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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 assessment1
Apple Falling on Newton’s Head
Sir Isac Newton
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• 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 itssurface is the same as if there were two “pointmasses” separated by the Earth’s radius, RE
(~6370 km)
To avoid dependence on the surface mass, mS, 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
Acceleration due to gravity
On equating both laws as they represent F, g can be obtained as
Acceleration due to gravity
• 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.
Acceleration due to gravity
• 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).
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Rock Densities
Earth’s Mass
Earth’s interior made of denser material;e.g. mostly iron in the core [meteorite studies]
Can you calculate Density and Mass of the Earth?
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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]
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Gravity Units
gravity variations anomalies are VERY SMALL
δ, very small quantity/differenceΔ, considerable difference
Gravity Units: milliGals
In the example: the buried sphere anomaly is 0.1 mGal
Calculate gravity effect
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Gravity Units
gravity variations anomalies are VERY SMALL
δ, very small quantity/differenceΔ, considerable difference
Gravity Units: milliGals
In the example: the buried sphere anomaly is 0.1 mGal
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Gravity Anomalies of Specific Bodies (3)
sphere & horizontal cylinder at different depths
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Gravity Anomalies of Specific Bodies (2)
buried sphere
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Worden Gravity Meter
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Lecoste Romberg gravimeter
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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)
Magnitude of “g”
At equator (g) = 978.0318 cm/sec2
At poles (g) = 983.152 cm/sec2
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
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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):
For a gravity survey < 10x km variation proportional to distance:
λ = latitude of station
λ = latitude where survey is carried out
only the N-S distance matters, and as g increases towards the poles, the correctionis ADDED to all measurements on the equator side of the base station, and SUBTRACTEDfrom all measurements on the polar side of the base station, in order to cancel this decrease
Corrections to Observed Gravity
• The various corrections are:1. Instrumental drift2. Latitude3. Free air 4. Bougeur5. Elevation6. Tidal effect7. Terrain/Topography8. Eotuos
9. Isostatic
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Data Reductions - Corrections (3)
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)
ρ = density in Mg/m3 h = slab thickness (m)
(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
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Data Reductions - Corrections (4)
Topographic corrections:
combined elevation correction (free-air & Bouguer)
ρ = density in Mg/m3 h = slab thickness (m) positive above datum
Free-air correction: is ADDED, to correct for the reduction of g at greater heights in the airBouguer 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 +
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
Terrain Correction
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)
Tidal Effect
Tidal Correction
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 Dry
(a) Sedimentary 1.54-2.30 1.98-2.70
(b) Igneous rocks 2.24 3.17
(c) Metamorphic 2.60 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.
Nettleton’s Method
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 & Residual
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
Graphical Method
Example of a regional-residual gravity anomaly separation using graphical smoothing
Graphical Method
Topography & Gravity Anomaly
2-D Gravity (Contours)
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.
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Gravity map of Oman
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Gravity Anomalies of Specific Bodies (1)
irregular body
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Gravity Anomalies of Specific Bodies (4)
sheets (dykes or veins)
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Gravity Anomalies of Specific Bodies (5)
horizontal sheet/slab
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Gravity Anomalies of Specific Bodies (6)
horizontal half-sheet/half-slab
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Gravity Anomalies of Specific Bodies (6)
horizontal layers offset by vertical faulting
Δρ = ρ1 - ρ1 = 0
Δρ = ρ5 – ρ5 = 0
Δρ = ρ3 – ρ1
Δρ = ρ3 – ρ3 = 0
Δρ = ρ2 – ρ1
Δρ = ρ2 – ρ3
Δρ = ρ4 – ρ3
Δρ = ρ4 – ρ5
Δρ = ρ2 – ρ1
Δρ = ρ2 – ρ3
Δρ = ρ4 – ρ3
Δρ = ρ4 – ρ5
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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 thenormal (background field) value
[from Chapter 2][from Chapter 2]
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Data Reductions - Corrections (2)
Eötvös correction:needed only if gravity is measured on a moving vehicle such as a ship, and arises because themotion produces a centrifugal force, depending upon which way the vehicle is moving
v = vehicle speed (km per hour) λ = latitudeα = direction of travel measured clockwise from north
only the E-W motion matters; the correction is POSITIVE, for motion from east to west
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Residual & Regional anomalies
from Chapter-2from Chapter-2
residual anomaly = observed field – regional field
signal vs. noise residual anomaly vs. regional field/anomalyconcepts depending on the survey target interest
backgroundvalue of g
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signal vs. noise residual anomaly vs. regional field/anomalyconcepts depending on the survey target interest
from Chapter-2from Chapter-2
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Inversion/Inverse Modelling: trying to deduce the form of a causative body from the anomaly
problems: non-uniqueness (multi-solutions) [noise, measuring errors, resolution]
Forward Modelling: (previous) 2D, 21/2D, 23/4D, 3D
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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)
anda half-slab with a positive density contrast from one side of a fault produces the same anomaly with a half-slabfrom the other side of the fault with a similar but now negative density contrast
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Modelling
GRAVITY anomaly is: (1) measured at or near the surface, and (2) represents a physical andnot a direct geological quantity, thus we need two further steps:
MODELLING & INTERPRETATION
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
a Model is a simplification of the geology
Forward Modelling: 2D, 21/2D, 23/4D, 3D
Model: # Causative body:simple shape irregular shapeabrupt boundaries gradational boundariesuniform physical properties gradational phys. properties
but simplification is not always a drawback, as it may emphasise the essential features
from Chapter-2from Chapter-2
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Depth rules
shape of body is known (a-c)
to find d (depth of buried body)
half-width:half the width athalf the peak height
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Gravity SurveyingSatellite Radar Altimetry
Free-air Gravity Anomaly
Earth’s gravitational field (Geoid)[measurements from GRACE orbital satellite]
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Gravity Surveying
Land surveys
Marine surveys(together with seismic reflection data collection)
Airborne surveys
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SEISMIC VELOCITY vs DENSITY
Nafe & Drake diagram
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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
base station
instrument drift
base stationdrift curve
Theoretically computed tidal variations
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Gravity force and Gravity Potential
Gravitational Force
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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
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
“point mass” = bodies so small in extent compared to their separation that all parts of one mass are closely same distance from the other
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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]
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.