Gravity anomalies - Gravity and Magnetic Exploration- Principles, Practices, and Applications.

Rθ

θ ^θ

(a) (b)

a cos

a V

a = centrifugal acceleration = angular velocity

V = platform velocity V = east–west platform velocity

ETV

ω ω

g

= 2 V ω cos(_θ) cos(α) + (V²/ R)

FIGURE 6.13 (a) Vector diagram of E ötv ös effect. (b) Platform direction of motion used in evaluating the E ötv ös effect (Equation 6.30).

and the velocity of the gravity measurement platform,V, in the east–west direction is

Vew=Vcosα, (6.27)

whereαis the heading of the platform with respect to east (Figure 6.13).

The change in angular velocity due to this motion is

ω=Vcosα/Rcosθ. (6.28)

As a result, the change in the vertical acceleration of the platform is

av=2Rωcos²θ(Vcosα)/Rcosθ=2V ωcosθcosα.

(6.29) In addition there is a centrifugal acceleration,V²/R, in the vertical direction due to the outward acceleration of the platform as it moves over a curved surface which is independent of the Earth’s rotation and the direction of motion. Thus, the total E¨otv¨os acceleration is

getv=2V ωcosθcosα+(V²/R). (6.30) Using an average radius of the Earth and neglecting the elevation of the platform, and expressing the velocity in km/hr and the gravity correction in milligals, the E¨otv¨os correction is

getv=4.040Vcosθcosα+0.00121V², (6.31) of ifV is in knots,

getv=7.487Vcosθcosα+0.004V². (6.32) This correction is subtracted from measurements on a westward-directed platform and added to measurements on an eastward-moving platform. The second term on the right of Equation 6.31 is generally neglected for marine

measurements because it is negligible in comparison with other sources of measurement errors, but it is evaluated in airborne measurements where this term is large because of the greater velocity of airborne measurements. However, airborne gravity gradiometer observations are not subject to this effect because the differential accelerometer measurements effectively cancel out linear as well as rotational acceleration gradients associated with the motion of the platform.

A related correction that may be necessary if the observation platform is subject to strong horizontal acceler- ations is the Browne correction (e.g. Browne, 1937;

LaCosteet al., 1967). This correction that seeks to elim- inate the effect of the horizontal acceleration by multiply- ing the observed gravitygobs by the cosine of the angle of deﬂectionξ(Figure 2.3) between truegNand observed gravity is

gBro=gobscos(ξ)≈gobs−(gNξ²/2). (6.33) The increased accuracy of navigation achieved with GPS has markedly decreased errors due to velocity and heading determination of moving platforms. One promis- ing use is to utilize GPS velocities to compute the E¨otv¨os correction directly, rather than differentiating the standard GPS position information over time. Thus, GPS has greatly improved the corrections and accuracy of gravity measurements from mobile platforms.

6.4 Gravity anomalies

As described in Section 6.3, gravity measurements are subject to a wide range of effects apart from the subsurface conditions that are the objectives of gravity surveys. The sources of these variations and methods of determining corrections for them have been detailed in the previous section. The application of these corrections leading to the calculation of a variety of the gravity anomalies is the subject of the following discussion.

6.4.1 Fundamental elements

There are two fundamental components in the calculation of the gravity anomaly, namely the observed and theoretical gravity at the observation point. The gravity anomaly is deﬁned as the difference between the observed and the theoretical or predicted vertical acceleration of gravity. The observed gravity is determined by converting the measurement of the change in position of a mass in a gravimeter between a base location and the observation station into units of gravitational acceleration through a calibration constant. This measured value is modiﬁed for the relative

146 Gravity data processing

change in gravity from the base to the station due to the time variation in gravity as measured by the gravimeter during the time interval between the base and station measurements, the drift of the meter, and the E¨otv¨os effect for measurements made on a moving platform.

Finally, the drift-corrected measurements related to the base station are adjusted to a common base or to a particular gravity datum.

The result of these computations is the observed gravity at a station, relative to an arbitrary datum, or an absolute value tied to an internationally recognized base station that has an established absolute gravity value.

The other component of the gravity anomaly is the theoretical or modeled gravity at the station. This is the gravity at the station taking into account variations over the surface of the Earth due to planetary and terrain factors and in specialized conditions additional corrections. Several gravity anomalies as described below have been deﬁned on the basis of an array of corrections considered in calculating the theoretical gravity value (e.g.HeiskanenandVen- ing Meinesz, 1958;Scheibe andHoward, 1964).

All anomalies have speciﬁc uses, but the most useful anomaly for general exploration purposes is the complete Bouguer gravity anomaly or some variation of it. Some anomalies, such as the Helmart and Faye gravity anomalies which take into account the latitudinal, height, and terrain effects of the Earth in the modeled gravity, are used exclu- sively by geodesists in the study of the size and shape of the Earth.

