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Gravity surveying...

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m1m2
!!!
F =G
2
r
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Gravity - Truong Quoc Thanh

8/17/2016


Introduction
Gravity surveying…
 Investigation on the basis of relative variations in the

Earth gravitational field arising from difference of
density between subsurface rocks

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Application
 Exploration of fossil fuels (oil, gas, coal)
 Exploration of bulk mineral deposit (mineral, sand, gravel)
 Exploration of underground water supplies
 Engineering/construction site investigation


 Cavity detection
 Glaciology
 Regional and global tectonics
 Geology, volcanology
 Shape of the Earth, isostasy
 Army

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Structure of lecture
1. Basic theory
2. Density of rocks
3. Gravity of the Earth
4. Measurement of gravity
5. Gravity reduction
6. Gravity anomalies and interpretation
7. Microgravity: a case history
8. Conclusions

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1. Basic Theory

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Gravity?...Rocks?...Huh?
 We all feel and see gravity’s effects:
Weight (i.e. a force) & downward acceleration

 The gravitational force depends on the rocks below us
If the rocks beneath the surface change with location, we

expect that the gravitational force we experience will also
change.
If we measure small changes in gravitational forces we
can use this information to make inferences about the
rocks below us.
 This is the essence of gravity surveying
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Newton’s Law of Gravitation
 Sir Isaac Newton (1642-1727):

Consider two point masses that are a distance r apart.
Newton’s theory of gravitation predicts that they will attract each
other with a force F that is given by:
F =G
m1

force

r

m1m2
r2

force

m2

The quantity G is called the gravitational constant (or “big G”) but
is actually a small number. Newton deduced this equation from
observing the motion of planets and moons in the solar system. The
units are as follows:
F
Newton (N) r
metres (m)
m
kg
G = 6.67 x 10-11

Nm2kg-2
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Gravity - Truong Quoc Thanh

8/17/2016


Newton’s Law of Gravitation
 Newton’s law applies to point sources of mass
m1

force

r

force

m2

 Earth isn’t a point source
 Non-point sources are treated as the sum of the forces of many

small parts of the body

 Because force is a vector, the vector sum must be calculated
 This can become computationally intense for odd shapes

 For some simple geometric shapes the result is simple…
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Gravity - Truong Quoc Thanh

8/17/2016


Newton’s Law of Gravitation
 For the gravitational attraction due to a hollow shell or

uniform sphere:
 The force is the same as that of a point source of the same

mass located at the center of the sphere
 This is true only outside the sphere.
 At the center, the gravitational force must be zero (vectors all cancel

due to symmetry)

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Gravity - Truong Quoc Thanh

Because Earth is approximately spherically symmetrical
(recall this from the seismology chapter?), we can treat
the Earth as a point source of mass located at the center
of the Earth!

8/17/2016



Acceleration Due to Gravity
 Because Earth acts like a point source of mass…

 We can use Newton’s universal gravitation eq to calculate gravitational force

on a small mass, ms, on the surface of the Earth

F =G

M E ms
2
RE

To consider acceleration…
 Insert Newton’s second law to solve for the force on the small body…

F = ms g = G

M E ms
2
RE

 The small body's mass cancels out, and we are left with the equation for the

acceleration due to gravity, g.

g =G

ME
2

RE

 This relationship implies that all bodies on earth’s surface should experience

the same falling acceleration
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Gravity - Truong Quoc Thanh

8/17/2016


Acceleration Due to Gravity
 Let us now consider that m1 is the Earth and m2 is a small

object that we are going to drop. This equation tells us that the
acceleration does not depend on the mass of the object
being dropped
 This was proved by Galileo who allegedly dropped masses
from the leaning tower of Pisa in Italy. This result says that a
small mass and a large mass will fall with the same
acceleration.

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Gravity - Truong Quoc Thanh

8/17/2016



Acceleration Due to Gravity
 Because Earth acts like a point source of mass…

So long as the mass of the Earth is spherically-symmetrical…
 The acceleration due to gravity is independent of an object’s properties (mass,
etc…)
 So far as gravity is concerned, the Earth could be:
 Hollow in the middle
 Made of cheese and gold

(So long as the mass was held constant)

 We know that Earth isn’t exactly spherically-symmetrical…
 E.g. lateral changes in rock type
 Topography
 Mantle plumes

 We should therefore be able to measure small changes in g and determine

subsurface rock properties that involve variations in mass.
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The mass of Earth
 Although the mass of Earth is not practical to measure, it


can be easily calculated:

g =G

ME
2
RE

 G = universal gravitational constant = 6.672 x 10-11 m3kg-1s-2
 g = 9.81 ms-2 (can be measured for falling objects)
 RE = can be measured from surveys or geometric observations
 Eratosthenes (~200 B.C.) ~ 6370 km

 ME = 5.97 x 1021 Mg or 5.97 x 1024 kg

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Gravity - Truong Quoc Thanh

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Units of gravity
• 1 gal
= 10-2 m/s2
• 1 mgal = 10-3 gal = 10-5 m/s2
• 1 µgal = 10-6 gal = 10-8m/s2 (precision of a
gravimeter for geotechnical surveys)
• Gravity Unit: 10 gu = 1 mgal
• Mean gravity around the Earth: 9.81 m/s2 or

981000 mgal

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8/17/2016


Keep in mind…
…that in environmental geophysics, we are
working with values about…

0.01-0.001 mgal ≈10-8 - 10-9 g !!!

