gravity studies -...
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Gravity studies
As part of GEO-DEEP9300
Maaike Weerdesteijn
11-11-2019
Courtesy: NASA Courtesy: red-leaf Courtesy: Airbus/GFZ Courtesy: macrovector Courtesy: EHT
Table of content
• History of gravity studies
• Gravity theory
• Measurement techniques
• Earth material characteristics
History of gravity studies
The first theories: Newton
• Gravity field reflects mass distribution and shape of the Earth
• Newton: shape of the Earth is an oblate body which had swollen in the direction of the equator
History of gravity studies Gravity theory Measurement techniques Earth material characteristics
1642-1727
The first theories: J. Cassini
• J. Cassini: shape of the Earth is longer along the north-south axis based on triangulation survey in France
• Curvature of the Earth from the distance and latitude difference between the end points of a meridian arc
History of gravity studies Gravity theory Measurement techniques Earth material characteristics
Pallikaris et al. (2009)
1677-1756
Newton vs. J. Cassini
• The French Academy of Sciences sent out a mission to find the truth• Bouguer to the equator in Ecuador
• Maupertuis to the pole in Lapland
• Meridian arc measurements close to the equator and close to the pole
At pole
- Meridian arc longer for fixed latitude difference
- Smaller curvature: Earth flattened at poles
• The Earth is flattened at the poles: Newton was right
History of gravity studies Gravity theory Measurement techniques Earth material characteristics
The first gravity measurements
• Huygens: Dutch geophysicist
• Invention of precise clock pendulum for gravity measurements• Pendulum has same period when
hung from its center of oscillation as when hung from its pivot
• Distance between the two points was equal to the length of a simple gravity pendulum of the same period
• Acceleration of gravity function of pendulum’s period, length, and amplitude
History of gravity studies Gravity theory Measurement techniques Earth material characteristics
1629-1695
The first seaborne gravity measurements
• Previous pendulum required stable platform
• Prior to 1920: only continental measurements
• 73% of Earth’s gravity field unknown
• Vening Meinesz: Dutch geophysicist / geodesist• Invention of gravimeter with multiple pendulums
• Mean periods of two pendulums
• The mean not affected by horizontal disturbances
• Seaborne gravity measurements
• Increased Earth coverage
Courtesy: Utrecht University archive
History of gravity studies Gravity theory Measurement techniques Earth material characteristics
1887-1966
Gravity theory
Gravitational attraction
• Newton’s law of gravitation• 𝐺 = 6.673 · 1011 Nm2kg-2
• 𝐅𝟏 = −𝐺𝑚1𝑚2
𝑟212 𝐞𝟐𝟏
• Newton’s second law of motion• 𝐅𝟏 = 𝑚1𝐚𝟏
• Acceleration of 𝑚1 due to itsattraction by 𝑚2
• 𝐚𝟏 = −𝐺𝑚2
𝑟212 𝐞𝟐𝟏
• Acceleration of attracted point mass is independent of its mass
• Gravitational field 𝐠 𝐫
• Gauss’s law: 𝛷 = −4𝜋𝐺𝑀, 𝑀 = σ𝑖𝑚𝑖
• Gravitational field of a spherically symmetric body
• 𝐠 𝐫 = −𝐺𝑀𝐫
𝐫 3
History of gravity studies Gravity theory Measurement techniques Earth material characteristics
Gravitational potential
History of gravity studies Gravity theory Measurement techniques Earth material characteristics
• Gravitation is a vector field: 𝐠 𝐫 = −𝐺𝑀𝐫
𝐫 3
• Gravitational potential: 𝛻V 𝐫 = 𝐠 𝐫 V 𝐫 =𝐺𝑀
𝐫
• The gravitational potential at point P VP is the work done to bring a unit mass from infinity to P
• On a gravitational equipotential surface the gravitational potential VP is constant
Courtesy: physbot
A rotating Earth: centrifugal potential
History of gravity studies Gravity theory Measurement techniques Earth material characteristics
Gravitation ≠ gravity!
