Gravity, Expl.ravity

Gravity Technique
1- Newton’s Law and Gravity Force
2- Some calculations
3- Gravity variation
4- Gravity data processing
a) Corrections
b) Derivations
5- Density of rocks
6- Bulk density assessment

1

Apple Falling on Newton’s Head

• this simplifies the calculation for any body consisting of concentric layers [seismology  true for the Earth]
• each layer can be shrunk to a central “point mass”, and thus the Earth behaves as if all its mass ME was
concentrated at the center

Earth’s attraction for a small mass, mS, at its
surface is the same as if there were two “point
masses” separated by the Earth’s radius, RE
(~6370 km)

To avoid dependence on the surface mass, m S, as well as on the Earth; we use the acceleration, g,
[falling rate; all masses fall with the same acceleration if they are dropped in a vacuum, which eliminates the air resistance]

that the force produces: force of attraction = mass x acceleration



g (“little gee”) = acceleration due to gravity
4

Acceleration due to gravity
On equating both laws as they represent F, g can be
obtained as

• That is,
…(
g = GM/(R2 )
3)
Where g- is the gravitational force
G- is the universal gravitational const.
(6.67 x 10-11 m³ /Kg s2 )
M-is the mass of the earth in Kg
Thus, theoretically, the gravity of acceleration is
the same/constant all over the earth.

• That is,
…(
g = GM/(R2 )
3)
Where g- is the gravitational force
G- is the universal gravitational const.
(6.672 x 10-11 m³ /Kg s2 )
M-is the mass of the earth in Kg
Thus, theoretically, the gravity of acceleration is
the same/constant all over the earth.

Universal constant of gravity
• G- is the universal gravitational const. Its
value is 6.672 x 10-11 m³ /Kg s2 in SI
(1)
is 6.672 x 10-8 cm³ /gm s2 in CGS (2)
is 6.672 x 10-8 m³ /Mg s2 in CGS (3)
•

Exercise: Derive (2) and (3) from (1).

Rock Densities
Earth’s Mass

Can you calculate
Density and Mass
of the Earth?

Earth’s interior made of denser material;
e.g. mostly iron in the core [meteorite studies]

9

Rock Densities
Earth’s Mass

G = 6.672 x 10-8 m3/Mg s2 [found in a laboratory by measuring
the tiny force between two masses]
g  found by timing the acceleration of a dropped mass
RE  6370 km
thus,

ME ≈ 5.97 x 1021 Mg

or

5.97 x 1024 kg

Earth’s density = 5.5 Mg/m3 or 5½ times that of water (1.0 Mg/m3)

Earth’s interior made of denser material;
e.g. mostly iron in the core [meteorite studies]

10

Gravity Units

δ, very small quantity/difference
Δ, considerable difference

Calculate
gravity
effect

gravity variations  anomalies are VERY SMALL

Gravity Units:  milliGals
In the example: the buried sphere anomaly is 0.1 mGal

11

Gravity Units

δ, very small quantity/difference
Δ, considerable difference

gravity variations  anomalies are VERY SMALL

Gravity Units:  milliGals
In the example: the buried sphere anomaly is 0.1 mGal

12

Gravity Anomalies of Specific Bodies (3)
sphere & horizontal cylinder at different depths

13


buried sphere

14

Lecoste Romberg gravimeter

16

Measuring Gravity: Gravimeter
body weight = mg
[differs from place to place,
space, Moon]

mass, m, is the same everywhere

sensitive gravimeters  0.01 mGal (10-8 g)

17

Magnitude of “g”
At equator
At poles

(g)
(g)

= 978.0318 cm/sec 2
= 983.152 cm/sec 2

Normal value of (g) = 980 cm/sec2
Thus, the difference in “g” between equator & poles is
approximately 5 cm/sec2
In Geophysics we adopt the unit as “Gal” where

1 cm/sec2 = 1 Gal

Data Reductions - Corrections (2)

Latitude correction:

Earth equatorial bulge  equatorial diameter > polar diameter
(due to centrifugal forces)
g (at equator) < g (at poles)

International Gravity Formula (for a gravity station):