Unfortunately there is confusion regarding gravity anomalies because it is not unusual for different nomen- clature to be used for the same anomaly. For example, the term Bouguer gravity anomaly may be used without the adjectives complete or simple. A complete Bouguer gravity anomaly has terrain corrections applied; a simple Bouguer gravity anomaly does not. Furthermore, geophysicists and geodesists differ in their use of the term

“anomaly.” Geodesists use the term gravity disturbance rather than anomaly when the vertical datum for reduction purposes is the ellipsoid, and only use the term gravity anomaly when the modeled gravity at a station is based on the geoid as the vertical datum. In contrast, in exploration geophysics, the term anomaly is used regardless of the vertical datum. The term anomaly is tightly woven into the geophysical culture and semantics so that it is unlikely to change. To geophysicists, the anomaly refers to anything left over in observation after removing the known gravity effects. As a result, the North American Gravity Data Committee recommends that the term ellipsoid precede the anomaly when the vertical datum is the ellipsoid in geophysical studies (NAGDC, 2005).

Generally, gravity anomaly values are understood to be in scalar form without regard to direction, based on the assumption that the measured gravity is perpendicular to the Earth’s ellipsoid. However, the gravity component traditionally measured is the vertical component of the Earth’s gravity vector which is normal to the geoid. The difference between the directions of the normal to the geoid and to the ellipsoid is referred to as the deflection of the vertical (Figure 2.3). The deflection of the vertical can be determined from the geoid model and the resulting change in the gravity measurement calculated due to its projection onto the normal to the ellipsoid. In actual- ity, the difference between the scalar value measured in gravity and the value normal to the ellipsoid is negligible because the deflection of the vertical is typically a few arcseconds, reaching as much as an arcminute at the extreme. The projection results in a relative change of up to 1−cos(1)=4.23×10⁻⁸. As a result, considering the approximate absolute value of gravity on the Earth’s surface, 980 Gal, the maximum error from the assumption that the measurement is perpendicular to the Earth’s ellipsoid is around 0.04 mGal over distances measured in hundreds of kilometers.

The deﬁnition of the gravity anomaly as the difference between the observed and theoretical or modeled gravity at an observation site should be kept in mind in calculating and interpreting anomalies. If the anomaly is calculated by subtracting the theoretical model as determined by the summation of planetary, terrain, and geological effects from the observed gravity, there is less chance for an error in determining the sign of the gravity effect than by the arithmetically equivalent method of correcting (reducing) the observed gravity value to a level datum, normally sea level. Furthermore, the computed gravity anomaly is the value at the precise site of the station, not at the datum level. There is no standard downward-continuation reduction procedure that corrects an anomalous gravity value to a height datum. This is particularly important in dealing with surveys in rugged terrain where shallow anomalous masses are of interest. In this case, the anomaly values calculated from the surface measurements will change rapidly and signiﬁcantly over the elevation range, owing to the proximity of the sources.

Airborne gravity gradiometer observations consist of differential measurements of accelerometers which effectively cancel linear as well as rotational gradients related to motion of the platform and any regional and planetary gradients. As a result, processing of the observations into an interpretable form is greatly simpliﬁed once the instrumentation and platform issues have been effectively handled. However, these observations are subject to

6.4 Gravity anomalies 147

h>0

h=d>0 SL

d>0

σ^r σ^sw

Land

Ocean Ocean

Land Land

(b) Underground site

h<0

d>0

(a) Land site

(d) Ocean bottom (c) Ocean surface

σ^r

σ^r σ^r

σ^sw

FIGURE 6.14 Parameters for calculating gravity anomalies at (a) land surface (Eqs. 6.35 and 6.38), (b) underground (Eqs. 6.42 and 6.43) (c) ocean surface (Eqs. 6.44 and 6.45), and (d) ocean bottom (Eqs. 6.46 and 6.47) stations. The location of the station site is indicated by the ﬁlled circle. SL is sea level and the remaining symbols are deﬁned in the text.

signiﬁcant error from terrain which must be removed from the observations.

6.4.2 Classes of gravity anomalies

Gravity anomalies of interest to exploration of the Earth can be divided into three classes, based on the mod- els assumed in the calculation of the theoretical gravity, which is subtracted from the observed gravity to pro- duce the anomaly. The primary class incorporates only analytically determined planetary considerations into the model, so they are called planetary anomalies. The second type applies additional effects from known or postulated subsurface geological conditions into the model. These anomalies are referred to as geological anomalies. The third type consists of filtered anomalies which result from the removal of arbitrary gravity effects caused largely by unknown sources that are empirically or analytically determined and by filtering to enhance particular attributes of the spatial pattern of the gravity anomalies. The anomalies become more arbitrary moving from planetary to geological to filtered anomalies, but they become more useful in identifying and resolving gravity effects of interest to explorationists. In the following sections these anomalies and their uses are described.