• Acceleration of gravity = gravitational acceleration + centrifugal acceleration
• 𝐠 𝐫 = 𝐚𝐠𝐫𝐚𝐯 𝐫 + 𝐚𝐜𝐞𝐧𝐭 𝐫
• 𝐚𝐜𝐞𝐧𝐭 𝐫 = ω2𝐩 𝐫
• Centrifugal potential
• 𝛻Z 𝐫 = 𝐚𝐜𝐞𝐧𝐭 𝐫 Z 𝐫 =𝜔2
2𝐩2
Courtesy: P. Ditmar
Gravity potential
History of gravity studies Gravity theory Measurement techniques Earth material characteristics
• Gravity potential = gravitational potential + centrifugal potential
• W = V + Z
• Total acceleration of a mass at the Earth
• 𝐠 𝐫 = 𝛻W 𝐫
Equipotential surfaces and geoid
History of gravity studies Gravity theory Measurement techniques Earth material characteristics
• Vertical direction of gravity at a point: plumb line, unit vector 𝐧
• Constant W: equipotential surface
• Surface of the oceans approximately coincides with an equipotential surface
• Mean sea is an surface equipotential surface: geoid
Courtesy: P. Ditmar
Finding the geoid on land
History of gravity studies Gravity theory Measurement techniques Earth material characteristics
• Geoid coincides with mean sea surface, but how about on land?
• Orthogonal trajectory to the equipotential surface: line of force
• Gravity vector is tangential to line of force
• Distance H along a line of force: from point P at Earth’s surface to the geoid
• Orthometric height
Courtesy: P. Ditmar
Reference ellipsoid
• Geoid surface W 𝐫 can be approximated by an ellipsoid of revolution
• Ellipsoid level surface: reference ellipsoid
• Difference between geoid and ellipsoid surface: geoid height N
• Approximate gravity potential such that ellipsoid is equipotential surface
• Normal gravity potential U 𝐫
• 𝛻U 𝐫 = 𝛄(𝐫): normal gravity vector
EGM96 model
History of gravity studies Gravity theory Measurement techniques Earth material characteristics
Geoid heights and deflections of the vertical
• Point P above reference ellipsoid
• Normal projection of point P on ellipsoid: point Q
• Distance between point P and Q: ellipsoidal height h
• Deviation between plumb line and
ellipsoidal normal: deflection of the vertical• ξ: deflection in North-South direction
• η: defection in East-West direction
History of gravity studies Gravity theory Measurement techniques Earth material characteristics
Courtesy: P. Ditmar
Disturbing potential
• Relation between the geoid height N, the orthometric height H and the ellipsoidal height h: 𝑁 = ℎ − 𝐻
• Difference between gravity potential at geoid W and at ellipsoid U• Disturbing or anomalous potential T: T 𝐫 = W 𝐫 − U 𝐫
• T can be related to geoid height N: 𝑁 =𝑇
𝛾is Bruns formula
• Decomposition of gravity field W into normal field U and anomalous field T practical• U is large but can be described by very
limited number of parameters
• T is irregular but small:
linear approximation often sufficient
History of gravity studies Gravity theory Measurement techniques Earth material characteristics
Courtesy: P. Ditmar
Gravity disturbance and gravity anomaly
• Gravity disturbance vector: 𝛿𝐠 = 𝐠 − 𝛄
• Gravity disturbance: 𝛿𝑔 = 𝐠 − 𝛄 = 𝑔 − 𝛾
• 𝛻T 𝐫 = 𝛿𝐠 𝐫
• 𝛿𝑔 ≈ −𝜕𝑇
𝜕𝑛
• Obtaining gravity disturbance practically• 𝐠 : measured• 𝛄 : computed• Precise ellipsoidal height needs to be known
• Nowadays: from GPS
• Before GPS: computation of gravity anomalies
History of gravity studies Gravity theory Measurement techniques Earth material characteristics
Courtesy: P. Ditmar
Gravity disturbance and gravity anomaly
• Gravity anomaly: Δ𝑔 = 𝑔𝑃 − 𝛾𝑄
= 𝑔𝑃 − 𝛾𝑃 + 𝛾𝑃 − 𝛾𝑄
= 𝛿𝑔𝑃 + 𝛾𝑃 − 𝛾𝑄
• After derivations I won’t bore you with…
