbit determination. Truth-model simulations are used to test the performance of the algorithm, and the effects of differencing intervals and orbital heights are analyzed. A semi-simulation study using actual gravity gradient data from the Gravity field and steady-state Ocean Circulation Explorer（GOCE） combined with simulated starlight refraction measurements is further conducted, and a three-dimensional position accuracy of better than 100 m is achieved.

Tectonics of the Solomon Sea Basin from Vertical Gravity Gradient and Seismic Data

The Solomon Sea Basin is a Cenozoic back-arc spreading basin within the convergence system of the Pacific and Indo-Australian plates.Against the background of subduction polarity reversal,the current Solomon Sea Basin gradually formed a rhombic morphology with the subduction of the basin along the New Britain Trench and the Trobriand Trough.By analyzing the vertical gravity gradient,natural earthquake and seismic reflection data,this study determines the structural characteristics of the Solomon Sea Basin.It was found that the tectonics of the basin are characterized by the original expansion structure within the central part in addition to the structure induced by the latest subduction along the basin margin.The original spreading structure of the basin presented an east–west linear graben and horst controlled by normal faults during the basin expansion period.As a result of the subduction and slab-pull of the Solomon Sea Basin,extensional structure belts parallel to the New Britain Trench formed along the basin margin.

Joint inversion of gravity and vertical gradient data based on modified structural similarity index for the structural and petrophysical consistency constraint

Joint inversion is one of the most effective methods for reducing non-uniqueness for geophysical inversion.The current joint inversion methods can be divided into the structural consistency constraint and petrophysical consistency constraint methods,which are mutually independent.Currently,there is a need for joint inversion methods that can comprehensively consider the structural consistency constraints and petrophysical consistency constraints.This paper develops the structural similarity index(SSIM)as a new structural and petrophysical consistency constraint for the joint inversion of gravity and vertical gradient data.The SSIM constraint is in the form of a fraction,which may have analytical singularities.Therefore,converting the fractional form to the subtractive form can solve the problem of analytic singularity and finally form a modified structural consistency index of the joint inversion,which enhances the stability of the SSIM constraint applied to the joint inversion.Compared to the reconstructed results from the cross-gradient inversion,the proposed method presents good performance and stability.The SSIM algorithm is a new joint inversion method for petrophysical and structural constraints.It can promote the consistency of the recovered models from the distribution and the structure of the physical property values.Then,applications to synthetic data illustrate that the algorithm proposed in this paper can well process the synthetic data and acquire good reconstructed results.

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.

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.

The absolute gravity measurement by FG5 gravimeter at Great Wall Station,Antarctica

Gravity measurement is of great importance to the height datum in Antarctica. The absolute gravity measurement was carried out at Great Wall Station, Antarctica, using FG5 absolute gravity instrument. The gravity data was processed with corrections of earth tide, ocean tide, polar motion and the atmospher, and the RMS is within -3×10^-8 ms^-2. The vertical and horizontal gravity gradients were measured using 2 LaCoaste ＆ Romberg （LCR） gravimeters. The absolute gravity measurement provides the fundamental data for the validation and calibration of the satellite gravity projects such as CHAMP, GRACE and GOCE, and for the high accuracy geold model.

Distribution law and susceptibility of geohazards across a gradient belt of the Western Sichuan Plateau

Across a gradient belt of the Western Sichuan Plateau,geohazards have seriously limited economic and social development.According to incomplete statistics,15,673 geohazards have been recorded in the study area.In order to mitigate the threat of geohazards to human engineering activities in the region,an overall understanding of the distribution pattern of geohazards and susceptibility assessment are necessary.In this paper,a gradient belt of the Western Sichuan Plateau and its zoning criteria were defined.Subsequently,on the basis of relief amplitude,distance to faults,rainfall,and human activities,three indicators of endogenic process were introduced:Bouguer gravity anomaly gradient,vertical deformation gradient,and horizontal deformation gradient.Thereafter,the distribution patterns of geohazards were investigated through mathematical statistics and ArcGIS software.By randomly selecting 10,449 hazards,a geohazard susceptibility map was generated using the Information Value(IV)model.Finally,the IV model was validated against 5224 hazards using the Area Under Curve(AUC)method.The results show that 47.6%of the geohazards were distributed in the zone of steep slope.Geohazards showed strong responses to distance to faults,human activities,and annual rainfall.The distribution of geohazards in the gradient belt of the Western Sichuan Plateau is more sensitive to vertical internal dynamics factors(such as vertical deformation gradient and Bouguer gravity anomaly gradient)without any apparent sensitivity to horizontal internal dynamics factors.The areas of high and very-high risk account for up to 32.22%,mainly distributed in the Longmenshan and Anning River faults.According to the AUC plot,the success rate of the IV model for generating the susceptibility map is 76%.This susceptibility map and geohazard distribution pattern can provide a reference for geological disaster monitoring,preparation of post-disaster emergency measures,and town planning.

