Understand the gravity

1. PART I
GRAVITY METHOD

2. Potential Field Theory
The Universal Law of Gravitation
 The gravitational method is a study of the distribution of
Newton's attraction force F caused by all masses of the
earth. One part of this force provides the uniform motion
of a body around the rotation axis of the earth, and the
other part originates the weight force.
 The Universal Law of Gravitation States each particle of
matter in the universe attracts all others with a force
directly proportional to its mass and inversely proportional
to the square of its distance of separation.
 Gravitational attraction obeys a simple differential
equation.

3.  The phenomenon of attraction of masses is one of the
most amazing features of nature, and it plays a
fundamental role in the gravitational method.
 Everything that we are going to derive is based on the
fact that each body attracts other.
Fields
A field is a set of functions of space and time.
The gravitational attraction of the earth and the
magnetic field induced by electrical currents are
examples of force fields.

4. Gravitational Attraction and Potential
Newton's law of gravitational attraction stated: The
magnitude of the gravitational force between two
masses is proportional to each mass and inversely
proportional to the square of their separation.
 In Cartesian coordinates (Figure below), the mutual
force between a particle of mass m centered at point
Q = (x', y', z') and a particle of mass mo at P = (x,y,z) is
given by:

5.  Consider the following figures
m
r
Fig. Masses m and mo experience a mutual gravitational
force which is proportional to m, and .
By convention, unit vector is directed from the
gravitational source to the observation point, which in this
case is located at test mass mo.

6. Where,
G- Newton's gravitational constant( )
is a unit vector directed from the mass m to the
observation point P.
 If we let mass mo be a test particle with unit magnitude, then
dividing the force of gravity by mo provides the gravitational
attraction produced by mass m at the location of the test
particle:
 In the International System, abbreviated SI and mks system of
units, m and mo have units of kilograms, distance is in meters,
and gravitational attraction is reported in .

7.  is directed from the source to the observation point.
 g is force divided by mass, it has units of acceleration and is
sometimes called gravitational acceleration.
 We will use the terms attraction and acceleration
interchangeably in reference to g.

8.  In the cgs system of units, mass has units of grams,
distance is in centimeters, and gravitational attraction is
reported in units of .
 The cgs unit of acceleration is often referred to as the Gal
(short for "Galileo"), where 1 Gal = 1 , and the
geophysical literature commonly reports gravitational
attraction in units of mGal ( ).

9. Equi-potential Surfaces
 The potential of vector field F is defined as the work
function or as its negative depending on the convention
used.
 If particles of like sign attract each other (e.g., gravity
fields), then and the potential equals the work
done by the field.
 If particles of like sign repel each other (e.g.,
electrostatic fields), then , and the potential
equals the work done against the field by the particle.
In the latter case, the potential is the potential
energy of the particle; in the former case, is the
negative of the particle's potential energy.

10.  For a conservative vector field F can be expressed as the
gradient of a scalar , called the potential of F, and
conversely F is conservative if .
 It was asserted that such potentials satisfy Laplace's
equation at places free of all sources of F and are said to be
harmonic.

11.  Note that any constant can be added to without
changing the important result.
 This constant is chosen generally so that approaches 0
at infinity. In other words, the potential at point P is given
by:
 The value of the potential at a specific point, therefore, is
not nearly so important as the difference in potential
between two separated points.

12.  An equipotential surface is a surface on which the potential
remains constant; that is
 Field lines at any point are always perpendicular to their
equipotential surfaces and, conversely, any surface that is
everywhere perpendicular to all field lines must be an
equipotential surface.
 No work is done in moving a test particle along an
equipotential surface. Only one equipotential surface can
exist at any point in space.
 The distance between equipotential surfaces is a measure
of the density of field lines; that is, a force field will have
greatest intensity in regions where its equipotential
surfaces are separated by smallest distances.

