Gravity Gradient Tensor Research Articles

Tesseroids: Forward-modeling gravitational fields in spherical coordinates

We have developed the open-source software Tesseroids, a set of command-line programs to perform forward modeling of gravitational fields in spherical coordinates. The software is implemented in the C programming language and uses tesseroids (spherical prisms) for the discretization of the subsurface mass distribution. The gravitational fields of tesseroids are calculated numerically using the Gauss-Legendre quadrature (GLQ). We have improved upon an adaptive discretization algorithm to guarantee the accuracy of the GLQ integration. Our implementation of adaptive discretization uses a “stack-based” algorithm instead of recursion to achieve more control over execution errors and corner cases. The algorithm is controlled by a scalar value called the distance-size ratio ([Formula: see text]) that determines the accuracy of the integration as well as the computation time. We have determined optimal values of [Formula: see text] for the gravitational potential, gravitational acceleration, and gravity gradient tensor by comparing the computed tesseroids effects with those of a homogenous spherical shell. The values required for a maximum relative error of 0.1% of the shell effects are [Formula: see text] for the gravitational potential, [Formula: see text] for the gravitational acceleration, and [Formula: see text] for the gravity gradients. Contrary to previous assumptions, our results show that the potential and its first and second derivatives require different values of [Formula: see text] to achieve the same accuracy. These values were incorporated as defaults in the software.

USING THE ANALYTIC SIGNAL METHOD OF GRAVITY GRADIENT TENSOR (GGT) TO DETERMINE THE LOCATION AND DEPTH OF THE FAULTS IN THE PRE-CENOZOIC BASEMENT ROCKS OF THE RED RIVER TROUGH

USING THE ANALYTIC SIGNAL METHOD OF GRAVITY GRADIENT TENSOR (GGT) TO DETERMINE THE LOCATION AND DEPTH OF THE FAULTS IN THE PRE-CENOZOIC BASEMENT ROCKS OF THE RED RIVER TROUGH

Gravity gradient tensor due to a cylinder

We have derived closed-form expressions for the gravity gradient tensor above a vertical semi-infinite cylinder. We first derived the solution in the cylindrical coordinate system where, due to the azimuthal symmetry of the cylinder, there are only three nontrivial independent components of the tensor. The expressions in the Cartesian coordinate system can then be obtained through a coordinate transformation. For general use, we have also developed a solution to the gravity gradient tensor due to a semi-infinite cylinder with an arbitrary orientation through rotation between a user-defined coordinate and the cylinder-affixed Cartesian system. We have verified the validity of the solutions numerically through comparison with the results from approximating the cylinder with a set of small prisms and by comparison with the analytical solution for a semi-infinite line source.

A regional background model for the Arabian Peninsula from modeling satellite gravity gradients and their invariants

Gravity gradients of the GOCE (Gravity field and steady state Ocean Circulation Explorer) mission are available and provide free available data with global homogeneous coverage. The spatial resolution is about 80km which makes it possible to construct regional models of large or remote areas. We study the benefit of using this data for compiling a 3D regional density model of the lithosphere with emphasis on the crustal structure for the Arabian Peninsula. We keep the data at satellite level to avoid noise amplification. The initial density model was set up with only a few seismic constraints and modified by forward modeling. Forward models benefits from the use of the full gravity gradient tensor and the invariants. The misfits between the measured and calculated field are more easily recognized in the gravity gradient tensor than in the gravity field alone. Different shape and location of anomalies are visible in the different components. Hence, modeling is better controlled as geometries which fit one component do not necessarily fit all other components. Regional models are important for a general understanding of any area and as a background for local interpretations. Here, we explain how to construct density models from satellite gravity gradients and show the advantage of such an approach by comparison to earlier studies based on either seismic or gravity data.

Interpreting the direction of the gravity gradient tensor eigenvectors: Their relation to curvature parameters of the gravity field

Rotating the gravity gradient tensor about a vertical axis by an appropriate angle allows one to express its components as functions of the curvatures of the equipotential surface. The description permits the identification of the gravity gradient tensor as the Newtonian tidal tensor and part of the tidal potential. The identification improves the understanding and interpretation of gravity gradient data. With the use of the plunge of the eigenvector associated with the largest eigenvalue or plunge of the main tidal force, it is possible to estimate the location and depth of buried gravity sources; this is developed in model data and applied to FALCON airborne gravity gradiometer data from the Canning Basin, Australia.