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2. Density of rocks

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Densities of Common Rocks
 Rocks have a range of densities*
 But in general, rock density does not widely vary
 What does this say about typical gravity anomaly sizes?
Type

Rock

Density

Unconsolidated

Sand

1400-1650 kg/m3

Sedimentary

Salt

2100-2600

Limestone

2000-2700

Shale

2000-2700


Granite

2500-2800

Basalt

2700-3000

Quartzite

2600-2700

Gneiss

2600-3000

Galena

7400-7600

Pyrite

4900-5200

Magnetite

4900-5300

Igneous

Metamorphic
Ore

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Densities of Common Rocks
Type

Rock

Density

Unconsolidated

Sand

1400-1650
kg/m3

Sedimentary

Salt

2100-2600


 Rocks in the phreatic / vadose

Limestone

2000-2700

zone
 Fractured / unfractured rocks

Shale

2000-2700

Granite

2500-2800

Basalt

2700-3000

Quartzite

2600-2700

Gneiss

2600-3000

Galena


7400-7600

Pyrite

4900-5200

Magnetite

4900-5300

 Which has higher / lower density?
 Surface / deep rocks
 Weathered / unweathered rocks

• Which rock types make good
targets for gravity surveying?
• Which don’t
• Why

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Gravity - Truong Quoc Thanh

Igneous

Metamorphic

Ore


8/17/2016


Densities of Common Rocks
 Igneous

rocks
are
generally
more
dense
owing to minimal porosity.
 Why Mafic rocks are more
dense than felsic rocks?

Type

Rock

Density

Unconsolidated

Sand

1400-1650
kg/m3

Sedimentary


Salt

2100-2600

Limestone

2000-2700

Shale

2000-2700

Granite

2500-2800

Basalt

2700-3000

Quartzite

2600-2700

Gneiss

2600-3000

Galena


7400-7600

Pyrite

4900-5200

Magnetite

4900-5300

Igneous

Metamorphic

Ore

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Gravity - Truong Quoc Thanh

8/17/2016


Densities of Common Rocks
 Variation in porosity is the main cause of density variation in

sedimentary rocks. Thus, in sedimentary rock sequences, density
tends to increase with depth, due to compaction, and with age, due
to progressive cementation.
 Most igneous and metamorphic rocks have negligible porosity, and

composition is the main cause of density variation. Density
generally increases as acidity decreases; thus there is a progression
of density increase from acid through basic to ultrabasic igneous
rock types.

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Gravity - Truong Quoc Thanh

8/17/2016


Densities of Common Rocks
 Density is commonly determined by direct measurements

on rock samples. A sample is weighed in air and in water.
The difference in weights provides the volume of the
sample and so the dry density can be obtained. If the rock
is porous the saturated density may be calculated by
following the above procedure after saturating the rock
with water.
 The density of any particular rock type can be quite
variable. Consequently, it is usually necessary to measure
several tens of samples of each particular rock type in
order to obtain a reliable mean density and variance.
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3. Gravity of the Earth

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Gravity - Truong Quoc Thanh

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3.1 Shape of the Earth: spheroid
• Spherical Earth with R = 6371 km is an approximation!
• Rotation creates an ellipsoid or a spheroid

Re − R p
Re

1
=
298

Re= 6378 km and Rp= 6357 km. The ratio of flattening is
approximately 1/298. Since a point on the Equator is further from
the center of the Earth than the poles, gravity will be slightly
stronger at the Equator.
If we move from pole to the equator, the gravity will
decrease 6480 mgal
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Gravity - Truong Quoc Thanh

8/17/2016


3.2 Centrifugal acceleration and potential
The rotation of the Earth also causes gravity to vary.
Imagine you are standing at the North Pole. The rotation of
the Earth will not change g, all that will happen is that you
rotate once a day.
Now imagine you are at the Equator. If we could
increase the rotation rate of the Earth enough, you would be
ultimately be thrown into space (i.e. become weightless).
Thus rotation makes gravity weaker at the equator.

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Gravity - Truong Quoc Thanh

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3.2 Centrifugal acceleration and potential
𝐦𝐦𝐯𝐯 𝟐𝟐
𝐦𝐦𝐦𝐦 =
𝐝𝐝

At equator d = R = 6378km
V = ω 2 R2
=> g = ω2R

c

If we move from pole to the equator, the gravity will
decrease 3373 mgal
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Gravity - Truong Quoc Thanh

8/17/2016


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