Overview of Chapter 1: General theory of gravity Typical densities of rocks Newton s law, Geoid, variation with latitude Technical aspects: How does a gravimeter work? Measuring gravity and required corrections: instrument drift, tides, free-air, Bouguer- and terrain corrections, latitude correction, isostatic correction, Eötvös-effect Interpretation approach Case-studies EPS435-Potential-01-01
Gravity surveying measures variations in the Earth s gravitational field caused by differences in density of sub-surface rocks. Although referred to as the gravity-method it is actually the difference in acceleration due to gravity that is measured. The gravity-method is typically used in hydrocarbon exploration, for regional geological/geophysical studies, isostatic compensation studies, detection of subsurface cavities (archeo-geophysics), and in geodesy (shape of the Earth and other planets). While gravity was used in hydrocarbon exploration mainly in the early 20 th century, technological developments especially in the early 1980-ies pushed limits of accuracy to a point where airborne gravity surveys became routine. This has allowed aircraft-borne gravimeters to be used over otherwise inaccessible terrains. EPS435-Potential-01-02
Theory The basis on which the gravity method depends is encapsulated in two laws derived by Sir Isaac Newton, namely his universal law of gravitation and his second law of motion: Newton s Universal Law of Gravitation states that the gravitational force is proportional to the product of the two masses (M, m) attracting each other as well as inversely proportional to the distance (d) squared between them: r F = GMm 2 d Where G is the gravitational constant 6.67 10-11 N m 2 kg -2. (EQ 1.1) EPS435-Potential-01-03
Theory Newton s second law of motion states that a force is equal to mass (m) times acceleration (g): F = m g (EQ 1.2) Combining the two basic equations yields another simple expression: G M m F = 2 d G M g = 2 d = m g (EQ 1.3) Let s substitute d with the radius of the Earth (R) to describe acceleration on the surface of the Earth due to the mass of the earth itself; Thus the acceleration expressed with equation 3 is only dependent on the radius and the mass of the Earth. EPS435-Potential-01-04
Theoretically this means that the acceleration g should be constant over the Earth. In reality, however, gravity varies from place to place because the Earth has the shape of a flattened sphere (i.e. R is not constant), rotates, and has an irregular surface topography and variable mass distribution (especially near the surface). The shape of the Earth is a consequence of the balance between gravitational and centrifugal forces causing a flattening to form an oblate spheroid, mathematically referred to as ellipse of rotation. It is further helpful to define a hypothetical gravity surface, called the geoid. The shape of the surface that the oceans would form if left undisturbed by wind, tides etc. is defined as the geoid. Irregular mass distributions in the Earth disturbs (warps) the geoid so that it is not identical to the ellipse of rotation. EPS435-Potential-01-05
axis of rotation Exaggerated view of the difference between a sphere and the ellipse of rotation. Geoid Warp Sphere Ellipse of rotation Excess Mass Ellipse of rotation Example of the effect of excess mass: the geoid is warped upward relative to the ellipse of rotation. EPS435-Potential-01-06
Gravity units: The first measurements of the acceleration g due to gravity was made by Galileo in a famous experiment where he dropped weights from the top of the tower of Pisa. The normal value of g is 9.81 kg/s 2. In honor of Galileo the unit of acceleration due to gravity is 1 cm/s 2 = 1 Gal. Since the introduction of SI units, acceleration due to gravity has the official unit of 1 g.u. (gravity unit) = 0.1 mili Gal (mgal). EPS435-Potential-01-07
Variation of gravity with Latitude The value of acceleration due to gravity varies over the surface of the Earth not only because of sub-surface mass variations, but also because of the Earth s shape. As the polar radius (6357 km) is 21 km shorter than the equatorial radius (6378 km), the points at the poles are closer to the Earth centre and therefore the value of gravity is greater by about 0.7% than at the equator. But the Earth also rotates resulting in an additional centrifugal force, which is maximum at the equator. The rotational velocity at the equator is 1674 km/h and reduces to 0 km/h at the poles. The centrifugal acceleration g (rotational velocity ω squared times distance to the rotational axis g = ω 2 d) reduces the value of gravitational acceleration. EPS435-Potential-01-08
Centrifugal acceleration and variation of gravity with latitude (after Reynolds, 1997). Resultant gravity as an effect of centrifugal and gravitational acceleration (from Reynolds, 1997). EPS435-Potential-01-09
In 1930, the International Union of Geodesy and Geophysics (IUGG) defined a reference formula for gravity as function of latitude. This formula was again revised in 1967, referred to as GRS67 the Geodetic Reference System - 1967. The IUGG has set g 0 (reference gravity) to: g 0 (ϑ) = 9.78031846 [1+0.005278895 sin 2 ϑ 0.000023462 sin 4 (ϑ)] m/s 2 (EQ 1.4) If gravity data from before 1967 need to be compared to more recent data, a difference term needs to be applied: g(1967) g(1930) = (-172 + 136 sin 2 ϑ) μm/s 2 (g.u.). (EQ 1.5) EPS435-Potential-01-10
