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PX266 Geophysics (2010/11)
Lecture 7 Handout – Earth’s Gravity and Large Scale Anomalies
Dr. Gavin Bell Variations in surface gravity of the Earth tell us something about its internal structure, at least within a few hundred km of the surface. Accurate local surveys can be made readily using gravimeters, and over the last few years accurate satellite tracking (the GRACE satellites) has provided beautiful global mapping of Earth’s sea-level equipotential surface (the geoid ) and sea-level acceleration due to gravity. We account for the major deviations from global uniformity (Earth’s rotation and shape) in the reference spheroid and reference gravity formula, and remaining deviations are quite tiny (roughly ±100m in geoid height and ±50mgal in field). In local surveys, we must make further corrections, principally for measurement height and known density structures.
Reference gravity formula g g ( 0 ) 1 sin^2 sin^4 = 5.279 10 -3 = 2.346 10 -5 g = 9.780 ms-2^ g = 9.832 ms- (equator) (poles) The potential giving rise to this surface (sea-level) field accounts for the oblate spheroid shape of the Earth and its rotation. Its sea-level equipotential surface is the reference spheroid.
Large scale gravity anomalies Our definition – the difference between measured gravity and calculated gravity based on the R.G.F. with additional corrections. In the equations below, go is the local observed value of gravity, gF is the free-air correction, gB is the Bouguer correction, and gT is the terrain correction.
Free-air anomaly g (^) F go g (^) (^) gF Our “basic” Bouguer anomaly g (^) B g (^) o g (^) (^) g (^) F gB “Full” Bouguer anomaly g (^) B g (^) o g (^) (^) gF g (^) B gT
Bouguer correction for an infinite slab g B 2 Gt
Free-air correction ^ gF^ ^ elevation^ 0.31 mgal m^1
Please note – knowing the sign convention is much less important than knowing which way corrections should work, e.g. “we are higher up so gravity is less, therefore we add a free-air correction to our measured gravity” or “we are over the ocean and water is less dense than rock, so we add a Bouguer slab correction to simulate the presence of rock where there is really water”.
Measuring gravity anomalies Option 1: trundle around the surface with a GRAVIMETER. The idea is to measure g very accurately and this can be done using a well calibrated mass-on-a-spring (measure the extension of the spring and calibrate against a known value of g ). Vibration isolation is very important! Calibration can be done against an absolute measurement of g by accurate timing of a free-falling mass in an interferometer. Relative gravimeters are readily portable so can be used in surveys.
Option 2: measure gravity from space using satellites.
“You must feel the Force around you; here, between you, me, the tree, the rock, everywhere, yes. Even between the land and the ship .”
NASA’s GRACE satellites measure tiny changes in their relative positions ~ 220 km apart as they orbit the Earth, by microwave ranging. Position changes arise from one satellite experiencing a slightly higher or lower gravitational field than the other, allowing the field to be inferred accurately.
The GRACE satellites (images from NASA web site).
GRACE maps the whole Earth field, hence the geoid, every 30 days and has been operating since 2002. Each orbit improves the accuracy of the map (see below).
The ESA’s GOCE satellite was launched in 2009 into a low orbit (260km, needs ion propulsion drive to compensate for drag). It uses 6 pairs of sensitive accelerometers configured as a “gradiometer” to directly measure gravitational changes to 1 part in
Note – the approximation of an infinite slab clearly doesn’t work so well where there is rapid lateral change of density, e.g. edge of plateau, cliffs, steep mountains, etc. The terrain correction , computed from a model structure, accounts for these effects.
Exercise – Bouguer and free-air anomalies for “plateau” density models
Assume the plateau is 500km across and 3 h = 2km, r = 1.33km. Let sub = 3.0×10^3 kgm- , 1 = 2.75×10^3 kgm-3^ and 2 = 2.85×10^3 kgm-3. Why can we ignore terrain corrections at P? Calculate the free-air and Bouguer corrections at point P for each model and give their sign (do we add or subtract gravity from the measurement?).
Further study Question 7 on a simple geopotential and derivation of its surface field. Questions 8 and 9 on accuracy and looking up time dependence in gravity surveys.
You might like to work out the derivation of gravity due to an infinite slab (in the exam this formula would be given – you don’t need to memorize it).
Make sure you have a general idea how gravity measurements are made (gravimeters and satellites). Make sure you understand how to use the free-air and Bouguer slab corrections (especially what value of or to use). We only consider simple slab-like geometries in this course.
2 sub
h
Density Model B
r
1 sub
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Density Model A P
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