Gravitational Gradient Tensor Research Articles

Modified Parker's Method for Gravitational Forward and Inverse Modeling Using General Polyhedral Models

AbstractWe propose a new modified Parker's method for efficient gravitational forward modeling and inversion using general polyhedral models. We have made several important modifications to the classical method, including: (a) The new method is now applicable to a general polyhedron represented by triangulated surface or tetrahedral mesh, and with arbitrarily variable 3D density distribution. (b) An optimal Fourier‐domain sampling strategy is used to improve the numerical accuracy of the new algorithm significantly. (c) A simple and effective automatic layering technique is introduced to accelerate the convergence rate of Parker's method. The method is demonstrated using both synthetic and real polyhedral models, including a sphere model approximated by a polyhedron, two asteroids, a digital elevation model in the Himalaya region, and the Yucca Flat basin model in Nevada. The numerical results show that, compared with analytical solutions of polyhedral models in the space domain, the modified Parker's method can improve the computational efficiency by several orders of magnitude while obtaining almost the same simulation results. The difference is well below existing instrumentation error level. By embedding the new forward algorithm into an iterative process, it can be used for fast inversion of density interfaces. Our new method is suitable for the efficient modeling and inversion of gravitational potential, gravitational vector, and gravitational gradient tensor caused by polyhedral models with a large number of faces, representing geological abnormal bodies, asteroids, and single or multilayer density interface models with triangulated surfaces.

Технология определения параметров гравитационного поля Земли по градиентометрическим измерениям 6. Вычисление компонент тензора гравитационного потенциала в земной пространственной прямоугольной системе координат

This is the sixth one in a series of articles describing the technology of determining the Earth’s gravitational field parameters through gradiometric measurements performed with an onboard satellite electrostatic gradiometer. It provides formulas for calculating the components of the gravitational potential tensor in a geocentric spatial rectangular earth coordinate system in order to convert them into a gradiometric one and obtain a free term for the equations of correcting gradiometric measurements when determining the parameters of the Earth’s gravitational field. The components of the gravitational gradient tensor are functions of test masses accelerations measured by accelerometers and relate to the gradiometer coordinate system, while the desired parameters of the Earth’s gravitational field model relate to the Earth’s coordinate one. The components of the gravitational gradient tensor are the second derivatives of the gravitational potential in rectangular coordinates. The calculated values of the gravitational potential tensor components in the earth’s spatial rectangular coordinate system are obtained through double differentiation of the gravitational potential formula. Basing on the obtained formulas, an algorithm and a program in the Fortran algorithmic language were developed. Using this program, experimental calculations were performed, the results of which were compared with the data of the EGG_TRF_2 product.

Recursive Analytical Formulae of Gravitational Fields and Gradient Tensors for Polyhedral Bodies with Polynomial Density Contrasts of Arbitrary Non-negative Integer Orders

Exact computation of the gravitational field and gravitational gradient tensor for a general mass body is a core routine to model the density structure of the Earth. In this study, we report on the existence of closed-form solutions of the gravitational potential, gravitational field and gravitational gradient tensor for a general polyhedral mass body with a polynomial density function of arbitrary non-negative integer orders that can simultaneously vary in both horizontal and vertical directions. Our closed-form solutions of the gravitational potential and the gravitational field are singularity-free, which implies that the observation sites can have arbitrary geometric relationships with polyhedral mass source bodies. However, weak logarithmic singularities exist on the edges of polyhedra for the gravitational gradient tensor. A simple prismatic mass body with polynomial density contrast varying in the vertical direction and a complicated dodecahedral mass body with quartic-order density contrasts were tested to verify the accuracy of the newly derived closed-form solutions. For the gravitational potential, gravitational fields and gradient tensors, our closed-form solutions are in excellent agreement with previously published analytical solutions and Gaussian numerical quadrature solutions.

