Unformatted text preview: 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 = 5.27910-3 = 2.34610-5 g= 9.780 ms-2
(equator) g= 9.832 ms-2
(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
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 g F Our “basic” Bouguer anomaly g B go g g F g B “Full” Bouguer anomaly g B g o g g F g B gT Bouguer correction for an infinite slab g B 2 Gt Free-air correction g F 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 1013. This translates to measuring ~100 km wide gravity anomalies to an accuracy of
1 mgal and similar sized geoid height anomalies to 2 cm.
Example Free-air and Bouguer slab corrections
Measuring at sea
There is no free-air correction in the case below: measurements are at sea level. At
sea, we know there is a slab of relatively low density water between the measuring
point (a ship) and the unknown density structures (beneath the sea bed / land) in
which we are interested. Gravity is a little bit weaker than expected and so the
Bouguer correction adds on some gravity to the measured value according to g B 2 Gd rock water Land Measuring at sea rock water d unknown Measuring on a plateau
The free-air correction adds to measured gravity (measured is lower than at sea level
due to increased height). However, there is a slab of rock of known density between
us and sea level not accounted for by the free air correction which increases gravity,
so the Bouguer correction is negative. The density of air is negligible. g B 2 Gd rock g F h d 0.31 mgal m1 Measuring on plateau
Land height h
above sea level rock d unknown
Note – in the exam, it’s easy to make a ‘mistake’ with the sign of a gravity correction:
as long as you explain qualitatively what to expect and how you have corrected you
will get the credit. 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
Density Model A P 1 h sub P Density Model B h sub 2 r Assume the plateau is 500km across and h = 2km, r = 1.33km. Let sub = 3.0×103kgm3
, 1 = 2.75×103kgm-3 and 2 = 2.85×103kgm-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
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. ...
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This note was uploaded on 09/12/2011 for the course ECON 102 taught by Professor Gavinbell during the Spring '11 term at LSE.
- Spring '11