Gravity surveying is sensitive to rock density. In the following typical values for various types or rocks and minerals are given. However, laboratory measurements of rock/sediment/mineral density are in error relative to the true in situ densities. Rock samples are typically altered either by drying (loss of moisture) or by release of the in situ stress. Sedimentary rocks: There are at least 7 factors affecting density of sedimentary material. The values given in the brackets are the average change in density possible due to this physical factor: Composition (35%), cementation (10%), age (25%), depth of burial (25%), tectonic stresses (10%), porosity and pore-fluid type (10%). EPS435-Potential-01-11
Densities of common geologic materials (data from Telford et al., 1990; table from Reynolds, 1997) EPS435-Potential-01-12
Variation in density for different rock types (data from Telford et al., 1990; table from Reynolds, 1997) EPS435-Potential-01-13
Igneous and metamorphic rocks: Igneous and metamorphic rocks tend to be denser than sedimentary rocks, but there is considerable overlap. Density typically increases with silica content, so basic igneous rocks are denser than acid ones. Similarly, plutonic rocks tend to be denser than the volcanic equivalent. Crystal size Fine-grained (volcanic) Coarse-grained (plutonic) Silica content Acid Intermediate Basic Rhyolite Andesite Basalt 2.35 2.70 g/cm 3 2.40 2.80 g/cm 3 2.70 3.30 g/cm 3 Granite Syenite Gabbro 2.50 2.81 g/cm 3 2.60 2.95 g/cm 3 2.70 3.50 g/cm 3 EPS435-Potential-01-14
As gravity surveying is sensitive to density variations, the table to the left shows some material with commercial value, for which the gravity method may be used for exploration purposes. From Reynolds, (1997) EPS435-Potential-01-15
Measurement of Gravity Absolute and relative gravity Determination of acceleration due to gravity in absolute terms requires very careful experimental procedures, normally only achieved in laboratories. Two methods of measurement are used: the falling body and a swinging pendulum. A network of gravity stations has been established worldwide where absolute gravity is being determined. The network is referred to as the International Gravity Standardization Net 1971 (IGSN-71). It is thus possible to tie any regional gravity survey to absolute values by reference to the IGNS-71. However, it is the much easier measured relative variation in gravity that is of interest and value in exploration. EPS435-Potential-01-16
In gravity exploration surveys first a base station is selected (which can be related to an IGSN-71 absolute gravity station) 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. Spacing of gravity stations is critical so the subsequent interpretation of the data (recall sampling theorem). In regional surveys, stations may be located with a density of 2 3 per km 2, whereas hydrocarbon exploration surveys require 8 10 stations per km 2. In localized surveys where high resolution of shallow features is required, stations may be spaced 5 50 m apart and in micro-gravity surveys (cavity detection) stations may be as close as 50 cm from each other. For a gravity survey to achieve an accuracy of ±0.1 mgal, the latitudinal position must be known within 10 m and the elevation to within 10 mm! EPS435-Potential-01-17
The Canadian Gravity Standardization Net (CGSN) provides a nationwide gravity datum defined by more than 5000 control stations systematically distributed throughout Canada. The adjustment of the CGSN is based on the International Gravity Standardization Net (IGSN71), therefore the datum definition is considered accurate to several tens of microgals. http://www.geod.nrcan.gc.ca/products-produits/ggns_e.php EPS435-Potential-01-17a
Gravity meters: No single instrument is capable of meeting all requirements of every survey, so there is a variety of devices which serve different purposes. In 1749, Pierre Bouguer found that gravity could be measured with a swinging pendulum. By the nineteenth century, the pendulum was in common use to measure relative variations in gravity. The principle is simple: Gravity is inversely proportional to the square of the period of oscillation (T) and directly proportional to the length of the pendulum (L): g = 4π 2 L/T 2 (EQ 1.6) Portable pendulum systems were still used until the 1930s and had an accuracy of about 1 mgal. EPS435-Potential-01-18
Gravity meters: Since the 1930s, the pendulum instruments were replaced by instruments based on springs and small mass elements. Gravimeters are sophisticated and delicate spring balances from which a constant mass is suspended. The weight of the mass is the product of the mass and the acceleration due to gravity. The greater the weight acting on the spring, the more the spring is stretched. The amount of extension (δl) of the spring is proportional to the extending force, i.e. the excess weight of the mass (δg). [recall: weight = mass times acceleration!] EPS435-Potential-01-19
The constant of proportionality is the elastic spring constant κ. The relationship is also known as Hooke s law. As the mass is constant, variations in weight are caused by changes in gravity. By measuring the extension of the spring, differences in gravity can then be determined. As the variations in g are small (1 part in 10 8 ), the extension of any spring will also be extremely small. For example, for a 30 cm long spring, changes in length of the order of 30 nano-meters need to be measured. Consequently, gravimeters use some form of system to amplify the movement so that it can be measured accurately. Hooke s Law: extension of a spring δl = m δg / κ (EQ 1.7a) change in gravity δg = κ δl / m (EQ 1.7b) EPS435-Potential-01-20