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 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


• 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).

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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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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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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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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

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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irregular body

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sheets (dykes or veins)

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horizontal sheet/slab

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horizontal half-sheet/half-slab

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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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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


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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


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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.

gravity, expl.ravity

Technology