13. Determination of the Gravitational Field
 Gravitational field caused by masses located beneath the
earth's surface. If the Earth were a perfect sphere with no
lateral inhomogeneities and did not rotate, g would be the
same everywhere and obey the formula:
 This is not the case, however; the Earth is inhomogeneous
and it rotates. Rotation causes the Earth to be an oblate
spheroid with an eccentricity .
 The polar radius of the Earth is approximately 20 km less
than the equatorial radius, which means that g is
approximately 0.4% less at equator than pole. At the
equator, g 5300 mGal (milliGals.)

14.  Fig. Difference between a sphere and an Ellipse of rotation
(Spheroid)
 The best fitting spheroid is called the reference spheroid,
and gravity on this surface is given by the International
Gravity Formula (the IGF), 1967 and is called the normal
field (theoretical), which is perpendicular to the spheroid
and can be described by the formula:

15.  Because the force of gravity varies from place to place
about the earth, equipotential surfaces surrounding the
earth are smooth but irregular.
 An equipotential surface of particular interest is the geoid,
the equipotential surface described by sea level without
the effects of ocean currents, weather, and tides.
 The geoid at any point on land can be thought of as the
level of water in an imaginary canal connected at each end
with an ocean.

16. Gravitational potential
 The gravitational potential is the potential energy of a unit
mass in a field of gravitational attraction.

17.  The best fitting spheroid is called the reference spheroid,
and gravity on this surface is given by the International
Gravity Formula (the IGF), 1967 and is called the normal
field (theoretical), which is perpendicular to the
spheroid and can be described by the formula:
 Where:

18.  The geoid is an equipotential surface corresponding to
mean sea level.
 On land it corresponds to the level that water would reach
in canals connecting the seas.
 The geoid is a conceptual surface, which is warped due to
absence or presence of attracting material. It is warped up
on land and down at sea.

19.  The shape of the geoid is influenced by underlying masses;
 it bulges above mass excesses (e.g., mountain ranges or
buried high-density bodies) and
 is depressed over mass deficiencies (e.g., valleys or buried
low-density bodies).
 Because the geoid is an equipotential surface, the force of
gravity at any point on the geoidal surface must be
perpendicular to the surface, thereby defining "vertical"
and "level“
 Because of the complexity of internal density variations, it
is customary to reference the geoid to a simpler, smoother
surface.
 By international agreement, that equipotential surface is
the spheroidal surface that would bound a rotating,
uniformly dense earth at each point.

20.  Fig: The relationship between the geoid, the spheroid,
topography and anomalous mass.

21.  Because of the complexity of internal density variations, it
is customary to reference the geoid to a simpler, smoother
surface. By international agreement, that equipotential
surface is the spheroidal surface that would bound a
rotating, uniformly dense earth.
 Differences in height between this spheroid and the geoid
are generally less than 50 m and reflect lateral variations
from the uniform-density model.
 Because of the competing forces of gravity and rotation,
the spheroid very nearly has the shape of an ellipse of
revolution and, consequently, is called the reference
ellipsoid.
 If the equatorial radius a and polar radius c and often is
expressed in terms of the flattening (f) parameter

22. The earth is nearly spherical, of course, with
flattening of only 1/298.257, and this fact will permit
several simplifying approximations in the following
derivations. The force of gravity on the earth is due
both to the mass of the earth and to the centrifugal
force caused by the earth's rotation.
The total potential of the spheroid, therefore, is the
sum of its self-gravitational potential and its
rotational potential ,

23. Theoretical Gravity
 The ellipsoid is defined and refined by international
agreement through the International Association of
Geodesy (IAG) and its umbrella organization, the
International Union of Geodesy and Geophysics (IUGG).
 Three international systems have been sanctioned in this
way:
 The first internationally accepted reference ellipsoid was
established in 1930, and its associated parameters provided
the 1930 International Gravity Formula,

