Gravity gradiometry studies variations (anomalies) in the Earth's gravity field by measuring the spatial gradient of gravitational acceleration. This gradient is described by a 3x3 tensor (the gravity gradient tensor), represented by the Jacobian matrix of the acceleration vector. Gravity gradiometry has units of s−2, reflecting its dimension of reciprocal time squared. It is widely used by mineral and oil prospectors to detect subsurface (bedrock) density anomalies, aiding in targeting deposits. The technique also assists in mapping the water column density for bathymetry (water depth) and supports inertial navigation through gravity compensation using gravimeters.
Gravity gradient
Gravity measurements are a reflection of the earth's gravitational attraction, its centripetal force, tidal accelerations due to the sun, moon, and planets, and other applied forces. Gravity gradiometers measure the spatial derivatives of the gravity vector. The most frequently used and intuitive component is the vertical gravity gradient, Gzz, which represents the rate of change of vertical gravity (gz) with height (z). It can be deduced by differencing the value of gravity at two points separated by a small vertical distance, l, and dividing by this distance.
G z z = ∂ g z ∂ z ≈ g z ( z + ℓ 2 ) − g z ( z − ℓ 2 ) ℓ {\displaystyle G_{zz}={\partial g_{z} \over \partial z}\approx {g_{z}{\bigl (}z+{\tfrac {\ell }{2}}{\bigr )}-g_{z}{\bigl (}z-{\tfrac {\ell }{2}}{\bigr )} \over \ell }}The two gravity measurements are provided by accelerometers which are matched and aligned to a high level of accuracy.
Units
The unit of gravity gradient is the eotvos (abbreviated as E), which is equivalent to 10−9 s−2 (or 10−4 mGal/m). A person walking past at a distance of 2 metres would provide a gravity gradient signal approximately one E. Mountains can give signals of several hundred Eotvos.
Gravity gradient tensor
Full tensor gradiometers measure the rate of change of the gravity vector in all three perpendicular directions giving rise to a gravity gradient tensor (Fig 1).
Let V {\displaystyle V} be the gravitational potential field (defined up to an additive constant). The gravitational field vector field is − ∇ V {\displaystyle -\nabla V} (more properly, it is a differential one-form), and the gravity gradient tensor field is the second derivative Γ := − ∇ 2 V {\displaystyle \Gamma :=-\nabla ^{2}V} , a differential two-form.
In general, a differential two-form in R 3 {\displaystyle \mathbb {R} ^{3}} has 9 free variables, but because − ∇ 2 V {\displaystyle -\nabla ^{2}V} is symmetric, it has only 6 free variables. Furthermore, by the Poisson equation, Tr Γ = 4 π G ρ {\displaystyle \operatorname {Tr} \Gamma =4\pi G\rho } , so in free space, Tr Γ = 0 {\displaystyle \operatorname {Tr} \Gamma =0} , resulting in only 5 free variables. In particular, this means that when the equipment performing the gradiometry is in air or vacuum, which is almost always the case, the full gravity gradient tensor Γ {\displaystyle \Gamma } requires measuring only 5 numbers.1
Comparison to gravity
Being the derivatives of gravity, the spectral power of gravity gradient signals is pushed to higher frequencies. This generally makes the gravity gradient anomaly more localised to the source than the gravity anomaly. The table (below) and graph (Fig 2) compare the gz and Gzz responses from a point source.
| Gravity (gz) | Gravity gradient (Gzz) | |
|---|---|---|
| Signal | G M z ( r 2 + z 2 ) 3 / 2 × 10 5 [ mGal ] {\displaystyle {GM\,z \over (r^{2}+z^{2})^{3/2}}\times 10^{5}\;\left[{\text{mGal}}\right]} | G M ( r 2 − 2 z 2 ) ( r 2 + z 2 ) 5 / 2 × 10 9 [ E ] {\displaystyle {GM(r^{2}-2z^{2}) \over (r^{2}+z^{2})^{5/2}}\times 10^{9}\;\left[{\text{E}}\right]} |
| Peak signal (r = 0) | G M z 2 × 10 5 {\displaystyle {GM \over z^{2}}\times 10^{5}} | 2 G M z 3 × 10 9 {\displaystyle {2GM \over z^{3}}\times 10^{9}} |
| Full width at half maximum | 1.53 z {\displaystyle 1.53\,z} | ≈ z {\displaystyle \approx z} |
| Wavelength (λ) | 3.07 z {\displaystyle 3.07\,z} | 2 z {\displaystyle 2\,z} |
Conversely, gravity measurements have more signal power at low frequency therefore making them more sensitive to regional signals and deeper sources.
Dynamic survey environments (airborne and marine)
The derivative measurement sacrifices the overall energy in the signal, but significantly reduces the noise due to motional disturbance. On a moving platform, the acceleration disturbance measured by the two accelerometers is the same so that when forming the difference, it cancels in the gravity gradient measurement. This is the principal reason for deploying gradiometers in airborne and marine surveys where the acceleration levels are orders of magnitude greater than the signals of interest. The signal to noise ratio benefits most at high frequency (above 0.01 Hz), where the airborne acceleration noise is largest.
Applications
Gravity gradiometry has predominately been used to image subsurface geology to aid hydrocarbon and mineral exploration. Over 2.5 million line km has now been surveyed using the technique.2 The surveys highlight gravity anomalies that can be related to geological features such as Salt diapirs, Fault systems, Reef structures, Kimberlite pipes, etc. Other applications include tunnel and bunker detection3 and the recent GOCE mission that aims to improve the knowledge of ocean circulation.
