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.

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.

Noise filtering of full-gravity gradient tensor data

In oil and mineral exploration, gravity gradient tensor data include higher- frequency signals than gravity data, which can be used to delineate small-scale anomalies. However, full-tensor gradiometry （FTG） data are contaminated by high-frequency random noise. The separation of noise from high-frequency signals is one of the most challenging tasks in processing of gravity gradient tensor data. We first derive the Cartesian equations of gravity gradient tensors under the constraint of the Laplace equation and the expression for the gravitational potential, and then we use the Cartesian equations to fit the measured gradient tensor data by using optimal linear inversion and remove the noise from the measured data. Based on model tests, we confirm that not only this method removes the high- frequency random noise but also enhances the weak anomaly signals masked by the noise. Compared with traditional low-pass filtering methods, this method avoids removing noise by sacrificing resolution. Finally, we apply our method to real gravity gradient tensor data acquired by Bell Geospace for the Vinton Dome at the Texas-Louisiana border.

Full gravity gradient tensors from vertical gravity by cosine transform

We present a method to calculate the full gravity gradient tensors from pre-existing vertical gravity data using the cosine transform technique and discuss the calculated tensor accuracy when the gravity anomalies are contaminated by noise. Gravity gradient tensors computation on 2D infinite horizontal cylinder and 3D ＂Y＂ type dyke models show that the results computed with the DCT technique are more accurate than the FFT technique regardless if the gravity anomalies are contaminated by noise or not. The DCT precision has increased 2 to 3 times from the standard deviation. In application, the gravity gradient tensors of the Hulin basin calculated by DCT and FFT show that the two results are consistent with each other. However, the DCT results are smoother than results computed with FFT. This shows that the proposed method is less affected by noise and can better reflect the fault distribution.

3D density inversion of gravity gradient data using the extrapolated Tikhonov regularization

We use the extrapolated Tikhonov regularization to deal with the ill-posed problem of 3D density inversion of gravity gradient data. The use of regularization parameters in the proposed method reduces the deviations between calculated and observed data. We also use the depth weighting function based on the eigenvector of gravity gradient tensor to eliminate undesired effects owing to the fast attenuation of the position function. Model data suggest that the extrapolated Tikhonov regularization in conjunction with the depth weighting function can effectively recover the 3D distribution of density anomalies. We conduct density inversion of gravity gradient data from the Australia Kauring test site and compare the inversion results with the published research results. The proposed inversion method can be used to obtain the 3D density distribution of underground anomalies.

Edge enhancement of gravity anomalies and gravity gradient tensors using an improved small sub-domain filtering method

In order to enhance geological body boundary visual effects in images and improve interpretation accuracy using gravity and magnetic field data, we propose an improved small sub-domain filtering method to enhance gravity anomalies and gravity gradient tensors. We discuss the effect of Gaussian white noise on the improved small sub-domain filtering method, as well as analyze the effect of window size on geological body edge recognition at different extension directions. Model experiments show that the improved small sub-domain filtering method is less affected by noise, filter window size, and geological body edge direction so it can more accurately depict geological body edges than the conventional small sub-domain filtering method. It also shows that deeply buried body edges can be well delineated through increasing the filter window size. In application, the enhanced gravity anomalies and calculated gravity gradient tensors of the Hulin basin show that the improved small sub-domain filtering can recognize more horizontal fault locations than the conventional method.

Selecting the optimum location of the corner using gravity gradient method

The conventional gravity gradient method to plot the geologic body location is fuzzy. When the depth is large and the geologic body is small, the Vzz and Vzx derivative errors are also large. We describe that using the status distinguishing factor to optimally determine the comer location is more accurate than the conventional higher-order derivative method. Thus, a better small geologic body and fault resolution is obtained by using the gravity gradient method and trial theoretical model calculation. The actual data is better processed, providing a better basis for prospecting and determination of subsurface geologic structure.