24.  The advent of satellites provided a breakthrough in the
accuracy of various geodetic parameters, and a new
ellipsoid was adopted in 1967 called Geodetic Reference
System 1967, thereby providing the 1967 International
Gravity Formula,
Most recently the IAG has adopted Geodetic
Reference System 1980, which eventually led to the
current reference field, World Geodetic System 1984;
in closed form it is given by;

25. The Geoid
 The reference ellipsoid is the equipotential surface of a
uniform earth, whereas the geoid is the actual
equipotential surface at mean sea level. Differences in
height between these two surfaces rarely exceed 100 m and
generally fall below 50 m.
 The shape of the geoid is dominated by broad undulations,
with lateral dimension of continental scale but with no
obvious correlation with the continents; they apparently
are caused by widespread mantle convection.
 Gravity anomalies, are referenced to the reference ellipsoid
but involve various corrections relative to sea level (the
geoid).

26. The gravity survey
Introduction:
 Gravity surveying measures variations in the Earth’s gravitational
field caused by differences in the density of sub-surface rocks.
 Areas of applications of gravity method:
 Hydrocarbon exploration
 ƒ
Regional geological studies ƒ
 Isostatic compensation determination
 ƒ
Exploration for, and mass estimation of, mineral deposits ƒ
 Detection of sub-surface cavities (microgravity) ƒ
 Location of buried rock-valleys ƒ
 Determination of glacier thickness
 ƒ
ƒ
Monitoring volcanoes.
 Shape of the earths (geodesy)

27.  Gravity survey : Measurements of the gravitational
field at a series of different locations over an area of
interest.
Measurements of gravity
 An absolute gravity measurement is the measuring
actual value of g by the speed of a falling mass using a
laser beam. It is expensive, heavy, and bulky.
 A second type of gravity field measurement is the
relative changes in g between two locations.
 The relative gravity measurements are tied to the
absolute gravity network.

28. Measurement of gravity on land
A. bsolute Gravity
Requires very careful experimental procedures and normally
only under taken under laboratory.
Two methods of measurements are used;
i. The falling body and
ii. Swinging pendulum
 Absolute gravity values were determined by the network
referred as the International Gravity Standardization net
(IGSN 71).
B. Relative Measurements
 In gravity exploration it is not normally necessary to
determine the absolute value of gravity, but it is relative
variation that is measured.

29.  Relative gravity measures the relative changes in g
between two locations.
 The relative gravity measurements are thereby tied to
the absolute gravity network.
 Lateral density changes in the subsurface cause a
change in the force of gravity at the surface.
 The intensity of the force of gravity due to a buried
mass difference (concentration or void) is
superimposed on the larger force of gravity due to the
total mass of the earth.

30.  A base station (related to IGSN 71) is selected and a
secondary network of gravity stations is established.
 All gravity data acquired at stations occupied during the
survey are reduced relative to the base station. If there is no
need for absolute values of g to be determined, the value of
gravity at a local base station is arbitrarily determined as
zero.
 Relative gravimeters are used, which have a nominal
precision of 0.01 mGal. It requires a lot of skill and great
care to use them well. The results are measurements of the
differences in g between stations.
 There are two basic types of gravimeter:
i. Stable gravimeters
ii. UnStable gravimeters

31. The following factors must be considered in designing a
survey:
1. If it is desired to tie the survey to others, the network must
include at least one station where absolute g is known.
2. The station spacing must fit the anomaly scale.
3. The heights of all stations must be known or measured to
10 cm.
4. Latitudes must be known to 10 m.
5. Topography affects the measurements, thus it is best to
locate the stations where there is little topography.
6. Access is important, which often means keeping stations to
existing roads or waterways if there are no roads.
7. In the design of the gravity survey, station spacing and
accuracy are most important.

