Earth's Gravity Gradient Research Articles

Global marine gravity gradient tensor inverted from altimetry-derived deflections of the vertical: CUGB2023GRAD

Abstract. Geodetic applications of altimetry have largely been inversions of gravity anomaly. Previous studies of Earth's gravity gradient tensor mostly presented only the vertical gravity gradient (VGG). However, there are six unique signals that constitute the gravity gradient tensor. Gravity gradients are signals suitable for detecting short-wavelength topographic and tectonic features. They are derived from double differentiation of the disturbing potential and hence are susceptible to noise amplification which was exacerbated by low across-track resolution of altimetry data in the past. However, current generation of altimetry observations have improved spatial resolutions, with some better than 5 km. Therefore, this study takes advantage of current high-resolution altimetry datasets to present CUGB2023GRAD, a global (latitudinal limits of ±80°) 1 arcmin model of Earth's gravity gradient tensor over the oceans using deflections of the vertical as inputs in the wavenumber domain. The results are first assessed via Laplace's equation, whereby the resultant residual gradient is virtually zero everywhere. Further analysis at local regions in the Arctic and south Indian oceans showed that Txy, Txz and Tyz are the most dominant gravity gradients for bathymetric studies. This proves that bathymetric signatures in the non-diagonal tensor components are worth exploiting. Bathymetric coherence analysis of Tzz over the Tonga Trench showed strong correlation with multibeam shipboard depths. This study proves that current generation of altimetry geodetic missions can effectively resolve Earth's gravity gradient tensor. The CUGB2023GRAD model data can be freely accessed at https://doi.org/10.5281/zenodo.10511125 (Annan et al., 2024).

Improvement of density inversion efficiency with projected gradient method: Application to FTG data

With the development of instruments for measuring Earth's gravity gradient tensor, full tensor gravity (FTG) data can be obtained validly and accurately. The highly precise performance of FTG data ensures that they can play an important role in many domains. Generally, FTG data can be acquired from airborne platform, and large-scale data can be obtained by long term observations. For this type of data, research methods for large data processing are very important. Three-dimensional data inversion is an effective and significant method for the quantitative interpretation of FTG data, which has received much attention. Large-scale data inversion is inevitable, and may consume large amounts of time with increase in the amount of data and model parameters. The main aim of this study is to improve the calculation efficiency of FTG data inversion compared with general conjugate gradient (CG) method. Accordingly, an improved gradient method with projections onto a convex set (the projected gradient method) was applied to accelerate the recovery rate of the sub-surface rock density, and a corresponding preconditioning factor was introduced to further improve the calculation efficiency of inversion. This new projected gradient method can improve the calculation efficiency of large amounts of FTG data, which is very necessary for practical application of FTG data. The application of synthetic FTG data with Gaussian noise and the real data demonstrated the feasibility and application value of the proposed inversion method.

Ultra-low frequency active vibration control for cold atom gravimeter based on sliding-mode robust algorithm

An ultra-low frequency vibrational noise isolation apparatus from external vibration can be a critical factor in many fields such as precision measurement, high-technology manufacturing, scientific instruments, and gravitational wave detection. To increase the accuracies of these experiments, well performed vibration isolation technology is required. Until recently the cold atom gravimeter has played a crucial role in measuring the acceleration due to gravity and earth gravity gradient. The vibration isolation is one of the key techniques in the cold atom gravimeter. To reduce the vibrational noise caused by the reflecting mirror of Raman beams in the cold atom gravimeter, a compact active low-frequency vibration isolation system based on sliding-mode robust control is designed and demonstrated. The sliding-mode robust control active vibration isolation method is used to solve the vibration problem of Raman mirror in the cold atomic gravimeter. The purpose of vibration control is that the controller enables the system to be at zero state as the system states are away from the equilibrium due to vibration disturbance. In this system, the mechanical setup is based on a commercial passive isolation platform which only plays a role at higher frequency. A sliding-mode robust control subsystem is used to process and feed back the vibration measured by a seismometer which can measure the velocity of the ground vibration. A voice coil actuator is used to control and cancel the motion of a passive vibration isolation platform. The simulation and experiment results of vibration isolation platform show, on the one hand, that the vibration noise power spectral density decreases by up to 99.9%, and that the phase noise in cold atom interferometry produced by vibration decreases by up to nearly 85.3% compared with the results of the passive vibration isolation platform. On the other hand, compared with the lead-lag control method, the vibration noise power spectral density decreases by up to 83.3% and the phase noise in cold atom interferometry produced by vibration decreases by nearly 40.2%. Therefore, the sliding-mode robust control has the advantages of less tuning parameters, strong anti-interference ability, and more obvious vibration isolating effect.