Geophysics Bright Spots

Three Bright Spots shine in the May-June issue of Geophysics. The first is a new approach to elastic full-waveform inversion (FWI). The second takes a new look at some long-period, long-duration events associated with hydraulic fracturing operations. The third extends a classic theorem of Poisson to relate the gravity gradient tensor and a given magnetic field component (given a common source of each field).

3D inversion of full gravity gradient tensor data using SL0 sparse recovery

We present a new method dedicated to the interpretation of full gravity gradient tensor data, based on SL0 sparse recovery inversion. The SL0 sparse recovery method aims to find out the minimum value of the objective function to fit the data function and to solve the non-zero solution to the objective function. Based on continuous iteration, we can easily obtain the final global minimum (namely the property and space attribute of the inversion target). We consider which type of tensor data combination produces the best inversion results based on the inversion results of different full gravity gradient tensor data combinations (separate tensor data and combined tensor data). We compare the recovered models obtained by inverting the different combinations of different gravity gradient tensor components to understand how different component combinations contribute to the resolution of the recovered model. Based on the comparison between the SL0 sparse recovery inversion results and the smoothest and focusing inversion results of the full gravity gradient tensor data, we show that SL0 sparse recovery inversion can obtain more stable and efficient inversion results with relatively sharp edge information, and that this method can also produce a stable solution of the inverse problem for complex geological structures. This new method to resolve very large full gravity gradient tensor datasets has the considerable advantage of being highly efficient; the full gravity gradient tensor inversion requires very little time. This new method is very effective in explaining the full gravity tensor which is very sensitive to small changes in local anomaly. The numerical simulation and inversion results of the compositional model indicates that including multiple components for inversion increases the resolution of the recovered density model and improves the structure delineation. We apply our inversion method to invert the gravity gradient tensor survey data from the Vinton salt dome, Louisiana. The results of inversion are in good agreement with the known formation in the region, which supports the validity of our method.

Inversion of gravity gradient tensor data: does it provide better resolution?

Inversion of gravity gradient tensor data: does it provide better resolution?

Estimating rock-vector magnetization from coincident measurements of magnetic field and gravity gradient tensor

Poisson’s theorem relating components of the magnetic field to components of the gradient of the gravity vector assuming a common source has been cast into a general form. A given magnetization distribution in the terrain or in the underlying crust is propagated into the corresponding magnetic field through the gravity gradient tensor. Conversely, measured magnetic field anomalies and measured gravity gradient tensor anomalies can be used to estimate the unknown magnetization vectors without knowledge of the geometry of the sources. We have tested the method on recently acquired data over a greenstone belt in Northern Sweden. The topographic relief was sufficiently variable to dominate the measured gravity gradient tensor. In practice, we have concentrated on areas where the norm of the gravity gradient tensor reached a maximum so that there was a better chance of identifying isolated sources with well-defined density and magnetization. We have surrounded the selected points by a small window and used all the data lying within that window to estimate the magnetization vectors. We have compared the estimated amplitudes and directions of magnetization with those measured from selected rock samples in the area and found a relatively modest agreement. We have interpreted this as a result of two effects: (1) Measured magnetizations are generally lower than those estimated by this method, and we believe that this is related to the fact that the collection of samples in the field is biased because of a small number of outcrops in most parts of the area. (2) This analysis is biased toward high-amplitude magnetic anomalies; i.e., the estimation procedure works best for high-amplitude magnetic anomalies, in which case, the influence of neighboring anomalies is reduced. The estimated magnetization directions show a strong dominance of remanent magnetization over induced magnetization in agreement with laboratory measurements on rock samples from the area.

