Section 2: Gravity Surveying
Gravity surveys measure the acceleration due to gravity, g. Average value of g at Earth’s surface is 9.80 ms-2.
Gravitational attraction depends on density of underlying rocks, so value of g varies across surface of Earth.
Density, r, is physical parameter to which gravity surveys are sensitive.
Newton’s Law of Gravitation
Gravity surveying is based on Isaac Newton’s Universal Law of Gravitation, described in Principia Mathematica in 1687.
Newton’s Universal Law of Gravitation
The force of attraction between two bodies of known mass is directly proportional to the product of the two masses and inversely proportional to the square of the distance between them.
where M and m are the masses of the two bodies, r the distance separating them, and G=6.67 x 10-11 Nm2kg-2 is the gravitational constant.
On the Earth:
For a mass on the surface of a uniform spherical Earth of mass M and radius R, the gravitational attraction on a small mass m is given by:
g m is the weight of the mass, and g is the acceleration due to gravity, or just gravity. g = 9.80 ms-2 on average.
Units of Gravity
Galileo made the first measurement of the acceleration due to gravity by dropping objects from the leaning tower of Pisa.
1 Gal = 1 cm s-2
Modern gravity meters are extremely sensitive and can measure g to within 1 part in 109. (Equivalent to measuring the distance from the Earth to the Moon to within 1 metre).
1 mGal = 10-3 Gal = 10-3 cm s-2
10 g.u. = 1 mGal
Both mGal and g.u. are commonly utilised in gravity surveying.
What is the value of g in mGal?
g = 9.80 ms-2 = 9.8 x 106 mm s-2 = 9,800,000 g.u.
= 980,000 mGal
Accuracy of Gravity Measurement
On land: ± 0.1 g.u.
At sea: ± 10 g.u. (due to motion of ship)
Shape of the Earth
If the Earth were a uniform sphere g would be constant. However, gravity varies because the density varies within the Earth and the Earth is not a perfect sphere.
REquator = RPole + 21 km = 6378 km
· The Earth’s shape is described mathematically as an ellipse of rotation, or an oblate spheroid.
The sea-level surface, if unaffected by tides or winds, is called the geoid.
Geoid vs. Ellipse of Rotation
The difference between the average geoid at a latitude around the earth shows the effect of the long wavelength variations.
There are also extremes in the geoid over India, -80 m, and the western Pacific, +60 m, relative to ellipse of rotation.
Variation of Gravity with Latitude
Gravity is 51860 g.u. greater at the poles than at the equator. The acceleration due to gravity varies with latitude due to two effects:
For a uniform ellipse of rotation, the measured gravity is the resultant of the gravitational attraction vector and the centrifugal acceleration vector.
International Gravity Formula
The angle f defines the geographic latitude.
In 1743, Clairaut deduced a formula that expressed the variation of gravity with latitude.
This has been incorporated into the International Gravity Formula:
where g0 is the gravity at sea level at the equator and f the latitude.
The most recent standard derived from the IGF is the 1967 Geodetic Reference System (GRS67), given by:
g0 = 9.78031846 m s-2
a = 0.005278895
b = 0.000023462
Note that there is an older 1930 standard, and surveys that use this will exhibit differences from the 1967 standard unless corrected (see Reynolds Box 2.4).
Densities of Geological Materials
We must know the density of typical subsurface rocks accurately to interpret gravity data. The table below gives common ranges and avergae values for density in Mg m-3.
Densities of Sedimentary Rocks
Sedimentary rocks exhibit the greatest range of density variation due to factors such as:
Typically the contrast between adjacent sedimentary layers is less than 0.25 Mg m-3.
Density is increased by depth of burial:
Densities of Igneous Rocks
Igneous rocks tend to be denser than sedimentary rocks, with the density controlled primarily by silica content:
The range of density variation tends to be less than in sediments as porosities are typically lower.
Measurement of Gravity
There are basically two type of gravity measurement:
Measured under laboratory conditions using careful experiments employing two possible methods:
Used to provide absolute values of g at network of worldwide sites such as National Physical Laboratory in UK or National Bureau of Standards in USA. (International Gravity Standardisation Net 1971, IGSN 71).
In most applications, only the variation of gravity relative to a base station (which can often be related to IGSN 71) is necessary.
Gravity readings are recorded at secondary stations such that the difference relative to the base station is well known.
The spacing of gravity stations varies:
On land, achieving an accuracy of ±0.1 mGal (or 1g.u.) requires that:
Many different instruments for measuring ralative gravity
Pendulum-Based Gravity Meters
Gravity first measured using a pendulum by Pierre Bouguer in 1749. Method commonly used up to 1930s in hydrocarbon exploration.
Period of a Pendulum
Gravity is inversely proportional to the square of the period of oscilation, T, of a swinging pendulum:
where L is length of pendulum.
If pendulum swung under identical conditions at two locations, relative change in g can be found:
Gravimeters, essentially a mass suspended from a sophisicated spring balance, have been used to measure relative gravity since 1930s.