Gravity gradiometers
Lockheed Martin gravity gradiometers
During the 1970s, as an executive in the US Dept. of Defense, John Brett initiated the development of the gravity gradiometer to support the Trident 2 system. A committee was commissioned to seek commercial applications for the Full Tensor Gradient (FTG) system that was developed by Bell Aerospace (later acquired by Lockheed Martin) and was being deployed on US Navy Ohio-class Trident submarines designed to aid covert navigation. As the Cold War came to a close, the US Navy released the classified technology and opened the door for full commercialization of the technology. The existence of the gravity gradiometer was famously exposed in the film The Hunt for Red October released in 1990.
There are two types of Lockheed Martin gravity gradiometers currently in operation: the 3D Full Tensor Gravity Gradiometer (FTG; deployed in either a fixed wing aircraft or a ship) and the FALCON gradiometer (a partial tensor system with 8 accelerometers and deployed in a fixed wing aircraft or a helicopter). The 3D FTG system contains three gravity gradiometry instruments (GGIs), each consisting of two opposing pairs of accelerometers arranged on a spinning disc with measurement direction in the spin direction.
Other gravity gradiometers
Electrostatic gravity gradiometer This is the gravity gradiometer deployed on the European Space Agency's GOCE mission. It is a three-axis diagonal gradiometer based on three pairs of electrostatic servo-controlled accelerometers. ARKeX Exploration gravity gradiometer An evolution of technology originally developed for European Space Agency, the Exploration Gravity Gradiometer (EGG), developed by ARKeX (a corporation that is now defunct), uses two key principles of superconductivity to deliver its performance: the Meissner effect, which provides levitation of the EGG proof masses and flux quantization, which gives the EGG its inherent stability. The EGG has been specifically designed for high dynamic survey environments. Ribbon sensor gradiometer The Gravitec gravity gradiometer sensor consists of a single sensing element (a ribbon) that responds to gravity gradient forces. It is designed for borehole applications. UWA gravity gradiometer The University of Western Australia (aka VK-1) Gravity Gradiometer is a superconducting instrument which uses an orthogonal quadrupole responder (OQR) design based on pairs of micro-flexure supported balance beams. Gedex gravity gradiometer The Gedex gravity gradiometer (AKA High-Definition Airborne Gravity Gradiometer, HD-AGG) is also a superconducting OQR-type gravity gradiometer, based on technology developed at the University of Maryland. iCORUS gravity gradiometer The iCORUS gravity gradiometer is a strapdown airborne gravity gradiometer, based on technology developed at iMAR Navigation in Germany. Quantum Technology gravity gradiometers Quantum Technology gravity gradiometers based on atom interferometry are currently under development by a number of universities world wide and are beginning to be used in practical applications.4See also
- Gravity-gradient stabilization – Method for the stabilization and the orientation of various spacecraft
- Gravity map
- Robert L. Forward#Forward Mass Detector – American physicist and writer (1932–2002)
- Accelerometer – Device that measures proper acceleration
- Equivalence principle – Hypothesis that inertial and gravitational masses are equivalent
- Gravimetry – Measurement of the strength of a gravitational field
- Eotvos (unit) – Unit for gravitation gradients
- Instrumentation – Measuring instruments which monitor and control a process
- Pedersen, L. B.; Rasmussen, T. M. (December 1990). "The gradient tensor of potential field anomalies: Some implications on data collection and data processing of maps". Geophysics. 55 (12): 1558–1566. Bibcode:1990Geop...55.1558P. doi:10.1190/1.1442807. ISSN 0016-8033.
- Jekeli, Christopher (2020), "Gravity, Gradiometry", Encyclopedia of Solid Earth Geophysics, Encyclopedia of Earth Sciences Series, Springer, Cham, pp. 1–18, doi:10.1007/978-3-030-10475-7_80-1, ISBN 978-3-030-10475-7
- Veryaskin, Alexey V. (2018). "1. Gravity gradiometry". Gravity, Magnetic and Electromagnetic Gradiometry: Strategic Technologies in the 21st Century. IOP Concise Physics Ser. San Rafael: Morgan & Claypool Publishers. ISBN 978-1-68174-700-2.
External links
- Advances and Challenges in the Development and Deployment of Gravity Gradiometer Systems Archived 2015-06-26 at the Wayback Machine
- GOCE mission payload
References
Jekeli, Christopher (2011), "Gravity, Gradiometry", Encyclopedia of Solid Earth Geophysics, Encyclopedia of Earth Sciences Series, Springer, Dordrecht, pp. 547–561, doi:10.1007/978-90-481-8702-7_80, ISBN 978-90-481-8702-7 978-90-481-8702-7 ↩
Gravity Gradiometry Today and Tomorrow (PDF), South African Geophysical Association, archived from the original (PDF) on 2011-02-22, retrieved 2011-06-27 https://web.archive.org/web/20110222012752/http://www.sagaonline.co.za/2009Conference/CD%20Handout/SAGA%202009/PDFs/Abstracts_and_Papers/difrancesco_paper1.pdf ↩
Using Gravity to Detect Underground Threats, Lockheed Martin, archived from the original on 2013-06-03, retrieved 2013-06-14 https://web.archive.org/web/20130603074229/http://www.lockheedmartin.com/us/mst/features/2010/100714-using-gravity-to-detect-underground-threats-.html ↩
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