Gravity exploration
2.3 The Earth’s gravity field
the subsurface are a few examples of local applications of
the gravity method.
In general, the gravity method is implemented in drill- hole, terrestrial, marine, airborne, and satellite survey envi- ronments. It is based on measurement of perturbations in the Earth’s gravitational field caused by lateral variations in density of subsurface units. The component of gravity which is derived from these density variations is extracted from the measured signal and interpreted in geologic terms by inversion. The long history of development has pro- duced an effective process for using the gravity method in subsurface exploration.
2.3 The Earth’s gravity field
The gravity method is a passive exploration method in that it is based on a natural and ever-present gravitational force field. It requires no active energy source, as is nec- essary with seismic and most electrical exploration meth- ods. This is an obvious advantage, but also a disadvantage because the field cannot be modified to suit the particular application. And although the Earth’s gravity field is ever- present, it varies both spatially and temporally to distort the changes in the anomaly fields caused by local subsur- face conditions of interest in gravity surveying. Thus, to produce interpretable measurements, these variations must be removed from the observed data.
2.3.1 Gravitational force
The gravity force field is caused by the fundamental phe- nomenon of gravitation that attracts bodies toward each other. Specifically, a mass in the presence of another body, like the Earth, has energy due to the gravitational attraction which is called gravitational potential. This energy causes objects to accelerate towards each other if they are free to move. The force of gravity between objects on the surface of the Earth is not observed because their mass is so much smaller than the Earth’s, and thus their attractive effect on each other is negligible.
It is the mass of an object, a property determined by its volume, atomic content, and the packing of the atoms, that is the source of the gravitational field. In geophysical considerations, mass is replaced by the product of density (in other words the mass per unit volume) and the vol- ume of the object. Thus, density is the operative material physical property for the gravity method. Gravitational force is proportional to the product of the masses of the attracting bodies (Figure 2.2). It is not affected by the pres- ence of material between the masses, temperature, physical state, or other environmental conditions. In the case of the
m2
+ +
Newton’s First Law: Fg∝ (m1m2)/r2 m1 Fg
r Fg
FIGURE 2.2 Illustration of two bodies attracted toward each other with a gravitational forceFgproportional to the product of their massesm1andm2and inversely proportional to the square of the distancerbetween the centers of the masses.
Earth, the gravity force field is dependent on the attraction between the mass of the Earth and the mass of an object.
The weight of an object, or the gravitational force on it, is the attraction between the Earth and the object. Thus, the mass of an object is invariant, but its weight is dependent on the attractive body (here, the Earth). On the surface of the Moon, an object has the same mass as on the Earth, but its weight is one-sixth of that on the Earth because the mass of the Moon is only about 1.25% of that of the Earth and its radius is about a quarter of the Earth’s radius.
In the seventeenth century, Newton showed that grav- ity force is inversely dependent on the distance between objects. In the case of sources whose dimensions are small with respect to the separation, e.g. the planets revolving around the Sun, the force is inversely proportional to the square of the distance between the center of the objects.
Thus, if the distance between bodies is doubled, the force on them is reduced to a quarter. Newton showed from Kepler’s observations of the motion of the planets around the Sun that the radial attractive force of the Sun on the planets is
Fg= 4π2
r2 m, (2.1)
where the vectorFgis the attractive force, the scalarmis the mass of the planet, andris the displacement vector describing the distance between the center of the Sun and the planet. Furthermore, because action and reaction are equal and opposite by Newton’s Third Law, the planets must exert a similar force on the Sun. Thus,
Fg=Gm1m2
r2 , (2.2)
which is Newton’s universal law of gravitation where, in SIu,Fis the force of attraction in newtons (N) between masses m1 and m2, in kilograms, that are separated by a distance rin meters. The universal gravitational con- stant Ghas the value (6.674 215±0.000 092)×10−11
24 The gravity method
TABLE 2.1 Parameters commonly used in the gravity method in equivalent CGS and SI systems of units.
Gravity parameter CGSu SIu
Universal gravitational constant 6.674×10−8cm3/g·s2 6.674×10−11m3/kg·s2
Force of attraction 105dynes newton (N)
Gravitational acceleration cm/s2 10−2m/s2
milligal (mGal) 10−5m/s2 microgal (μGal) 10−8m/s2
Density g/cm3 103kg/m3(metric tonne)
(m3×kg−1×s−2) or (N×m2/kg2) in SIu (Gundlach and Merkowitz, 2000). This constant can be deter- mined by a variety of field and laboratory methods, with most recent precise determinations made in the laboratory by measuring the force between two masses. An additional result of Newton’s studies is that the attraction on the sur- face of the Earth is equivalent to the situation where the mass of the Earth is concentrated at its center.