λ = latitude of station

For a gravity survey < 10x km



variation proportional to distance:
λ = latitude where survey is carried out

only the N-S distance matters, and as g increases towards the poles, the correction
is ADDED to all measurements on the equator side of the base station, and SUBTRACTED
from all measurements on the polar side of the base station, in order to cancel this decrease
19

Corrections to Observed Gravity
• The various corrections are:
1. Instrumental drift
2. Latitude
3. Free air
4. Bougeur
5. Elevation
6. Tidal effect
7. Terrain/Topography
8. Eotuos
9. Isostatic


Topographic corrections:

1) Free-air correction: g is reduced if measured in the air (B) above datum (A) because it is measured
further from the center of the Earth

(0.3086 mGal per meter rise)

(2) Bouguer correction: g is increased if measured at C (plateau/hill/mountain) above datum (A) because
of the additional pull of the additional thickness, h, of rock below

effect of infinite sheet/slab 

ρ = density in Mg/m3
h = slab thickness (m)

21


Topographic corrections:

ρ = density in Mg/m3
h = slab thickness (m)
positive above datum

combined elevation correction 
(free-air & Bouguer)

Free-air correction: is ADDED, to correct for the reduction of g at greater heights in the air
Bouguer correction: is SUBTRACTED, to correct for the additional pull of the intervening slab
(3) Terrain correction: not an infinite slab (as Bouguer correction assumes) but terrain (topographic/
morphological) differences (D) that reduce g by sideways and partly upwards pulls (H)
or by removing downwards pulls (V)

BOUGUER ANOMALY



+

22

7. Terrain Correction (TC)
• Neither EC nor BC takes care of hills & valleys
and hence to account for these, TC is
necessary.
• TC removes the effect of topography to fulfill the
Bouger approximation.
• Computation TC is practically difficult.
• Hammer chart is used to accomplish this.

7. Terrain/Topographic
Correction

Always positive
Requires detailed info on elevation around station, not just at station
Size of terrain correction depends on relief and its proximity to station

6. Tidal Correction
• As water in ocean/sea responds to the
gravitational pull of both moon & sun, the same
way, the solid earth behaves. Earth tides change
the value of (g) which can be estimated by
repeated measurements at the same station
over a period of time ( a minimum of 12 hrs) as
in the case of drift.
• Range: 0–3 g.u (Refer to published table)

Importance of Density
•

Density/density contrast plays very significant role in gravity method.

•

Gravity anomaly depends on the difference in density contrast
between body/structure of rock and its surroundings. (guest & host)

•

For a body of density ρ1 in a material of density ρ2 , the density
contrast ∆ ρ = (ρ2 - ρ1 ). The sign of gravity anomaly g(x) depends
upon the sign of ∆ ρ

•

Density varies with respect to:
(a) Depth
(b) Age
(c) Porosity/pore fluid/fracture/joints &
(d) Dry/wet conditions

Density of Rocks
• Average of density (gm/cc) in various rocks:
Wet
(a) Sedimentary
(b) Igneous rocks
(c) Metamorphic

Dry

1.54-2.30
2.24
2.60

1.98-2.70
3.17
3.37

Density of Various Earth Materials
–
–
–
–
–
–
–
–
–
–
–

Material Density (gm/cm^3)
Air
~0
Water
1
Sediments
1.7-2.3
Sandstone
2.0-2.6
Shale
2.0-2.7
Limestone
2.5-2.8
Granite
2.5-2.8
Basalts
2.7-3.1
Metamorphic Rocks
2.6-3.0

Density Determination
• Nettleton’s Method:
A reasonably satisfactory estimate of density of near
surface may be estimated by this method which
needs a representative gravity profile.
• The gravity data are reduced to produce Bouger
gravity profile assuming various values of density
for corrections.
• Among the resultant Bouger gravity profiles, the
smoothest one which reflects the topography least
corresponds to the approximately correct density.

Regional + Residual = Bouger
Gravity Anomaly
• Bouger Anomaly = Regional + Residual.
• Depending upon our objective whether our
interest is deep seated larger structures or
shallow depth smaller structures, we have to
proceed
for further processing and
interpretation of our data.