Planetary gravity anomalies

Free-air and Bouguer gravity anomalies are planetary gravity anomalies based on theoretical model of the gravity at a station that incorporate effects caused by varying

gravity over the surface of the Earth due to its rotation and changing radius, and the topography of the Earth and the related effect of the atmosphere. The Bouguer gravity anomaly is particularly valuable because it includes the gravitational effect of the mass within the included topography which greatly enhances the identiﬁcation and study of subsurface mass variations.

Examples of scenarios for calculating free-air and Bouguer gravity anomalies and their relevant equa- tions are presented in the following descriptions and in Figures 6.14 and 6.15.

(A) Free-air gravity anomaly

The free-air gravity anomaly takes into account the latitudinal change in gravity on the Earth’s best-ﬁtting ellipsoid represented by GRS80 and the vertical change in gravity between the reference datum and the observation height assuming that the gravity station is located in free air, hence the name free-air anomaly. In this anomaly, the interven- ing space between the observation and the height datum is assumed to have no mass and no gravitational effect. This is the most basic of anomalies used in geologic studies because unlike other anomalies no assumptions are made about the Earth’s masses. For this reason Woollard (1966) referred to this anomaly as the natural anomaly. It has also been referred to in the past as the total gravity anomaly. The free-air anomaly does not take into account the effect of terrain departures from the height of the observation, but in some geodetic applications it is incorporated into the calculation. The resulting anomaly is sometimes

148 Gravity data processing

d>0 d>0

d>0 h>0

h>0

h >0 h>0

h>0 SL

σfw σfw

σice

σ^r

Land

Land Ice

Lake

Land

Lake (b) Lake bottom (a) Lake surface

σr GS

σr

σ

FIGURE 6.15 Parameters for calculating gravity anomalies at (a) lake surface (Eqs. 6.48 and 6.49), (b) lake bottom (Eqs. 6.50 and 6.51), (c) ice cap (Eqs. 6.52 and 6.53), and (d) airborne (Eqs. 6.54 and 6.55) stations. The location of the station site is indicated by the ﬁlled circle.

SL is sea level and the remaining symbols are deﬁned in the text.

called the Faye anomaly. The free-air gravity anomaly gFAAis calculated from

gFAA=gobs−gmod, (6.34)

wheregobs is the observed gravity and gmodis the theoretical gravity model at the observation site taking into account the change in gravity with latitude,gθand height, gh, above the vertical datum. When the free-air anomalies are referenced to sea level (geoid) as a datum, the height (h) is appropriately referred to as elevation, rather than the generic term height which refers to the vertical distance between a point and a horizontal datum. Thus, because gmod=gθ−gh, the equation is

gFAA=gobs−gθ+gh×h. (6.35) The free-air anomaly is unsuitable for most terrestrial geological problems except under special circumstances because it contains the gravitational effect of uncompen- sated topographic masses which generally obscure the effect of subsurface masses in exploration surveys. In marine applications, it is sometimes useful where bathymetric effects are negligible or where the sea bottom is very far from the sensor.

The free-air anomaly is the difference between actual gravity and the theoretical gravity on the ellipsoid that best ﬁts the physical surface of the Earth. As such thegFAA

is a measure of the mass of the subjacent Earth including the topographic masses. The term subjacent acknowledges that the gravity effect of equivalent masses increases with their proximity to the observation site. If there were per-

fect isostatic equilibrium regardless of the size of the topographic feature, this anomaly would be zero everywhere when averaged over an area. In fact, when the continent- wide values of the free-air gravity anomaly are plotted against the station height, the values are distributed around the zero anomaly value regardless of height, conﬁrming the isostatic equilibrium hypothesis on a regional basis.

Regional free-air anomalies may exist where the stress ﬁeld of the Earth maintains the geological structure in a non-isostatic condition. Examples include basins, ore bod- ies, and other small geologic structures that may be held in the crust in a non-isostatic condition by the strength of the lithosphere. Subduction zones also are characterized by prominent local free-air anomalies because the inherent strength of the Earth’s lithosphere is sufﬁcient to maintain the local mass imbalance. In this case, the Earth does not deform elastically in a vertical sense. The island of Hawaii is another example where the load of the volcanic pile that makes up the island deforms the surrounding ocean crust in a series of concentric rings, detectable as free-air anomalies. The free-air anomalies here tend to detect geologic features out of vertical isostatic balance. The mass that can be supported without deformation depends on the nature of the rocks involved and their temperature.

