Components Of Gravity Gradient Tensor Research Articles

Modeling Gravity Gradients from Surface Gravity Anomaly Data

Gravity gradients are useful to characterize near mass anomalies since they are much more sensitive to short wavelength anomalies than gravitational accelerations. Estimating gravity gradients from surface gravity data is based on numerical implementations of solutions to geodetic boundary value problem for determination of disturbing potential. One of methods to solve this problem is least-squares collocation which is basically based on data and a defined covariance function. This study deals with estimating gravity gradient tensor components from along track surface gravity anomaly data. The Least-Squares Collocation solution is based on a stationary local covariance function defined for the disturbing potential which allows upward continuation of the observations to a desired altitude. The modeling method is evaluated in using Earth Gravitational Model 2008(EGM2008) and real airborne gravity gradiometry data collected over Southern Texas, Oklahoma region. The results show that modeled gravity gradients estimated in both on the ground and at a certain altitude have basically good agreement with EGM08 gradients. Modeled gradients including horizontal components in the east-west direction exhibit some discrepancies in comparison to the airborne gradiometry data, which may be attributed to some measurement errors in the gradient data.

Fast 3D forward modeling of a potential field based on spherical symmetry of gravitational potential

A novel approach is developed to implement the forward modeling of a potential field caused by prismatic grids on gridded observations with higher efficiency and less memory. Based on the spherical symmetry of gravitational potential, we summarize the parity, symmetry, and interconversion relations from derivatives and higher derivatives of gravitational potential, which reveal that there are vast repetitive calculation and storage redundancies in conventional forward modeling of a potential field for prismatic grids, especially for cubic grids. These properties help to reduce not only the size of a single kernel matrix but also the number of necessary kernels in multicomponent forward modeling of a potential field, such as a gradient tensor field or joint inversion of a potential field. Several experiments on the synthetic and the realistic Bishop models are presented to illustrate how these properties are used in the forward modeling of gravity and magnetic gradient tensors. Based on the proposed properties, a 2D fast Fourier transform is performed to accelerate the discrete deconvolution of the kernel and the model parameters. With the new method, one single kernel can be reduced to half the number of model parameters, and the memory requirement and time cost of our method can be reduced by 1/8 and 1/2, respectively, compared with previous works. In forward modeling of six gravity gradient tensor components, only two kernels need to be computed and stored, which are 1/3 of the conventional method. These examples indicate that, with high precision, our method requires less memory and less time than conventional methods. It shows potential in the fast large-scale inversion of gravity, magnetic, and gradient tensors with limited hardware resources.

Using gravity gradient component and their combination to interpret the geological structures in the eastern Tianshan Mountains

SUMMARY As early as the Palaeozoic era, the Tianshan Mountains constituted a vast tectonic convergent zone within the Eurasian Continent; by this time, they had already experienced multiple intense crustal movements. The Indian Plate is currently advancing northwards by more than 50 mm yr–1; it affects the Tianshan Mountains even though they are located more than 1000 km away. However, many of the geological issues of the Tianshan Mountains, especially in eastern Tianshan, which remains an active tectonic zone today, are not yet fully understood. There have been few studies into the seismogenic characteristics of this intense seismic zone. Here, the full gravity gradient tensor (GGT) within the eastern Tianshan Mountains was calculated using a stable spatial frequency domain algorithm, based on high-spatial resolution (7.2 arcsec or approximately 200 m) Global Gravity Model plus (GGMplus) surface gravity data. In addition, the features of the geological structures in the eastern Tianshan Mountains were interpreted using different combinations of GGT components. Moreover, the possibilities of using different combinations of GGT components to identify the distributions and strikes of faults was discussed regarding the potential earthquake rupture risks in this study area. The results show that (1) the distributions and strikes of the main fault zones in the study area are highly consistent with the linear geological features within the mountains, as extracted from a combination of different GGT components, including terrain effects. (2) Removing the terrain effects revealed that the different components of GGT (derived from the complete Bouguer gravity anomaly) vary gently, showing few linear geological features that could be extracted. This implies that satellite gravity data are not sufficient to analyse the characteristics of small underground geological structures (spatial scales less than ∼10 km). (3) Comparing the spatial distributions of fault traces depicted at different ages with the linear geological features of the study area revealed that discovering more unknown faults increased the coincidence ratio of linear geological features to fault traces in the area (from 27 to 40 per cent). In conclusion, the findings of this study represent a valuable reference for further revealing the seismogenic characteristics of this area, where there is a lack of detailed surface fault structure measurements due to its inaccessibility. The proposed method could also recognize faults in other similar areas with harsh conditions that are not feasible for ground-based surveys, such as forests and glaciers.

