Review questions
3.4 Reduction of gravity data
3.4.4 Variations in gravity due to the Earth’s rotation and shape
The Earth’s gravity field is stronger at the poles than at the equator by about 51,000 gu as a result of several effects. The Earth’s rotation produces a centrifugal acceleration directed outwards and perpendicular to its axis and is strongest at the equator. As a result the Earth’s shape is roughly ellipsoidal (Fig. 3.14a), with the equatorial radius (Requator) greater than
the polar radius (Rpole). As a consequence, the Earth’s surface
gets progressively closer to its centre of mass at higher latitudes. This proximity effect is greater than the attraction at the equator from the extra mass between the surface and the centre due to the larger equatorial radius, as expected
from Eq. (3.3). The centrifugal acceleration opposes the
acceleration due to gravity, so gravity is less at the equator (Fig. 3.14a). At progressively higher latitudes, the centrifugal acceleration decreases and becomes increasingly oblique to the direction of gravity. It is zero at the poles.
The surface of equal gravitational potential (seeSection 3.2) corresponding with mean sea level is known as the geoid. It is an undulating surface mainly influenced by variations in the distribution of mass deep within the Earth. It defines the horizontal everywhere and is an important surface for surveying. Numerous global and local geoids have been com- puted, a recent example being EGM96 (Lemoine et al.,1998). The direction of a suspended plumb line is the direction along which gravity acts (thefield lines in Fig. 3.2a). It is directed downwards towards the Earth’s centre of mass and is perpendicular to the geoid. It defines the vertical (seeFig. 3.15a).
The undulating geoid is not a convenient surface to represent the Earth for geodetic purposes. Instead a smooth ellipsoid is used that is a‘best-fit’ to the geoid (Li and Göetze, 2001). It is known as the reference ellipsoid, also referred to as the reference spheroid, and is the ideal- ised geometric figure to which all geographical locations are referenced in terms of their geographic coordinates,
Day (2005) Perth Toronto 1 gu a) Moon Attraction of celestial bodies Sun Decreased gravity Increased gravity b) Earth <REquator RPole >REquator
1st June 2nd June 3rd June 4th June
Figure 3.13 Earth tides. (a) Schematic illustration of the gravitational effects of the Sun and the Moon on gravity measurements on the Earth’s surface. (b) Earth tides at Toronto (northern hemisphere) and Perth (southern hemisphere) over a 4-day period in June 2005. Note the approximate 12-hour periodicity.
latitude and longitude (Fig. 3.15a). There have been a number of determinations of the best-fitting reference ellipsoid. The geocentric Geodetic Reference System 1980 (GRS80) spheroid (similar to the World Geodetic System 1984 (WGS84) spheroid) is used as the datum for positioning and gravity surveying. The separation between the WGS84 spheroid and the undulating geoid is shown in Fig. 3.15b. Typically the separation is just a few tens of metres.
The variation in gravity with latitude (ϕ) is defined in terms of gravity on the surface of the spheroid, and is known as the normal or theoretical gravity (gϕ). It can be calculated using the International Gravity Formula, which is periodically updated. For the case of the GRS80 spher- oid, the Gravity Formula 1980 given by Moritz (1980) is:
gϕð1980Þ¼ 9,780,326:7715 1+ 0:001 931851353sin 2ϕ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10:006 694 380 022 9sin2ϕ p gu ð3:13Þ Normal gravity across the Earth is shown inFig. 3.14b.
3.4.4.1 Latitude correction
The variation in gravity due to the difference in latitude between the survey station and the survey base station is compensated with the latitude correction. When latitude and the absolute gravity are known at all the survey sta- tions, the latter obtained by tying the base station to a permanent absolute gravity mark or when making absolute gravity measurements, the latitude correction is the normal gravity calculated for that location (Eq. (3.13)). It is sub- tracted from the drift, Eötvös and tide-corrected reading. Since latitude is spheroid-dependent it is important that position is determined according to the correct spheroid (see Featherstone and Dentith,1997).
Alternatively, the latitude correction can be obtained from the latitudinal gradient of normal gravity. At the scale of most mineral exploration activities, the change in normal gravity is sufficiently smooth and gradual that it appears as a very small linear increase in the direction of the nearest geographic pole. The latitude gradient a)
Stationary Earth Rotating Earth
Decreasing gravity Constant gravity REquator RPole RPole= REquator R Pole< REquator REquator RPole 180°W 90°N 60°N 30°N 30°S 60°S 90°S 0° 180°E 120°E 60°E 120°W 60°W 0° Gravity (gu) b) 9,810,000 9,820,000 9,800,000 9,790,000 (9,780,327) 9,820,000 9,830,000 9,790,000 9,832,187 9,780,327 9,840,000 9,810,000 9,780,000 9,810,000 9,800,000 9,830,000 Ce ntr ifu g a l fo rce
Figure 3.14Gravity variations due to rotation and shape. (a) Schematic illustration of the effects of the Earth’s rotation and shape on gravity. (b)
Variation of normal gravity with latitude calculated from the International Gravity Formula 1980. Units are gu.
represents the change in gravity with north–south distance from a base station and is given by:
gϕðNSÞ ¼ 0:00812 sin 2ϕ gu=mðNSÞ ð3:14Þ
whereϕ is the latitude (negative for southern hemisphere) of the base station. This formula is accurate enough for most exploration applications for distances up to about 20 km north and south of the base station (preferably located central to the survey area). It is useful where the latitudes of the gravity stations are unknown. The latitude correction at a gravity station is simply the north–south latitude gradient multiplied by the north–south distance (in metres) between the station and the base station. Since gravity increases towards the poles (Fig. 3.14b), the correction is subtracted for stations located on the pole side of the base station and added for stations on the equatorial side.
Station location needs to be known to about 10 m north–south in order to calculate the latitude correction to an accuracy of 0.1 gu.