重力勾配テンソルを用いた構造境界抽出のための半自動解釈手法

重力勾配テンソルを用いた構造境界抽出のための半自動解釈手法

Computation of the gravity field and its gradient: Some applications

New measuring instruments of Earth's gravity gradient tensors (GGT) have offered a fresh impetus to gravimetry and its application in subsurface exploration. Several efforts have been made to provide a thorough understanding of the complex properties of the gravity gradient tensor and its mathematical formulations to compute GGT. However, there is not much open source software available. Understanding of the tensor properties leads to important guidelines in the development of real three dimensional geological models. We present a MATLAB computational algorithm to calculate the gravity field and full gravity gradient tensor for an undulated surface followed by regular geometries like an infinite horizontal slab, a vertical sheet, a solid sphere, a vertical cylinder, a normal fault model and a rectangular lamina or conglomerations of such bodies and the results are compared with responses using professional software based on different computational schemes. Real subsurface geometries of complex geological structures of interest are approximated through arrangements of vertical rectangular laminas. The geological application of this algorithm is demonstrated over a horst-type structure of Oklahoma Aulacogen, USA and Vredefort Dome, South Africa, where measured GGT data are available.

Error analysis of a new planar electrostatic gravity gradiometer for airborne surveys

Moving-base gravity gradiometry has proven to be a convenient method to determine the Earth’s gravity field. The ESA mission GOCE (Gravity field and steady-state Ocean Circulation Explorer) has enabled to map the Earth gravity field and its gradients with a resolution of 80 km, leading to significant advances in physical oceanography and solid Earth physics. At smaller scales, airborne gravity gradiometry has been increasingly used during the past decade in mineral and hydrocarbon exploration. In both cases the sensitivity of gradiometers to the short wavelengths of the gravity field is of crucial interest. Here, we quantify and characterize the error on the gravity gradients estimated from measurements performed with a new instrument concept, called GREMLIT, for typical airborne conditions. GREMLIT is an ultra-sensitive planar gravitational gradiometer which consists in a planar acceleration gradiometer together with 3 gyroscopes. To conduct this error analysis, a simulation of a realistic airborne survey with GREMLIT is carried out. We first simulate realistic GREMLIT synthetic data, taking into account the acceleration gradiometer and gyroscope noises and biases and the variation of orientation of the measurement reference frame. Then, we estimate the gravity gradients from these data. Special attention is paid to the processing of the gyroscopes measurements whose accuracy is not commensurate with the ultra-sensitive gradiometer. We propose a method to calibrate the gyroscopes biases with a precision of the order $$10^{-8}$$ rad/s. In order to transform the tensor from the measurement frame to the local geodetic frame, we estimate the error induced when replacing the non-measured elements of the gravity gradient tensor by an a priori model. With the appropriate smoothing, we show that it is possible to achieve a precision better than 2E for an along-track spatial resolution of 2 km.

Rosborough approach for the determination of regional time variability of the gravity field from satellite gradiometry data

It is well-known that Rosborough approach, based on spherical harmonics, can be efficiently applied to determine the unknown global gravitational field parameters in the framework of the so-called space-wise approach. Usually, for regional gravity field modeling radial basis functions instead of spherical harmonics are used as basis functions. Using the strong relationship between radial basis functions and spherical harmonics, the Rosborough counterparts of radial basis functions are developed. Based on these regionally supported Rosborough basis functions, all components of gravity-gradient tensor are expressed as series in regionally supported Rosborough basis functions. In a simulation study the feasibility of regional static and time-variable gravity field recovery from satellite gradiometry data using regionally supported Rosborough functions is tested.

Taylor iteration downward continuation method for gravity gradient tensor data

Gravity gradient tensor downward continuation can highlight the anomaly characteristics of the shallow, reflect the characteristics of abnormal body horizontal density variation. Nowadays, most gradient tensor downward continuation studies are focusing on spatial domain, and the current obstacles are slow computation speed and small downward continuation height in frequency domain research. In this study, we showed a new method to downward continuation of gravity gradient data using Taylor iteration method. The general Taylor iteration method formula was deduced, and we proved the convergence of the approach. Modeling experiment showed a good fit to large point spacing. The analysis of the iterations radio showed that the Taylor expansion item is correlated with convergence speed. Furthermore, the application to the actual data of America Vinton, LA demonstrated that Taylor iteration method has a good effect to actual data.