As weight of mass (mass x gravity) increases, the spring is stretched.
Amount of extension of spring, dl, is proportional to extending force.
In gravimetry, extending force is change in gravity, dg, and spring constant, k, is known:
Variations in g are small, so need to meaure very small values of extension. For 30 cm long spring, change in length is ~3x10-8m (30 nm, which is less than wavelength of light at ~500 nm).
Mechanism for amplifying spring extension required so it can be measured.
Stable gravimeters consist of a mass at end of a beam, which pivots on a fulcrum, and is balanced by a tensioned spring.
Changes in gravity affect weight of mass, which is balanced by restoring force of spring.
Beam is pivoted on main spring. A beam of light is reflected from the mass to a photoelectric cell. Deflection of mass, displaces light beam and changes voltage in circuit.
Retensioning auxiliary spring restores beam to null position, i.e. same position at which all measurements made.
Stable Gravimeters Using Electrical Amplification
Some gravimeters, including the common Scintrex CG-3, use the small extension of the mass to change the capcitance in an electric circuit.
Mass is in form of a bobbin with two metal plates suspended between two other metal plates.
Scinntrex CG-3 Gravimeter
CG-3 operates on same principle, but uses feedback circuit to control current to plates that restores mass to null position.
Unstable (Astatic) Gravimeters
In a stable system, mass will return to equilibrium position after small disturbance. In unstable system, mass continues to move.
Unstable gravimeters use mechanical instability to exaggerate small movement due to change in gravity.
LaCoste-Romberg Unstable Gravimeter
Worden Unstable Gravimeter
Corrections to Gravity Observations
Instrument Calibration Factor
Reading on gravimeter must usually be multiplied by calibration factor to get value of observed gravity, gobs.
Reduction to Geoid
Before interpretation, raw gravity data must be corrected to common measurement datum such as mean sea level (geoid).
Difference between the observed anomaly and the value of the International Gravity Formula, e.g. GRS67, at the same location is the Gravity Anomaly with which we work.
Summary of Gravity Data Corrections
Several corrections must be applied to observed gravity to obtain sea level reference and anomaly:
Gravimeters are very sensitive instruments. Temperature changes and elastic creep in springs cause reading to change gradually with time.
Drift is monitored by repeating reading at same station at different times of day, perhaps every 1-2 hours, to produce a drift curve.
Instrument drift correction for each station can be estimated from drift curve.
Drift curve for survey shown above. For reading taken at 12:30 hrs, observed gravity reading should be reduced by d.
Drift values are typically < 10 gu per hour. Larger values indicate an instrument problem.
Solid Earth responds to pull of Sun and Moon just like oceans, but movement is much less.
Pull of Sun and Moon large enough to affect gravity reading. Changes gobs with period of 12 hours or so.
Earth tide corrections can be corrected by repeated readings at same station in same way as instrument drift.
Correction can also be made using published tables, e.g. Tidal Gravity Corerctions for 1991.
Gravity anomaly values obtained by subtracting theoretical value of gravity defined by International Gravity Formula, gf
Local Latitude Correction
Approximate correction can be applied to small-scale surveys (<100 km), not tied in to absolute gravity network through base station readings.
gu per km
Corrects for reduction in gravity with height above geoid, irrespective of nature of rock below.
Free-air correction is difference between gravity measured at sea level and at an elevation, h, with no rock in between.
Free-air Correction = 3.086 g.u./m = 3.086 h g.u.
Free Air Anomaly
Free-air Anomaly is obtained after application of latitude and free-air corrections.
Free-air correction takes no account of rock mass between measurement station and sea level.
Bouguer correction, dgB, accounts for effect of rock mass by calculating extra gravitational pull exerted by rock slab of thickness h and mean density r.
where r is in Mg m-3 and h is in metres.
where h is the water depth.
BA is Free-air Anomaly after subtraction of Bouguer correction.
Bouguer correction quite sensitive to value of density used.
With 250 m elevation, an 0.1 Mg m-3 error in density will produce an error of 10gu.
The Bouguer Anomaly should show no correlation with topography, but an incorrect choice of density will reflect the topography.
Nettleton developed method of calculating BA using various trial density values. Correct density corresponds to BA correction that shows no correlation with topography.
Elevation is 76 m (250 ft). Correct density is 2.3 Mg m-3
Bouguer correction assumes subdued topography. Additional terrain corrections must be applied where measurements near to mountains or valleys.
If station next to mountain, there is an upward force on gravimeter from mountain that reduces reading.
If station is next to valley, there is an absence of the downward force on gravimeter assumed in Bouguer correction, which reduces free-air anomaly too much.
In both cases, terrain correction is added to Bouguer Anomaly.
Example of terrain correction
Terrain corrections can be computed using transparent template, called a Hammer Chart, which is placed over a topgraphic map.