Topographic effects up to gravitational curvatures of tesseroids: A case study in China

Topographic effects on gravity field modeling are important for geodesy, geophysics and related geosciences. In this study we evaluate the gravitational effects of tesseroids in spherical coordinates, including the gravitational potential (GP), gravity vector (GV), gravity gradient tensor (GGT) and especially the gravitational curvatures (GC). With the adaptive discretization stack-based algorithm by Gauss-Legendre quadrature approach, the optimized distance-size ratio values (D) of the GC components are analyzed. Numerical experiments demonstrate that the difference percentage values of the GC components (e.g., Vxxz, Vyyz and Vzzz) are larger at the range of D ∈ [0; 10] compared to those of the GP, GV and GGT components (i.e., V, Vz, Vzz). Different distance-size ratio values D = 6, 7, 14, 30, 35, 41 and 50 for the GC component Vzzz are recommended to reach the 0.1% threshold error at corresponding computational heights 260, 150, 50, 10, 8, 6 and 4 km. Moreover, the forward modeling for the gravitational effects up to GC of tesseroids based on the ETOPO1 model in China is investigated. The GC functionals could help to extend the knowledges of interior structures of the Earth and other planetary objects.

Derived the complete gravity gradient tensor from the vertical component of gravity by a cosine transform technique

A cosine transform technique has been developed to calculate the complete gravity gradient tensor from preexisting vertical gravity data which provides an alternative determination of the gravity gradient tensor components. Gravity gradient tensor components are computed for a three-dimensional buried Y-type dyke model. A comparison between the discrete cosine transform(DCT) results and forward(or theoretical) calculating gradient components from the 3D model shows that the root-mean-square error for each component, between the two results, is at most 3.94E. In addition, according to another comparison between this DCT technique and FFT method, there is a relatively larger error between the FFT derived and model derived, but the results of DCT derived coincide well with the model derived except for several data of the boundary, indicating that our technique is more efficient than traditional FFT method.

Source Parameter Estimation of Indian Ocean Earthquake from observation of GRACE Gravity Gradient Tensor

In this study, the focal mechanism of Indian Ocean earthquake is determined by combination of GRACE gravity gradient change and analytical model of Okubo (1992). To do so, the co-seismic gravity gradient change resulting from Indian ocean earthquake is derived using monthly solution of GRACE level 02 data after removing the contaminations from hydrological effects and post-seismic creep. Since, De-striping methods lead to reduction and distortion of a co-seismic gravity change signal, it is suggested to compute a set of gravity gradient tensor components e.g. ΔVxx and ΔVxz, because they are more sensitive to small-scale signals and they reduce the striping errors without need to any de-striping filtering. Then the computed gravity gradient components are compared with the gravity gradient components from Okubo model (1992), which accounts the focal mechanism of the earthquake. By the way, the nonlinear inversion algorithm is constructed and solved by Genetic algorithm to find the best value of fault parameters for Indian Ocean earthquake. For better constraining the fault parameters in the inversion process, a sensitivity analysis is also conducted which reveals that the selected model is highly sensitive to a strike angle, dip angle, length, width and average slip, although it is less sensitive to a depth of a fault. The ultimate optimal estimate of the fault parameters shows a good correspondence compared with some existing slip models obtained by various constraints or via inversion of seismic observations.

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.

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.

Evaluating the utility of gravity gradient tensor components

Gravity gradiometry allows individual components and combinations of components to be used in interpretation. Knowledge of the information content of different components and their combinations is therefore crucial to their effectiveness, and a quantitative rating of information level is needed to guide the choice. To this end, I use linear inverse theory to examine the relationship between the different tensor components and combinations thereof and the model parameters to be determined. The model used is a rectangular prism, characterized by seven parameters: the prism location [Formula: see text], [Formula: see text]; its width [Formula: see text] and breadth [Formula: see text]; the density [Formula: see text]; the depth to top [Formula: see text]; and thickness [Formula: see text]. Varying these values allows a variety of body shapes, e.g., blocks, plates, dykes, and rods, to be considered. The Jacobian matrix, which relates parameters and their associated gravity response, clarifies the importance and stability of model parameters in the presence of data errors. In general, for single tensor components and combinations, the progression from well to poorly determined parameters follows the trend of [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], to [Formula: see text]. Ranking the estimated model errors from a range of models showed that data sets consisting of concatenated components produced the smallest parameter errors. For data sets comprising combined tensor components, the invariants of the tensor produced the smallest parameter errors. Of the single tensor components, [Formula: see text] gave the best performance overall, but those single components with strong directional sensitivity can produce some individual parameters with smaller estimated errors (e.g., [Formula: see text] and [Formula: see text] estimated from [Formula: see text]).