Magnetic susceptibility as a tool for mineral exploration (Case study: Southern of Zagros Mountains)

Magnetic susceptibility has been extensively used to determine the magnetic properties of rocks for different applications, such as hydrocarbon or mineral explorations. This magnetic quantity can be directly measured in an accurate but time-consuming operation, or it can be mathematically approximated using a reliable procedure to achieve a desired accuracy. The Poisson theory is one of the most well-known approaches which provide a meaningful relationship between the earth’s gravity and magnetic fields to derive the magnetic susceptibility. In this approach, the reliability and efficiency of the derived magnetic susceptibility depends on the method of computation of the gravity gradient tensor. We investigated two different methods of determination of gradient tensor; different distance method and Fourier transform technique. From the investigation, the Fourier transform method was more consistent with the geological features which led to more reliable information required for mineral explorations. The performance of the Poisson theory, the different distance method, and the Fourier transform was investigated in the coastal Fars, in Iran. This was highly disposing for geological and mineral features. Salt domes in the study area were detected and results compared with the available geological map.

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.

Improved curvature gravity gradient tensor with principal component analysis and its application in edge detection of gravity data

Edge detection is one of the common processes in the interpretation of gravity data, and it gives a view of the earth which highlights boundaries of geological bodies. Many mathematical techniques have been proposed for edge detection including the curvature gravity gradient tensor method (CGGT). Eigenvalues of the curvature gravity gradient tensor can well delineate the edges of some geological bodies. However, it cannot be applied to complex gravity data with both positive and negative anomalies. Moreover, the CGGT method is also very sensitive to noise. In view of limitations of the method, we proposed an improved CGGT method by incorporating the principal component analysis (PCA) into the CGGT formula. The improved method can be utilized to outline the edges of causative bodies in more general cases and is insensitive to noise. The method was tested on synthetic gravity data and actual gravity data recorded from a metallic mineral deposit area in the middle-lower reaches of the Yangtze River in China. All of the results have shown effectiveness of the proposed method.

Edge detection of gravity field using eigenvalue analysis of gravity gradient tensor

In this paper, eigenvalues of the full gravity gradient tensor (GGT) are used to detect edges of geological structure. First, the solving of GGT eigenvalues is discussed; then a new edge detection method is proposed by using the eigenvalues of GGT. Comparing with the pervious edge detection method based on curvature gravity gradient tensor (CGGT), the full gravity gradient tensor contains more independent gradient components that are helpful to detect more subtle structures of the sources. The proposed method is applied to the synthetic data with and without noise to determine the locations of the edges of the mixed positive/negative contract density bodies. It has also been tested on real field data. All of the experimental results have shown that the newly proposed method is effective for edge detection.

Gravity gradient and magnetic terrain effects for airborne applications — A practical fast Fourier transform technique

We have implemented a practical fast Fourier transform technique for fast and approximate calculation of terrain effects for airborne measurement of the gravity gradient tensor and the total magnetic field. The calculations proceed in two steps. Starting from a digital terrain model (DTM), we first calculate the fields on a plane surface lying above the highest point of the terrain in the selected area. This calculation can be made arbitrarily accurate by including a sufficiently large number of terms in Parker’s well-known Fourier transform technique. The second step involves a downward continuation of the fields to a draped surface describing the positions of the airborne measurements. The inherent instability of downward continuation through the level of the highest terrain is compensated for by low-pass filtering the calculated fields on the plane surface prior to downward continuation. We use a Gaussian filter with cutoff wavenumbers well below the Nyquist wavenumber corresponding to a wavelength equal to the distance between flight lines. Tests on synthetic data as well as on real data from a DTM from northern Sweden demonstrated that the method works well and provides a low-pass-filtered version of the true terrain effect.

重力勾配テンソルの固有ベクトルを用いた断層あるいは構造境界の傾斜角の推定

重力勾配テンソルの固有ベクトルを用いた断層あるいは構造境界の傾斜角の推定