State-Transition-Algorithm-Based Underwater Multiple Objects Localization With Gravitational Field and Its Gradient Tensor

Recently, several techniques using gravitational data such as gravitational field and its gradient tensor have been developed to localize underwater multiple objects. However, performances of these existed techniques largely rely on proper selections of the initial values and, thus, are likely trapped into local minima. To deal with this issue, a global optimization algorithm, named as the state transition algorithm (STA), is investigated to localize multiple objects with both gravitational field and gravitational gradient tensor data sets in this letter. Using a heuristic random search strategy, the proposed algorithm features good global search capability and avoids the dependence of using proper initial values. To assess the performance of the proposed method, different models which contain three and four underwater objects are tested. The experimental results demonstrate that the proposed method is promising with sound stability and strong antinoise ability for dynamic localization problem with multiple objects.

The gravity field model IGGT_R1 based on the second invariant of the GOCE gravitational gradient tensor

Based on tensor theory, three invariants of the gravitational gradient tensor (IGGT) are independent of the gradiometer reference frame (GRF). Compared to traditional methods for calculation of gravity field models based on the gravity field and steady-state ocean circulation explorer (GOCE) data, which are affected by errors in the attitude indicator, using IGGT and least squares method avoids the problem of inaccurate rotation matrices. The IGGT approach as studied in this paper is a quadratic function of the gravity field model’s spherical harmonic coefficients. The linearized observation equations for the least squares method are obtained using a Taylor expansion, and the weighting equation is derived using the law of error propagation. We also investigate the linearization errors using existing gravity field models and find that this error can be ignored since the used a-priori model EIGEN-5C is sufficiently accurate. One problem when using this approach is that it needs all six independent gravitational gradients (GGs), but the components $$V_{xy}$$ and $$V_{yz}$$ of GOCE are worse due to the non-sensitive axes of the GOCE gradiometer. Therefore, we use synthetic GGs for both inaccurate gravitational gradient components derived from the a-priori gravity field model EIGEN-5C. Another problem is that the GOCE GGs are measured in a band-limited manner. Therefore, a forward and backward finite impulse response band-pass filter is applied to the data, which can also eliminate filter caused phase change. The spherical cap regularization approach (SCRA) and the Kaula rule are then applied to solve the polar gap problem caused by GOCE’s inclination of $$96.7^{\circ }$$ . With the techniques described above, a degree/order 240 gravity field model called IGGT_R1 is computed. Since the synthetic components of $$V_{xy}$$ and $$V_{yz}$$ are not band-pass filtered, the signals outside the measurement bandwidth are replaced by the a-priori model EIGEN-5C. Therefore, this model is practically a combined gravity field model which contains GOCE GGs signals and long wavelength signals from the a-priori model EIGEN-5C. Finally, IGGT_R1’s accuracy is evaluated by comparison with other gravity field models in terms of difference degree amplitudes, the geostrophic velocity in the Agulhas current area, gravity anomaly differences as well as by comparison to GNSS/leveling data.

CO-SEISMIC GRAVITY GRADIENT CHANGES OF THE 2006–2007 GREAT EARTHQUAKES IN THE CENTRAL KURIL ISLANDS FROM GRACE OBSERVATIONS

Abstract. GRACE satellites (the Gravity Recovery And climate Experiment) are very useful sensors to extract gravity anomalies after earthquakes. In this study, we reveal co-seismic signals of the two combined earthquakes, the 2006 Mw8.3 thrust and 2007 Mw8.1 normal fault earthquakes of the central Kuril Islands from GRACE observations. We compute monthly full gravitational gradient tensor in the local north-east-down frame for Kuril Islands earthquakes without spatial averaging and de-striping filters. Some of gravitational gradient components (e.g. ΔVxx, ΔVxz) enhance high frequency components of the earth gravity field and reveal more details in spatial and temporal domain. Therefore, co-seismic activity can be better illustrated. For the first time, we show that the positive-negative-positive co-seismic ΔVxx due to the Kuril Islands earthquakes ranges from − 0.13 to + 0.11 milli Eötvös, and ΔVxz shows a positive-negative-positive pattern ranges from − 0.16 to + 0.13 milli Eötvös, agree well with seismic model predictions.

