gravity method of exploration

1. MTU
College of Natural and Computational
Science
Geology Department
Course: Geophysics(Geol3091)

2. LECTURE OUTLINES
CHAPTER 1 THE EARTH’S GRAVITY
Newton's law, gravity
Gravity potentials and acceleration
Gravity of the Earth
Earth’s shape and composition
Normal gravity and gravity anomalies
Observed gravity and geoid anomalies
Flexure of the lithosphere and the viscosity of the mantle
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3. 6/2/2023
Chapter 2
Isostasy
Mechanics of isostasy
Isostasy and oceanic lithosphere
Isostasy and continental lithosphere
3

4. CHAPTER 3. SEISMICITY
Waves through the earth
Earthquake seismology
Refraction and reflection seismology
Seismic tomography
Global seismicity distribution
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5. 6/2/2023
CHAPTER 4.
GEOMAGNETISM
Origin of earth’s magnetism and magnetic
field
Magnetism and plate motions
Magnetization of rocks and
paleomagnetism
Magnetic anomalies 5

6. CHAPTER 5
The Sources Of Internal And External Heat And Their
Applications Conductive Heat Flow
 Calculation of simple geotherms
 Worldwide heat flow: total heat loss from the Earth
 Oceanic heat flow
 Continental heat flow
 The adiabatic and melting in the mantle
 Metamorphism: Geotherms in the continental crust
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7. 6/2/2023 7

8. Geophysics is the application of physical principles
and methods to problems in Earth Sciences. Essentially, as
the word suggests, geophysics is the application of method
of physics to the study of the Earth. The rocks does not
differ only by their macroscopic or microscopic properties
studied field geologists or petrologists. They also differ by
their chemical and physical properties.
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9. Hence as the rocks differ according to their origin,
structure, texture, etc. they also differ by their density,
magnetization, resistivity, etc. The bad news is that the
physical properties do not always clearly correlates with
geological classifications and do not necessarily easily
translates into the geological terms.
What does this mean? Lets take the density as an
example. We have a rock sample and we have
measured the value of density to be 2.60 g/cm3.
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10. According to this value we could assume that the rock sample
could be, e. g. a limestone, some shale, compact sand- stone,
rhyolite, phonolite, andesite, granite, possibly some kind of
schist and many others.
The wide range of possible rock types suggests that the physical
properties does not directly refer to the geological classification.
This is a principal problem of geophysics, however, as we will
see later, there are ways to overcome this problem.
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11.  So, what we can conclude from this
example? The geophysics is a kind of proxy
in our attempts to study the geological
structures.
 It does not “talk” to us in geological
terms and we have to interpret obtained
physical parameters in a geological sense.
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12. Moreover, the centrifugal force interpretation is not
unique as we have seen in our example. The successful
interpretation is based on experiences of an interpreter and
on the a priori knowledge of the geological environment
studied.
 In the terms of our example if we know that we are
working in the crystalline complex we can mostly likely
leave sedimentary rocks out of our interpretation.
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13. There is no clear distinction between general and
applied geophysics.
General geophysics methods are typically applied to
solve academic questions, whereas applied geophysics
can be characterized by the application of geophysical
methods for commercial purposes (“making money)
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14. Chapter one
Newton's "derivation" of the inverse square law of gravity
From observations of the night sky, it was clear to Newton (and
many before him) that there must be some form of attraction
between the earth and the moon, and the sun and the planets that
caused them to orbit around the Sun.
Yet, it was not at all clear that the same force of attraction could
be responsible for the behavior of falling bodies near the surface
of the earth.
Newton postulated that one force called gravity was responsible
for both motions. His problem, then, was to attempt to determine
the force law.
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15. His method involved a comparison of the motions of the moon
around the earth and an object (e.g., apple) falling towards the
earth. Since neither the apple or the moon are moving in a
straight line at constant speed, they must each be undergoing an
acceleration.
Would it be possible to determine the nature of the force
responsible for that acceleration? In order to do so, he must first
determine the accelerations of the moon and the apple.
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16. Inertial and Gravitational Mass
There are two kinds of mass define by the m in
their equations:
Inertial (F=ma) and gravitational ( g=Gm/r2 ).
should the two mass (m) be the same ? but we now
know that the two mass kinds are the same
(relativity). a detail is that force, momentum, and
velocity are vectors in 3-space.