The gravitational force of the Earth cannot be specified or measured independently of mass, so the acceleration of a mass falling in response to the gravity field is used to describe the gravitational force. The accelerationaof a freely falling mass in the Earth’s gravitational field is related to the gravitational forceFgthrough Newton’s sec- ond law
Fg=m1a, (2.3)
which gives the force acting on the massm1 due to the presence of another massm2. In the Earth’s gravitational field, the force on the bodym1is exactly the same as if it were being accelerated at the rate
a=Fg/m1=Gm2
r2 , (2.4)
wherem2in the terrestrial situation is the mass of the Earth.
Therefore, the attraction of the Earth can be considered the force per unit mass and is equivalent to the acceleration caused by the free fall of the mass in the gravitational field of the Earth. The gravitational accelerationais the quantity measured in geophysical exploration where it is commonly defined by the special symbolgor
g=Fg/m1=Gm2
r2 . (2.5)
This equation holds only for homogeneous or radially stratified spherical bodies and equivalent compact sources.
The relationship changes from this simple inverse square law when the acceleration is integrated over more complex sources.
The force of gravitational attraction is always positive, that is all objects are attracted towards each other, but varia-
tions in gravity may be negative in geophysical exploration as a result of a lower than normal mass due to horizontal variations in density within the Earth. These negative val- ues are not a result of repulsion between objects, but simply a lesser attraction than normal. This concept emphasizes that what is measured and analyzed in the gravity method of geophysical exploration is the relative spatial difference in gravity.
2.3.2 Gravity units
Gravitational acceleration or simply gravity is measured in N/kg or m/s2in SIu. Table 2.1 presents the equivalent units of common parameters used in the gravity method in both CGSu and SIu. In geophysical exploration, the gal is used for gravitational acceleration and is equivalent to 0.01 m/s2 in SIu or 1 cm/s2in CGSu. The gal honors Galileo Galilei who in the late sixteenth and early seventeenth centuries pioneered the investigation of the motion of the planets around the Sun and the nature of gravitation. However, the gal is a large unit compared with the changes in the grav- itational acceleration caused by variations in the Earth’s subsurface masses. As a result, the unit milligal (mGal) or one-thousandth of a gal is used in geophysical exploration, while in engineering geophysics and other high-sensitivity studies the microgal (μGal=0.001 mGal) is frequently used as the unit because of the very small magnitude of many of the significant anomalies. Another unit sometimes used in gravity exploration in the petroleum industry is the gravity unit (g.u.), which is equivalent to 0.1 mGal. The use of the g.u. should be avoided because of possible con- fusion with the unit milligal commonly used in gravity exploration.
The Earth’s gravity field includes spatial variations due to the size, shape, and rotational properties of the Earth and temporal variations related to the differen- tial gravity effects of the Moon and Sun on the Earth.
Every gravity observation at or near the Earth’s sur- face includes these normal gravity effects, which in the
2.3 The Earth’s gravity field 25
ξ Angle of geodetic
latitude Tangent to ellipsoid
Ellipsoid
(Geoid – ellipsoid) = N (geoidal undulation)
N
Geoid
Topography
Ellipsoidal flattening = f = (a – b)/a = 1/298.26
Sea level
Bath ymetr
y
Ellipsoid
Equatorial radius = a = (b + 21 km)
Polar radius = b = (a – 21 km)
Angle of geocentr ic latitude
Angle of
Vertical (
⊥⊥ geoid) (⊥ Ellipsoid) (through center of ear
th)
Equipotential
surf ace
(geoid)
geodetic latitude Angle of
“Deflection ofver tical”
astronomic latitude
FIGURE 2.3 Gravimeters measure vertical gravity (g) relative to the geoid (dashed line), which is the equipotential surface represented by mean sea level in the oceans. Normal gravity (gN) is defined perpendicular to the oblate ellipsoid or spheroid of revolution (full line) that is a mathematical approximation of the geoid. Geocentric latitude, which is also called co-latitude, is referenced to the center of the Earth, whereas astronomic latitude is measured relative to the true local vertical defined by the geoid. The deflection of vertical is the angular difference between the verticals observed on the geoid and defined by the ellipsoid, whereas height differences (N) between geoid and ellipsoid are geoidal undulations. Adapted fromS h e r i f f(2002) with surfaces not to scale.
case of spatial variations and some temporal variations are large compared with many geological gravity effects.