• Spherical approximation of fundamental equation of physical geodesy:
•𝜕𝑇
𝜕𝑟+
2
𝑅𝑇 + Δ𝑔 = 0: gravity anomalies Δ𝑔 disturbing potential T geoid heights N
History of gravity studies Gravity theory Measurement techniques Earth material characteristics
Courtesy: P. Ditmar“physical”difference
“geometrical”difference
Gravity anomalies in geophysics:free-air anomaly
History of gravity studies Gravity theory Measurement techniques Earth material characteristics
• Gravity anomalies determination: gravity field must be measured at geoid
• Gravity measured at point with orthometric height 𝐻
• Neglect gravity disturbance with height
• Gravity disturbance at observation point: 𝑔𝐻 − 𝛾𝐻 = 𝑔𝑃 − 𝛾𝑃
• From previous slide: Δ𝑔 = 𝛿𝑔𝐻 + 𝛾𝑃 − 𝛾𝑄
• Combine: Δ𝑔 = Δ𝑔𝐹 = 𝑔𝐻 − 𝛾𝑄 +𝜕γ
𝜕𝑛𝐻
= 𝑔𝐻 − 𝛾𝑄 + 3.086 · 10−6𝐻
Courtesy: P. Ditmar
Gravity anomalies in geophysics:Bouguer correction
History of gravity studies Gravity theory Measurement techniques Earth material characteristics
• Free-air anomaly: sensitive to topography near observation point
• Bouguer correction: applied to suppress topography influence
• Remaining signal related to underground density variations
Courtesy: F. Chambat
• Simple correction: relief is approximated by an infinite layer with thickness equal to the orthometric height H of measurement point.
• Complete correction: takes into account the terrain height variations relatively to the adopted Bouguer plate using a DTM
Measurement techniques
Influence of natural phenomena on gravity
Tomoda (2010)
History of gravity studies Gravity theory Measurement techniques Earth material characteristics
• 1 gal = 1 cm s-2
Ground-based measurements: absolute
• Measures acceleration of object freely falling in vacuum
• Falling object equipped with reflector
• Motion monitored by laser interferometer
• < 5 microgal standard deviation
• Compact dimensions
• Contains moving parts:
• Labour-, time-, and power-consuming
History of gravity studies Gravity theory Measurement techniques Earth material characteristics
Courtesy: Scintrex CG-6 Autograv
Ground-based measurements: relative
• Spring: precisely measures weight of a proof mass• Gravimeter needs to be installed such that vertical axis is parallel to plumb line• Major drawback: large instrumental drift
• Super-conducting: super-conducting test mass levitating in magnetic field• Weak vertical gradient: small gravity changes large displacements• Displacements measured by capacitive displacement sensor• Most sensitive and stable instrument
• Moving platform: gravimetric data supplemented with measurements of kinematic platform accelerations• Derived from GPS data• High-quality receiver essential on marine and airborne platforms
History of gravity studies Gravity theory Measurement techniques Earth material characteristics
Satellite measurements
• Low Earth Orbit (LEO) High resolution
• Near-polar orbit good Earth coverage
• Equation of motion of a satellite: ሷ𝐫 = 𝐠 + 𝐚𝐧𝐠
• Gravitation can be found if ሷ𝐫 and 𝐚𝐧𝐠 are known• ሷ𝐫: from precise orbit determination (POD) using GPS
• 𝐚𝐧𝐠: from on-board accelerometers
History of gravity studies Gravity theory Measurement techniques Earth material characteristics
Rieser et al. (2010)
Satellite measurements: CHAMP
• CHAMP: Challenging Minisatellite Payload
• Launched in 2000, died in 2010
• Gravity and magnetic field measurements
• Boom accommodates magnetometers• Far away to not be affected by onboard
instrumentation