32. Design of Survey
The two most critical factors of survey design are:
 Accuracy
 Spatial distribution of the data required
The other most important being interpretation
needs must be the availability of accurate density
values for the rock units in the study area (samples
are to be collected from the exposed units at the time
of surveys)

33. Gravity surveying
 The station spacing used in a gravity survey may vary
from a few metres in the case of detailed mineral or
geotechnical surveys to several kilometres in regional
reconnaissance surveys.
 During a gravity survey the gravimeter is read at a base
station at a frequency dependent on the drift
characteristics of the instrument.
 At each survey station, location, time, elevation/water
depth and gravimeter reading are recorded.

34. Regional and Local Gravity Survey
Regional Survey
Regional and local gravity surveys are performed for
geodetic, geophysical and geodynamic purposes
Regional surveys attempted with station separation
of 1-5 km
The desired accuracy of Bouguer anomalies is
0.05-0.1mGal
It is a reconnaissance survey 100km

100km Regional
Survey

35. Local Survey
Performed to solve special geological problems
 determine the structures (tectonics)
Depth of crystalline basement 10km
Local sediment basins
Salt domes 10km
O explore deposits, etc
Station separation is 50m-500m
Accuracy of anomalies 0.01-0.05 mGal

Local Survey

36. Gravity Networks
1. Global Absolute gravity Networks
2. National gravity networks
Regional Gravity
Networks
Local Gravity
Networks

37.  Global Gravity Networks
Network of globally distributed absolute gravity
base station with high precision
Station separation 100-1000km
Established by international cooperation
Purpose:
To maintain temporal gravity changes, which
evolve over a long period
To serve as control station for regional gravity
networks

38.  Regional Gravity Networks
Established as national gravity networks
Local gravity network
Mostly established for geophysical and geodesy
purpose with station separation of 0.1-10km
Regional and local gravity surveys are exhaustively
performed with spring gravimeters.
The object of the survey is to measure the values of
gravity at observation sites.
Spring gravimeter measure the relative g values Δg
between the observation sites

39.  The purpose of the survey is to determine the absolute
value of gravity g at each observation site, in order to
infer the results with other gravity surveys performed
all other world (accessible gravity base station)
If non of such base station is close to the survey area, it
is necessary to determine, the absolute(reference)
value of gravity at located base station using
gravimeter refered to the global IGSN71 system

40.  It is important to realize that no amount of computer
processing can compensate for poor experiment design.
This wise adage applies for all geophysics, and not just
gravity surveying. Linear features may be studied using one
or more profiles, two-dimensional features may require
several profiles plus some regional points, and for some
special objectives, e.g., determining the total subsurface
mass, widely-spaced points over a large area may be
appropriate.

41. Gravimetric Survey on land
 A gravity survey is conducted by making gravimeter readings at many
locations in an area of interest.
 The measurements differ with respect to:
 their spatial separation
 Station separation
 Accuracies defined
 The gravity measurements should be tied to :
 Global gravity reference systems or
 Gravity networks established to create global, regional and local arrays
of gravity control
 In addition to gravimeter reading , other measurements and
observations such as;
o Latitude and longitude: to describe the position of each observation
and to calculate theoretical gravity at each station.
o Elevation: to reduce each observation site to the geoid

42. o Observation time of each site: to determine correction for gravity drift.
Gravimeter Observation Sites
 Choice of observation point depend on:
 Geologic features that are particular interest and
 The accessibility of the area

43. Method
The following field procedure is usually adopted:
1. Measure a base station,
2. Measure more stations,
3. Re-measure (Reoccupying) the base station approximately
every two hours.
If the survey area is large, time can be saved by establishing
a conveniently sited base station to reduce driving. This is
done as follows:
Measure: base 1 –> new base station –> base 1 –> new base
station –> base 1.
 This results in three estimates of the difference in gravity
between base 1 and the new base station. From this, gravity
at the new base station may be calculated.

44.  The new base station can then be re-measured at two-
hourly intervals instead of base 1. This procedure may also
be used to establish an absolute base station within the
survey area if one is not there to start with.
 During the survey, at each station the following
information is recorded in a survey log book:
 the time at which the measurement is taken,
 the reading, and
 the terrain, i.e., the height of the topography around the
station relative to the height of the station.
 Transport during a gravity survey may be motor vehicle,
helicopter, air, boat (in marshes), pack animal or walking.
In very rugged terrain, geodetic surveying to obtain the
station heights may be a problem.