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.

Compact cold atom gravimeter for field applications

We present a cold atom gravimeter dedicated to field applications. Despite the compactness of our gravimeter, we obtain performances (sensitivity 42 μGal/Hz1/2, accuracy 25 μGal) close to the best gravimeters. We report gravity measurements in an elevator which led us to the determination of the Earth's gravity gradient with a precision of 4E. These measurements in a non-laboratory environment demonstrate that our technology of gravimeter is enough compact, reliable, and robust for field applications. Finally, we report gravity measurements in a moving elevator which open the way to absolute gravity measurements in an aircraft or a boat.

Effect of Crustal Density Structures on GOCE Gravity Gradient Observables

We investigate the gravity gradient components corrected for major known anomalous density structures within the Earth's crust. Heterogeneous mantle density structures are disregarded. The gravimetric forward modeling technique is utilized to compute the gravity gradients based on methods for a spherical harmonic analysis and synthesis of a gravity field. The Earth's gravity gradient components are generated using the global geopotential model GOCO-03s. The topographic and stripping gravity corrections due to the density contrasts of the ocean and ice are computed from the global topographic/bathymetric model DTM2006.0 (which also includes the ice-thickness dataset). The discrete data of sediments and crust layers taken from the CRUST2.0 global crustal model are then used to apply the additional stripping corrections for sediments and remaining anomalous crustal density structures. All computations are realized globally on a one arc-deg geographical grid at a mean satellite elevation of 255 km. The global map of the consolidated crust-stripped gravity gradients reveals distinctive features which are attributed to global tectonics, lithospheric plate configuration, lithosphere structure and mantle dynamics (e.g., glacial isostatic adjustment, mantle convection). The Moho signature, which is the most pronounced signal in these refined gravity gradients, is superimposed over a weaker gravity signal of the lithospheric mantle. An interpretational quality of the computed (refined) gravity gradient components is mainly limited by a low accuracy and resolution of the CRUST2.0 sediment and crustal layer data and unmodeled mantle structures.

DC cancellation as a method of generating a t2-response and of solving the radial position error in a concentric free-falling two-sphere equivalence-principle experiment in a drag-free satellite

This paper presents a new method for doing a free-fall equivalence-principle (EP) experiment in a satellite at ambient temperature which solves two problems that have previously blocked this approach. By using large masses to change the gravity gradient at the proof masses, the orbit dynamics of a drag-free satellite may be changed in such a way that the experiment can mimic a free-fall experiment in a constant gravitational field on the earth. An experiment using a sphere surrounded by a spherical shell both completely unsupported and free falling has previously been impractical because (1) it is not possible to distinguish between a small EP violation and a slight difference in the semi-major axes of the orbits of the two proof masses and (2) the position difference in the orbit due to an EP violation only grows as t whereas the largest disturbance grows as t3/2. Furthermore, it has not been known how to independently measure the positions of a shell and a solid sphere with sufficient accuracy. The measurement problem can be solved by using a two-color transcollimator (see the main text), and since the radial-position-error and t-response problems arise from the earth's gravity gradient and not from its gravity field, one solution is to modify the earth's gravity gradient with local masses fixed in the satellite. Since the gravity gradient at the surface of a sphere, for example, depends only on its density, the gravity gradients of laboratory masses and of the earth unlike their fields are of the same order of magnitude. In a drag-free satellite spinning perpendicular to the orbit plane, two fixed spherical masses whose connecting line parallels the satellite spin axis can generate a dc gravity gradient at test masses located between them which cancels the combined gravity gradient of the earth and differential centrifugal force. With perfect cancellation, the position-error problem vanishes and the response grows as t2 along a line which always points toward the earth. In the practical case where the cancellation is not perfect, the t2-response can hold for about 104 to 106 s (depending on the cancellation accuracy), and the position error is also suppressed proportional to the accuracy of the cancellation. Experience with a prior drag-free satellite indicates that this cancellation may be accomplished with an accuracy of at least 10−2 and possibly 10−4 to 10−6, and this ameliorates the measurement problem sufficiently that equivalence-principle tests with accuracies from 10−19 to 10−23 g may be possible. t2-response times between 105 and 106 s are equivalent to a very tall (0.1–10 AU) drop tower with an effective value of g equal to about 3.4 m s−2 which is 3/7 of the value of g at the orbital altitude.