17.  Inertial mass is define as m = F/a. this mass can be
measured by applying a force to an object and
measuring its acceleration.
 This mass is thus a measure of the inertia of an object.
and, inertia is the fact that masses remain at rest or in
straight-line constant velocity motion unless acted upon
by an unbalanced force. and a force is defined as a
change in an object velocity vector (momentum:
p=m*v)
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18. gravitational mass is defined as m = g(r)*r2/ g.
this mass can be measured by measuring the
gravity field at a distance r and knowing the
value of big-g. this mass is thus a measure of
the gravitational acceleration field made by all
objects (mass).
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19. Newton’s Law of Gravity
1 2
Fg
Fg
M1 = Mass of object 1 (grams)
M2 = Mass of object 2 (grams)
D = Distance between objects (cm)
G = 6.67x10-8 cm3/sec2/gm
D

Fg 
GM1M2
D2
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20. Newton’s Law of Gravity
Galileo’s Experiment
Lead Wood

Fg,lead 
GMEarthMlead
REarth
2

Fg,wood 
GMEarthMwood
REarth
2

alead 
Fg,lead
Mlead

GMearthMlead
REarth
2
•
1
Mlead

GMearth
REarth
2
awood 
Fg,wood
Mwood

GMearthMwood
REarth
2
•
1
Mwood

GMearth
REarth
2
Earth
Fg,lead
Fg,wood
alead
awood
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21. Newton’s Law of Gravity
Mass = Amount of Material
Weight = Force Due to Gravity
90 kg
200 lbs 200 lbs
Earth
30 lbs 30 lbs
Moon
0 lbs 0 lbs
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22. Newton’s Law of Gravity
Orbits - Moon is constantly in free fall, as are all orbiting
objects
- “Always falling to earth, but never getting closer!”
0.14 cm
1 km
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23. Newton’s Law of Gravity
Kepler’s 3rd Law
A
S
P
In order for planet P to travel in a circle around the Sun…
Central Force = Gravitational Force

MpVp
2
A

GMS Mp
A2

Vp
2
A

GMS
A2
Vp
2

GMS
A
Velocity depends only on distance,
not on mass of planet
Velocity decreases w. greater distance
23

24. Newton’s Law of Gravity
Kepler’s 3rd Law

Vp
2

GMS
A

Vp 
distance
time

2A
P

4 2
A2
P2

GMS
A

4 2
A3
P2
 GMS

4 2
A3
 GMS P2

4 2
G
A3
MS
 P2

P2

4 2
G
1
MS
A3
A
C
P
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25. Newton’s Law of Gravity
Kepler’s 3rd Law

P2

4 2
G
1
MS
A3
In general, in units of cm, sec, gms
(1yr)2

4 2
G
1
Mo
(1AU)3
For the Earth

P2

1
MS
A3 In general, in units of AU, years, solar
masses
P2
 A3
For objects orbiting the Sun (Mc = 1 Mo)
A
C
P
1 Solar Mass
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26. Gravitational force law and Inertial force Law
1 2
2 1
1 2
2 2
1
1 2 1
2 2
2
2 1 2
2 2
( ) ( )
( ) ( )
( ) ( )
on
on
m m
m m
m m m
F r G Newtons kg
r s
m m
g r G F m g
r s
m m
g r G F m g
r s

  
  
•The two Forces (F) is the magnitude of the
force applied by mass 1 on mass 2 AND the
force of mass 2 on mass 1.
•The magnitudes of the two force is equal and
opposite in direction.
•This unbalanced force will cause each mass
to accelerate in inverse proportion to its mass:
a = F/m.
•The little mass accelerates much more than
the big mass, but the forces magnitudes are
the same!
So when I drop a ball, the ball
accelerates down at 9.8 m/s*2…yes.
And, the earth accelerates upwards
towards the balls mass….yes.
Why don’t we notice the earth’s upward
acceleration?