They must be removed from gravity measurements to iso- late the gravity effects of subsurface targets for analysis.
The next two subsections provide general views of the terrestrial gravity field and its principal variations over the Earth; these are developed in greater detail in later chapters as needed to address the acquisition, processing, and interpretation of gravity data for Earth exploration purposes.
2.3.3 Spatial variations
The normal gravitational accelerationgN of the Earth is modeled in terms of a mathematical oblate ellipsoid that
best fits the geoid (Figure 2.3). The geoid is defined as a hypothetical surface from which topographic heights and ocean depths are measured. It is represented by sea level in the oceans and its continuation into the continents. It is parallel to the surface defined by a spirit level, and the net acceleration on the surface of the Earth is directed per- pendicular to it. It is often described as the gravitational equipotential surface which most closely corresponds to the shape of the Earth. That is, the summation of the poten- tial of the gravitational attraction and centrifugal force of the Earth’s rotation is constant on this surface which most closely corresponds to the ocean’s surface. The gravita- tional accelerations due to these forces are not constant on this surface because the force is a function of the spatial rate of change of potential (e.g., Equation 3.1).
26 The gravity method
-- -
-
-
+
+
+
90°
60°
30°
0°
–30°
–60°
90°
60°
30°
0°
–30°
–60°
–90°
0° 30° 60° 90° 120°
EGM96 Geoid heights (m)
–100° –80° –60° –40° –20° 0° 20° 40° 60° 80° 100°
150° 180° 210° 240° 270° 300° 330° 360°
0° 30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330° 360°
FIGURE 2.4 Earth Gravity Model-08(EGM08) geoidal undulation map of the Earth. Courtesy of the National Geospatial-Intelligence Agency (NGA). A colored version of this map is available at www.cambridge.org/gravmag.
The maximum deviation of the geoid from the refer- ence ellipsoid is minor compared with the radius of the Earth, about±100 m, as illustrated by the geoidal undu- lation map of Figure 2.4. This is only roughly one part in 64,000 of the average radius of the Earth. As a result the deflection of the true vertical (the angleξ in Figure 2.3), which is the angular difference between perpendiculars to the geoid and the reference ellipsoid, is extremely small, reaching maximum values of little more than one minute of arc.
Normal gravity varies from the equator to the poles of rotation by about 0.5%, changing by slightly more than 5 gal from approximately 978 gal at the equator to 983 Gal at the poles. The source of this planetary variation is two- fold. First, and more important because it accounts for more than 3 of the 5 Gal, is the change in the centrifu- gal force over the Earth’s surface caused by the Earth’s rotation around its axis as shown in Figure 2.5. This is a pseudo-force that results from the rotation of the coordi- nate system that is fixed to the Earth and whose origin is the center of mass of the Earth. Note that Equation 2.5 is for a non-rotating body. The centrifugal force imparts an acceleration to a body that is equal to the product of the square of the angular velocity of the Earth and the perpendicular distance between the axis of rotation and the body (radius of gyration). This acceleration is insepa- rable from the gravitational acceleration of the Earth and
b a
Geoid Ellipsoid
Components of the normal gravity field gN
gE
gC gN
FIGURE 2.5 The globally best-fitting ellipsoid (solid line) to the geoid (dashed line) with equatorial and polar axesaandb, respectively, is the reference surface for defining the rotating Earth’s normal gravitygNwhich is the vector sum of the mass effects of the EarthgEand the centrifugal forcegCdue to the Earth’s rotation about its polar axis – i.e.gNis the gravity effect of a homogeneous non-rotating Earth over the surface of the ellipsoid.
is vectorially added to the gravitational acceleration. The centrifugal acceleration effect is maximum at the equator because it is directly opposite to the gravitational acceler- ation and the radius of gyration is a maximum. In contrast, the centrifugal effect is zero at the poles because the radius of gyration is zero.