Analytical Formulas for Underwater and Aerial Object Localization by Gravitational Field and Gravitational Gradient Tensor

Object localization techniques have significant applications in civil fields and safety problems. A novel analytical formula is developed for accurate underwater and aerial object real-time localization by combining gravitational field and horizontal gravitational gradient anomalies. The proposed method enhances the accuracy of object localization and its excess mass estimation; it also effectively avoids the possible numerical instability and the singularity in the previous works. Finally, a synthetic underwater object navigation model was adopted to verify its performance. The results show that our newly developed method is more practical than existing methods.

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 ultra-high 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 GPS-leveling data for the frequency band of the degree between 20 and 160.

Effects of space weather on GOCE electrostatic gravity gradiometer measurements

We examine the presence of residual nongravitational signatures in gravitational gradients measured by GOCE electrostatic gravity gradiometer. These signatures are observed over the magnetic poles during geomagnetically active days and can contaminate the trace of the gravitational gradient tensor by up to three to five times the expected noise level of the instrument (\(\sim \)11 mE). We investigate these anomalies in the gradiometer measurements along many satellite tracks and examine possible causes using external datasets, such as interplanetary electric field measurements from the ACE (advanced composition explorer) and WIND spacecraft, and Poynting vector (flux) estimated from equivalent ionospheric currents derived from spherical elementary current systems over North America and Greenland. We show that the variations in the east-west and vertical electrical currents and Poynting vector components at the satellite position are highly correlated with the disturbances observed in the gradiometer measurements. The results presented in this paper reveal that the disturbances are due to intense ionospheric current variations that are enhanced by increased solar activity that causes a very dynamic drag environment. Moreover, successful modelling and removal of a high percentage of these disturbances are possible using external geomagnetic field observations.

Determining the Gravitational Gradient Tensor Using Satellite-Altimetry Observations over the Persian Gulf

With the advent of satellite altimetry in 1973, new scientific applications became available in oceanography, climatology, and marine geosciences. Moreover, satellite altimetry provides a significant source of information facilitated in the geoid determination with a high accuracy and spatial resolution. The information from this approach is a sufficient alternate for marine gravity data in the high-frequency modeling of the marine gravity field quantities. The gravity gradient tensor, consisting of the second-order partial derivatives of the gravity potential, provides more localized information than gravity measurements. Marine gravity observations always carry a high noise level due to environmental effects. Moreover, it is not possible to model the high frequencies of the Earth's gravity field in a global scale using these observations. In this article, we introduce a novel approach for a determination of the gravity gradient tensor at sea level using satellite altimetry. Two numerical techniques are applied and compared for this purpose. In particular, we facilitate the radial basis functions (RBFs) and the harmonic splines. As a case study, the gravitational gradient tensor is determined and results presented in the Persian Gulf. Validation of results reveals that the solution of the harmonic spline approach has a better agreement with a theoretical zero-value of the trace of the Marussi gravitational gradient tensor. However, the data-adaptive technique in the RBF approach allows more efficient selection of the parameters and 3-D configuration of RBFs compared to a fixed parameterization by the harmonic splines.

Mass-density Green's functions for the gravitational gradient tensor at different heights

Mass-density Green's functions for the gravitational gradient tensor at different heights

Gravitational gradient changes following the 2004 December 26 Sumatra-Andaman Earthquake inferred from GRACE

SUMMARY It has been demonstrated that the Gravity Recovery And Climate Experiment (GRACE) spaceborne gravimetry data are capable of observing coseismic gravity changes resulting from great earthquakes, such as the 2004 December 26 Sumatra–Andaman event (Mw 9.1–9.3). Here, we show for the first time that refined deformation signals from the 2004 December 26 Sumatra–Andaman Earthquake (Mw 9.1–9.3) together with the 2005 March 28 Nias earthquake (Mw 8.6) can be revealed by deriving the full gravitational gradient tensor from GRACE monthly gravitational field. The GRACE-inferred coseismic gravitational gradient changes agree well with coseismic slip model predictions. Since the high-frequency contents in gravitational field variation can be amplified by deriving the gravitational gradients, the GRACE-derived coseismic gravitational gradient changes clearly delineate the fault lines, locate significant slips, and better define the extent of the coseismic deformation.