27. Result:
The considerable inward directed
force of gravitational attraction
of the earth on the moon
balances the centrifugal force of
the moon orbiting around the
earth. The gravitational force
bends the straight line trajectory
(inertial law) into a circle.
The force also cause the
ocean tides as the earth
spins daily underneath the
moon (moon takes 28 day
to go around the earth).
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28. 6/2/2023 28

29. Inverse square laws result from spherical area scaling
The area of a spherically symmetric field increase as the radius squared;
therefore, for the field energy to be conserved (as it must), then the field
energy must decrease as the radius squared: e.g., the gravity law, coulomb
electrical force law.
1/r*2 fall-off
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30. Calculate
1 2
2 1
1 2
2
1
1 2 1
2 2
2
2 1 2
2 2
( )
( ) ( )
( ) ( )
on
on
m m
m m
m m
F r G N
r
m m
g r G F m g
r s
m m
g r G F m g
r s

 
 
Q. How can the gravity fields for the two masses
be different, BUT the forces on each mass has
the same magnitude (albeit opposite direction) ?
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31. Calculating gravity for general shapes
Problem: how to calculate gravity field between a point mass (m2)
and a general mass distribution on left side (i.e., earth, asteroid)?
Easy, divide the general shape into little squares (2-d) or cubes (3-
d) and label them (I and J) and then add up the vector forces
applied by all the little cubes on mass m2.
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32. Calculating gravity of perfect spheres
When an object’s geometry can be
approximated as a sphere, we integrate
(add-up) the sphere’s gravity using
spherical shells that extend over a small
radial distance.
In doing this integration, we find that the
symmetry of the sphere makes the sphere’s
gravity field to be the same as if ALL the
mass was concentrated at the center of the
sphere!!
One requirement: the ability to treat a sphere’s
mass distribution as a point is true ONLY IF the
ms mass is outside the radius of the sphere.
This theorem found by Newton greatly simplifies
the mathematics by treating the Me spherical
mass as a point!
Easy to apply to solar system as all the masses are
near spherical. On the other hand, it is the Earth’s
non-spherical ellipsoidal shape that makes the
moon recede from the earth about 3.8 cm/yr and
the earths rotation to slow by 0.002 s a day.
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33. Calculate gravity directly above a spherical
mass anomaly
5 2
1 10 /
mGal m s


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34. Densities of Rocks and liquids
Notice we are not using the MKS mass units of kilo-grams (kg) for
density.
Mass is being calculated in mega-grams which is a thousand (103 )
grams.
Note that in general the substances density increase as follows: liquids,
unconsolidated sediments, sediments, igneous/metamorphic rocks,
minerals/ores. Density is defined by the Greek letter rho as:
mass
volume
 
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35. 6/2/2023 35
 Estimates of earth’s
mass and volume give
a whole earth density
of ~5.5 g/cm3
 Typical rocks at the
surface of the Earth
have a density of 2.0-
2.5 g/cm3

36. Calculating gravity at different points on surface
To calculate the gravity effect of the
irregular body above at point P1 , the
body is divided into small squares
(parcels) and the many gravity
vectors from all the parcels are
added up to get the total gravity. Do
the same analysis to get the gravity
at the other points.
An important detail. The gravity field is a vector
quantity (has magnitude and direction). When
measuring the gravity of the above situation, both
the ‘pull of rest of Earth’ and the ‘total pull of
excess mass of body’ are measured. Note that with
respect to point P, the Earth and the ‘body’ pull at
different directions. When can ignore this detail,
because the Earth’s pull is so much greater, and just
assume we are measuring the ‘vertical component
of Fb
‘ .
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37. Gravity field of a sphere on the 2-d surface
If we measured the gravity at every point in the 45 km square plane
and reduce it to bouguer gravity, this is what the gravity field would
look like for a buried spherical mass.
What is the sign of the mass difference between the spherical mass
anomaly and the background material ?
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38. Gravity Anomalies
Gravity interpretation for subsurface density is inherently ambiguous.
Many possible density distributions can lead to identical anomalies.
Geological insight must be applied in gravity interpretation.

