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Page 1: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

Gravity studies

As part of GEO-DEEP9300

Maaike Weerdesteijn

11-11-2019

Courtesy: NASA Courtesy: red-leaf Courtesy: Airbus/GFZ Courtesy: macrovector Courtesy: EHT

Page 2: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

Table of content

• History of gravity studies

• Gravity theory

• Measurement techniques

• Earth material characteristics

Page 3: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

History of gravity studies

Page 4: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

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

Page 5: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

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

Page 6: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

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

Page 7: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

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

Page 8: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

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

Page 9: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

Gravity theory

Page 10: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

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

Page 11: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

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

Page 12: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

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

Page 13: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

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

Page 14: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

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

Page 15: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

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

Page 16: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

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

Page 17: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

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

Page 18: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

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

Page 19: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

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

Page 20: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

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

Page 21: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

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

Page 22: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

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

Page 23: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

Measurement techniques

Page 24: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

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

Page 25: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

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

Page 26: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

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

Page 27: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

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)

Page 28: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

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

Page 29: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

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

Page 30: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

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

Page 31: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

Earth material characteristics

Page 32: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

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

Page 33: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

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

Page 34: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

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

Page 35: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

Gravity gradiometry

History of gravity studies Gravity theory Measurement techniques Earth material characteristics

Ebbing et al. (2018)

Page 36: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

Gravity studies

As part of GEO-DEEP9300

Maaike Weerdesteijn

11-11-2019

Courtesy: NASA Courtesy: red-leaf Courtesy: Airbus/GFZ Courtesy: macrovector Courtesy: EHT

Page 37: Gravity studies - folk.uio.nofolk.uio.no/clinton/classes_oslo/GEO-DEEP9300_H19/lectures/Gravity.pdf · History of gravity studies Gravity theory Measurement techniques Earth material

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