45. Reduction of observations (Correction to Gravity Observations)
 The importance to reduce a gravity measurement made on
or near the earth's surface to an anomaly value is to know
that reflects density variations in the crust and upper
mantle.
 This involves a long series of operations with a well-
established tradition.
 These operations account for the mass, shape, and spin of a
"normal" earth, elevation of the measurements above sea
level, tidal effects of the sun and moon, motion of the
instrument, gravitational effects of terrain in the vicinity of
the measurement, and effects of isostasy.
 It is necessary to make many corrections to the raw gravity
readings to obtain the gravity anomalies that are the target
of a survey. This is because geologically uninteresting
effects are significant and must be removed.

46.  The difference between the value of observed gravity (
gobs ) and the determined either from International
Gravity Formula/ Geodetic Reference System is known as
gravity anomaly.
Various types of Corrections;
1. Instrument Drift
2. Latitude correction
3. Elevation (Free Air) correction
4. Bouguer correction
5. Terrain corrections

47. 1. Instrument Drift
 A geophysical instrument will usually not record the same
results if read repeatedly at the same place. This may be
due to changes in background field but can also be caused
by changes in the instrument itself, i.e. to drift.
 Drift correction is often the essential first stage in data
analysis, and is usually based on repeat readings at base
stations
 Gravity measurements, even at a single location, change
with time due to Earth tides, meter drift, and tares.
 The instrumental drift can be determined simply by
repeating measurements at same stations at different
times of the day, typically every 1-2 hours.

48.  The difference between successive measurements at
stations are plotted to produce a drift curve. Observed
gravity values from intervening stations can be corrected
by subtracting the amount of drift from the observed
gravity value.
 For example from the figure Below the value of gravity
measured at an outlying station at 12:30 hour should be
reduced by the amount of d.

49.  A graph is plotted of measurements made at the base
station throughout the day. Drift may be non-linear, but it
has to be assumed that it is be linear between tie backs for
most surveys. The drift correction incorporates the effects
of instrument drift, uncompensated temperature effects,
solid Earth and sea tides and the gravitational attraction of
the sun and moon

50. 2.Latitude correction
 The latitude correction is made by subtracting the
theoretical gravity ( ) calculated using International
Gravity Formula from the observed value ( ). This is
needed because of the ellipticity of Earth.
 Gravity is reduced at low latitudes because of the Earth’s
shape and because of rotation:

51. 3. Elevation (Free Air) correction
 Gravity measurements over land, however, must be
adjusted for elevation above or below sea level.
 Let represents the attraction of gravity on the geoid.
The value of gravity a small distance above the geoid is
given by a Taylor's series expansion:

52.  SBA=gobs-gfa±gbc-gN
 CBA=gobs-gfa±gbc-gt-gN

53.  Dropping high-order terms and rearranging the remaining
terms gives:
 If we assume that the earth is uniform and spherical, then
 The last term of this equation accounts for the difference in
elevation between and .

54.  It is known as the free-air correction ( ) because it is the
only elevation adjustment required if no masses were to
exist between the observation point and sea level. Using
values of g and r at sea level provides:
Where is height above sea level.
 Application of the free-air correction provides the free-air
anomaly given by;
 It is necessary to correct for the variable heights of the
stations above sea level, because g falls off with height. It is
added:

55. 4. Bouguer correction
 Gravity surveying is sensitive to variations in rock density,
so an appreciation of the factors that affect density will aid
the interpretation of gravity data. This accounts for the
mass of rock between the station and sea level. It has the
effect of increasing g at the station, and thus it is
subtracted.
 The formula for the Bouguer correction on land is:
BC = 2πρGh
where = height above sea level and = density. This is
also the formula for an infinite slab of rock.