Regional and Residual
• Region Anomaly: The component of gravity anomaly
having longer wavelength (low frequency) which are
due to sources with larger dimension particularly
deep seated structure such as a basin/geo syncline
etc.
• Residual Anomaly: The component of anomaly
having short wavelength (high frequency) which are
due to smaller structures such as anticline/salt
dome etc.

Regional and Residual Separation
• Isolation/extraction/separation of regional and
residual can be done basically by filtering either
by High Pass (HP) filter or Low Pass (LP) filter.
• In regional studies, the anomalies from features
of small lateral extent may be removed so as to
bring out larger scale structures more clearly.

Methods of Separation
• There are several methods by which the
separation of regional & residual can be isolated
which are either:
• (a) Graphical
(b) Polynomial fitting
• (c) Moving average
• (d) Derivatives
• (e) Upward continuation
• (f) Wavelength filtering

Regional & Residual Methods
• Graphical method:
Regional is estimated
from
plotted gravity
profiles/contour maps of the observed gravity (gobs )
data.
• Polynomial fitting:
Here, the observed “g” are used to compute
mathematically discernable surface by least squares
and this surface is considered as Regional trend.

Graphical Method

Example of a regional-residual gravity anomaly separation using graphical smoothing

Bouger & Residual Gravity
Anomalies

Upward & Downward Continuation
•

By knowing field at one elevation, one can compute the
field at a higher elevation (upward continuation- UWC) or
lower elevation (downward continuation-DWC) is know as
continuation methods.

•

UWC: Transformation of observed “g” on the surface to
some higher surface.

•

DWC: Transformation of observed “g” on the surface to
some lower surface .

Upward Continuation
• Calculation of the field at an elevation higher
than that at which the field is known/measured.
It is used to smooth out near surface effects.
• Or it is a filter operation that tends to smooth the
original data by attenuation of short wavelength
anomalies relative to their long wavelength
counterparts.

UWC & DWC
•

The purpose of the downward continuation filter is to
calculate the magnetic/gravity field with the measurement
plane closer to the sources. In this way the anomalies will
have less spatial overlap.

•

Easily distinguished from one another. This process also
increases the amplitude of the anomalies. Care must be
taken because in addition to the amplitude of the anomalies
increasing the amplitude of any noise present will also
increase.

•

Short wavelength signals are from shallow sources and
therefore must be removed to prevent a high amplitude and
short wavelength noise in the data.

•

A similar effect is achieved using the upward continuation,
except that the measurement plane is further from the
sources, and fewer side effects are produced.


irregular body

50

sheets (dykes or veins)

51

horizontal sheet/slab

52

horizontal half-sheet/half-slab

53

horizontal layers offset by vertical faulting

Δρ = ρ1 - ρ1 = 0
Δρ = ρ2 – ρ1

Δρ = ρ2 – ρ1

Δρ = ρ3 – ρ1
Δρ = ρ2 – ρ3
Δρ = ρ2 – ρ3
Δρ = ρ3 – ρ3 = 0

Δρ = ρ4 – ρ3
Δρ = ρ4 – ρ5

Δρ = ρ4 – ρ3
Δρ = ρ4 – ρ5

Δρ = ρ5 – ρ5 = 0

54

[from Chapter 2]
Data Reduction:
converting the readings (measurements)
into a more relevant/useful form

• “raw” readings/measurements, not always ready
for further calculations & modelling
• instrumentation influences need to be corrected
• datum plane/surface levelling:
all observations refer to the datum plane
• positioning (lat-long-height)
modern improvements with GPS

Geophysical Anomaly:
the measured value in relation to the
normal (background field) value

55


Eötvös correction:

needed only if gravity is measured on a moving vehicle such as a ship, and arises because the
motion produces a centrifugal force, depending upon which way the vehicle is moving

v = vehicle speed (km per hour)
λ = latitude
α = direction of travel measured clockwise from north

only the E-W motion matters; the correction is POSITIVE, for motion from east to west

56

Residual & Regional anomalies
residual anomaly = observed field – regional field
from Chapter-2
background
value of g

 signal vs. noise
 residual anomaly vs. regional field/anomaly
concepts depending on the survey target interest

57

 signal vs. noise
 residual anomaly vs. regional field/anomaly
concepts depending on the survey target interest
from Chapter-2