The second cause of the planetary change in gravity over the Earth’s surface is the change in the radius from
2.3 The Earth’s gravity field 27 the equator to the poles. The gradual change in the radius,
amounting to a maximum of 21 km, results in an ellipsoidal cross-section as shown in Figures 2.3 and 2.5. In particular, the Earth’s shape is that of an oblate spheroid in which the equatorial radii are equal and greater than the polar radius. This is the shape transcribed by the revolution of the elliptical cross-section of the Earth. The shape of the Earth is described by the geometrical (or polar) flattening (f) given by the ratio of the difference in the equatorial and polar radii to the equatorial radius which is approximately 1/298 (Figure 2.3). This minor flattening is verified by viewing the apparent circular cross-section of the Earth from space. The flattening of the Earth is an equilibrium condition between gravitational forces attempting to make the body spherical and rotational forces that are trying to flatten it out. The flattening of 1/298 reflects the increasing density of the Earth with depth as a result of compositional, temperature, and pressure variations, as well as lateral (non-radial) variations in the density of the Earth. The fact that the Earth’s shape can be produced by an ellipsoid shows that there is no planetary longitudinal variation in gravity. As a result, the planetary field can be described by an equation that only considers a latitudinal function.
The absolute value of gravity on the surface of the Earth has been the subject of considerable interest, but unfortunately measurements were not very accurate prior to recent improvements in electronics and in the accuracy of the measurement of time and length, necessary parame- ters in the determination of gravity. The theoretical or nor- mal gravity accounting for the mass, shape, and rotation of the Earth on the best-fitting terrestrial ellipsoidal surface is the 1980 Geodetic Reference System of the International Union of Geodesy and Geophysics (Moritz, 1980b).
Additional details on this standard and its use in geophys- ical studies are given in Chapter 6.
2.3.4 Temporal variations
The Earth’s gravity field varies over a broad range of peri- ods and amplitudes from a variety of internal and external sources. Fortunately for the exploration of the Earth, most periods and amplitudes of these changes do not seriously conflict with gravity observations. Fluctuations generally do not exceed amplitudes of 1 mGal (most are only a small percentage of that), and their periods are long with respect to gravity observations. Time variations of gravity origi- nate from planetary sources, such as wobbling of the axis of rotation of the Earth, which affects centrifugal acceler- ations; from local and regional atmospheric pressure per- turbations that vary the attraction of the atmosphere; from natural and anthropogenic changes in the subsurface mass
associated with transport of fluid and gas in subsurface pore and fracture space, and movement of magma related to igneous events and volcanic activity; and from tidal effects due to varying positions of the sun and moon rela- tive to a location on the surface of the Earth. Furthermore, the tides can lead to changes in surface elevation that will also affect surface gravity measurements. The principal temporal variation in gravity in terms of interference with exploration gravity measurements is the tidal effect of the sun and moon and related solid-Earth tides which change the radius of the Earth. The maximum tidal effect is only roughly 3×10−5% of the gravity attraction on the surface of the Earth or 0.33 mGal over an approximately 1-day period. The combined effect of the gravitational attrac- tion of the sun and the moon and the associated change in surface elevation can be calculated from standardized equations. Chapter 6 considers gravity temporal variations more fully in terms of their sources, characteristics, and elimination from observations during exploration survey- ing.
2.3.5 Measurement
The two observable quantities in gravity are the so-called
“bigG” and “littleg,” representing respectively the gravi- tational constant and the acceleration of gravity or gravity force per unit mass. The measurement of littleghas been long exercised both geophysicists and geodesists. Geode- sists measure gravity to determine the shape of the Earth, and geophysicists use the measurements to predict changes in the subsurface Earth and study geophysical phenom- ena. The requirements in the accuracy of gravity measure- ments are generally in the milligal range, but an accuracy of 1μGal or better has been the target of an increasing number of measurements for near-surface studies, geode- tic measurements, and drillhole gravity surveys. Obtaining these levels of precision is a remarkable achievement con- sidering that the Earth’s gravity field is roughly 1000 gal, which means the measurement of one part in a billion of the total field is obtainable.
Measurements of gravity may be either absolute or rel- ative. Absolute measurements have been made only on a limited basis because they are unnecessary for explo- ration purposes and until recently have been more difficult and time-consuming to make and have had a lower accu- racy than relative measurements. However, with absolute gravimeters that use laser interferometry to measure dis- tance and atomic clocks to measure the time for an object to free fall or rise and fall (ballistic instruments), it is possi- ble to obtain absolute gravity in the microgal range within a few minutes with an instrument roughly comparable