39. Gravity anomalies of a sphere & a cylinder
The y-axis is the gravity anomaly (mgal) and the x-axis is distance from the center of the
sphere/cylinder (m). These graphs are cross-section through these 3-d objects. The cylinder
extends to +/- infinite in and out of the page. This is why, for the same depth object and
mass anomaly, the cylinder and sphere anomalies are different.
Important: Note that the peak amplitude reduces and the anomalies ‘half-width’ widens as
the anomaly is placed deeper. This is just a consequence of gravity being an ‘inverse
square law’.
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40. Gravity anomalies of dipping narrow mass sheets
Note: three gravity
effects:
 As the depth to the top of the
anomalies increases the peak
amplitude of the anomaly
decreases.
 As the sheet anomaly dips to one
side, the anomaly’s peak value
moves to that side.
 As the sheet anomaly dips to one
side, the anomaly has a long-tail
on that side, and a short tail on the
other side.
From these effects, we can determine
the dip of the sheet anomaly.
What is the sign of the mass anomaly?
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41. Gravity effects of half-sheets
Note two effects:
1. The deeper sheet has a smaller
peak anomaly.
2. The deeper sheet has a wider
anomaly half-width.
Figure (a) shows a stratigraphic section with
different layer densities that has be offset by a
vertical fault.
Figure (b) shows the layer densities changed into
density contrasts so that we can easily calculate the
total gravity anomaly associated with the variable
vertical distribution of density. From this the
relative motion of the fault can be determined.
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42. Gravitational potential
By virtue of its position in the gravity field g due to mass M, any
mass m has gravitational potential energy. This energy can be
regarded as the work W done on a mass m by the gravitational force
due to M in moving m from ref to r where one often takes ref =
∞.The gravitational potential U is the potential energy in the field
due to M per unit mass. In other words, it’s the work done by the
gravitational force g per unit mass. Negative
sign is
absolutely
essential.
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43. 6/2/2023 43

44. 6/2/2023 44

45. Shape of the earth
Terms to know:
•Spherical radius
•Equatorial radius
•Polar radius
•Reference ellipsoid
•Geoid (Gravitational Potential)
Non-spherical shape and centrifugal
force makes the gravity vary as IGF
equation.
Also, the earth rotates once a day
around its spin axis to make forces.
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46. Misconceptions about crust, lithosphere, asthenosphere*
(1) Crust (compositional). The crust is the residue from
melting the mantle. Crustal thicknesses is 10 km
(oceans) and 30-80 km (continents).
(2) Mantle (compositional). The mantle +crust+core =
chondrite meteorites.
(3) Core (compositional). Made of liquid/solid iron mostly.
(4) Lithosphere (strength). Also, called a plate. The strong
layer that slides over the asthensophere. The lithosphere
is strong because it is colder.
(5) Asthenosphere (strength). The mantle just below the
lithosphere is often weaker than the deeper mantle due to
pressure and temperature effects.
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47. 6/2/2023 47

48. 6/2/2023 48
 Normal gravity and gravity anomalies
The mean value of gravity at the surface of the Earth is approximately
9.80m/s2, or 980,000 mgal.
The Earth’s rotation and flattening cause gravity to increase by roughly
5300 mgal from equator to pole, which is a variation of only about
0.5%. Accordingly, measurements of gravity are of two types.
The first corresponds to determination of the absolute magnitude of
gravity at any place; the second consists of measuring the change in
gravity from one place to another.

50. Geoid
Locus (surface) of points with equal gravity
potential approximately at Mean Sea Level.
Every day heights are relative to geoid (MSL)
physically exists.

51. Ellipsoid – a mathematical construct used to approximate
the geoid.
Fit it to the geoid:
A single ellipsoid can’t fit perfectly everywhere due to the
irregular nature of the geoid.

52. Viscous mantle (asthenosphere) response to loading

53. Earth model: an elastic lithosphere (crust), that does not
flow (but does break), over a ‘fluid’ asthenosphere/mantle