Predicted performance of the superconducting gravity gradiometer on the Space Shuttle

A high-resolution global gravity map would provide tremendous benefits to the Earth sciences. Mapping the lateral fluctuations of a planet's gravity field is one of the few means of probing its interior. For the Earth, a high-resolution gravity map would also reveal dynamic ocean current information presently hidden in existing sea surface height data. The use of gradiometers for gravity mapping has inherent advantages over other techniques and has the potential of providing the highest spatial resolution for global surveys from orbit. Extremely sensitive superconducting gravity gradiometers (SGGs) have been developed for space applications under NASA support. The present laboratory model has demonstrated a sensitivity almost two orders of magnitude better than any other gradiometer (< 20 mE Hz −1 2 , where 1 mE ≡ 10 −12 m s −2 m −1 ≈ 10 −13 g E m −1). In preparation for a dedicated gravity mapping mission, we have proposed a flight test of a spaceworthy version of the instrument in the SHOOT (Superfluid Helium On Orbit Transfer) dewars, which flew on the Shuttle in 1993. We have examined potential error sources for the instrument operating in the Shuttle environment. Of the largest error sources, most are motion-related. We obtained the motion of the Orbiter in six degrees of freedom (d.o.f.) by examining the SAMS (shuttle acceleration measurement system) and the CAS (calibrated ancillary system) data sets. Using this information and the measured error coupling coefficients of the laboratory SGG, we can predict the resolution with which the Earth's gravity gradient signal can be recovered on this Shuttle test flight. Although the Orbiter provides a much worse environment for the SGG than a dedicated satellite, our analysis shows that during quiet periods on the Orbiter, the ability to recover gravity data will approach the limit imposed by the instrument thermal noise.

Performance of a superconducting gravity gradiometer

We report the laboratory operation of an rf SQUID-based superconducting gravity gradiometer designed to measure from an aircraft, off-diagonal components of the earth's gravity gradient tensor. A detailed numerical model of the multi-mode electromechanical system shows excellent agreement with the experimental observations. We have demonstrated a common mode rejection ratio to linear accelerations of 180dB, as well as 60dB of active thermal compensation below 0.01Hz. Short term measurements show the instrument to be SQUID noise limited at 0.5Eo¨/√Hz in a frequency band from 50mHz to 1Hz. After correcting for the remaining effects of thermal fluctuations and flux creep the noise below 50mHz has a 1/f characteristic which is also dominated by the SQUID.

Laboratory tests of a mobile superconducting gravity gradiometer

We describe a gravity gradiometer designed to measure off-diagonal components of the earth's gravity gradient tensor, and intended for airborne geophysical exploration. The instrument consists of an orthogonal pair of mass quadrupoles pivoted about a common vertical axis. Each quadrupole is a rectangular bar of niobium having square cross section and with integrally machined flexural micro-pivots. Superconducting niobium pancake coils form the basis of the SQUID based transducers and provide supplementary magnetic springs to enable matching of mechanical parameters.

Motions in fluids caused by microgravitational acceleration and their modification by relative rotation

The primary reason for conducting many materials science experiments in space is to minimize or eliminate undesirable effects that might result because of convective motions in fluids which are driven by buoyancy effects. Of particular interest are the low-frequency accelerations caused by the earth's gravity gradient field, spacecraft attitude motions, and atmospheric drag, since it is known that these can give rise to sustained fluid motion. In order to gain a limited understanding of the effects of these accelerations their magnitude and orientation have been examined, and the Stokes' motion of a spherical particle in a fluid has been calculated for various types of spacecraft attitude motions. In addition, the effect of slowly rotating the experimental system relative to the spacecraft (in order to increase the residence time of the particle in the neighborhood of its initial position) is assessed.

The measurement of the gravitational constant in an orbiting laboratory

We propose to measure the gravitational constantG by putting in an orbiting laboratory a known mass of very high density and by tracking the motion of a small test mass under the gravitational influence of the primary mass. We analyse the different sources of perturbation; the consideration of the Earth's gravity gradient leads us to conclude that, if the laboratory is in a low Earth orbit, we cannot get stable satellite-like orbits of the test mass, but we must study only a process of ‘gravitational scattering’. In order to maximize the time of interaction it is proposed to use the ‘practical stability’ of a collinear equilibrium point of the system Earth-primary mass, by putting the test mass as close as possible to the stable manifold of an equilibrium point. This method will allow the determination of the value ofG within a few parts over 105, as shown by some computer simulations of the experiment taking into account also some unknown perturbation and random noise.