56.  The free-air correction and theoretical gravity ignore mass
that may exist between the level of observation and sea
level. The Bouguer correction accounts for this additional
mass.
 The simple Bouguer correction approximates all mass
above sea level with a homogeneous, infinitely extended
slab of thickness equal to the height of the observation
point above sea level.
 Free-air anomalies are strongly correlated with terrain.
 Simple and complete Bouguer anomalies over continental
areas are strongly negative. This happens because the
Bouguer correction has removed the effects of normal crust
above sea level but has left the effects of deeper masses that
isostatically support that crust.

58. Tidal Correction
 Earth-tides caused by the sun and moon are of sufficient
amplitude to be detected by gravity meters as time-varying
gravity.
 The effect is both time- and latitude-dependent; it is
greatest at low latitudes and has a strong periodic
component with period on the order of 12 hours.
 The tidal effect never exceeds 0.3 mGal, a small quantity
in comparison to other corrections to observed gravity.
 Nevertheless, tidal effects should be accounted for in high-
precision surveys.
 It may be appropriate in less precise surveys to assume that
the tidal effect is linear over periods of several hours and to
remove the tidal effect along with other temporal
adjustments.

59.  For example:
 most gravity meters used in gravity surveys produce
readings that drift slightly over the course of a day's
fieldwork.
 This problem usually is treated by reoccupying certain
observation points at various times during the day,
assuming that drift has been linear between the repeated
measurements, and subtracting the linear drift from all
other readings.
 The tidal effect can be considered part of the instrumental
drift.

60. Eotvos Correction
 the attraction of the earth at a point fixed with respect to
the earth is reduced by the centrifugal force related to the
earth's rotation.
 It stands to reason that the angular velocity of an observer
moving east is greater than for an observer remaining
stationary with respect to the earth's surface, and
consequently gravitational attraction will be slightly
reduced for the moving observer. Likewise, gravitational
attraction will be slightly increased for an observer moving
in a westerly direction.
 This motion-related effect, called the Eotvos effect, must be
accounted for in gravity measurements made on moving
platforms, such as ships or aircraft. The Eotvos correction is
given by

62. Terrain corrections
 The effect of terrain is always to reduce observed g. This is
true for a mountain above the station and a valley below the
station, which both cause g to be reduced.
 Terrain corrections are done by hand using a transparent
graticule, or by computer if a digital terrain map is
available.
 The graticule is placed on a map and the average height of
each compartment estimated. A “Hammer chart” is then
used to obtain the correction.
 This chart gives the correction for a particular distance
from the station. It has been worked out assuming a block
of constant height for each compartment.

63. A graticule

64. Interpretation of gravity anomalies
 Bouguer anomaly fields are often characterized by a broad,
gently varying, regional anomaly on which may be
superimposed shorter wavelength local anomalies.
 In gravity surveying it is the local anomalies that are of
prime interest and the first step in interpretation is the
removal of the regional field to isolate the residual
anomalies.
 It is necessary before carrying out interpretation to
differentiate between two-dimensional and three-
dimensional anomalies.
 Two-dimensional anomalies are elongated in one
horizontal direction so that the anomaly length in this
direction is at least twice the anomaly width.

65.  Anomalies may be interpreted in terms of structures which
theoretically extend to infinity in the elongate direction by
using profiles at right angles to the strike.
 Three-dimensional anomalies may have any shape and are
considerably more difficult to interpret quantitatively.
 Gravity interpretation proceeds via the methods of direct
and indirect interpretation.
 Direct interpretation provides, directly from the gravity
anomalies, information on the anomalous body which is
largely independent of the true shape of the body.

66.  Indirect interpretation involves four steps (model
calculation is performed):
 Construction of a reasonable model.
 Computation of its gravity anomaly.
 Comparison of computed with observed anomaly.
 Alteration of model to improve correspondence of
observed and calculated anomalies and return to step