If gravimeter is in moving vehicle such as ship or plane, it is affected by vertical component of Coriolis acceleration, which depends on speed and travel direction of vehicle.
where f is geographic latitude and V is vehicle speed in knots.
If no lateral density variations in Earth’s crust, Bouguer Anomaly would be the same, i.e. Earth’s gravity at the equator at geoid.
Bouguer anomaly positive over oceans, negative over mountains.
If crust is floating on mantle like an iceberg in ocean, total mass summed over any vertical column is the same. (Isostasy)
Airy proposed that crust is thicker beneath mountains and thinner beneath the oceans.
Excess mass under the oceans from a shallower, high density mantle. Mass deficiency beneath mountains due to crustal root.
Pratt proposed that observation could be explained by lateral changes in density within a uniform thickness crust.
Airy Isostatic corrections can be applied to remove these long wavelength variations, isolating upper crustal anomalies.
Regional and Residual Anomalies
Bouguer Anomaly maps contain:
Residual must be separated for interpretation. Smoothed trend is fitted to BA graphically or by computer.
Example of Bouguer Anomaly and regional field:
Interpretation of Gravity Anomalies is Non-Unique
As with interpretation of many geophysical data, interpretation of gravity is ambiguous.
Residual anomaly from a 600m radius sphere of 1.0 Mg m-3 at 3 km depth also be produced by each of the bodies shown.
Interpretation can proceed by forward modelling or inversion methods.
Calculating Gravity Anomalies of Simple Bodies
Gravity anomaly of a body can be calculated by summing contribution of its component elements using computer.
For simple bodies, anomaly can be calculated simply:
Sphere or Point Mass
For point mass at distance r, gravitational attraction given by:
Gravimeter only measures vertical component, so anomaly is projection onto vertical:
Since attraction due to sphere is same as that of same mass at centre, this is formula for anomaly of buried sphere. (3-D case)
Can integrate above formula to get result for horizontal cylinder(2-D case):
where m is mass per unit length.
Gravity Anomalies of Spheres and Cylinders
Anomaly shape can be plotted using formula. Anomaly from sphere decays more quickly than that of horizontal cylinder.
Vertical cylinder has different shape with steeper flanks:
Depth Estimation by Half-Width Method
Using the formulae for the anomalies due to various bodies, it is posible to estimate the limiting depth of a body.
Limiting depth is maximum depth at which top of body could occur to produce anomaly. (Body could be shallower).
Half-width, X1/2 , is the distance from the centre of an anomaly at which amplitude has decreased to half its peak value.
If anomaly is spherical:
If anomaly is horizontal cylinder:
If anomaly is vertical cylinder:
If anomaly is thin steeply dipping sheet:
Depths are overestimates as based on centre of mass of body.
Depth Estimation by Gradient-Amplitude Method
Can obtain estimates of limiting depth from maximum slope also.
If value of maximum slope, Dg’max , estimated:
For 3-D body:
For 2-D body:
Anomalous Mass is difference between mass of anomalous body and mass of body replaced with host rock.
If Dgi is value of residual anomaly in segment,
Total Anomalous Mass given by:
Actual mass can be determined simply from anomalous mass if densities of anomaly and host rock are known:
where r1 is density of body, and r0 density of host rock.
Application to Salt Domes
Average density of salt, 2.2 Mg m-3, is less than most sediments in a basin, so salt often rises in diapir due to its bouyancy.
Makes good target for gravity surveys, and will show up as a bullseye anomaly.
Example: Mors salt dome, Denmark
Studied for radioactive waste disposal. Dots are gravity stations.
From profile across anomaly:
Assuming salt dome represented by sphere, limiting depth, i.e. depth to centre of mass = 4.8 km
Assuming density contrast of –0.25 Mg m-3, radius of sphere estimated at 3.8 km. So top of salt at 1 km depth.
Best model obtained by forward modelling of gravity data:
Compare inferences from gravity with best model derived from all methods, including seismic reflection and drilling.
Application to Massive Sulphide Exploration
Massive sulphide ore bodies have high densities due to minerals present.
Can show as gravity high in residual anomaly.
Example: Faro Pb-Zn deposit, Yukon
Gravity proved to be best geophysical technique to delimit deposit
Tonnage of 44.7 million estimated from gravity, which compares with drilling estimate of 46.7 million.
Application to Detection of Underground Cavities
Buried cavities due to old mine workings can be a significant hazard!
Result of catastrophic failure of roof of ancient flint mine in chalk.
Cavities can be good target for micro-gravity due to high density contrast between a void, or rubble-filled void, and host rock.
In practice, many anomalies are greater than predicted by theory.
Example: Inowroclaw, Poland
Karst caverns in subsurface composed of gypsum, anhydrite, dolomite, and limestone. Develop towards surface and destroy buldings.
Density contrasts are around -1.8 Mg m-3 and -1.0 Mg m-3 for void and rubble-filled void.
Rubble-filled void should not have been detectable from calculated anomaly.