Non-singular expressions for the vector and the gradient tensor of gravitation in a geocentric spherical frame

The traditional expressions of the gravitational vector (GV) and the gravitational gradient tensor (GGT) have complicated forms depending on the first- and the second-order derivatives of associated Legendre functions (ALF), and also singular terms when approaching the poles. This article presents alternative expressions for the GV and GGT, which are independent of the derivatives, and are also non-singular. By using such expressions, it suffices to compute the ALF to two additional degrees and orders, instead of computing the first and the second derivatives of all the ALF. Therefore, the formulas are suitable for computer programming. Matlab software as well as an output of a numerical computation around the North Pole is also presented based on the derived formulas.

Satellite gradiometry using a satellite pair

The GRACE mission has substantiated the low–low satellite-to-satellite tracking (LL-SST) concept. The LL-SST configuration can be combined with the previously realized high–low SST concept in the CHAMP mission to provide a much higher accuracy. The line of sight (LOS) acceleration difference between the GRACE satellite pair, the simplest form of the combined observable, is mostly used for mapping the global gravity field of the Earth in terms of spherical harmonic coefficients. As an alternative observable, a linear combination of the gravitational gradient tensor components is proposed. Being a one-point function and having a direct relation with the field geometry (curvature of the field at the point) are two noteworthy achievements of the alternative formulation. In addition, using an observation quantity that is related to the second-instead of the first-order derivatives of the gravitational potential amplifies the high-frequency part of the signal. Since the transition from the first- to the second-order derivatives includes the application of a finite-differences scheme, the high-frequency part of the noise is also amplified. Nevertheless, due to the different spectral behaviour of signal and noise, in the end the second-order approach leads to improved gravitational field resolution. Mathematical formulae for the gradiometry approach, for both linear and higher-degree approximations, are derived. The proposed approach is implemented for recovery of the global gravitational field and the results are compared with those of LOS acceleration differences. Moreover, LOS acceleration difference residuals are calculated, which are at the level of a few tenths of mGal. Error analysis shows that the residuals of the estimated degree variances are less than 10−3. Furthermore, the gravity anomaly residuals are less than 2 mGal for most points on the Earth.

A «tidal» magnetic field?

t is shown that on the magnetization axis of a uniformly magnetized body of constant density the magnetic field intensity displays a «tidal» structure,i.e. the ratios among the differential magnetic field intensity in three orthogonal directions are the same as the ratios among the gravitational gradient tensor components pertaining to the same directions; it is also seen that the same characteristic ratios occur, both locally and non-locally, among the components of the magnetic field intensity and among the components of the gradient tensors of the two fields.

A prototype superconducting gravity gradiometer

We report the successful laboratory test of a single-axis gradiometer designed to measure a diagonal component of the earth's gravitational gradient tensor. It Consists of a pair of accelerometers mounted with their sensitive axes vertical and in line. The difference in displacement of the accelerometers is proportional to the component of the tensor gradient and is sensed via the modulated inductance of a superconducting coil coupled by a superconducting transformer into an RF biased SHE SQUID with energy sensitivity 4 × 10-29J/Hz. Rejection of in-line common mode accelerations is achieved by trimming the natural resonant frequency of each accelerometer: the restoring force acting on an accelerometer test mass is partly magnetic and can be trimmed by adjusting the persistent currents in a pair of force coils. A common mode rejection ratio exceeding 95 dB has been achieved in the presence of linear accelerations \sim 10^{-3} ms-2, and a laboratory generated gradient of 30 Eo rms has been detected with a signal to noise ratio of about 100. The dependence of this signal on the distance between source and detector has the expected Newtonian form. Under quiet conditions the background noise level of the instrument is at present 3 Eo/ \sqrt{Hz} . ( 1 Eo = 10-9s-2.) This is close to the practical limit achievable for such a single axis configuration: a three axis instrument for geophysical application is under development.