Characteristics and Tectonic Significance of the Gravity Field in South China

Based on gravity data processed with the matched filter, depth continuation and horizontal gradient we obtained the spatial distribution of the gravity field and made analyses of the tectonic framework of South China. Then, inversion was conducted for the depth to study the depth variation of the boundary between the crust and upper mantle, namely the Mohorovicic discontinuity (Moho). The results demonstrate that the Moho depth in South China ranges from 30 to 40 km, and the crust thins from west to east, 27-29 km under the continent margin and shallow sea. We think it possible that the Tanlu fault crosses the Yangtze River and extends southwards along the Ganjiang and Wuchuan-Sihui faults to the South China Sea, and that there is an E-W hidden structural belt along 24.5°-26°.

Bathymetry predicted from vertical gravity gradient anomalies and ship soundings

In this paper, the admittance function between seafloor undulations and vertical gravity gradient anomalies was derived. Based on this admittance function, the bathymetry model of 1 minute resolution was predicted from vertical gravity gradient anomalies and ship soundings in the experimental area from the northwest Pacific. The accuracy of the model is evaluated using ship soundings and existing models, including ETOPO1, GEBCO, DTU10 and V15.1 from SIO. The model＇s STD is 69. 481m, comparable with V15.1 which is generally believed to have the highest accuracy.

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.

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.

Gravity anomaly before the Leshan M5.0 earthquake?

The North-South Seismic Belt was analyzed using gravity observation data from 2011 to 2015, and the nontidal analysis results show that there was a nonlinear gravity change at both the Chengdu and Guza seismostations one month before the Leshan M5.0 earthquake.

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.

Lithosphere density structure beneath the eastern margin of the Tibetan Plateau and its surrounding areas derived from GOCE gradients data

A three-dimensional density model of the crust and uppermost mantle is determined by the inversion of a set of GOCE gravity and gradients residual anomalies beneath the eastern margin of the Tibetan Plateau and its surrounding areas. In our work, we choose five independent gravity gradients （Txx, Tzz, Txy, Txz, Tyz） to perform density inversion. Objective function is given based on Tikhonov regularization theory. Seismic S-wave velocities play the role of initial constraint for the inversion based on a relationship between density and S-wave velocity. Damped Least Square method is used during the inversion. The final density results offer some insights into understanding the underlying geodynamic processes： （1） Low densities in the margin of the Tibet, along with low wave velocity and resistivity results, yield conversions from soft and weak Tibet to the hard and rigid cratons. （2）The lowest densities are found in the boundary of the plateau, instead of the whole Tibet indicates that the effects of extrusion stress environment in the margin affect the changes of the substance there. The substances and environments conditioning for the earthquake preparations and strong deformation in this transitional zone. （3） Evident low-D anomaly in the upper and middle crust in the Lasha terrane and Songpan-Ganzi terrane illustrated the eastward sub-ducted of southeastern Tibet, which could be accounts for the frequent volcano and earthouakes there.

Temporal variation of gravity field prior to the Ludian Ms6.5 and Kangding Ms6.3 earthquakes

Using mobile gravity data from the central area of Sichuan and Yunnan Provinces, the relationship between gravity variation and earthquakes was studied based on the Ludian Ms6.5 earthquake that occurred on August 3rd, 2014, and the Kangding Ms6.3 earthquake that occurred on November 22 nd, 2014; the mechanism of gravity variation was also explored. The results are as follows：（1） Prior to both earthquakes, gravity variation exhibited similar characteristics as those observed before both the Tangshan and Wenchuan earthquakes, in which typical precursor anomalies were positive gravity variation near the epicenter and the occurrence of a high-gravity-gradient zone across the epicenter prior to the earthquake.（2） A relatively accurate prediction of the occurrence locations of the two earthquakes was made by the Gravity Network Center of China（GNCC） based on these precursor anomalies. In the gravity study report on the 2014 earthquake trends submitted at the end of 2013, the Daofu-Shimian section at the junction of the Xianshuihe and Longmenshan fault zones was noted as an earthquake-risk region with a predicted magnitude of 6.5, which covered the epicenter of the Kangding Ms6.3 earthquake. In another report on earthquake trends in southwestern China submitted in mid-2014, the Lianfeng, Zhaotong fault zone was also classified as an earthquake-risk region with a magnitude of 6.0, and the central area of this region basically overlapped with the epicenter of the Ludian Ms6.5 earthquake.（3） The gravity variation characteristics are reasonably consistent with crustal movements, and deep material migration is likely the primary cause of gravity variation.

An Expression for Gravity Generated by an Anomalous Geological Body and Its Application in Bathymetry Inversion

The gravity anomaly and vertical gravity gradient due to anomalous geological bodies are mainly computed by numerical methods,so it is difficult and time-consuming to use the gravity-geological method to invert seafloor topography.This paper addresses this issue by deriving an expression for gravity generated by a cylinder based on a series expansion.The choice of number for terms in the series is estimated by comparing with the numerical method,especially when the depth H=4000m,the accuracy of 1 mGal(1 Gal=10^(-2)m/s^(2))can be achieved when the series are 9.The expressions can be used to establish the relationships between the shape of an anomalous body and the generated vertical gravity and vertical gravity gradient,respectively.Finally,the potential applications of the expressions in inverting seafloor topography are illustrated by synthetic examples.