Dynamics of a Flexible Passive Space Array

A passive space communicator in synchronous orbit will consist of small aluminum beads chained together and aligned vertically by the Earth's gravity gradient. The analysis for small deviations from vertical uses a continuous-chain model, and pitch and roll frequencies and mode shapes are described with and without the attachment of substantial end masses. An end mass with appreciable moment of inertia can provide effective damping of the lowest-order, otherwise rigid and undamped, modes. The distortion of the chain by solar radiation pressure is estimated, assuming undesirable variations in the mass-to-area ratio of beads and end masses.

Theory of an experiment in an orbiting space laboratory to determine the gravitational constant

The paper analyzes an experiment in an orbiting laboratory to determine the gravitational constantG. A massive sphere, according to a suggestion of L. S. Wilk, is to have three tunnels drilled through it along mutually perpendicular diameters. The sphere either floats in the orbiting laboratory, with its center held fixed by means of external jets issuing from the spacecraft, or is tethered to the spacecraft. In either case it is free to rotate; in the second case this freedom would be achieved by a system of gimbals. Each tunnel contains a small test object, which is held on the tunnel's axis by means of a suspension system, perhaps electrostatic, and held at rest relative to the sphere by slowly rotating the latter by means of inertia reaction wheels, governed by a servomechanism. Fundamentally, one balances the gravitational forces on the test objects by centrifugal force, determines the latter by measuring the components of angular velocity, and calculatesG from the resulting balance. It is better to use three tunnels than one because their use minimizes the effects of the Earth's gravity-gradient. Many other measurements and corrections are required. The latter arise from Earth gravity-gradient, aerodynamic drag (with the tethered sphere), gravitational forces produced by the spacecraft itself, and the force reductions produced by the empty space in all three tunnels. After the consideration of these effects there is a presentation and discussion of the equations required to reduce the observations to obtainG. There then follow the extra equations, not needed in the reduction, that are required for a computer simulation to investigate the possible extraction of a test object and to aid in designing the servomechanisms. In Appendix B, I have devised another version of the experiment, in which the sphere is kept intact, but has short thin hollow ‘vestigial tunnels’ attached to the outside of the sphere, along perpendicular diameters. These external tunnels would contain the test objects and the suspension systems. The servomechanisms would then have to prevent collision of a test object with the sphere, as well as extraction. This second method could allow for some inhomogeneities in the sphere, would require no accurate drilling, and would make the suspension systems more accessible for construction and adjustment.

CAPTURE OF A PASSIVELY STABILIZED SATELLITE BY EARTH'S GRAVITY FIELD

I is currently high in passive stabilization of spacecraft, particularly satellites of the earth. Of the several techniques of passive stabilization, use of earth's gravity field is receiving perhaps most attention. In this technique, the satellite's configuration is so arranged that torques are produced on the satellite by the gradient of earth's gravity field, restoring the satellite toward one of its equilibrium positions if displaced from equilibrium. Under the action of these torques, the satellite librates or oscillates about one of the equilibrium positions. If the satellite also contains some damping mechanism, the oscillations will decay until they fall within a small steady-state deadband about one of the equilibrium positions. The satellite is so configured that in its equilibrium position an antenna, for instance, is directed to earth. The classic configuration that is torqued by earth's gravity gradient is a dumbbell. The dumbbell is oriented in pitch and roll but not yaw, where the yaw axis is the local vertical. Another configuration, one now being widely studied and developed, consists of a central body from which long thinwalled tubes uncoil once the satellite is in orbit. Because of their great length (hundreds of feet for some applications), the tubes endow the satellite with the necessary inertia to develop from the gravity field high restoring torques. If the tubes extend from the central body to form a cross having its two members at right angles and of unequal length, restoring torques in yaw, as well as pitch and roll, are developed. Yaw restoring torques are also developed if the rods extend to form an X, that is, with two members of equal length intersecting at an angle different from 90°. A typical mission profile of such a gravity oriented satellite consists of the following: 1) Injection into orbit from a final propulsion stage either spin-stabilized such as Scout or containing its own attitude stabilization such as Agena. In the former instance the satellite, after separation from the propulsion stage, is itself spinning. In the latter instance the satellite is slowly tumbling after separation. 2) Conversion of the spinning or tumbling of the satellite to oscillation about the local vertical. Oscillation about the local vertical, when seen from inertial space, is a steady rotation of the satellite plus a superimposed oscillation. The rotation is about an axis normal to the orbital plane. The rotation is at a rate eaual to the satellite's orbital rate, wnich is much slower than the rate at injection. As seen from inertial space, then, the spinning or random tumbling at injection must be converted to a much slower rotation about an axis having a particular direction in space. 3) Decay of the satellite's oscillation. The oscillation decays to a small deadband whose width is determined by the