The size of surface and crustal features that are in isostatic equilibrium varies between continents and oceans and within them as well (Watts, 2001). Topographic features over the North American continent of the order of 100 km and larger are generally in isostatic equilibrium (e.g.Woollard, 1966;Dorman andLewis, 1972).

6.4 Gravity anomalies 149 However, this size varies considerably. For example, sig-

niﬁcantly smaller areas are in isostatic equilibrium in the western USA where the crust is thinner and the lithosphere hotter than in the more stable cratonic areas of the continent.

In contrast to the plot of regional values of the free- air anomalies as a function of station elevation, the graph of local values ofgFAAversus station elevation, assuming negligible regional anomalies and terrain effects, has a positive slope which is equal to the quantity 2π Gσ in the mass correction Equation 6.14, whereσ is the density of the surface topography. This provides a simple means of approximating the density of near-surface materials of a region where the assumptions are appropriate.

Although the free-air anomaly is usually not helpful for the interpretation of local gravity studies, it is useful in regional, continental-scale studies. It can be used on a regional basis as a simpliﬁed isostatic anomaly and for interpretation of airborne and satellite survey results that are observed at high altitudes compared with the terrain relief. In both cases, the effect of local mass variations due to topographic mass irregularities is averaged out of the results. In addition, free-air anomalies are widely used in interpretation of surveys in marine areas where computational substitution of rock for water will lead to large changes in gravity anomaly level that may distort the results regionally, or where anomaly distortions are likely to be associated with local bathymetry as a result of a mismatch between the density of the substitute rock in the mass correction and the density of the bathymetric features. This is why global or continental gravity surveys that include both continents and adjacent oceans typically use Bouguer gravity anomalies over land areas and free-air anomalies in marine areas. Where the values are adjusted to a sea level datum, the Bouguer and free-air anomalies are equivalent at the ocean shoreline so that the contouring of anomaly values is continuous across this boundary. How- ever, the anomalies will differ slightly at the shoreline if the Bouguer gravity anomalies incorporate terrain effects, as in the complete Bouguer gravity anomaly described below, or if the reduction of the anomalies is referenced to the Earth’s ellipsoid.

(B) Bouguer gravity anomaly

The Bouguer gravity anomaly is the most frequently used of the gravity anomalies in surveys of continental and near-shore marine areas. It differs from the free-air gravity anomaly by including the gravitational acceleration of the mass between the site of the observation and the datum level of the survey. This attraction is calculated using the horizontal (or Bouguer) slab equation (Equation 6.14) and

modifying it for the deviation of the surface relief from the horizontal slab.

Bouguer gravity anomalies usually do not correlate with local topography unless these features are related to structural or stratigraphic variations that cause density variations below the datum level. On the contrary, the relationship of these anomalies to broad topographic variations that are isostatically compensated is negative. The negative linear relationship has been used to study isostatic relationships within the Earth (e.g.Simpsonet al., 1986;Chapin, 1996). In high mountains, the value of Bouguer anomalies may reach minus 400 mGal or less, in deep oceans the average value is positive 300 mGal or more, and in continental areas averaging an elevation of around 300 m, the Bouguer gravity anomalies are typically around minus 30 mGal. The trend of the relationship between Bouguer gravity anomalies and the height of the stations for continental-scale data sets can be predicted based on the Bouguer slab equation. However, the mean density of the slab speciﬁed by this trend is less than the density contrast between the surface and the atmosphere, which is typically assumed to be 2,670 kg/m³. As pointed out byChapin(1996), the slope of the trend is the mean Bouguer slab effect using the density difference between the surface–atmosphere and the crust–mantle density con- trasts because the gravity effects from both density con- trasts are height-dependent.

(1) Simple and complete Bouguer gravity anomalies.

The theoretical gravity used in the calculation of the Bouguer gravity anomaly for surface land stations is gmod=gθ−gatm−gh×h+gind+gm

×h−gter+gcurv. (6.36)

The respective atmospheric (gatm), curvature (gcurv), and indirect (gind) effects generally are applied only under specialized conditions, and thus may or may not be included in calculating the Bouguer gravity anomaly with parameters shown for the examples in Figures 6.14 and 6.15.

When the terrain correction and the specialized terms of the theoretical gravity equation are not used, the simple Bouguer gravity anomaly,gSBA, is calculated for land surface stations (Figure 6.14a) by

gSBA=gobs−gmod=gobs−gθ+gh

×h−2π Gσr×h. (6.37)

When the theoretical gravity also takes into account the terrain correction, the complete Bouguer gravity anomaly,

Dalam dokumen Gravity and Magnetic Exploration- Principles, Practices, and Applications. Willam J. Hinze, Ralph R. B. Von Frese, Afif H. Saad. 2008 (Halaman 159-171)