Effect of gravity gradient in weak equivalence principle test

A high accuracy test of the weak equivalence principle（WEP） is of great scientific significance no matter whether its result is positive. We analyze the gravity gradient effect which is a main systematic error source in the test of WEP.The result shows that the uncompensated gravity gradient effect from the coupling term of the dominated gravity gradient multipole moment component q_（21） and the relative multipole field component Q_（21） contributes to an uncertainty of 1×10^（-11） on the E otv os parameter. We make a Q_（21） compensation to reduce the effect by about 20 times, and the limit of the test precision due to this coupling is improved to a level of a part in 10^（13）.

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.

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.

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.

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.

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.

Solving Three Types of Satellite Gravity Gradient Boundary Value Problems by Least-Squares

The principle and method for solving three types of satellite gravity gradient boundary value problems by least-squares are discussed in detail. Also, kernel function expressions of the least-squares solution of three geodetic boundary value problems with the observations {Γ zz },{Γ xz , Γ yz} and {Γ xx -Γ yy ,2 Γxy}are presented. From the results of recovering gravity field using simulated gravity gradient tensor data, we can draw a conclusion that satellite gravity gradient integral formulas derived from least-squares are valid and rigorous for recovering the gravity field.

A fast-imaging method for airborne gravity gradient data based on tensor invariants

Airborne gravity gradient data contain additional short-wavelength information about the buried geological bodies.This study develops a fast interpretation method based on the gravity gradient data for the sources’spatial location and physical property parameters.This study analyzes the advantages of the source parameter inversion method based on tensor invariants.It proposes a normalized fast-imaging method based on tensor invariants to quickly estimate the spatial location parameters of sources through the local maximum value position of the imaging results.First,the tensor invariant characteristics and the imaging method’s effect in a simple model are analyzed using a theoretical model.Second,to analyze the imaging method’s application effect in complex model conditions,the method’s applicability is quantitatively analyzed using the data added with noise,superimposed anomalies of adjacent sources,and anomalies of deep and shallow geological bodies.The theoretical model’s simulation results show that the model’s imaging results in this study have satisfactory performance on the spatial position estimation of the sources.Finally,the method is applied to the gravity anomaly data corresponding to the Humble salt dome.The imaging results can effectively estimate the distribution of the salt dome’s horizontal and depths,verifying the practicability of the method.

Evaluating gravity gradient components based on a reweighted inversion method

In gravity gradient inversion,to choose an appropriate component combination is very important,that needs to understand the function of each component of gravity gradient in the inversion.In this paper,based on the previous research on the characteristics of gravity gradient components,we propose a reweighted inversion method to evaluate the influence of single gravity gradient component on the inversion resolution The proposed method only adopts the misfit function of the regularized equation and introduce a depth weighting function to overcome skin effect produced in gravity gradient inversion.A comparison between different inversion results was undertaken to verify the influence of the depth weighting function on the inversion result resolution.To avoid the premise of introducing prior information,we select the depth weighting function based on the sensitivity matrix.The inversion results using the single-prism model and the complex model show that the influence of different components on the resolution of inversion results is different in different directions,however,the inversion results based on two kind of models with adding different levels of random noise are basically consistent with the results of inversion without noises.Finally,the method was applied to real data from the Vinton salt dome,Louisiana,USA.

An iterative Wiener filtering method based on the gravity gradient invariants

How to deal with colored noises of GOCE （Gravity field and steady - state Ocean Circulation Explorer） satellite has been the key to data processing. This paper focused on colored noises of GOCE gradient data and the frequency spectrum analysis. According to the analysis results, gravity field model of the optima] degrees 90-240 is given, which is recovered by COCE gradient data. This paper presents an iterative Wiener filtering method based on the gravity gradient invariants. By this method a degree-220 model was calculated from GOCE SGG （Satellite Gravity Gradient） data. The degrees above 90 of ITG2010 were taken as the prior gravity field model, replacing the low degree gravity field model calculated by GOCE orbit data. GOCE gradient colored noises was processed by Wiener filtering. Finally by Wiener filtering iterative calculation, the gravity field model was restored by space-wise harmonic analysis method. The results show that the model＇s accuracy matched well with the ESA＇s （European Space Agency） results by using the same data,

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.

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.

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.