58

Forward Modelling: (previous) 2D, 21/2D, 23/4D, 3D
Inversion/Inverse Modelling: trying to deduce the form of a causative body
from the anomaly
problems: non-uniqueness (multi-solutions) [noise, measuring errors, resolution]

59

problems: non-uniqueness (multi-solutions)

observed gravity anomalies depend only on lateral density differences or contrasts
thus:
a density contrast of (2.6 Mg/m3–2.5 Mg/m3) produces the same anomaly as a contrast of (2.4 Mg/m3–2.3 Mg/m3)
and
a half-slab with a positive density contrast from one side of a fault produces the same anomaly with a half-slab
from the other side of the fault with a similar but now negative density contrast

60

GRAVITY anomaly is: (1) measured at or near the surface, and (2) represents a physical and
not a direct geological quantity, thus we need two further steps:

from Chapter-2

MODELLING & INTERPRETATION

M odelling

Modelling is:
1) Constructing a 2D or 3D physical Earth model
with dimensions and material properties
2) Calculate the GRAVITY anomaly
produced by the model
3) Compare the observations with the model
4) Iterative (trial-and-error) improve the model

Forward Modelling: 2D, 21/2D, 23/4D, 3D
a Model is a simplification of the geology
Model:
#
simple shape
abrupt boundaries
uniform physical properties

Causative body:
irregular shape
gradational boundaries
gradational phys. properties

but simplification is not always a drawback, as it may emphasise the essential features

61

Depth rules

shape of body is known (a-c)

to find d (depth of buried body)

half-width:
half the width at
half the peak height

62

Gravity Surveying
Satellite Radar Altimetry

Earth’s gravitational field (Geoid)
[measurements from GRACE orbital satellite]

Free-air Gravity Anomaly

63

Gravity Surveying
Land surveys

Airborne surveys

Marine surveys
(together with seismic reflection data collection)

64

SEISMIC VELOCITY vs DENSITY

Nafe & Drake diagram
65

Data Reduction in Gravity Surveying
Gravity surveying: uses the lateral variations in g (gravity anomalies) to investigate
the densities and structures in the subsurface

Instrumental Effects & Corrections (1)
Conversion of readings: gravimeter springs differ slightly from one to another, so they give slightly different readings for the same
change of g; readings are converted into true values using conversion tables provided by manufacturers

Drift:
(1) instrumental: due to slow creep of the spring
(2) periodical: due to tidal distortion of solid Earth (up to 0.3 mGal throughout the day)
base station

base station
drift curve

base station

base station

instrument drift

Theoretically computed tidal variations

66

Gravity force and Gravity Potential

67

Gravity Surveying or Prospecting:
• is a method to detect lateral variations/differences (in density) of subsurface rocks
• thus, useful for finding buried bodies and structures [e.g. igneous intrusions, basins filled with
less dense rocks, faults] on scales ranging from few meters to tens of kilometers across

Newton’s Gravitation Law
• [discovered gravitation after being hit by a falling apple]
• similarly, gravitational force is responsible for holding planets in their orbits around the Sun
• in fact, all bodies attract one another

“point mass” = bodies so small in extent compared to
their separation that all parts of one mass
are closely same distance from the other

Universal gravitational constant (“big gee”) G = 6.672 x 10 -8 m3/Mg s2,
when m1, m2 are measured in Mg (tonnes), and r in meters

69

Newton’s Gravitation Law & “extended bodies”
extended bodies = treated as an assembly of many small masses: the force on each component mass is
calculated and these are added together

Such calculations can be complicated if forces are not parallel, but some shapes are easy to evaluate:

spherical shell (thin, hollow sphere) => attraction is exactly the same, as if all its mass was concentrated
or shrunk to its center [true for bodies outside the shell # at its center ca. zero due
to symmetry and counteracting pulling forces]

70

Newton’s Laws
•

The very basis on which the gravity method
depends on TWO laws of Newton.

Universal law of gravitation ULG
The force of attraction between any two bodies of
known masses is directly proportional to the
product of the two masses and inversely
proportional to the square of the distance between
their centers of masses.

Gravity, Expl.ravity

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