Evaluating the utility of gravity gradient tensor components

Gravity gradiometry offers multiple single components and possible combinations of components to be used in interpretation. Knowledge of the information content of components and their combinations is therefore crucial to their effectiveness and so a quantitative rating of information level is needed to guide the choice. To this end we use linear inverse theory to examine the relationship between the different tensor components and combinations thereof and the model parameters to be determined. The model used is a simple prism, characterized by seven parameters: the prism location, xc, yc, its width w and breadth b, the density I?, the depth to top z, and thickness t. Varying these values allows a wide variety of body shapes, e.g., blocks, plates, dykes, rods, to be considered. The Jacobian matrix, which relates parameters and their associated gravity response, clarifies the importance and stability of model parameters in the presence of data errors. In general, for single tensor components and combinations, the progression from well- to poorly-determined parameters follows the trend of I?, xc, yc, w, b, z to t. Ranking the estimated model errors from a range of models shows that data sets consisting of concatenated components produce the smallest parameter errors. For data sets comprising combined tensor components, the invariants I1 and I2 produce the smallest model errors. Of the single tensor components, Tzz gives the best performance overall, but those single components with strong directional sensitivity can produce some individual parameters with smaller estimated errors (e.g., w and xc estimated from Txz).

Some aspects of interpretation of tensor gradiometry data

Aspects of the interpretation of measured data on the gravity gradient tensor (GGT) are examined. The problem is posed in relation to the great progress achieved in recent years in the development of instrumentation and the method of GGT measurements on mobile carriers. In our opinion, the new methods of measurement and the new data obtained with their help require the development of new methods of interpretation of potential fields. The paper addresses two methods taking advantage of simultaneous measurements of all components of the GGT and anomalies of the gravitaty field V z. It is shown that the joint analysis of all GGT components can provide independent constraints on the noise level in various components. The method of tensor deconvolution proposed in the paper is a tensor analogue of the Euler method. The method is based on the calculation of invariants and is, therefore, stable with respect to the orientation uncertainties of the measuring system. The method provides means for estimating the structural index and, therefore, is particularly effective in the treatment of fields that contain isometric and/or elongated anomalies. The calculation of invariants and the tensor ratio can also be used for the development of procedures enabling automatic estimation of the axis strike azimuths of elongated anomaly-forming bodies.

Understanding the structural setting in the Southern Apennines (Italy): insight from Gravity Gradient Tensor

The Lucania Apennines form the frontal part of the Apennines in Southern Italy. This orogenic belt was formed in response to late Miocene–Early Pleistocene shortening and allochthon emplacement toward the northeast, and was subsequently affected by extensional faulting which migrated to northeast ahead of the thrust system. As a result, contraction structures are found associated to Pliocene–Early Pleistocene thrust-top basins in the NE part of the area, whereas extensional faults shape Middle Pleistocene–Holocene basin on the SW sector. The main structural grain of the belt is NW–SE, but E–W structures are widespread in the area as a result of phases of non-coaxial thrusting and strike-slip faulting. The analysis, one by one, of the components of the Gravity Gradient Tensor (GGT) has proved to yield a fine image of the structural setting of the investigated area. GGT is a second rank tensor containing the second spatial derivatives of the gravity potential and in this paper is used instead than the more traditional gravity data. The Tzz component allows an accurate description of the location and of the shape of basins and other structures, but different components of the GGT provide even more detailed insights for such structures. In particular, we found that the Tzy and Tyy locate well those structures trending closer to the E–W direction for both thrust-top and extensional basins, and their termination against the main NW–SE structures. The combined use of components of GGT provides a finer definition of the anomaly sources particularly if a good knowledge of their strike and depth is independently estimated with other geological and geophysical investigations.

The complete gravity gradient tensor derived from the vertical component of gravity: a Fourier transform technique

A technique has been to developed to determine the complete gravity gradient tensor from pre-existing vertical gravity data using the fast Fourier Transform (FFT). Since direct measurement of the entire gravity gradient tensor is generally unavailable, our technique provides an alternative determination of the gravity gradient tensor components. Traditionally, derivatives of vertical gravity ( g z,x , g z,y , and g z,z ) have been the only gravity gradient tensor components that have been computed directly. Gravity gradient tensor components are computed for four different, three-dimensional (3-D), idealized horst-and-graben models, with varying depths to the horst. Comparing the FFT results with calculated gradient components from the 3-D models shows that the RMS error for each component, between the two results, is at most ∼3.3 Eötvos Units. In addition, measured gravity gradient components from an airborne survey over the Wichita Uplift and Anadarko Basin region of southwest Oklahoma compare favorably with the FFT-derived results using available vertical gravity data. No error analysis was attempted between the two results due to a low signal-to-noise ratio in the measured data. Our technique offers a novel way to transform and visualize the available data, and it also offers an inexpensive and previously unavailable subsurface mapping capability.