History of gravity studies Gravity theory Measurement techniques Earth material characteristics
Courtesy: Astrium GmbH
Satellite measurements: GOCE
• GOCE: Gravity Field and Steady-State Ocean Explorer
• Launched in 2009, died in 2013
• Gravity gradiometry: measuring spatial gradients• Field measured at 2 closely located points
• Drag-free control system• measurements of accelerometers used to control ion
thrusters non-gravitational forces acting in along-track direction compensated almost entirely
• Nearly free-fall motion of satellite maintained
History of gravity studies Gravity theory Measurement techniques Earth material characteristics
Courtesy: ESA
Satellite measurements: GRACE and GRACE-FO
• GRACE(-FO): Gravity Recovery And Climate Experiment (Follow-On)
• Launched in 2002, died in 2017: designed lifespan of only 5 years!
• Successor GRACE-FO launched in 2018
History of gravity studies Gravity theory Measurement techniques Earth material characteristics
Courtesy: Airbus/GFZ
• Twin satellites in the same orbit 220 km apart
• Distance between satellites continuously measured• Distance changes due to acceleration from both
satellites by gravity anomalies
• GRACE allows for quantifying mass transport in the Earth system, mainly on the Earth’s surface: GIA and climate change related
Earth material characteristics
Spectral representation of Earth’s gravitational potential• Spherical harmonics
• 𝑉 𝑟, θ, λ = 𝑓(𝑟, θ, λ, 𝐺𝑀, 𝑅,𝑚, 𝑙, 𝐶𝑙𝑚 , 𝑆𝑙𝑚)
History of gravity studies Gravity theory Measurement techniques Earth material characteristics
Spectral analysis: lithosphere
History of gravity studies Gravity theory Measurement techniques Earth material characteristics
• Ground-based and satellite measurements
• Low degrees 1-9 removed from data in right figures
Root (2017)
• Ground-based: high degrees dominate observations
• Satellite height: structures that correlate with lithosphere appear
Spectral analysis: mantle viscosity
History of gravity studies Gravity theory Measurement techniques Earth material characteristics
• Glacial isostatic adjustment and Earth’s viscosity structure from GRACE
• For a “known” ice history and with geoid rates from GRACE: estimation of mantle viscosity and lithospheric thickness
• No joint inversions:
• Mantle viscosity within 30-40%
• Lithospheric thickness within 15-20%
• Joint inversions: improved estimates
Gravity gradiometry
History of gravity studies Gravity theory Measurement techniques Earth material characteristics
Ebbing et al. (2018)
Gravity studies
As part of GEO-DEEP9300
Maaike Weerdesteijn
11-11-2019
Courtesy: NASA Courtesy: red-leaf Courtesy: Airbus/GFZ Courtesy: macrovector Courtesy: EHT
References
• Pallikaris, A., et al. (2009). New meridian arc formulas for sailing calculations in navigational GIS. International Hydrographic Review.
• Tomoda, Y. (2010). Graviy at sea – A memoir of a marine geophysicist. Proceedings of the Japan Academy, Series B Physical and Biological Sciences, 86, 769-787.
• Rieser, D., et al. (2010). Refining regional gravity field solutions with GOCE gravity gradients for cryospheric investigations. Proceedings ESA Living Planet Symposium. ESA SP-686.
• Ebbing, J., et al. (2018). Earth tectonics as seen by GOCE - Enhanced satellite gravity gradient imaging. Scientific Reports.
• Root, B. (2017). Gravity field constraints on the upper mantle of Northwestern Europe. PhD dissertation. Delft University of Technology